9353507 Kaput They prose to build, test and perform close clinical analyses of students working with a series of engaging simulations and activities designed to link students' real experience with motion, fluid flow and other continuously variable quantities, first to graphical and then to more formal representations. the ultimate intent is to help democratize access to the central idea of calculus by providing the conceptual and technological foundation for a global reorganization of the core quantitative mathematics curriculum from the middle school onward. We aim to treat calculus not as a capstone course for the intellectual and professional elite that follows upon a long series of prerequisites, but as a mainstream strand in the curriculum for ALL students. While we initially design for and work with those students who are now alienated from mathematics in unacceptably high numbers, we plan to create exciting activities appropriate for all students beginning in the middle grades and continuing into university calculus, where versions of our materials can serve as a front-end. For students will be able to drive a "MathCars" simulation to create position and velocity (vs. time) graphs that can then be downloaded to an inexpensive hand-held device to do further analysis, e.g., curve- fitting, slope or area approximating, etc. They will also be able to recreate their school bus trip home by inputting velocity data, for example, estimated during the actual trip. They could then specify the motion of one of two simulated vehicles to be that of the bus and 'follow; the bus in the second, using only position graphs, say, instead of the usual windshield view. Or they could follow the bus displaying its position graph while getting feedback only on their own velocity. They will also be able to run a variety of experiments with 'ToyCars' on a two lane track, specifying their motion algebraically, and eventually examine other regular motion, including harmonic and other periodic motion. Using clinical research methods we plan to examine in considerable detail the relations between simulation-based learning and learning involving physical MBL devices, differences in conceptual impact of different concrete contexts such as motion vs. fluid flow, and the gradual process of transforming understandings mediated by concrete and graphical representations to understandings encoded in formal algebraic representations. We will integrate our work with the Memirovsky Change and Variation Project at TERC, which is investigating learning in physical contexts. A second level of classroom based testing with underrepresented populations at several sites in the last 18 months of the project will follow the research with individuals and pairs, using enrichment modules in both middle and high school classes at all levels, as well as university precalculus and calculus classes. ***
