__Principal Investigators__Ricardo Nemirovsky,

*TERC*

Jim Kaput,

*UMass Dartmouth*(Co-PI)

__Funding__National Science Foundation

2002 - 2004

Award: REC-0087573

__Abstract__The aim of this project is to investigate new approaches to nurture and cultivate the mathematical imagination of all students. Mathematics as a science to imagine-with is not incompatible with memorizing the multiplication tables, number facts, or shortcuts to operate fractions, but it changes what these memories are part of. It is about imagining space and time: shapes, patterns, or trajectories; it is about envisioning how things could be; it is about discriminating the finite and infinite, the discrete from the continuous, and the possible from the impossible. The main conjecture of this is that cultivating mathematical imagination is deeply related to enriching bodily action/perception. The proposed research intends to investigate this thesis through a series of studies with high school students and pre-service teachers.

It is widely acknowledged that an essential goal of pre-service teacher education involves reconstituting prospective teachers' mathematical knowledge to serve as a foundation for teaching. Indeed, mainstream pre-service teacher education makes creative use of manipulatives to achieve this revisiting of important ideas in ways that also inform pedagogy—how to teach those ideas. Thus, for example, the geoboard is commonly used to revisit rational number ideas, to help deepen understandings of area and perimeter, and, as importantly, to build teachers' understanding of the roles of manipulatives—how to relate the physically-based actions to mathematical notations, how to use them to diagnose confusions (e.g., between area and perimeter, or between a fraction-as-operator and fraction-as-quantity), how to integrate their use productively and pragmatically in real classroom situations, etc.

However, the pre-service grade 6-12 teachers of the next few years will have students who will be living into the 22^{nd} century, and the kinds of mathematics that those students will need to learn and use will reach far beyond the mathematics commonly taught in schools today. While we cannot predict with certainty the details of that knowledge and how it will be instantiated in curriculum materials or in peoples' lives, we have very good reason to expect that, at least in terms of the Mathematics of Change and Variation, that this mathematics will include dynamical systems and the different representations, behaviors, and phenomena that they are used to describe and understand (Casti, 1996; Hall, 1992; Kaput & Roschelle, 1998; Prigogine, 1999). Furthermore, understanding and being able to deal with the complex relationships among physical, simulated and notational systems is likewise an important intellectual skill of this new century. And, as argued above, these interactions
may be fundamental to the kinds of learning that all students will need to experience. Hence teachers' appreciation of these concrete roots of the learning of content needs to be rooted in their personal experience as learners of both content and as learners of the processes of learning and teaching.

Therefore, we propose to study in detail the means by which a "21^{st} century manipulative" can serve a role analogous to that played by manipulatives such as the geoboard today. Our prototypical candidate, based on an existing body of work and development, is the Bouncing Car, a small car sliding on an inclined plane and colliding with a moving piston on the lower end of the plane. This device provides a manipulable and visually explicit version of a nonlinear dynamical physical system studied by Tufillaro, Abbott and Reilly (1992), which involved a ball bouncing on a vibrating table (e.g., a ball-bearing bouncing on a vibrating speaker). The amplitude and the frequency of the oscillating piston can be modified as it moves. Based on a motion sensor, a computer generates real-time graphs of various kinds for both the car and the piston. These graphs can display various data (velocity or position vs. time, phase space, etc.) which can be displayed on the same screen if desired.

The Bouncing Car is a physical device which can exhibit a wide range of dynamic behaviors, while its inner workings are visible and open to examination and control and its motion is symbolized with real-time graphs generated on a computer screen. These three features, coupled with the fact that these systems are physical and tangible, are critical to students' engagement with the science and mathematics of system dynamics (Correia, 1997; Noble, et al., 1996).

Math in Motion Project funded by the National Science Foundation, Grant #REC-0087573.