Abstract 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.
"We believe that seeing mathematical experience as necessarily individual, social, and physical all at once is a step toward grasping the complexity inherent in this and all human experience. Our own emerging understanding of Norman and Luke's mathematical experience is grounded in our gradual noticing of how the same trip enacted in the different environments involves different questions and ways of acting. While using the Cuisenaire rods, one can ask, "Where are you now?", because at any moment both boys have reached a certain position on the ruler by flipping the rods. In Moment 2 we see that while using a table of numbers, one must define when "now" is, because the table represents in one place events that took place at many different times. One must also define "who" is moving according to the table, because the same table of numbers could represent different actors, as we saw when Luke put himself into the story he told about the table. In this way Luke used his story about the table to link the Cuisenaire rods environment, where he was in charge of moving, to this static table of numbers describing the trip taken by an imaginary boy and girl. Thus, while we came to understand the differences between the different environments, and the mathematical value of these differences, which we had been led to "forget" by our own education; we watched the boys develop ways of linking the environments together."