### Students' Mathematical Thinking and Learning

Understanding of how students think and talk about mathematics has the potential to inform instruction so that it builds upon students' prior knowledge to establish increasingly sophisticated mathematical thinking. I, along with several co-authors from the University of Illinois, have investigated the ways in which students' mathematical thinking is intertwined with their understanding of contextual elements during work on real-world problems. Specifically, we have studied how students' conceptions of reflective symmetry can provide a context for examining connections with perpendicular bisector; how geometric similarity can be considered in the context of shadow puppets or for scaling abstract figures; and how a problem about locating a new grocery store in a community elicits different ways of using knowledge of geometry along with knowledge of the community. We have also examined the different types of mathematical reasoning skills that students use on open-ended problems in algebra and geometry.