The Role of Tools and Bodily Activity in Mathematical Learning and What This Suggests for Characterizations of Knowing

 

Abstract

As part of a larger study investigating students' learning about central ideas and methods in dynamical systems, I examine how tool use and bodily activity combine in mathematical learning and how this combination suggests an alternative characterization to knowing. Through examination of a series of open-ended interviews with three undergraduate students in differential equations as they work with a physical tool called the "water wheel", I elaborate on a form of knowing referred to as knowing-with that is different than knowing-how and knowing-that. In relation to knowing-with, I highlight qualitatively different forms of being the tool that contribute to developing particular views and sensitivities to important mathematical ideas. As such, knowing-with appears to be an essential and often neglected aspect of emerging fluencies with tools and ideas in mathematics education. Finally, I illustrate how the construct of knowing-with is useful for characterizing students^Y experience with symbolic tools, like rate of change for example, as well as with physical tools like the water wheel.