UNDERSTANDING MATHEMATICAL EXPERIENCE

We discuss the activities of two 5th grade boys working together during two days of mathematics class, on a problem of representing a motion along a linear path Over the two days, the boys represent the motion in three different mathematical environments: at the blackboard using a table of positions and stepsizes over time; at the computer using a computer simulation of two people walking; and at their desks moving Cuisenaire rods along a meter stick. In this paper we ask the question: "How does one describe and grasp others' experiences". In attempting to answer this question, we find that it is essential to understand experience as simultaneously individual, social, and physical; and to be aware that what we see in students' experiences is necessarily related to what we come to see in ourselves. Theoretical Framework Theories in mathematics education often distinguish themselves by their focus of study. While often combining elements of several perspectives at once, some tend to highlight mental structures (Steffe et al., 1983), others socio-cultural environments (Walkerdine, 1988), and others the interaction with representational objects (Kaput, 1991). Some of the current debates center on whether the field should make complementary use of differing points of view (Cobb, 1992; Bauersfeld, 1992) or make a choice among them (Lerman, 1996). Rather than supplementing one focus with another or making a choice among pre-defined possibilities, we attempt in this paper to describe students' experiences in a classroom, recognizing that: 1) Experiences are simultaneously individual, social, and physical (Goodwin, 1993; Ochs et al, 1994; Meira, 1995) and that 2) What we see in students' experiences is necessarily related to what we come to see in ourselves (Confrey, 1991; Ball, 1996). This raises the question: "How does one describe and grasp others' experiences?" In this paper, we do not provide general answers to this question, but instead grapple with it by attending closely to what two students - Norman and Luke - actually say and do, and how we come to understand it. We will trace how, for example, the boys' individual ways of counting, short conversations among themselves and with the teacher, and the manipulation of physical objects all influence how they see, talk, and act within three different mathematical environments. We will also describe