Simcalc Abstract

Abstract: Simulations for Calculus Learning (Jim Kaput, UMass Dartmouth) This Project will help build a technological and curricular foundation for democratizing access to calculus for all students ranging from the middle grades to introductory university calculus. It will support the building of flexible simulation software beginning with linear motion simulation based on car-driving and graphical representations of that motion on the "dashboard." This base-level simulation, carefully constructed to help map the phenomenologically rich experience of motion in an automobile to graphical representations of that motion, would be extended in two dimensions: (1) the semantic dimension - to fluid flow, to relative motion involving two cars, to more abstract motion-forms such as hannonic motion and its variants, and to two and three dimensional motion; (2) the representational dimension - from graphical to numerical and symbolic representations, where the simulation is increasingly "driven" by formally defined functions. An open system design aims toward wide-scale use and connectivity to other software and hardware systems. In particular, we will design for use of the simulation software on computers sufficiently powerful as to provide a realistic driving experience, while also enabling the users to export their resulting data to analytic environments on inexpensive computers, including hand-held devices, perhaps out of school. Additional analysis of the simulation-generated data (curve-fitting, slope and area estimation, etc.) could be done on the inexpensive device, and, furthermore, data defined on the inexpensive device could be used to drive the simulation. A research component linked to Nemirovsky's Mathematics of Change and Variation Project at TERC will study the processes by which students at the different levels build or fail to build the conceptual structures that constitute understanding of elementary differential and integral calculus at increasing levels of complexity and formality using the different components of the system. We will compare learning with the simulation and learning with MBL, examine understanding of scaling and negative quantities, and, most importantly, we will test the assertion that calculus can be meaningfully learned before, during, and after algebra - indeed, that mathematizing change may be an important context for learning algebra, just as it was the context for the development of much algebra historically. The Project will engage under-represented student populations from the outset at each school level. Activities, challenges and explorations will be designed to engage and excite as wide a range of the population as possible, especially those who have been alienated from school mathematics. It is expected that younger students will spend much more time with the base-level of the system, while older students, while they may begin at the same level, will progress more quicldy to the extensions. The system will be configurable to enable such accelerated movement by more mature students. Indeed, we will test the system for use as a "front-end" for a university calculus course, and possibly in a physics course. The Project clearly embodies unconventional approaches to confront central problems in mathematics education. It does not pretend to nibble at the margins. The PI will use his extended experience and contacts

in the US and elsewhere to employ the very best advice available from among experts in the mathematical sciences, education, science and science education (particularly in kinematics), computer science and the psychological sciences. He will likewise use his experience and contacts to ensure the broadest availability of Project products to commercial and academic interests and NSF projects.

--.