Figurative and Operative Imagery: Essential Aspects of Reflection in the Development of Schemes and Meanings

Our purpose in this chapter is to clarify the role of imagery in students’ mathematical learning and reasoning. In pursuit of this goal, we address interdependencies among imagery, reflection, scheme, and meaning as theoretical constructs and illustrate their interdependence by examples. Our motive for this expanded charge is that while the notion of schemes and scheme development is sometimes discussed in studies of students’ mathematical learning, the role of imagery in that process is often neglected, yet it is central to the development of productive mathematical meanings. We build on Glasersfeld’s (Radical constructivism: A way of knowing and learning. Falmer Press, 1995) notion of re-presentation to define images as recalled experience and explain the distinction between figurative imagery and operative imagery as resting on an image’s degree of necessity to the subject’s in-the-moment reasoning. Examples are drawn from a seventh-grader’s construction of a scheme to play a generalized game of Nim and a mathematics educator’s projection of his reasoning to a level of operative schemes to resolve a confusion arising from his assimilation a problem at a figural level. In all, we highlight three arguments: (1) imagery, as re-presentations of experience, includes far more than visualization; (2) a main function of imagery in students’ mathematical learning is that students form images of having reasoned; and (3) imagery, as a construct, does not stand alone. It is useful only to the extent that it allows us to focus on the contexts of students’ schemes and meanings and to employ reflective abstraction as a construct for explaining and investigating students’ mathematical learning.