mathematical entities through symbolizing them, not by observing them as _ such. Analogously to the case of literature, many mathematical conversations are hybrids encompassing empirical and pure possibilities, but the difference between them often becomes conspicuous to the participants in the course of analysis or disagreements. device consists of two laser pointers held up at 90° from each other (see Fig. 4). The lase: ights reflect on two side panels set up along perpendicular axes (see vertical panel on the eft side of the student in Fig. 4 along one of the axes). The idea is to move the device tracing a certain figure (for instance a triangle, shown with black lines in Fig. 4) whils keeping the orientation of the laser pointers always perpendicular to the side panels or which their light shines. In this way, the points of light shining on the two panels reflect < parametric decomposition, say along X and Y axes, for the path of the traced figure. Ir other words, each point traced on the flat figure (e.g. the triangle shown in Fig. 4) get: decomposed in X and Y coordinates made visible by the laser lights on the two sid panels. The students worked in teams of three, each with a laser pointer device and ar open folder standing vertical for the XY side panels. They studied the correspondenc« between the lines traced on the flat sheet of paper and the trajectory of the two laser light: on the side panels. The class was able to see only the movement of the points of light shining through the translucent panels; in other words, the class did not see which figure Tracy was tracing but the moving points of light she generated with the laser pointer device on the two side panels. The ensuing class discussion was about the sort of triangle that corresponded to the motion of the lights they had seen. As can be seen from Fig. 6, the triangle had been an equilateral one oriented in such a way that the tracing of the first side (labeled ‘1’ in Fig. 6) corresponded to both lights moving toward the vertex of the side panels, the second side left one light still and moved the other one away from the vertex, and the tracing of the third side of the triangle moved one light toward the vertex and the other one away from it. The students discussed several ideas, such as that the triangle had only one side parallel to the XY panels: the one whose trace had left one of the lights still. I I In Utterance 2 (Fig. 7) Carlene re-positioned the imaginary panels for them to coincide with the edges of her desk, so that her index fingers, “touching” the position of each light, moved along these edges. The physicality of the desk’s edges facilitated the marking of the lights’ movement for Carlene: it made their path more salient and better defined. In terms of Hutchins’ (2005) analysis, the edges of the desk functioned as opportunistic “material anchors” that helped Carlene stabilize the events she imagined. Noble (2007) analyzed how material anchors generate sensory-motor feedback for the user, offering a kind of feedback that participates in the thinking itself. In this case, the edges of the desk might have guided her fingers, in a perceptuo-motor sense, as Carlene traced the motion of the laser lights, letting her feel that the lights followed a clearly defined linear path. Fig. 7 Utterances 1-3 The motions of the laser light Carlene referred to in Utterances 2 and 3 (Fig. 7) were similar to the ones generated by side ‘3’ of the triangle traced by Tracy in front of the class, as seen from her position (see Fig. 6). In other words, Carlene moved both lights from her left side to her right side, which is how she, and the rest of the students, had seen the lights moving when side ‘3’ had been traced (Fig. 6). In Utterances | and 2 Carlene looks at her own desk as if watching the motion of the lights from left to right, as she had seen them moving on the large panels in front of the class. In Utterance 3 she lifts her gaze toward Apolinario Barros, to whom she is directing her talk. VeTIeX, WHICH IS We pommel aimed DY We Ment Nand MOVING along a OISECUNS UNE, INO Wat ner whole right hand stresses directionality and aim of the motion, which is something that changes in Utterance 5: now her hands contract, letting the index fingers indicate the position of the points of light moving on the side panels. Carlene looks at her moving hands, the right one in Utterance 4 and both hands approaching the vertex in Utterance 5. Listeners usually fix their gaze on the face of the speaker, except when the speaker looks at something away from the listeners themselves. Carlene’s look at her moving hands might have been a way of emphasizing for herself and for her listeners, the most critical aspects of her utterances. In Utterances 6-8 (Fig. 10) Carlene illustrates a second case. Now the line is oriented “backwards”. Her left arm is again marking one of the side panels, as in Utterance 4, but now the right hand aims at her elbow. Instead of approaching the vertex located in the “forward” extreme of the table, it approaches the opposite side which is “backward” with respect to her. Her right hand is extended to stress this backward directionality for the imaginary line. She then extends her index fingers to indicate the corresponding motion of the laser lights, breaking their temporal simultaneity in a sequence similar to Utterances 2 and 3. In Utterance 7 Carlene makes apparent the intricate and multilayered aspects of hand motion: she describes the movement as “up” (“‘pa riba”), which reflects the natural motion of the arm forward (i.e. as we move the hand away from us in the forward direction we tend to move it up also). The class had differentiated three cases for the motion of the laser lights corresponding to the three sides of the triangle traced by Tracy. Carlene articulated two of them corresponding to the sides ‘1’ and ‘3’ of the triangle used by Tracy (see Fig. 6). Side 2 was Fig. 11 Utterance 9 In Section 6, we analyzed three utterances in which Carlene enacted juxtaposed displacements (See Fig. 12). First she mirror-reflected the side panels, then she anchored the side panels on the edges of her table, a third displacement was the translation of each of the lights’ movements preserving the left/right orientation they had had from her point of view; the fourth displacement was temporal: enacting two simultaneous motions in sequence (U2)— (U3). In Section 3 we suggested the notion of “thought” as an interpretation of one or more utterances. We can use the present example to clarify this idea. What were Carlene’s thoughts that she expressed through her utterances? The most succinct description of her thoughts was produced by Mr. Barros in Utterance 9: “Oh, OK. So one time they go in the same direction. And another time they go in the opposite direction”. However, we avoid reifying thoughts as entities with causal powers that manifest themselves in contingent and contextualized utterances. What is actually experienced and felt are the utterances themselves, with their overt and covert aspects, each of them embedded in a “just past” (its origin) and an “about to happen” (its future-oriented aim). In our example, Carlene’s thoughts are interpretations of her utterances oriented to grasping their unity or overall “point”. Like all interpretations, they are open-ended descriptions that reflect a certain “take” on the interpreted phenomena. Mr. Barros’ interpretation was his way of making sense of Carlene’s utterances—an interpretation that coincided with Carlene’s own, as she indicated through her animated approval. At another moment Mr. Barros, Carlene, Tracy, or any of the other students, might offer a complementary or alternative account for her thoughts. There is not an ultimate thought to be ascertained. By re-contextualizing Carlene’s utterances, the question of what she thought remains open. ee a ae ee ce en a ee ee a, a ee re. (a. a ae
