Teachers' knowledge shapes decisions they make while planning instruction and while teaching. Knowledge of student thinking (e.g., typical solution paths, difficulties) appears to be an especially powerful influence on teachers' practices and a productive site for professional development. These issues, however, have not been examined in many content domains. The research reported here is an initial step towards building a comprehensive picture of college mathematics instructors' knowledge of student thinking in calculus. In particular, we seek to understand what knowledge of student strategies and difficulties is held by mathematics graduate students. More specifically, we focus on understanding the nature of graduate students' knowledge about student thinking for problems that rely (implicitly) on understanding function and slope in service of understanding ideas related to derivative. Comparisons are made between participants' knowledge of student thinking and relevant findings from research on student thinking. These findings can inform decisions made about the extent and focus of professional development that is needed for graduate students as they learn to teach. Implications for practice and design of professional development are discussed.

This discussion of engineering education across K-20 examines curricular initiatives in order to examine how university engineering can impact K-12 education and vice versa. In this case, the unique position of engineering as it is more fully entering the K-12 curriculum and the emergence of new levels of scholarship in the field of engineering education have contributed to an increase in engineering education research. In addition, partnerships can connect university faculty in engineering with K-12 school systems to provide opportunities for development at all levels. The introduction of challenging content from an engineering perspective includes emerging areas of research such as nanotechnology that are new to engineering and hence new to education as well. Challenging content also can imply ensuring all students have access to the fundamental content required to enter an engineering program. Initiative objectives include reducing achievement gaps, recruiting underrepresented populations into engineering, and developing challenging curriculum for K-20 settings.