How do Your Students do When you Ask them to generalize? Do they reason numerically or geometrically? Can they make connections among representations? Algebraic reasoning and representations are foundational for middle-grades students as they prepare to enter high school and pursue future studies. Many policymakers suggest an “algebra for all” approach (Silver 1997). Although there is no consensus on what this algebra might look like, many agree that understanding patterns, relations, and functions and representing and analyzing mathematical situations with variables are important components of algebra (NCTM 2000). Studying patterns in middle school builds a foundation for more formal investigations with functions in high school.
Angela S. Krebs and Eileen L. Kaller
In recent years, elementary school preservice teachers often have a fieldwork experience before student teaching. However, the quality of these experiences varies greatly (Wilson, Floden, and Ferrini-Mundy 2002). What supports a good fieldwork experience? Certainly we want students to be taught in classrooms in which they are asked to reason, represent, and communicate. At the University of Michigan–Dearborn, we strive to find these sorts of placements—first, by working with districts using reform-based materials and, second, by asking local district leaders to identify exemplary teachers. Moreover, our future teachers have experienced inquiry-based lessons in the mathematics and science courses they take as university students. Even with this careful design, when we observe classrooms in the field we find that future teachers focus on surface aspects rather than on mathematical thinking. This experience concurs with the findings of Moore (2003) and Putnam and Borko (2000), who found that novices focus on management and procedures, not on learning. So we asked ourselves, How do we sharpen the future teachers' focus?
Judith Flowers, Angela S. Krebs and Rheta N. Rubenstein
Details problems and instructional approaches intended to promote preservice teachers' understanding of the reasoning underlying whole number multiplication.