Geometry as Objective Science in Elementary School Classrooms: Mathematics in the Flesh (Routledge International Studies in the Philosophy of Education)

no ah: no:> (0.47) ((has begun to walk toward chalkboard, turns to Ben)) hum? ah no ((shakes head)) (0.43) no? ((picks up the cube)) i know (0.43) yes? (0.41) yes because you can feel the edges ((touches 2 vertices of cube in Mrs. Turner’s hand, Fig.)) (1.46) what ´about them. (1.10) they have vertices ((touches 3 vertices, Fig.)) =they have vertiCE:S, yes? ((Baveneet nods)) (1.08)

and (0.65) they have ((gets up, takes cube in lH, “feeling”/ “caressing” around one face, Fig.)) (2.07) <

i dont know> (0.86)

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((:B)) hOW did you know; im looking at your model here. ((points to her model))

96 Geometry as Objective Science in Elementary School Classrooms “You can feel the edges” (turn 48), she says while touching two of the vertices (see drawing). With falling intonation, Mrs. Turner utters, both a question and an invitation, “what about them?” After a pause, Bavneet continues, “They have vertices” and points to three vertices (turn 52). Mrs. Turner exactly repeats the words Bavneet has used and then, with rising intonation utters, “Yes?” After another considerable pause, Bavneet begins offering a response, “and they have,” and gets up to take the cube from Mrs. Turner. Her right hand index fi nger repeatedly moves back and forth over one of the faces, almost as if she were caressing it (turn 55); she exhibits a questioning facial expression. At the end of a long silence, Bavneet responds with very low speech volume—as if she were apologizing for the contradiction to what she has stated during turn 44—“I don’t know” (turn 55). This ends the interaction abruptly, as Mrs. Turner stops this interaction by shifting to her right and by directly addressing Ben, uttering what comes to be another question. Here, Bavneet tells us that the edges stand out as something special, appropriate to the question she has heard Mrs. Turner ask. But, as apparent in the latter’s query (turn 50), Bavneet’s utterance is not a response to what Mrs. Turner has intended asking; it might be a response if Bavneet had responded to the second question offered: “what about [the edges]?” Bavneet gives it a try, “they have vertices.” In fact, she gesturally points to three of the four vertices associated with the top square, thereby also making salient two or three of the four edges. But apparently this, too, is not enough, as Mrs. Turner merely repeats the constative statement and then, with rising intonation, offers an invitation to go on, “Yes?” (turn 53). Bavneet makes another attempt, but then gives up, “I don’t know”; and Mrs. Turner, in turning to Ben, gives up pursuing any further understanding what might have led Bavneet to announce that she knows something else about the cube. The edges do stand out in visual perception, and so do the vertices. Bavneet articulates this salience in moving her arm forward until her index finger touches one or the other vertex for a total of five times. But this salience is not the one sought after, which would be the one telling everyone that the mystery object is a cube. Vertices and edges are salient; and this fact is not the one that Mrs. Turner appears to question. For, in asking “what about them?,” she actively acknowledges the presence and salience of the edges; and she does the same for the vertices, both acknowledging and asserting their presence. But despite this salience, she is pursuing one that Bavneet realizes is hidden from her: “I don’t know.” Mrs. Turner turns to Ben, and, as seen in Chapter 6, allows the bringing forth of measurement in geometry. From the normative perspectives of a person concerned with enacting curriculum, this fragment may constitute an instant of missed opportunities for working out salience and how it pertains to the lesson at hand. There was an opportunity—at least as seen from the sidelines where the analyst stands—for teasing out why edges and vertices alone do not distinguish cubes from other parallelepipeds. It might have been possible to tie

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Bavneet’s articulations to the events in the group that had its turn before Ben’s, where during the modeling exercise there was a long debate about whether the mystery object is a rectangular prism or a cube (see Chapter 5). At this point, however, we do not know what is salient to Mrs. Turner while she talks to Bavneet. In the heat of the moment, she may have had a general feeling (sense) that Bavneet’s contribution is an interruption of the exchange that she has engaged in with Ben. What is salient in children’s perception is important, as it is precisely this perception that they use as a resource in making sense. What is salient is not the result of a “construction,” but rather something that is given to us—recall from Chapter 1 how our eyes do make three-dimensional cubes appear even though we are not aware of their activity. In one of my science education research studies, I had found out that the students in a 12th-grade physics course were divided—without knowing so—about whether there was movement or no movement in their teacher’s science demonstration (Roth, 2006). But those students who saw movement used the observation to make sense of the teacher’s subsequent explanations of the theory in the same way that those students who did not see movement used what they perceived to make sense of the teacher’s explanation. The result was conceptual confusion, because the same teacher explanation was heard (by some) as explaining movement and (by others) as explaining the absence of movement. Knowing what is salient for another person requires some careful attention to their expressions, and, perhaps, explicit engagement with them. As we see in the above fragment involving Chris, Mrs. Turner apparently is not aware what is salient to Chris and what he is talking about; and, in the present instance, that which is salient to Bavneet does not fit into the flow of the lesson and remains without conclusion. Salience also is the central feature in the next episode.

OF FACES AND EDGES (HOW THE EDGES OF A CUBE APPEAR) In this fragment from a lesson on edges, we notice how edges appear differently to a student than the teachers intended. Unlike what we see in the preceding sections, we observe in the present section the pedagogy that the teachers mobilize in interaction to make the features appear differently. A drawing of a pyramid with labels on the different parts—edges, vertices, and faces—prominently features at the chalkboard (Figure 4.1), just behind where Mrs. Winter has positioned herself. She interacts with one of the students, Thomas, who articulates for us how certain shapes stand out and how, to the fi nger, an edge appears triangular. They appear triangular to the fi nger, which, in moving along, can feel the edge. But Mrs. Turner and Mrs. Winter want him to see the edge as straight edge, as opposed to the one of the circular face of the cylinder, which is “curved [round].” Both teachers apparently work hard on getting something else appear for Thomas.

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Figure 4.1 A concept map of the lesson on vertices and edges is prominently displayed on a large sheet of paper taped to the chalkboard in front of the classroom.

After Mrs. Turner has solicited a vote on the nature of edges and articulated the results—“some people say all the edges are all the same” and— “edges are different”—she asks for a showing of hands of those who “had their hands up for some edges to be different from other edges” (turn 01). Thomas is one of the students who raises his hand, and, after a pause, Mrs. Turner calls on him. There is a long pause before Thomas responds to the “who” question by saying “everyone.” There is another pause, and then Mrs. Turner asks Thomas, “How would you describe this edge” while running her finger along the edge of the cube that is facing the class generally and Thomas particularly. There is yet another pause, and then, in a slow delivery, Thomas utters “well” (turn 08). Mrs. Turner reaches forward to hand the cube to Thomas, who is sitting in the first row of desks, asking him to “describe it.” Fragment 4.3a (Thomas) 01

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okay (0.63) wHO had their hands up for some edges to be different from other edges. (1.81) ((Mrs. T looks around the classroom, points to Thomas)) thomas. (2.62) everyone.

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(2.36) thomas; hOW would you describe this edge. ((Runs her fi nger along an edge of the cube)) (1.31) <well;> i ask you to run your finger along it; ((Mrs. T reaches the cube toward Thomas, who leans forward,)) (0.79) ((Thomas gets cube from Mrs. T)) how would you describe it. (0.58) ((Thomas intently looks at the cube, Fig.))

well (0.85) because um; (2.30) < because its like a sort of (0.60) vertex> (0.39) like (2.30) ((gets up from his seat with cube in hand)) like this one. ((points to the word vertices in the diagram on chalkboard with a pyramid on which the different parts of an object Fig.)) (0.52) yES. ((Fig.) (0.79) vertices? ((she points to the top vertex in the diagram)) yea. (0.36) like this on [mine ] [but your] instead of your if you start on one verticee; put that finger on that verticee. (0.82) your finger (0.26) point (0.62) one verticee (0.48) and rUN:it along that edge to the other verticee (0.77) okay? (1.24) whats that feel like; what kind of edge is that. ((Mrs. W runs his finger along the edge))

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Thomas intently looks at the cube, which he holds in both of his hands (turn 13). His interjections alternate with pauses, which can be heard as hesitation, and, after a particularly long one, he describes, in what we can hear to be a confident delivery, that he observes “it” as “like a sort of vertex” (turn 14). He then gets up and walks toward the chalkboard to point to the word “vertices” on the parts of a geometrical object exemplified in the pyramid (turns 16, 18). From pointing to the diagram, the hand moves down to one of the vertices of the cube (turn 16) and then points back up to the “vertices” in the diagram (turn 18). Mrs. Turner responds affirmatively, and Mrs. Winter utters with rising intonation, “Vertices?,” to which Thomas provides an affirmative second turn (turn 21). He adds, “like this on mine.” However, this does not appear to be correct, as Mrs. Winter’s talk overlaps with his, beginning her turn with an adversative conjunction, “but your” (turn 22). The sense of adversity is strengthened when Mrs. Winter, in a second attempt, again marks opposition with the adverbial phrase “instead of your,” only to start a third time with an opposition-marking conditional, “if you start.” In this fragment, we observe how, after some initial hesitations, Thomas confidently articulates an observation about vertices. There is a form of generalization, whereby he categorizes the vertices of his cube in terms of the vertices on the drawing, which are vertices in general (though also the particular vertices of the figure in the drawing). Thomas talks about the vertices that stand out on his cube and then gets up and points to the term “vertices” (in the diagram), from which there are arrows to the different vertices of a pyramid. The generalization that we observe first is from the specific three-dimensional object in his hand, an imperfect cube, to a two-dimensional diagram. Second there is a generalization from a material object to an ideal object. Third, there is a generalization of a feature from the category of cubes to one that is also found in other categories. That is, in his pointing, there is a triple generalization. But this generalization meets an apparent opposition. Mrs. Winter orients toward Thomas and takes some time to grab the cube from him while she begins to talk. She asks him to put a finger on the vertex where she is placing her finger—i.e., she is placing her finger as a non-verbal instruction for where his finger is to go (turn 22, a). Thomas’s index finger hits the vertex, but as he raises his gaze to look toward the class, his hand slips off. Mrs. Winter follows with her hand and, when she has succeeded grabbing hold of the index finger (turn 22, b) utters “your finger.” She moves the finger to the vertex while uttering “point,” but the hand moves away again. Mrs. Winter pulls it back to the vertex (turn 22, c) and utters “one verticee.” She then pulls the finger along the edge to another vertex, while describing the process in words, “run it along the edge to the other verticee.” There is a pause, which she follows with an “okay?” marked by a rising intonation. During the speaking pause, she runs the finger again along the edge. There is another pause, and, while uttering with rising intonation toward the end, “What’s that feel like?,” she runs the finger along the edge for a third time. Mrs. Winter runs the finger yet another time along the edge while uttering, again with rising intonation, “What kind of edge is that?” She retracts her

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hand; and, in the pause that follows (turn 22), Thomas runs his finger twice more along the edge. An expression of consternation follows, and then, while gazing toward the floor, he scratches his head—as if he were asking, “What do you want from me?” (turn 23). Fragment 4.3b (Thomas) 22

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[but your] instead of your if you start on one verticee; put that finger on that verticee. ((Fig. a)) (0.82) your fi nger ((grabs fi nger, Fig. b)) (0.26) point (0.62) ((holds fi nger to vertex, Fig. c)) one verticee (0.48) and rUN:it along that edge to the other verticee (0.77) okay? (1.24) whats that feel like; what kind of edge is that. ((Mrs. W runs his fi nger along the edge))

(4.22) ((scratches his head, Fig.)) ((Mrs. W gets the cylinder)) kay, () where (0.57) all right thIS time; (0.68) put your finger OUT ((Thomas puts finger out, Mrs. W takes it and places it on the edge of the cylinder)) (1.12) what does thAT one feel like. ((moves his fi nger along the edge)) (0.88) it feels like um; (4.26) ((Mrs. W moves his fi nger repeatedly around the circumference)) ((Thomas has questioning look, Fig.)) does it feel the sAME or does it feel different; (0.32) feel different. (0.93) what is different about those two edges. (0.77)

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because um this one is round and this one is ap (0.48) isa square (1.63) ((Thomas looks up to Mrs. W, as if looking for confi rmation)) if you jUSt did ONE of them ((moves fi nger up and down an edge of the cube)) if you jUSt did ONE of them ((moves fi nger up and down an edge of the cube)) (0.79)((Thomas nods)) what would it (0.64) f::eel like. (1.19) feel like um this one (0.65) ((moves finger up and down the edge, Fig.)) is um () triangle, (1.83) ((Mrs. W gets a pyramid)) <

okay> that would be the triangle. (0.82) triangle and (0.57) and this one feels like um; (1.14) a triangle ((moves around the triangular face of the pyramid)) (0.95) right; not using a sh:ape word. (0.22) so we=re not using a tRIangle CIRcle or squARE. ((rhythmically counting, beat gesture))

Mrs. Winter turns around to get a cylinder from the tray of the chalkboard, makes a false start (“where”), then suggests “all right this time,” and fi nally asks Thomas to put his finger out gesturing (“put out the fi nger”). As before, she captures his index fi nger and, just as she says “what does that one feel like,” moves the fi nger around the circular edge of the object (turn 26). Thomas begins a response by repeating part of Mrs. Winter’s phrase, “it feels like.” He stops; and a long pause develops while Mrs. Winter moves his fi nger twice around the circumference. Gazing emptily in front of him, Thomas has raised his hand to his chin facing the cylinder in Mrs. Winter’s hand (turn 29). He appears lost. At this point, Mrs. Turner enters the conversation, beginning a questionresponse sequence, “Does it feel the same or does it feel different?,” to which the second part comes forth almost immediately, “feel different” (turn 32). There is a pause before Mrs. Turner offers another question, “what is different about those two edges?,” to which there is a defi nite second part clearly informed by what is in front of his eyes: “this one is round and this one is a square.” During the fi rst part, Thomas follows the circumference of the cylinder with his fi ngertip, then moves it to the cube and tracks the edge around the entire top square. During the ensuing

The Flesh, Distractions, and Mathematics 103 pause, Thomas’s gaze moves upward toward Mrs. Winter’s face, as if he were looking for confi rmation. But Mrs. Winter begins with a conditional, “if you just did one of them” and moves her finger down and up one of the cube’s edges. There is a pause, and another utterance with question structure, “What would it feel like?” accompanied by her finger running down and up the edge of the cube that faces Thomas. He completes the question-response pair by describing, “feel like this one is triangle.” As the utterance unfolds, he runs his fi nger five times down and up the edge closest to him on the cube that Mrs. Winter is tending toward him (turn 43). But, while turning toward the chalkboard tray, Mrs. Winter clamps the cylinder under her left arm, picks up a tetrahedron and makes a constative statement, “that would be the triangle” (turn 45). Thomas runs his fi nger up and down the cube’s edge while uttering again, “triangle,” then turns and moves his fi nger around one of the triangular faces of the tetrahedron twice spaced by a pause, while uttering, “and this one feels like a” and “triangle.” After a pause, Mrs. Winter acknowledges, “right,” and continues, “not using a shape word.” She elaborates, rhythmically beating down with her right hand while unfolding the fi ngers as if counting from one to three while articulating the shape words, “we are not using a triangle, circle, or square” (turn 49). (I return to this instance in Chapter 7.) In this second part of the fragment, Thomas provides a second, third, and fourth response to the original question about what he feels when running his fi nger along the edge. In the first of these three attempts, he runs the index fi nger around the circumferences of the circular and square faces and tells everyone listening—after being encouraged to state the differences in the tact—that the fi rst feels round and the second square. His hand in fact describes a curve in the fi rst case, while changing direction three times in the case of the square. His words are accurate descriptions of the movements that his hand accomplishes at the same time. In the second attempt, Thomas runs his fi nger repeatedly along the edge and then describes what he feels: The edge does feel (tri-) angular. But again, Mrs. Winter does not evaluate his response positively. They begin a third attempt, in which Thomas uses the same verbal description for what he feels moving up and down the edge of the cube and when following the triangular circumference of the tetrahedron. In this situation, the verbal description is accurate and consistent with his previous articulations. The edge of the cube feels triangular, and the circumference also feels like a triangle. But again, the response apparently is not appropriate, as Mrs. Winter tells him that they are not using shape words. After Mrs. Winter states that they are not using shape words and exemplifies the kinds of words not to be used (this performance is further analyzed in Chapter 7), a pause unfolds during which she runs her index fi nger three times down and up the edge of the cube. She offers up a question, “Can someone help us out?” but there is no response. She offers up another invitation to a question-response pair, “he said this one is”

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(turn 53) leaving an answer slot in a grammatically unfi nished sentence. But again, there is a pause. Mrs. Winter produces yet another invitation, initially providing a response for her second question, “What we said, round” and then uttering, while pointing to the edge of the cube, “What would that one be?” (turn 55). She calls on Alicia, who proffers a response, “straight,” which Mrs. Winter, after a pause, follows up with an invitation for indications of agreement (“Can we agree?”). The response is silence, so Mrs. Winter continues, “this one is straight, this edge is round” while moving her fi nger fi rst along the edge of a cube and then along the edge of the cylinder. Fragment 4.3c (Thomas) 49

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right; not using a shape word. (0.22) so we are not using a triangle circle or square. ((rhythmically counting, beat gesture)) (1.93) ((Mrs. W moves fi nger 3 times up and down the edge of the cube)) can someone help us out? (0.70) he said thIS one is ((moves fi nger around the circumference of cylinder)) (1.74) ((moves to look at his face)) however what we said; rOUNd (1.12) what would this one bE:? ((feels edge of cube)) (0.35) ^alICia. straight. (1.03) can we agree? (0.44) this one is straight ((moves fi nger along edge of cube)) (0.83) thIS () edge ((moves around cylinder)) () is round. (0.44) would you agree with that thomas? (0.38) .h uh yea. (0.44) so which one is the straight (0.80) edge. (0.37) this one ((moves fi nger down the face of the cube)) is it down hERE? showm put your fi nger down straight edge. (1.51) ((Thomas moves fi nger down the middle of the cube face, Fig.))

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now when yOU=re doing this we are going right through the face. (0.68) here is our strAIGht edge ((she grabs his fi nger and moves it along the edge of the cube, Fig.)) now go through the rOUNd edge ((Thomas puts finger on the circular edge of cylinder, moves it around)) <this one.> good for you. ((looks toward Cheyenne)) (1.03) cheyenne () you need to put that shirt properly ((Thomas begins to move back)) (0.70) jane nOW (0.68) um (1.19) ((bends down to ear of Thomas)) <

you can sit down> ((Thomas moves to his seat))

Mrs. Winter orients herself toward Thomas and asks him whether he can agree with the statement, which he affirms. But Mrs. Winter follows up asking him to identify the straight edge. Thomas’s hand moves forward, touches the face of the cube, and the index finger moves downward in a straight line while Thomas says, “this one.” But with a rising intonation and a grammatically formed question emphasizing a location “here,” Mrs. Winter utters, “Is it down here?” and then reiterates the request for showing a straight edge. It is not just a reiteration, but being reiterated, it also is an evaluation that the intervening response has been inappropriate. Thomas again moves his finger down the center of the face of the cube (turn 70). Mrs. Winter describes what he is doing as “going straight through the face.” She grabs his finger and places it on the edge of the cube; and while moving his finger down the edge, she articulates “here is our straight edge” and then invites him “to go through the round edge” (turn 71). With a very low volume—which we can hear as a timid attempt—Thomas says “this one” while pointing to the edge of the cylinder and then moving it around the edge. The response is immediate, embodying a positive evaluation, “Good for you.” There are some brief comments concerning appropriate classroom behavior (see comments in Chapter 7), and the beginning of a teacher-student exchange between Mrs. Turner and Jane. Bending down to bring her mouth near his ear, Mrs. Winter softly invites Thomas to sit down, which constitutes of the work of bringing this episode to a close and marking this closure.

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In the course of this fragment, Thomas has exhibited a lot of knowing in action and his responses are in many ways appropriate, articulating and expressing precise responses to the questions that are posed to him. But these responses are not the ones the teachers apparently want to hear, as evident in the repeated attempts at eliciting a different answer, attempts that come to an end when the apparently desired response has been given. The positive evaluations and the turning to a different topic are two markers that suggest that what has happened just before is appropriate, whereas the recognizable repetitions, rearticulated beginnings, and oppositive expressions all are markers that what has preceded does not constitute the desired response or course of action. For example, Thomas does in fact move his fi nger through a straight line, thereby enacting a movement that has the verbal expression “is straight” as an appropriate predicate, but in this lesson on edges, the teachers apparently want him to apply the predicate “is straight” to the edge. That is, the unfolding lesson emergently exhibits the intention to articulate some aspects at the expense of all those others that perceptually stand out. Moreover, there is also an intention to distinguish between the responses that might be appropriate here—and in other parts of the curriculum—from those that are appropriate for this lesson, which is one about edges. One of the achievements of this fragment, therefore, is to allow this intention to emerge for the students from the apparently chaotic display of geometrical knowledge, which, though appropriate generally, is not appropriate in this special context. In this fragment, therefore, teacher intentions come to be marked all over the classroom to be perceived publicly. This is so even when the teachers do not state their intention—children’s recognition of round (curved) and straight edges. Like anger, shame, hate, love, and so on, intentions are not hidden psychological facts but are “types of behavior or styles of conduct visible on the outside. These are over the face or in the gestures and not hidden behind them” (Merleau-Ponty, 1996, p. 67). Intentions can be heard and seen in the (discursive, practical) actions of the teachers and in the style of their conduct, where they are available to the children. But intentions to act emerge from the self-affection of the flesh; and only someone capable of the action can recognize intentions, that is, someone who has experienced the same self-affection of the flesh.6 In fact, without apparent intentions, children would not orient toward what we consider to be aspects of the world. This was quite clear in the education of deaf-blind children in the Soviet Union (see Chapter 3), who did not exhibit an orientational reflex when an object was placed in their hands. They did not automatically orient to the objects, “What is this?” or explore the objects through tactile investigation. Rather, these children learned to explore from the hands of their caretakers and educators, who held the children’s hand in their own. Children learned the intention to explore with their hands from the explorations of the hands of their educators.

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There are interesting linguistic dimensions to be coordinated with the perceptual experiences that emerge from this lesson. Thus, for example, the adjective round can be used with different senses, an important one being circumferential, pertaining to a circle. But in this way, Thomas has used the adjective when he describes what he feels to be a “circle.” Another sense is curvilinear, curved, and forming part of a circle; and it is in this sense that the adjective is to be used at this instant in this classroom. A similar differentiation might have been employed to get to the object of the lesson if the line had been described as a line, a straight line, or as linear. There is also a coordination of distinction to be made between what it feels to be touching an edge as distinguished to the feeling of the movement. The differences are not made explicit in this lesson fragment, and it is left to Thomas and the other students to figure out for themselves the difference.

TEACHING: IN FLESH (AND BLOOD) In this chapter, we see three students articulate aspects salient in their own worlds but that are not the ones that need to be standing out in the world of the mathematics curriculum that the two teachers have planned. But it is in and through the interactions that new aspects come to be disclosed to the children in a process essentially grounded on what they currently know. How do such changes in the lifeworld occur? How do the students come to disclose new aspects that add to, expand, and transform their own lifeworlds? The world as I come to perceive it in the course of my experiences, my lifeworld, consists of and is fi lled with things. Things touch other things, and the places where they touch are articulations, joints and distinctions simultaneously. Because the world and its things are externalities, they may appear differently depending on the context—like light may appear as a wave or particle, or value may appear as use-value or exchangevalue. Articulations apparent in my perception can be articulated, that is, named and distinguished in talk. Verbal articulations, therefore, mark and denote (material) articulations in and of our lifeworld. In the course of interacting with others in a world fi lled with things that others use and refer to, children’s lifeworlds change to become increasingly those inhabited by their older fellow beings. Therefore, it is inside his lifeworld, that is, inside his own fi eld of subjectivity, that Thomas develops who he is. “Outside that field, relegated to the external space of perception (and hence other) he has nothing, nor can there by anything that is his own” (Mikhailov, 2001, p. 27). He would be the isolated subject of constructivism, “if he were himself not one of the charged ‘items’ in a fi eld common to all, stuffed with the affective purposes of people’s dealings with one another and continuously recharged by them” (p. 27). In the school classroom, his peers, the books, the physical organization,

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and every other salient aspect is “stuffed with affective purposes” of real people acting in fl esh and blood. Throughout the three fragments with see Mrs. Turner and Mrs. Winter moving, scanting, and engaging with their entire beings in teaching this lesson. When Mrs. Winter reaches out to grab Thomas’s hand and fi nger, it is contact with a living human hand, not just contact between bodies. When Thomas moves up and down the edge of the cube, there is a particular feel that offers itself to him: It feels like a triangle. He would not have needed to reach out and touch and move repeatedly up and down the edge if he had already known what it feels like. In reaching out, ready to be touched in touching—contact always implies the touching with, touching and being touched—he opens himself up to the possible experiences that extend what he can anticipate. In contact, he is affected; and so it is precisely affect that we need to understand in teaching and learning. He has taken a risk in allowing himself to be touched by the edge of the cube. It is the edge that gives itself to him, as he gives himself to it, to allow a new feel to emerge that he can become conscious of. But when Mrs. Winter takes his hand, it is the hand of another human being moving his fi nger along the edge. It is a human hand, already shaped by culture, which now moves his fi nger along a culturally charged “item” in a cultural field common to them. The relevant features of this item have been identified culturally, but, as made salient in the act of teaching, they are not yet apparent in the perception of the student. Human hands exhibit intentions, as do human-produced “noises” (i.e., talk), and children are still in the process of coming to know the intentions that are immanent in actions of others. It is in the contact of another human hand that is contagious, his own hand movement becomes “contaminated,” and in being affected, his affect comes to be socialized. No longer is it just the cube that gives itself, it is the cube and feel given to him in the course of another person’s actions. The movement is not just any arbitrary one, but one inhabited by a reason and intention, even though these may yet have to be disclosed and become apparent to Thomas. He is not just moving his fi nger along the edge, but another human being (living body) moves it with him. His movement comes to be shaped by his teacher’s movements—it is apt to take up not only her intention but the collective knowing and intention that it exhibits and marks. In her hands, his hand is allowed to develop the intentionality for exploring the straight line of an edge rather than any other straight line. Intention and intentionality are not given: they require the self-affection of the flesh, which knows how to move prior to all linguistic forms of knowing of such movement. But these movements do not just come about. In the present instance, Mrs. Winter makes them happen with her own guiding hands that exhibit a particular intentionality to Thomas specifically but to anyone caring to observe generally. The voice that comes with the guiding hand is the voice of another human being. It is not just a mechanical sound: It is a voice addressing Thomas,

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with a particular rhythm, intonation, and various temporal features (see Chapter 7). It continuously modulates itself together with the sound envelope that we can hear as words. At times the intention of the teacher’s talk may not be apparent, and in such situations, we can observe the questioning or consternate expression in and through Thomas’s body. From a culturalhistorical activity theoretic perspective, this voice, as all the other things in this classroom, “are experienced as his own affective purposeful response to the plenitude and wholeness of beings as it addresses him, and as it is cognitively perceived by him and by all the ‘remote organs of his life’” (Mikhailov, 2001, p. 27, original emphasis). It is not just that he is/feels cognitively addressed, but the address is inherently affectively charged by the “affective thought of all those who address him with their own co-thinking and co-feeling, and to whom he addresses his own unique being” (p. 27). It is because of the senses that the flesh is outside of itself, where it is apt to be fashioned by the regularities that it encounters. There are regular features when the teacher speaks, and in the way she counts out the shapes not to be used while moving the hand/arm in beat fashion (see Chapter 7). But these regularities in themselves do not invite investigation of the geometrical object held in front of Thomas’s eyes. There is an investigative intention that children have to acquire. And this they do not do on their own. As the work of Alexander Meshcheryakov with deaf-blind children showed, the investigative intention is learned in the course of purposeful activity, which, in the present case, is the collective learning activity itself. It is a form of orientation that Thomas, Bavneet, and Chris learn in the enactment of the lesson. It is not just a situation between individual students and the teacher. For example, present in the classroom are not just Thomas and Mrs. Winter. They talk, but they do so not just for themselves. Their talk is directed toward the other, therefore involving Mrs. Turner and Thomas’s classmates as audience. In fact, in the course of this lesson fragment Thomas orients himself toward his classmates repeatedly. His gaze turned toward the class seven times during the 2 minute 37 second fragment; and it does so as if he were looking for his peers’ evaluation of what he has done and said. He also uttered “everyone” as a next turn to Mrs. Turner’s question about who has raised the hand regarding the existence of differences between “some edges” and “other edges.” In saying “everyone,” he articulates his perception of others and their responses to the question. What he says, shows, and does is not just for himself, but also for the others generally and Mrs. Winter particularly. And what he says is evaluated, something that he would find out in the course of this fragment if he were not already familiar with this kind of turn of events. In the lesson fragment, we observe how Thomas’s movements come to be shaped precisely because he has a body, which is open to the world to be fashioned. Because of this openness in and to society (corps social), this living body (corps) has the capacity and opportunity to be socialized. The

110 Geometry as Objective Science in Elementary School Classrooms expressions of others become his expressions; the intentions exhibited by the teachers become his intentions. This fashioning occurs as Thomas’s movements follow or are complementary to those of Mrs. Winter. There are like movements, such as when Mrs. Turner moves her fi ngers to the same vertices that Bavneet has touched immediately before or when Mrs. Winter moves her fi nger up and down the straight edge of a cube before and after Thomas has had repeated turns. There are also complementary movements, especially when the teacher’s hands and those of students are oriented to the same object, such as when Mrs. Winter tends the cube, cylinder, and tetrahedron toward Thomas, who works on and gestures in its direction; and such as when Mrs. Turner holds the cube in the direction of Bavneet, who expresses and exhibits the salience of the vertices. These constitute moments of a community of co-movers, and the two teachers precisely function to select among all the movements those that defi ne the good ones (both verbal and gestural) from those that are less appropriate. In our classroom, as in music making, “there is a social world, an organization of ways of doing such movements, and an organization of ways of regarding them” (Sudnow, 1979, p. 4). It therefore is not as we sometimes fi nd in the embodiment, enactivist, and constructivist literatures that the individual somehow develops new insights by metaphorical and metonymical extensions from their bodily schema. First, it is in and with their flesh that they come to inhabit a world that gives itself and that they come to enact. Here, it is precisely the flesh that comes to be fashioned. The children’s bodies become mathematical bodies not only learning movements but also acquiring intentions that underlie these movements designed to displace, feel, point to, or gesturally represent objects. In this way, the children acquire a general style of bodily movement, which comes to be of a complexity that is not easily described by formalisms such as the metaphorization of basic sensorimotor experiences in diagrams that embodiment theorists sometimes use. Rather, as has been said about reading, we “learn to deal with the essential shapes of these figures, acquiring gestural modes of looking—of timing, of reaching for places in contexts of places at eye, or reaching to traverse the text with just the right sort of coverage” (Sudnow, 1979, p. 17). It is precisely these “ways of ‘reading’ that transcend the idiosyncratic requirements of some particular physics arrangement” (p. 17).

5

Coordinating Touch and Gaze Re/Constructing a Mystery Object Mappings have still to be viewed phenomenologically more broadly, but this much can be said, that if mappings present themselves in any geometric context whatsoever, they are fi rst of all mappings of restricted parts of space, which can be indicated or filled by bodies. (Freudenthal, 1983, p. 231)

Many philosophers consider contact and tact as the condition for human forms of knowing to emerge. Whereas we can do without the other senses—beautifully exemplified in the life story of Helen Keller or of Meschcheryakov’s students who became university professors despite their deaf-blind nature—we could not ever do without touch. The other senses come to be coordinated with and enabled when we are with contact, and in contact. Yet, perhaps because we experience the world as one given to all of our senses simultaneously, we tend to be unaware that the senses are not inherently coordinated and that what we “see”—i.e., perceive and understand—is not necessarily what we know when we can merely touch. There is a mapping that we learn when we are in contact with, as Freudenthal writes in the introductory quote, “restricted parts of space, which can be indicated or filled by bodies.” Although I understand the nature of phenomenological investigations differently from Freudenthal, I too use this term to denote my investigations into the problem of mathematics and its relation to the flesh. We tend to assume that gaze and touch work in unison inherently. But this, as this chapter shows, is not the case. One lesson in particular brought out that this is not the case inherently, allowing us to reflect upon the relation between the different sense modalities. For this lesson, Mrs. Turner had placed “mystery” objects (cube, rectangular prism, cone, etc.) into shoeboxes. She had cut a round hole in one side of each box just large enough to allow pushing a hand inside. On the inside of the hole, she had taped a plastic bag so that the children could not “cheat” by looking inside; but reaching through the hole into the plastic bag allowed them to feel the mystery object. Whereas we might assume that it is an easy task to reach and feel the object inside the box to know what it is, it turns out not to be the case. In many student groups on my videotapes, students differ in what they feel when reaching into the same shoebox. For example, Nathan builds a pyramid. He is sure that he felt the bumps of the edges of a pyramid. But in the end, it turns out that there is a cone in the shoebox. Cheyenne, working in the same group, has a little flat

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tree-shaped model rather than a pyramid; and Geena has bits and pieces of plasticine rather than some recognizable geometrical object. Thus, students in the same group may differ in assessing what the mystery object is that they feel. In this chapter, I follow one group of students—Jane, Melissa, and Sylvia—in the process of building (a) model(s) for their mystery object. Their individual models differ initially and what they feel when reaching into the box and touching the mystery object comes to change over time. The purpose of this chapter, therefore, is to push a deeper understanding of the relationship between the sense of touch, the sense of vision, and gestures, on the one hand, and the language students have available for describing their experiences, on the other hand. Here, as in other chapters, I take the transcript to be a protocol of the activity through the lens of the children, who exhibit what is relevant to them for each other and to the occasional adult interacting with them. This is not to mean, however, that their inquiry is unmediated by the adult world, that is, by the current cultural-historical conditions and forms of relations. The objects in the classroom, the discourses of teachers and research assistants, and the language the children have available all mediate what the children can learn and how they learn it. The transcript I use, therefore, constitutes a protocol of the children’s effort to make it through the task. By proceeding in this manner, we eschew the kinds of troubles that characterize the work of Piaget and his coworkers, problems articulated, among others, by Hans Freudenthal.

ON MAPPING In the chapters of Part A, I show and make an argument for the grounding of mathematics in the flesh. I articulate how this position differs from that of the constructivists, to whom the role of the body is an epiphenomenon, a ground for arriving at an abstraction, which, for Piaget, was prefigured by a form of logic inherent in nature and our bodies. As a consequence, he “misses the extraordinary conceptual reality; infants think in bodily terms” (SheetsJohnstone, 2009, p. 368). Mind, to Piaget, is something metaphysical, and the process of abstraction allows us to move from embodied operations to the formal operations in the metaphysical realm; mind comes with the power to think in words or clear images. For embodiment theorists, mappings also do occur. Thus, fundamental experiences somehow are metaphorized— from Gr. meta-, between, and phérein, to carry—literally carried between two domains to take on linguistic/mathematical form in the second. Núñez (2009), for example, shows how during a mathematical proof on the chalkboard a mathematician uses gestures to express the source-path-goal schema that is consistent with the ε-δ definition of continuity. What we observe here is a correlation between a mathematical concept and a linguistically structured explanation; what we do not observe is a necessary condition of the body in the development of this concept. The schema, inherently, already is a

Coordinating Touch and Gaze 113 structure of mind and does not take us back to the original bodily experience, the fact that human beings, as other animate beings, think in movement. The existence of such a condition has to be shown, because everyday experiences often contradict mathematical structures and knowledge so that the question how mathematics can emerge from contradictory experiences.1 That is, in the embodiment literature we do not find studies that investigate how formal mathematics emerges even though mathematical structures contradict everyday experiences. In science education, for example, there is a whole body of literature concerning “misconceptions,” “alternate frameworks,” “naïve ideas,” and other categories concerned with elucidating fundamental everyday understandings and phenomenological primitives. The solution this literature has offered to teachers is this: Design strategies that “eradicate” or make children “abandon” their “naïve ideas.” But the real question is how everyday, mundane experience constitute the very ground of mature mathematical knowledge even though it might contradict the formal mathematics. I return to this question in Chapters 8 and 9. Even if eradication and abandonment were the phenomena at issue, these processes still would have to be mobilized by the person’s mind articulated to be a deficient one. In this chapter, I present a case from a task where groups of students build models of mystery objects inside shoeboxes that cannot be seen but only felt through an opening. That is, the children are engaged in producing/evolving models for something that they cannot see but only touch. This task is of the same nature as the one Piaget and his co-workers used to study the recognition of shapes by means of haptic perception. A child is presented with an object and touches or feels it but is not allowed to see it. In Piaget’s work, the children are asked to name, draw, or point out from a collection of objects the one it feels. He considers the task to require the translation of “tactile-kinaesthetic impressions from an invisible object into a spatial image of a visual kind” (Piaget & Inhelder, 1967, p. 18). There are other mappings possible, however, which do not require the “construction” of visual-spatial images. Thus, in the present instance, one child (Melissa) will come to recognize her first model as inappropriate while comparing original and model by means of simultaneously touching both (one with left, the other with right hand). Here, I focus on the investigation of one group—Jane, Melissa, and Sylvia. Melissa has three models available as a reference among which to select the one best corresponding to the one she can feel, including the one that she has formed herself. The three-dimensional objects are such that they require what in Piaget’s work constitutes a second form of the task, where the child, because of the complexity of the object, no longer can make an identification by simply touching it but is forced to engage in tactile exploration. It is after what he terms “third stage” around the age of 6.5 to seven years that the child supposedly achieves the synthesis of complex forms. Models are of special importance in geometry, because they function as cultural objects that allow us to apperceive the idealities of geometry

114 Geometry as Objective Science in Elementary School Classrooms without having to go through the original processes underlying the formation of their sense.2 Thus, for example, the cube as an ideal object with six precisely square faces of the same size either parallel to one another or at a 90° angle is a historical accomplishment. We do not have to return to these formation processes of the establishment of the idealities, much as we do not have to know about electricity, semi-conductors, and liquid crystals to use a computer with an LCD monitor. It is in this way that sensual models “are embodiments of sedimented significations in the methodical praxis of mathematicians” (Husserl, 1997a, p. 26, emphasis added). Interestingly, therefore, the present task requires children to make a model of a model, that is, their plasticine models are models of objects that are already models of the ideas of cubes, cylinders, spheres, pyramids, cones, and so on. In a reflection on the experience of cubes—which shares a lot with the phenomenological inquiry that I provide in Chapters 1 and 3—Freudenthal (1983) suggests that the “mental image of a cube seems to differ considerably from the visual one prescribed by the theory of perspective” (p. 242). Actual cubes always present themselves under a perspective, but this is not so for the ideal cubes in our imagination. The image of a cube “involves as much as one needs to recognize, to make, to produce, and to reproduce cubes. It includes six faces, though one cannot see more than three at a time and may be unsure about the actual number, four, or six, or eight” (p. 242). Based on this and similar reflections, Freudenthal articulates a strong critique of Piaget’s position on children’s development of geometrical understandings generally and space particularly. He suggests that Piaget “dogmatically interpreted [the] Erlanger program” (p. 231), which he characterizes as a bodice for the adult mathematician and as an oversized suit for the child. The aspects of this Erlanger program “function as blinkers” (p. 232) in Piaget’s laboratory investigation. According to Piaget, ontogenetic development proceeds along a trajectory characterized by the adjectives topological, projective, affine, similar, and congruent, or “from the poor to the rich structures, from large groups of automorphisms to small ones” (p. 232). A model is another material object that reproduces certain aspects of the thing it comes to stand for. It is a reproduction, which, for geometrical objects, is an important feature. They are to build a model from plasticine. The intended task structure is one of a mapping, where the students touch/ feel a three-dimensional object and build a model that is of the same kind but in which size, color, or position in space (i.e., on the table) do not matter. Such mappings belong into affi ne transformations of the type f: [x1,x 2 ,x3] → [α1x1, α2x 2 , α3x3], with the special condition in the current task that α1 = α2 = α3 (or something close to it) so that, for example, a cube remains a cube (rather than becoming a rectangular prism), a sphere remains a sphere (rather than becoming

Coordinating Touch and Gaze 115 an ellipsoid), and so on.3 Here I follow the discussion of the phenomenology of boxes and their transformations that Freudenthal presented in opposition to Piaget’s work. As the analysis featured below shows, each of the three girls in the group has a different model, two of which are “rectangular prisms,” that is, parallelepipeds in which all sides are rectangles (Sylvia, Jane), and one is a cube (Melissa). There is some debate with Melissa to assist her in touching/feeling that the mystery object is more like the rectangular prisms than like the cube she has built. But it is not until Mrs. Turner requires them to arrive at one and the same model—there is only one object in the box to be modeled—and the fact that other groups have already completed the task that the three girls push toward a common model. In the extended episode at the heart of this chapter, therefore, there are repeated occasions where a girl touches the mystery object again, sometimes holding her own or another model in the left hand while touching the mystery object in the shoebox with the other. Melissa comes to change what she senses following a particular instruction that leads her to directly compare her model and the mystery object. The entire episode—marked as such by the beginning of the task engagement to the point where Mrs. Turner moves into another activity, discussing the results of the children’s modeling tasks—is cut up into eight fragments. Three of these fragments precede the one in which Mrs. Turner joins the three girls and asks them to come up with one single model rather than the two (three) they have arrived at up to that point; there are two further fragments preceding the instant when the teacher announces to the entire class that students should read until the few groups are done who still work on their task.

“IT’S A CUBE . . . I’M MAKING MINE A CUBE” Jane, Melissa, and Sylvia are working together on this task. They have received their shoebox containing a mystery object. Just after they arrive at the big round table where they are going to work during this task and where they occupy just one quarter of its circumference (Figure 5.1), Sylvia takes her turn at reaching into the box to touch/feel the object it contains. When she is done, Jane pulls the box closer to her only to push it over to Melissa, who takes a turn at reaching into the box, where she touches/ feels around for about 11 seconds. After having worked her plasticine for a while—in the apparent attempt to make it more pliable—Jane also reaches into the box. Sylvia takes another turn, reaching deep into the box, and then announces that she feels it (turn 001).4 Melissa responds, “Feel it, eh?” and then continues to categorize it, “I have felt it’s a cube” (turn 002). Jane grimaces, as if questioning this description of the object as a cube, and Sylvia announces that “it is not a cube” (turn 004), a description Jane accepts by ascertaining the same fact, “I didn’t feel a cube” (turn 006), followed by a reconfi rmation on the part of Sylvia. A pause unfolds.

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Figure 5.1 Sylvia, Jane, and Melissa (from left to right) sit at a large round table where they investigate a mystery object hidden in a shoebox to build a model of it from plasticine.

Melissa says that she has felt a cube, and then announces a reason: She has checked the sides, “like that,” while showing how she had checked. Holding her cubical model between thumb and index fi nger of the right and left hands, a “caliper configuration,” she gesturally exhibits what she has done to check the shape (turn 009). She shows for each of the three dimensions the same distance between thumb and index fi nger. In fact, she exhibits at this instant symmetry with respect to 90° rotations around two of the three axes through the center of opposite sides. In response, Jane puts her left hand back into the box and Melissa instructs her to “feel around,” continuing after a pause, “to feel it.” Jane continues reaching into the box, and then announces, “if I feel the top it seems like its square, but if I feel the side, it seems like rectangle” (turn 015). Melissa disagrees with her, “I don’t feel a rectangle” (turn 016). The two descriptions contradict: Melissa firmly suggests that she has felt a cube, and shows how she has ascertained the shape, and Jane is equally convinced that the top is a square but “the side” is not. Fragment 5.1a 001 002 003 004 005 006 007

S: M: S: J: S:

i feel it feel it eh? i have felt its a cu:be () <hU:::::::ge.> (1) ((J grimaces, questioningly?)) no its not a cube ((shakes head while rH in box)) () i didnt feel a cube. me either.

Coordinating Touch and Gaze 008 009

010 011 012 013 014

M:

J: M: M:

015

J:

016 017

M:

117

(3) i did. (1) i checked the sides like that. ((Fig., caliper grip on each of 3 sides)) ((puts lH into box)) you should feel around. (2) to feel it (3) ((puts lH into box)) if i feel the top it seems like its square but if i feel the side it seems like rectangle i dont feel any rectangle. (2)

While Jane and Sylvia work on their models, Melissa accompanies her shaping of the cube with utterances, explaining that she is making a cube, exhibiting the cube-shaped plasticine object in her hand. Fragment 5.1b 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032

M: J: M: M: M: M: M: M:

i=m making mine like this. ((shapes it into cube)) i think its oblong. (2) i am making mine a cube (6) if its the if its the same. ((pounds on plasticine)) (2) to make a (??) (11) ((lifts ‘cube’, lets it fall)) h: .h: its alive (4) you see (2) basically the same basically basically (1)

Sylvia then announces what she thinks “it” to be, holding up her model of the mystery object. After a while, Lilian (the research assistant fi lming the group) utters, “What’s this?” to which Jane responds by suggesting that “it is a little bit flat”; and Sylvia shows her model again. But Melissa insists, uttering, while holding up her model, “big cube” (turn 038), and later continuing, “I think it is a cube” (turn 041); in both instances, she emphasizes the category name, “cube.” Lilian points out that “there is only one” and asks, “which one you think is in the box” (turn 039). She encourages the three girls to “touch it a little more” (turn 042).

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Fragment 5.1c 033 034 035 036 037 038

S: L: J: S: M:

039

L:

040 041 042 043 044

J: M: L: J:

045 046 047 048

049 050 051 052 053 054 055 056

L:

M: S:

they=re all like (3) now you feel it? ((holds hands fl at, rubs, Fig.)) feel it feel the side of it how fl at it is youll see (1) see how long it is. (2) its shaped like this oh:: (9) oh you cant feel it like that

M:

(14) ((after feeling for a long while, M turns her gaze to the “cube” ((pulls rH out)) i thi i still think it is a cube.

S:

J: J: M:

057 058

<i think> its this ((holds up her model)). (3) whats this its a little bit flat i think its like this ((holds her model)) i think its like this ((holds, stares at her “cube”)) (0.85) big=cUBe but there is only one; which one you think is inside the box. actually i think i think it is a cU:Be may be you guys need to touch it a little more (3) ((pushes box to M)) feel feel it ((M reaches inside the box)) (1)

Jane pushes the shoebox toward Melissa asking her to feel it (turn 044) and, as Melissa reaches into the box, asks whether she can feel it. Sylvia makes a hand gesture rubbing the palms of her hands together, while saying, “feel it, feel the side of it how fl at it is” (turn 048). Here, the verb “feel” can be heard in two ways, as an instruction on how to touch and as an instruction of what to feel in touching. She thereby invites Melissa to touch and to feel in a particular way, that is, consistent with her own model that lies in front and slightly to the right of her (turn 048). Moreover, in Sylvia’s gesture, we observe the parallelism that underlies the defi nition of parallelepipeds—facing faces are parallel. The gesture therefore captures—much better than the word “rectangular

Coordinating Touch and Gaze 119 prism”—the nature of the mystery object as it comes to be represented in and by Sylvia’s hands. Melissa apparently continues to touch/feel about. Jane watches her intently uttering “see how long it is,” and she continues, while pointing to her own model, “it’s shaped like this” (turn 052). There are longer pauses, while Melissa continues to feel and while the two others are shaping their plasticine, until Melissa announces, “I still think it is a cube” (turn 058). That is, at this point, after having had her right hand in the box for 47 seconds apparently touching/feeling the object inside, Melissa announces again that she thinks it is a cube.

“A TYPE OF RECTANGLE . . . I HAVE IT AS A CUBE . . . IT’S THE SAME EVERYWHERE” In response to Melissa’s announcement that she still thinks the mystery object is a cube, Sylvia proposes to check and reaches into the box. Operating the camera, Lilian asks Melissa why she thinks it is a cube (turn 060). Using the same iconic gesture as before, the caliper grip, Melissa says “it’s the same” and “it’s the same everywhere” while holding the cube between the thumb and index. As she talks, Melissa rotates the cube and exhibits the caliper configuration once for each of the three dimensions of the cube (turn 061). It is a cube, because the dimensions shown by the constant caliper configuration are the same everywhere. But her two peers disagree, announcing that they do not consider the object to be a cube (turns 067) or reaching into the box (turn 063). Fragment 5.2a 058 059

M: S:

((pulls rH out)) i thi i still think it is a cube. ((S picks the box, turns it, reaches in)) let me check

060 061

L: M:

062 063

S:

064 065

M:

why do you think it is a cube. cause like the same ((turns cube and has caliper grip with thumb/index Fig.)) (1) its the same () everywhere (2) where is it ((reaches into the box)) () i cant feel it now () ough (2) i say a cube ((gazing at cube, then at S, who still feels)) (3)

066 067 068

J: M:

i dont think its a cube. i think a cube. ((S still with lH in box))

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069 070

M:

(4) i need me to measure it ((tosses cube in hands, gazes at it))

Melissa suggests making a measurement (turn 070), but Sylvia invites her to touch the object again and to touch it in the way she concurrently shows with her hand gesture (turn 071), a caliper configuration. But in Sylvia’s caliper, the thumb comes to stand against the other four fi ngers, forming a plane. In this configuration, the gesture bears an iconic relation to her own model, where a large “top” area contrasts a rather low height of the rectangular parallelepiped (“rectangular prism”). Throughout the remainder of this Fragment 5.2b, Melissa insists on her description of the mystery object as a cube: “I have it as a cube. It’s a cube. I think it’s a cube” (085). But Melissa and Jane resolutely maintain that the mystery object is not a cube. Whereas some readers may get the sense from the transcription that Melissa is opposed merely for playful purposes, the intonations and the seriousness in which the three engage each other as available from watching the videotape contradicts such a hearing. Fragment 5.2b 071

S:

072

S:

073

M:

074 075 076

M: S:

077 078 079 080 081 082

M: S: J: M: L: S:

083

L:

why dont you come here touch () touch it (1) feel the flat parts ((rubs palms of hand as in turn 048, M reaching, gazing at M)) and you touch it like this ((makes the caliper, Fig.)) () youll see. <bt i still think its> ((reaches rH into the box, feels about)) (6) yea (1) its a cube ((reaching into box) <i dont think its a cube> he? i dont think its a cube i dont i DO why you dont think it its a cube. because it has flat faces () so it has rectangular parts ((picks up model, holds, turns)) like thi:s yea

Coordinating Touch and Gaze 121 084

S:

085

M:

like this ((holds model in lH, runs rH palm)) and it has squares here ((strikes with palms)) () a[nd ] [i th]ink a type of rectangle (1) a type of rectangle ((gazes at S)). () i have it as a cube. ((turning to J)) its a cube. () i think its a cube. ((squarely addresses J, who gazes back; J has hand in box, feeling))

In both the iconic caliper configuration and in the description of requiring a measurement, Melissa articulates yet unused resources: use of measurement. Measurement takes us in fact back to the beginning of geometry: It has been the phenomenological origin of geometry as a science. Yet Piaget—in the attempt to follow the Erlangen program of Felix Klein concerning topology as the organizing framework for the human conception of space—insisted on measurement as following spatial relations. But, to do justice to Piaget, the rotational symmetry Melissa exhibits to show that the three orthogonal edges are of the same length does fit the Erlangen program of characterizing geometrical knowledge in terms of group theory. However, in the caliper configuration that Sylvia produces, spatial relations is prefigured, contextualized by the configuration of the four fingers in the form of a plane surface, indicating the measurement to be taken as the height over the plane. How precisely measurement emerges from such gestures is the focus of Chapter 6. “I STILL THINK IT’S (NOT) A CUBE” Following Melissa’s last announcement that the mystery object is a cube, that she thinks it is a cube, Lilian asks her about the defining characteristics of a cube: “What does it have to have to be a cube?” Melissa delineates, “It has to have the same sides and faces” (turn 089) and, following a query about whether this is what she felt, says that she “felt all around it the same . . . as my cube” (turn 092). That is, she again expresses sameness under rotation or properties of the octahedral symmetry group. In response, Jane pushes the shoebox in front of her, an invitation that Melissa accepts by putting her hand back into it. Jane invites her to feel the side that is facing at her (turn 095) and Sylvia comments that “it can’t be a cube” (turn 096), a statement Jane confirms by affirmation. Melissa states again her opposing description. Fragment 5.3 086

M:

087 088

L:

[i th]ink a type of rectangle (1) a type of rectangle ((gazes at S)). () i have it as a cube. ((turning to J)) its a cube. () i think its a cube. ((squarely addresses J, who gazes back; J has hand in box, feeling)) (1) what does it have to have to be a cube.

122 Geometry as Objective Science in Elementary School Classrooms 089

M:

090 091 092

L: M:

093 094 095 096 097

M: J: S:

098

J:

099 100 101 102 103

M: S: S:

104 105

J: S:

106 107 108

L:

((gazing at cube, turns in hands, caliper grip)) has to have the same (2) sides. (1) <and faces.> (2) and did you (2) [feel ] [i felt] all () around it the same ((points to different faces)) (1) as my cube. (4) ((J pushes box to her)) lets see: ((rH into box)). ((lH into box)) feel feel this side thats pointing at you <it cant be a cube.> ((J works on plasticine, S & M look at the object in her hand)) <i still think it is not a cube> ((gaze at model in S’s hand)) (3) ((rH in box)) think i think its a cube. <i dont think so> (5) i think its this shape ((holds up a thin book, lH, rH covering lower part of it)) you cut it down a bit its like this ((places lH as if with knife at some part on the book, hidden by box)) (1) do you think its like this. (4)

After a longer pause, Sylvia holds up a thin booklet, and, covering its lower part with the four fingers of each hand, announces that the object looked like it. Jane agrees and adds that it, the mystery object, would look like the booklet if the bottom half were cut off from it (turn 103). Sylvia gestures such a cutting by holding the palm of her hand vertically (its normal vector parallel to the table), as if she had a knife in her hand with which she is cutting the booklet in about half. But Melissa still announces that she thinks it is a cube. In this fragment, we fi nd a continuation of the debate, Melissa insisting on a cube-shaped nature of the mystery object, and the other two holding that it has a different shape. Sylvia produces another model, and Jane affi rms that the mystery object has the shape indicated in the iconic gesture. In these gestures, they exhibit different symmetries than the one that Melissa articulates. The two elaborate this new model, which has approximately the same shape as Sylvia’s plasticine model (somewhat flatter). They have produced a transformation of the type f :[x1,x 2,x 3 ] → [α12 x1, α 22 x2, α 32 x3 ],

where the coefficients are all the same, but different from the one that characterizes the transformation into the plasticine model. It will turn out—in the

Coordinating Touch and Gaze 123 discussion that follows this episode and that I do not represent here (similar to the one where Bavneet talks about the cube as featured in Chapter 4)—that Sylvia’s model is identical in size when compared to the mystery object. That is, in her transformation α1 = α2 = α3 = 1. Here, Sylvia and Jane, together, have produced yet another transformation of the mystery object, this time into a model of a different material (plasticine) and of different size, but maintaining the same relation between the three dimensions. That is, although the alphas are the same within the first and second models, that is,

α11 = α 12 = α 13 and α12 = α 22 = α 32 they are different between the two models. But their relations continue to be the same so that α 11 α 12 α 13 = = α 12 α 22 α 32

“SO YOU STILL THINK IT’S A (SQUARE) CUBE?” Melissa now appears to pursue a different tack. She announces to be making pretty edges while pinching the edges of her cube and crimping them. She laughs. But Jane suggests putting her hand into the box and gives instruction to “try to fi nd the long face” so that she “can feel the square face”; and Jane continues that this would allow her to “feel each side.” Melissa exhales strongly, making a face, as if in exasperation. She appears to feel about with her right hand in the shoebox while Jane gazes intently at her face, as if in expectation of something to happen.5 Fragment 5.4a 109

M:

110 111

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112 113 114 115 116

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im making pretty edges ((pinches edges)) () hī hī hī: ()((edges with pinches)) i am making it like this ((J gazes at her object, then moves to get closer look)) () hī hī hī () .hhī (3) <

hold your hand in there> fi rst like try to fi nd it there try to feel the long face () so you can feel the square face. and then you can feel each side on it. () <<exasperated>.h h:::> (9) ((M feels about the box)) <

yea> () ((M feels, J gazes at her face, as if in expectation of something to happen))

124 Geometry as Objective Science in Elementary School Classrooms 117 118 119 120 121 122

J:

J: M:

feel it at its sides again. ((M gazes at her cube)) (8) ((M begins to fiddle with model)) so you still think its a cube? ((nods)) uh hm () i sort of like i still think its a cube. (9) ((J reaches into box, lH, shrugs shoulders))

Jane asks Melissa if she still thinks the object to be a cube, as if testing whether her “teaching” (scaffolding) has brought about a desired effect: for Melissa to recognize the mystery object as something different than the cube as that she is currently modeling it. Melissa responds that she “still think[s] it is a cube.” Jane reaches into the box, which moves a bit as she feels about, then shrugs her shoulders making a face as if in disagreement with Melissa. Fragment 5.4b 122 123 124 125 126 127 128 129 130

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131 132 133 134 135 136 137 138 139

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(9) ((J reaches into box, lH, shrugs shoulders)) i think its this ((brings model to mouth, J gazes at her)) (10) so its (2) flat ((pads it)) () <h::> so what else we supposed to do now well when you are fi nished you have to do a () read a book ((points to M’s object)) ((M rH in box)) okay (mea? un?) (2) she did ((pointing to another group)) (25) ((orientation to another group; S reaches into box)) i still think it is a square cube ((gazes at her cube, S still feels)) (7) <yes:> () what yes: () this ((rH holds up her model, lH feels, she gazes into space before her, J looking at her)) (3) ((approaches group)) have you all agreed what your object is?

ARTICULATING THE OBJECT/MOTIVE Up to now, the three girls have contrasted the different models that they have built. When Mrs. Turner arrives at their table, she articulates the differences between the models and suggests that there is only one object in the box. She points out that as a group they have to fi gure out which

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125

one of the three models most accurately corresponds to the object in the box. The fragment begins with Mrs. Turner’s question whether they agree on the nature of the mystery object and then invites the children to articulate what they “think” is in the shoebox. Melissa describes her model as a cube, Jane categorizes hers as a “rectangular prism,” and Sylvia says that she has forgotten the name (turn 150). After a pause, Mrs. Turner says that it “will be a rectangular prism as well” (turn 152), and thereby categorizes the two models the same. She does so again when she lines up all three objects, placing Sylvia’s and Jane’s models next to each other and setting Melissa’s a bit apart. She says, “this is a cube and these are rectangular prisms.” That is, although the two models have different proportions, suggesting different factors αi in the transformation—in one model at least, the three alphas are not the same—they are also members of the same category, “rectangular prisms.” Fragment 5.5 139

T:

140 141 142 143 144

J: M: J: S: T:

145 146

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147 148 149 150 151 152

J: T: S: T:

((approaches group)) have you all agreed what your object is? yea we we ((turns about toward T)) we think its this okay () okay take a piece of () ((pushes box aside)) lets take a look you all think it[s [cube] okay melissa thinks its a cube ((points in direction)) what do you think it is jane a rectangular prism ((:S)) and what do you think it is ((points)) (3) <

oh () i forget now> (3) this one will be a rectangular prism as well ((turns the object)) okay ((takes object from J)) bt lets take a look ((lines up all 3 objects)) (3) they all look like different types of something this is a cube ((points)) and these are rectangular prisms right? ((places S’s and J’s objects together, Fig.))

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153 154

M: T:

155

M:

156

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157 157 159

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160 161

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162

M:

.hh .hh ((reaches for box)) but there is only one object in here but we are trying to () i think we are () it all has () the same ((makes caliper grip)) ((has taken box)) i cant fi nd it just a minute (2) you dont really () oh there it is () okay (3) i=m gonna take this i think it is () let me just touch it to make sure () okay i touched it now (2) you as a group need to figure out what it feels more like () does it feel more like ((holds up S’s)) more like a rectangular prism like this ((picks up J’s)) like this or ((picks up M’s)) more like a cube ((holds it up)) i think it is a cube ((reaches for her object)) i (??) you have to come together and decide what its gonna be because its only one object in here ((pounds on box, gazes at S)) right? yea and as a group you have to say () WE think it is () because okay i=m really thinking its a cube ((gazes at her object))

Melissa laughs (turn 153), and—in response to the teacher’s statement that there is only one object in the shoebox—reiterates: “it all has the same.” At the same time, she repeats the caliper confi guration that she already has articulated before (turns 009, 061). That is, she focuses on and exhibits for others the rotational symmetry she feels, which is exhibited in her model. In the subsequent turn, Mrs. Turner then suggests to be touching the mystery object herself and invites the group members to come to a consensus, whether it “feels more like a rectangular prism or more like a cube” while holding up Sylvia’s and Melissa’s models, respectively. Melissa again announces that the mystery object is a cube. Sylvia says something (inaudible on the videotape); and Mrs. Turner, while hitting the shoebox, suggests that they have to “decide what it’s gonna be, because it’s only one object in here.” She continues by emphasizing that they have to come to a group decision (“as a group,” and “WE think”). In this case, Mrs. Turner marks the two rectangular prisms as different “more like this,” holding up Sylvia’s model, “or more like this,” holding up Jane’s model. Then she marks the two as different from Melissa’s model, which she denotes by the term “cube.” Mrs. Turner has already articulated previously that Sylvia’s model also is a rectangular prism, the same name that Jane has used for denoting her own model. That is, Mrs. Turner articulates that there are differences between the models, which thereby are different models of the same object.6 But because there is only one object in the shoebox, only one kind of model is allowed.

Coordinating Touch and Gaze 127 The teacher works with students so that the object/motive is changed from the one they have pursued, each making a model, to each making a model that is consistent with all the other models in this group. That is, much as the intentions underlying movement and sensing, which are the result of the auto-affection of the flesh, the intention embodied in the activity is not inherently given but is something that the children have to realize in and through their actions. Whether they actually do so is an empirical matter. More importantly, this collective intention actually gives sense to the goal-directed, intended actions (Leontjew, 1982). But it is only through such intended actions that the motives of activity come to be realized. In not distinguishing between the two types of rectangular parallelepipeds (“rectangular prisms”), Mrs. Turner articulates the motive of the activity to be a recognition as a parallelepiped rather than a particular transformation, one in which the proportion of the different dimensions is maintained. In her practice, it is the geometric category—rectangular prism versus cube—that is of importance rather than the particulars of the transformation. In fact, there is an interesting contradiction at the heart of this event. As can be seen in the image accompanying turn 152, there is a difference between Sylvia’s model and Jane’s model. In the end, and unbeknownst to all involved at the moment, Sylvia’s model is nearly identical with the mystery object in all dimensions. All three factors α of the transformation will be not only equal but also identical to 1. But in Jane’s model, the factors clearly are not identical, as her model differs from Sylvia’s not only in size but also in the relation of the different sides. That is, the multiplication factors α i were different from each other α 1J ≠ α 2J ; α1J ≠ α 3J ; α 2J ≠ α 3J .

This, too, is the case for Melissa. That is, α 1M ≠ α 2M ; α 1M ≠ α 3M ; α 2M ≠ α 3M .

But, in the discursive practices that Mrs. Turner exhibits, the factors are sufficiently similar in Jane’s case so that the resulting transformation still yields a “rectangular prism,” whereas in Melissa’s case, the factors for the three dimensions are clearly different leading to a cube. Now a cube also is a rectangular prism, though one of a special kind and with different symmetries, here exhibited in the gestures and rotations that Melissa performs. But in this classroom, there is a distinction between the two types of objects that is encouraged throughout the lessons (see, e.g., Chapters 8 and 9); and this distinction is one of the conditions for the students’ bodies to develop socialized flesh rather than individualsubjective mental constructions.

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“THE SAME EDGES, AND THAT FEELS SQUARE . . . I STILL THINK IT’S THE SAME AS MINE” Fragment 5.6 begins as the earlier fragments have ended. Melissa insists on the mystery object being a cube, as in her model, whereas Jane and Sylvia suggest otherwise. Jane then reaches back, even pushing both hands into the hole of the shoebox, saying that she “can feel it,” and describes what she can feel: “flat top squares and sides flat” (turn 173). She pushes the box over to Melissa, who again reaches into the shoebox with her right hand while holding on to her plasticine model in the left, turning it between her fi ngers. She appears to be asking whether it matters if the model is exactly the same (“right as thing as”), then suggests that “this is actually a flat cube,” and concludes that she “think[s] it’s the same” (turn 175). Jane responds after a while: “They are not the same” (turn 177). Here, Melissa offers a possibility for transforming her earlier description. On one hand, she still uses the word “cube” to classify the mystery object, but, on the other hand, she describes it as a “flat cube,” which is one of the descriptions that has been used previously in public discussions, including on the previous day when Chris was talking about the pizza box as an instance of a flat cube (see Chapters 1–3, 4). Fragment 5.6 162 163 164 165 166 167 168 169 170 171 172 173

M:

J: M: S: M: J:

174 175

M:

176 177 178 179

J: M: L:

180

M:

S: J: M:

i=m really thinking its a cube ((gazes at her object)) (1) i dont think like janes (??) (1) ((:S)) do you think we need it flattened i=m mak think it is this though (14) ((J reaches into box, feels)) ((???)) ((leans to her, bringing ear closer)) what? do you think its that one? ((points to J’s)) i think its this ((gazes at her own)) i can feel it ((gazes at M)) () with fl at top squares and sides fl at ((both hands in hole)) (3) ((J pushes box to M)) is it matter; has to be a right as thing as ((holds up cube)) because this is actually a flat cube ((gazes at cube, lH; feels in box with rH)) yea:: () .h ha ha () i think its the same (2) they are actually not the same sEE it has to be the same ((M gazes at L)) so what you do with your plasticine has to be like what you have in the box i feel i feel like the same and it is still flat

Coordinating Touch and Gaze 129 181 182 183 184

M:

185 186

L: M:

187 188 189 190 191 192 193 194

L: M: L: M: L: M:

J:

S:

(1) no it is not ((mutual gaze)) (3) and it has ((holds cube, gazes at her cube)) and it has the same the same what? the same edges. () ((turns her cube)) and () that feels () square (1) ((feels about)) and it has ((J approaches her object, M gazes at it, looks at her own)) (1) and this is ((gazes)) it has same it is a it is the same ((pushes back box)) all the faces are the same? uh hm did you check all around? ((nods vigorously twice)) and they feel all the same? i still think its the same s=mine (4) ((S reaches lH into box)) it isnt the same

At this point, Lilian enters the discussion saying that “it” “has to be the same,” where it is not evident that the three iterations of “the same” refer to the same entity. Lilian, having been present during Mrs. Turner’s instructions, repeats the conditions of this task: Students in a group have to have “the same” model—the same despite their differences. Melissa may have suggested that the two models are the same, because a flat cube still has something in common with a cube (a “flat cube” still is a cube modified by an adjective), but Jane figures them to be different. As the conversation unfolds, it becomes clear that Melissa is talking about the three sides of the object as being the same, as in her cube, which she is holding up in response to Lilian’s question (turn 185). Lilian asks whether all the sides are the same, and Melissa not only confirms this but also states that “it” “is the same” as expressed in the model that she has built. She then finds her assessment contradicted by Sylvia, who insists that “it isn’t the same” (turn 194). “WE DON’T CARE ABOUT THE SIZE IF IT’S THE SAME” In this fragment, both Sylvia and Jane reach into the box and comment on the fact that it, Melissa’s model, is not the same as the one they are presently touching—as Melissa understands what they are saying. Jane takes Melissa’s model from the latter’s hands, and then, while holding the model in her left hand, reaches into the box with her right hand (turn 200). Melissa suggests a conjunctive reason (“’cause”) concerning the size of her object, but Jane rejects the reason, as “we don’t care about the size” and adds, “it’s the shape” (turn 203). She places the open palm of her left hand on the top of the cube, then rotates the cube and feels the side now on the top. During this time, the other hand is in the box; Melissa gazes at her intently. Jane then announces, “It does

130 Geometry as Objective Science in Elementary School Classrooms not feel the same,” thereby describing in words for the others what she has done with her hands: compared the two objects. That is, she has touched/felt the object, rotated it, and has touched/felt it again. When she says it is not the same, she in fact articulates that the mystery object is not the same following a 90° rotation. Jane articulates evidence of the different rotational symmetries that the two objects belong to. In fact, we can see in Jane an articulation of an orientation with her entire body, one hand intending to feel the mystery object, the other hand feeling the model visibly in front of them. The attitude is one of investigation as part of which two objects—one of which is proposed as a model of the other—come to be compared. But Melissa insists, “It’s still the same as mine, it’s a cube” (turn 211). Her statement again meets the opposition of both Jane (turn 213) and Sylvia (turn 214), yet Melissa continues to insist, the vehemence of her response expressed in increased speech intensity and pitch (turn 215). At this point, all three continue to press on their models, working the plasticine, as if attempting to increasingly approximate the surface to that of an ideal object. Sylvie pulls the box again, reaches into it and apparently feels the object. Nearly a minute goes by in this manner, when all of a sudden Mrs. Turner announces that those who are finished should read while the others complete their task. Fragment 5.7 194 195 196 197 198 199 200

S: M: S: J:

201 202

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203

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204 205 206

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207

J:

it isnt the same (4) fEEL it ((reaching her object toward S)) (7) <no it is not the same> (1) see can i feel it ((grabs object, reaches into box rH, holds object in lH)) (1) cause its small ((gazes intently)) that one is smaller ((pointing to hers)) but we dont care about the size it is the shape (3) ((gazes in air, lH flat on cube, rotates it 90 degrees, feels new top, Fig.)) this one (4) ((pointing to top square)) it does not feel the same that it could be ((takes cube back, presses on top)) ((still with lH in box)) no not the bumps

Coordinating Touch and Gaze 208 209 210 211 212 213 214 215

M: M:

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131

(3) ((J pushes box back toward M)) cause its the same exactly as mine (1) its still the same as mine ((Jane gazes at cube)) (1) its a cube () actually it sort of () cause a cube is (?) ((:M)) yEA cubes arent ((J gazes at her)) (??) <yE::S it is>

In this fragment, Jane evolves a direct method of comparing the model with the original. Up to this point, the three students tended to have either worked on their object or reached into the shoebox to feel the mystery object. In doing both actions simultaneously, the body becomes a living (literal) link between the original and the model, the source and target domain. The mapping f :[x1,x 2,x 3 ] → [α12 x1,α 22 x 2, α 32 x 3 ]

is symbolized in the living body that makes a connection. But this living body is one only because of the tact that is spread throughout, making the feeling in the right hand and the feeling of the left hand the feeling of one and the same person. This unity of the tact is characteristic of the flesh. By feeling each side of both objects, Jane can sense the different multiplication factors that exist between what she has in her right hand, the mystery object, and what she feels in her left hand and has visually available. Whereas Piaget is worried about the size in children’s understandings of affi ne transformations (parallelograms, as implemented in the Nuremberg scissors), Jane here announces that size does not matter. It is the shape. Piaget considers it to be a shortcoming of the child who draws consecutive stages of the scissors with different lengths of the sides. But then he might marvel at these children concerned with preservation of the symmetry group, which an object of a particular shape has independent of its size. In this sense, the models Jane and Sylvia built are the same, for they both belong to the dihedral D2h symmetry group, whereas Melissa’s model exhibits the octahedral symmetry Oh.7 THE CUBE DISAPPEARS FROM THE HAND The teacher has suggested that several groups already finished their task. Only a few groups are still working at their models. Sylvia announces that they are not done but that they “still have to figure out” (turn 220). As before, Melissa repeats her position: “I think it is a cube” (turn 221), but hears that Jane is not convinced and responds, “I think it feels like one

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though” (turn 224). Sylvia then proposes to her a way of testing; she articulates what Jane has done before, that is, directly comparing the original with the model. Fragment 5.8 220 221 222 223 224 225

S: M: J: M: S:

226

M:

227 228

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229 230

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231

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232 233

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234 235 236 237

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guess we are not done yet we still have to figure out i think its a cube (1) ((looks at her model)) what () i will i think it feels like one though. okay. look what you can do. you can take your object and () and feel like with that one ((hits box l) and take it as you feel like its really the same. ((shakes each hand separately)) ((reaches for J’s object)) can i feel with this to see its the same? ((reaches into box, rH, holds J’s object, lH)) (2) no i (want?) you to take here in your hand ((holds her object)) (3) feel this part ((touches square face)) and go in here ((touches the box)). ((places object on narrow edge, turns it over 90 degrees at a time, touches with index fi nger; J watches intently)) (11) oh oh ((lifts gaze)) ((turns over J’s object)) (5) ((M makes bunny mouth, returns object to J, grabs her own)) i=m making it else ((begins to reshape her object)) (3) so you can just make it like this ((shows her own)) ((M makes her object to resemble that of J and S.))

Sylvia presents the instruction what to do not only by verbal means: “You can take your object and feel like with that one . . . and feel like it is really the same” (turn 225). But, while doing so, she holds her own model in the right hand and hits the shoebox with her left hand before making a “feeling” movement with it (Figure 5.1). In response, Melissa asks Jane for her model to feel with it and to see whether it is the same (turn 226). She holds Jane’s model in her left hand, while reaching with her right hand into

Coordinating Touch and Gaze 133 the shoebox. Sylvia instructs Melissa how to hold and manipulate Jane’s object using her own model as an example. Melissa touches different parts of the model with her left hand, intently gazing at it. Jane gives further instruction where to feel, directly pointing to the surface and touching it with her flat open palm (turn 230). Melissa turns the model over from resting with its large face on the table, then turns it so that one of its narrow edges comes to rest on the table. At this point she begins to rotate the object by 90°, placing her index finger on the narrow face (turn 230). She rotates the model another 90°, again placing her finger on the narrow face. She repeats these movements five more times so that she has nearly turned it over twice in its entirety. What she does is in fact exploring similarity under a particular kind of operation, 90° rotation. If the four narrow sides were the same, the mystery object/model would be a “flattened cube,” that is, a square prism with a dihedral D4h symmetry; if adjacent pairs are not the same, the resulting object/model would be cuboid with a dihedral D2h symmetry. Melissa raises her head, puckers her lips, returns Jane’s model, begins to grin, picks up her own model, squeezes it, then begins to work it into a different shape. She formulates8 the intent of her actions: “I’m making it else” (turn 235); and Sylvia suggests a little while later, “So you can make it like this” (turn 235). At this instant, the cube has emerged for her in her touch. She announces the surprise, first in the interjection, then in her face, and then by means of the action with which she begins to change her model. About one and one half minutes later, Melissa’s model has approximately the same shape as the models of her peers. In this final part of the episode, the relationship between Melissa and the natural object in the shoebox changes. Before that moment, she feels it and names and describes (verbally and gesturally) what she has felt as a cube and explains—using the “caliper” gesture—that she feels all sides to be the same. She states that a condition of the cube is that the edges and sides are the same. But what she senses changes when she directly “compares” what she has in her right hand, the mystery object, with the object she has in her left hand, Jane’s model. It is this object—which she can feel and see—that mediates her relation to the one she only feels, hidden from sight in the shoebox. That is, in this situation the ultimate shape of the mystery object that she holds in her hand does not emerge as the result of an immediate perception but it is one that is mediated by another artifact that another student had made in the hereand-now of the lesson. But the object they modeled is itself not a geometrically ideal one but a fabricated object that has been denoted by the category name “rectangular prism,” initially by Jane and confirmed by the teacher.

ON INTENTIONS AND REPRESENTATIONS By taking Jane’s model and reaching into the shoebox to engage in a comparison, Melissa exhibits willingness to open up and learn. She does exhibit surprise and changes her own model immediately after this expression. At

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a minimum, she has had to open up to an experience that she initially held to be other: feeling the mystery object as something else than she has done before. She opens up to be impressed, and the comparison constellation allows the nature of what she senses to change. In reaching into the shoebox, Melissa exhibits the capacity to move her arm, hand, and fi ngers to engage in touching. Her living/lived body already knows how to move, which is the precondition for intending to reach out and learn by sensing. But learning by sensing also means that the flesh is ready to be impressed in and by the contact with a specific section of the material world around her; and she is impressed by what she feels, affected by the way in which the object responds to her pressure. And the resulting affect is evidenced in the articulation of surprise. This shaping has occurred, involving culture in different places. On one hand, there is the mystery object, itself an objet trouvé that the teacher has placed within the box and reachable only through a plastic bag. Yet these objects in the shoeboxes are not quite objets trouvés, for in this classroom, they stand for the idealizations of geometry: these are the cubes, spheres, prisms, pyramids, and so on that scientific geometry is about. On the other hand, culture, in the form of a categorized fabricated object, Jane’s model, allows Melissa to learn to feel. It is as if Jane’s hand were guiding here, though it is “only” the rectangular prism that Jane has produced. Melissa’s feeling with the right hand is mediated by the left hand, itself touching an object that is the result of Jane’s touch. This object is not just some lifeless product, it is a product in which Jane’s sense has come to be crystallized and found objective form. This allows Melissa to learn coordinating the unseen object with one that is visible, and in the process, her sense of touch comes to be shaped. Something else happens in the episode, though not emphasized as such in my description. Throughout the entire time, the three girls work the surfaces of their plasticine models in apparent attempts to straighten out the bumps, to make them smooth, to shape and polish their surfaces so that the “human-made” models increasingly come to approximate the ideal geometrical world. This strikingly resembles the retrospective account of how the primal geometers polished artifacts to align their surfaces with the emerging ideas of ideal geometrical objects (Husserl, 1997a). In fact, the increasing capacity of working and polishing surfaces and the increasing idealization of their objects emerged hand-in-hand. My videotapes, therefore, exhibit the efforts of artisan-engineers already attuned to a science, geometry, which they are only in the process of learning, a science that deals in idealities rather than in objects with bumps and unshapely edges. Idealizations are the end product of continuous refi nement processes that work on entities of our mundane, everyday experiences. Tact, the sense of touch, is characteristic of the flesh rather than of bodies, as all flesh also has a body but not all bodies are flesh. Moreover, bodies do not have intentions, which are the results of auto-affection experienced by the flesh in movement. Without this intention underlying the reach of the hand to

Coordinating Touch and Gaze 135 contact the mystery object, the sensorimotor intentions that are so prominent in enactivist accounts would not be able to emerge and exist. As described in Chapter 3, such intentions do not exist in deaf-blind children, where they have to be actively taught. But such sensorimotor intentions do not merely arise in the auto-affection of the flesh; it is in understanding the collective motive, which gives intention to the activity as a whole, which gives sense to and calls for the intentioned sensorimotor exploration of the mystery object in the box. What the three girls learn is, therefore, the intentional motive in this form of activity that their participation in this lesson realizes. The episode presented in this chapter allows us to reflect on the position Piaget takes with respect to the recognition of shapes. Perception, to him, is the knowledge of objects that results from direct contact. The concepts of representation or imagination pertain to instances where some object is imagined or when it is present in parallel to perception. Representation itself “is a system of meanings or significations embodying a distinction between that which signifies and that which is signified” (Piaget & Inhelder, 1968, p. 17). The image is held to be an imitation that is internalized, resulting from motor activity, “even though its final form is that of a figural pattern traced on the sensory data” (p. 17).9 The mental image, because of this dual nature, tends to move back and forth between the two characteristics. We do not know what is in the mind of Melissa, whether she has any particular visual image in her mind that she looks at and compares to the mystery object. But she has built a representation, a cube, and she denotes the cube as cube, specifying its characteristics both in words and in measurement terms. That is, we see her compare a model (image) in the left hand with the target object in the right hand rather than forming an image in her mind. Even when she explicates why she thinks the mystery object to be a cube, she uses her own cube together with gestures as a means of articulating her “idea.” It is as material as the mystery object itself. Melissa does not merely indicate visual criteria for a cube, which might be articulated in the words “same edges” and “same faces,” but she also provides gestural representations of what is to be done to ascertain that an object is a cube. She produces the caliper configuration with her hand and moves it to the different edges corresponding to the x, y, and z axes in a mathematician’s spatial representation of her model. Because the edges are the same following 90° rotations of the model around two of these axes—it does not change under the corresponding transformation—the object has to be a cube. She suggests to have done it to the mystery object as well. Her articulations, which are consistent with the model directly available to vision and touch, turn out to be inconsistent with the mystery object once she has had the opportunity to directly compare it with another model of different shape. Piaget appears to be right, if only with respect to the importance of experiences that can be described in terms of symmetry groups. As the episode unfolds, we can observe the different types of representations that the three students evolve. At the very end, it becomes obvious that the models the three had built when Mrs. Turner fi rst approaches

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their table constituted a Euclidean congruent representation (Sylvia), an (approximate) similarity relation (Jane), and an affi ne relation (Melissa). Congruence allows change in position and orientation, whereas other characteristics must not change, preserving angles and lengths of geometrical objects. That is, Sylvia’s model embodies Euclidean congruence, which preserves the object under translations, rotations, and reflections. The other two models embody affine congruence, since the lengths of the original are not preserved in the models. The idea of congruence is itself a culturalhistorical achievement that arises with the limit figures of geometry, that is, with ideality, for in empirical praxis there is not absolute precision. Only ideal figures are exactly self-same: Only ideal cubes have exactly the same extensions, 12 edges and six faces. In the real lived in world, such objects of experience do not exist. The discovery that brought geometry into being was that these “elementary shapes, singled out in advance as universally available . . . because of the method that produces them, are intersubjectively and univocally determined” (Husserl, 1997a, p. 26). This method includes the provision of accounts. The objectivity of geometry, its intersubjective and unequivocal nature arises from the very methods already available in the children’s everyday manner of holding each other accountable. Even experiences are reproducible, even if mediation is required, such as when both Jane and Melissa use a second, visibly available object to compare it to the mystery object that they cannot see. In fact, in this comparison, the existing models serve precisely the same function as they do in scientific research: to be tested for their viability during experimentation. Iconic representation does not require identity of the relative dimension, yet the cube would be excluded as an iconic representation of the parallelepiped in the shoebox. Melissa’s model certainly is a symbolic representation, even including some iconic ones as well, for example, rectilinearity of faces, orthogonality of adjacent sides, and parallelism of opposing sides. This kind of reproduction falls under a class denoted by the adjective “compository” (Freudenthal, 1983, p. 244), because it is combination of iconic parts and accounts for flexibilities. It is as important to geometry for reproducing objects as other forms of reproduction. These different forms of representation do “not at all bear witness to defective mental objects” (p. 244). This episode develops over what might be considered a long period of time considering a relatively simple task. There is a temporality at work, which is produced in and through the actions of the children. In fact, it is their action that temporalizes the children and what they do. This temporality is of a different order than the reversible time of the physical sciences, and it is of a different pace than the one enacted by a knowing consciousness in which time does not really exist. It is the temporality of a search, where the outcome is not known, and, therefore, where there is no ultimate measure that would indicate to the children whether they are “on the right track.” This temporality of real sensuous labor of mathematics not only is absent in the theories of Kant, Piaget, and the (radical, social)

Coordinating Touch and Gaze 137 constructivists but also from that of embodiment and enactivist theorists. But it is a central aspect of my own way of thinking mathematical knowledge in terms of knowing in the flesh, which is an important reason for coming back to the question of temporality—as in rhythm, pace, frequencies—in Chapter 7.

6

Emergence of Measurement as the Realization of Geometry

Geometry, as the etymology of the term denotes, derives from the Greek words ge, earth, and metria, measuring. That is, a central aspect of proto-geometry has been the act of measuring—though subsequently, once the Greeks were doing geometry scientifically, measurement does not appear to have played a major role. Historically, however, geometry as mathematical science arose from the pre-scientific, intuitively given world, from the “first very primitively and then artistically exercised method of determination by surveying and measuring in general” (Husserl, 1997a, p. 26, original emphasis). The children featured in this book, however, have not been born into the same pre-scientific world. Theirs is a world in which idealities, embodied in and mediated by concrete cultural artifacts, is part of the everyday experience of growing up. They are not living in a pre-geometric world where kúbos (cube) was a die for playing with, kúlindros (cylinder) a roller, sphaîra (sphere) a ball, pyramis (pyramid) a royal tomb of Egypt, and kírkos (circle) a round or ring. At that time, all of these words pertained to real objects in the real world of the Greek. But, even though the children today do not rediscover geometry in the way that the ancient Greek did, they still do discovery work. It is measurement that allows the objects of the intuitively, non-objectively given world, to become intersubjectively available in the same way to every member of the collective. It is the art of measurement that—to paraphrase Husserl—practically discovers the possibility to choose certain empirical basis forms as measures and to use them to practically determine in an unequivocal manner their relation to other bodily forms. In this chapter, I describe how measurement emerges in this mathematics classroom at different places and times exhibiting the different functions of hand/arm movements just articulated. The emergence of measurement is tied to the practical, experienced need to be accountable for any claim, statement, or argument made. In this way, the children in the second-grade classroom reproduce geometry as an objective science precisely because measurement and accountability make objects intersubjectively available, that is, make them inter-objective. MEASUREMENT AND LIVED PRACTICAL WORK In the geometry lessons that are presented throughout this book, the issues at hand are not just related to shape or aspects of shape (vertices, edges,

Emergence of Measurement as the Realization of Geometry 139 faces) and their features (straight, curved). Rather, the metric aspects of geometry become characteristic features of the interaction between participants. They become issues, as we see in Chapter 5, precisely when teachers or peers ask students to render accountable this or that position they are taking. For example, in the preceding chapter, we observe repeated instances where others (Sylvia, Jane, Mrs. Turner, and Lilian) challenge Melissa to make her position on the nature of the mystery object—which she has articulated to be a cube in her model and in her talk—visible, rational, and reportable for the purposes at hand. Much of the three students’ activity consists in producing ways of accounting for the process of transformation between what they feel when they touch the mystery object and the different models they build thereof. Because they are asked to produce only one model—i.e., that all three girls featured in the preceding chapter have to have the same model—the activity structure encourages them to produce rational accounts for the transformations that they are produce. Measure can be understood as the conceptual tool by means of which two entities can be compared in numerical terms (Crump, 1990). Once such a tool exists, it can be applied to make numerical comparisons by assigning this unit “to every member of the class to which the measure is applicable” (p. 73). For example, the measure of length can be applied to anything that has at least one spatial dimension, a line or, rather, an idealized line. In the previous chapter, both Melissa and Sylvia use what I call the “caliper confi guration” to measure or to symbolize having measured the linear dimensions of the objects at/in the hand. The hand— being our principal organ of tact and contact, allowing us to manipulate and sense the world—may well be the original measurement instrument on which all forms of calipers have been modeled. Thus, the Kpelle in Liberia—at least during the 1960s when the research was performed— still use the handspan and multiples thereof as a means for measuring short distances (Gay & Cole, 1967). Whereas the handspan may be a fi xed unit, the distance between thumb and index fi nger may be modified to constitute the unit of comparison. In fact, any other fi nger could be used for that purpose: Whenever required in garden or other handiwork, I use the distances between right-hand thumb and the other fi ngers of an open, slightly spread hand as good approximations of four-, six-, eight-, and nine and one half-inch units (with an accuracy of 10% or better). The movable, “natural” caliper also emerges as a geometrical practice within this classroom generally in response to challenges concerning the size (length) of some object. We see such instances in the preceding chapter, when Melissa uses the “caliper configuration” to make a case for the mystery object to be a cube, as embodied in her model. Hand/arm movements (gestures) may have different functions. Some movements serve to operate upon the world. They do work and therefore are referred to in the French cognitive science literature as gestes ergotiques, translated as “ergotic gestures” in English translations of these papers. The

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adjective ergotic, deriving from the Greek word ergon (work), is used to refer to the transformative aspect that the hand movement has on the world. Other hand/arm movements have epistemic function because their function is to find out about the world—to feel surface characteristics, temperature, or to measure. My own research has shown that in the course of science investigations, hand movements change their character so that iconic hand movements (gesticulations) emerge from earlier ergotic and epistemic hand movements (Roth, 2003). During the task described in Chapter 5, the children in this class find out (learn) about a mystery object merely by means of contact and tact. Finally, there are hand/arm movements with symbolic functions, such as those movements used in pointing and descriptive (iconic) gestures. For example, Sylvia deploys an iconic gesture to indicate that the mystery object in their shoebox is low in vertical dimension—distance between thumb and palm—relative to the two other dimensions symbolized by the hand. The advantage of such a description—a continuity of hand/arm movements that have ergotic, epistemic, and symbolic function—derives from the fact that it allows for a continuity in the transformation of the hand movements: from doing work, and thereby exploring the world, or exploring the world alone, to using gestures to refer to aspects of the world, including, reflexively, to the hand movements that previously have changed the world or explored it. In fact, it is during the work that new movements are generated as variations on earlier ones, and as necessitated by the aspect of the world at (in) hand. In an auto-affection of the flesh, the new movements become a form of memory required for intention and subsequent symbolic use of the movement as a whole or parts thereof.1 As athletes know, fi nding the right movement may take a considerable amount of time and training, but once found, it will be remembered rather easily and for a long time: One does not easily forget how to ride a bicycle or how to row a skiff after having learned it. In the changeover from ergotic to epistemic and symbolic movements, the differences between consecutive states are undecidable and syncopic, in other words, it is impossible to say where one state ends and the next one begins. This is so because, from the perspective of the recipient, the hand may be doing one, the other, or both. That is, because the two types of movements cannot be distinguished, they are syncopal in nature. Thus, rather than theorizing a change in the type of hand/ arm movement, the same hand/arm movement is understood as taking on different functions. We thereby obtain a continuous trajectory of signs that begins with the auto-affection of the flesh, which then takes on sign function as the movements fi rst are used for epistemic purposes and fi nally for symbolic purposes. Moreover, the relations between the movements do not require embodied image schematic representation as it does in embodiment and enactivist theories. Here the movement represents itself, if it represents anything at all. The flesh auto-affects itself as it enacts the movements for a first time, giving it the capacity to intentionally effect the movements for symbolic purposes in subsequent situations.

Emergence of Measurement as the Realization of Geometry 141 THE CALIPER CONFIGURATION: BETWEEN HOLDING AND COMPARING In this first section, we return to Jane, Melissa, and Sylvie engaged in the task of building a model for the contents of the shoebox. Already at the very beginning of the episode, the different results that Jane’s, Melissa’s, and Sylvia’s explorations have yielded are made evident. Melissa announces that what she has felt is a cube, but her two teammates express doubt (Jane’s grimace) and oppositional views: “It is not a cube” (Sylvia), “I didn’t feel a cube” (Jane), and “me neither” (Sylvia). Responding to the opposition, Melissa provides an account of how she has arrived at the conclusion that the mystery object is a cube. She says, “I checked the sides like that” while taking her cubical model between thumb and index finger (turn 009). She rotates the cube—as can be gauged from the position of the grey-shaded face in drawing accompanying turn 009—and applies the caliper configuration to another dimension of her cube; she rotates it yet another time and exhibits the caliper configuration along the third dimension. In this configuration, the positions of the fingers resemble the caliper—a term whose origins are uncertain, but which some hypothesize to be the French word calibre, size of a bullet. The measurement device, however, has been in use for nearly 3,000 years and has been developed independently in southern Europe (Greece, Italy) and in China. Though coming in various shapes, as compass or Vernier caliper, these precisely resemble the configuration that Melissa’s fingers exhibit for us as the way in which she compares (“measured”) the three different dimensions of the mystery object as shown in her account of what she has done with the hand hidden from view. Fragment 6.1 (from Fragment 5.1a) 001 002

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i feel it feel it eh? i have felt its a cu:be () <hU:::::::ge.> (1) ((J grimaces, questioningly?)) no its not a cube ((shakes head while rH in box)) () i didnt feel a cube. me either. (3) i did. (1) i checked the sides like that. ((caliper grip on each of 3 sides, Fig.)) ((puts lH into box)) you should feel around.

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What we can observe here, in fact, is a movement from a simple perceptual discrimination that Jane and Sylvia offer—in saying, “I didn’t feel a cube”—to a substitution by a comparing operation. Melissa moves the hand, establishes a unit distance in opening the caliper to match the length of the fi rst dimension, and then takes this unit to the sides of the two faces orthogonal to the fi rst and orthogonal to each other. Here we can also see a transformation from something that originally may have served as a way of holding on to the object, a grip, and allowing it to emerge into a standard (sign) applied to other dimensions as well. Or, in other words, the same measure is consistent with the length of the edge after a 90° rotation of the object (cube). That is, Melissa also exhibits symmetry properties of the object under specific types of transformations. In this situation the caliper configuration—clearly that of the left hand—simultaneously serves to hold the cube, that is, it does work. The difference between the movement doing work and doing measure (epistemic movement) is undecidable and therefore syncopal in nature. Moreover, this sign can subsequently emerge, as we see below, into a gesture even in the absence of the object itself (here an edge with its end points). It is precisely when the same unit is applied repeatedly to an entity and associated with counts that true measurement will emerge. Initially, all three girls have made models, and for a short while, they simply state that the mystery object resembles their own. But then, after the research assistant present (Lilian) reminds them that there is only one mystery object inside the box and that therefore there can be only one model, fi rst ways of accounting for what they feel inside the box are produced. For example, Sylvia asks Melissa, who has her hand inside the box, “you feel it? Feel it, feel the side of it how flat it is, you’ll see” (Chapter 5, turn 048). In this way, Sylvia instructs Melissa how to go about noting what she has noted before, which has led her to build a flat model. It is precisely when you “feel the side of it,” that you can have the experience of the flatness. Of course, this description is as far from the feel as is the description “knead the dough until smooth” is from feeling the dough in the way required in a successful realization of a bread recipe. The recipe is but a rational account of what the baker has done with her hands. In the same way, Sylvia has made her actions of modeling—or, rather, the “feel of it”—rational and reportable for the purposes at hand. She thereby has exhibited the accountable nature of her actions. Repeatedly, Melissa provides rational accounts for what she has done, all of which are based on a comparison using a standard unit: “I checked the sides like that” + caliper configuration of three orthogonal edges (turn 009); “Cause like the same ((caliper grip)) it’s the same everywhere” (turn 061); “but we are trying I think we are—it all has the same” (makes caliper grip) (turn 152); and “It has the same edges, it is the same” (turns 184, 186). Melissa explicitly articulates measurement as the principle on which to base the account (turn 070). Sylvia, too, uses the caliper in an iconic way

Emergence of Measurement as the Realization of Geometry 143 to show how a shorter distance between the parallel caliper legs separates two long and wide faces. In this instance, it is not the more scientific (mathematical) measurement that wins out but the “feel” of the object in the hand, the nuance. But the simple articulation of the feel does not get the three girls out of the deadlock. Jane and Sylvia variously articulate the feel of the mystery object: “didn’t feel like a cube,” “oblong” (turn 019), “top square” (turn 015, 084, 111), “side rectangle” (turn 015, 082), “can’t be a cube” (turn 096), “flat” (turn 036, 048, 082, 125, 166), “not the same” (turn 194, 205), a double-handed gesture with a flat object separating the palms (turn 048), and “long” (turn 050). Rather, just as in the case of rock musicians communicating the feel of a particular musical phrase by playing it or by reminding each other of some other band’s playing the phrase, the turning point comes when Melissa engages in a direct comparison of the feel. The direct comparison later enables her to simultaneously feel the mystery object in the right hand and the model in the left hand, and, therefore, she could feel the difference as well.

DOING THE CALIPER: FROM EPISTEMIC TO SYMBOLIC MOVEMENT As long as students use the caliper grip to hold the object they measure and talk about, the difference between the ergotic and symbolic aspects remains undecidable, syncopic. Thus, we may not be able to decide—based on a videotape, photograph, or live event—whether a hand movement/position is to hold the object, to fi nd out something about it, or to symbolize holding or fi nding out. A certain form of abstraction does in fact occur when the hand produces a caliper shape in the absence of the object measured. Here, a differentiation has occurred, then, between the ergotic function of the grip or the epistemic function of the touch, on one hand, and the caliper configuration as a symbolic denotation of the measurement process, on the other hand. This, therefore, becomes the reference of some unspecified thing. It is measurement as such. In the following fragment from the same lesson as the preceding one, Ben’s group is asked to state the nature of the mystery object in the shoebox. In the course of the interaction with Mrs. Turner, the students generally—but Ben particularly—are asked to articulate evidence for the contention that the mystery object looks like the model they constructed: a cube. Initially the issue comes to be articulated as one in which the students are asked to distinguish their model from a rectangular prism. Part of the episode not reproduced here (turns 33–56, see Fragment 4.2) involves the teacher and Bavneet, before Mrs. Turner turns back to Ben and asks him to articulate why his model contains all squares rather than rectangles or oblongs. It is at that point that Ben articulates measurement. But let us look at the entire lesson fragment (excluding the exchange with Bavneet).

144 Geometry as Objective Science in Elementary School Classrooms Following Mrs. Turner’s question offer, “what makes you think it is a cube?,” Ben unfolds a response, punctuated by pauses and interjections (turns 05–09). While contrasting the “one side” with the “other,” his right hand produces two gestures that suggest sides orthogonal to each other, the first in a vertical direction to the desktop, the second parallel to it (turn 08). Uttering the final part of his statement with rising intonation—using the same gestures as Ben but in the reverse direction, first the one parallel to the desktop then the one orthogonal (turn 13)—Mrs. Turner questions its contents, and Ben affirms. Mrs. Turner makes three long steps, stretches out to catch hold of a rectangular prism on the chalk rest. She repeats, like a stutterer, “like a,” while returning to her previous position next to Ben’s table. Here she completes the first part of a question-response pair, “like a rectangular prism where one side is the same as the other?” (turn 13). As the utterance unfolds, Mrs. Turner first holds the parallelepiped out in front of her while she names it “rectangular prism,” and then touches fi rst one of the square sides with her right palm, then the opposing side with her left palm. Fragment 6.2a 01

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bens group. (0.20) ben ´what does your group think thIS is: in here. a cube a cube what makes you think its a cUBE? (0.92) um; (1.15) ((T shakes head, opens palm “inviting” explanation)) .hh that; (0.20) um; (1.43) it um, (0.21) one sides the same as the otHER. ((Fig.)) (0.95) ONE side is the same as the otHER? yea. (0.53) okay ((walks to right, picks up parallelepiped)) (0.57) like a like a like a (0.52) rectangular prISM where one side is the same as the other? ((touches facing “square” sides, Fig.)) yea. (0.86) (:B) no; =so why is it not a rectangular prISM then. (0.25)

Emergence of Measurement as the Realization of Geometry 145 19 20 21 22

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they are all the same(0.71) theyr whats all the [same.] [i: th]ink its the same. ((turns toward object at chalkboard, points, Mrs. T follows his fi nger with her gaze)) (1.31) ((Mrs. T walks to objects, picks the cube)) um because um the sides are all the same; (0.89) ((Mrs. T returns to her position)) okay (0.43) so all the faces, yea are all the same? ((:b)) do you agree with that? ya. ((Mrs. T gazes at Joel)) yap. ((Mrs. T’s fi nger points to Joel)) (0.27) okay (0.19) and (1.03) anYthing else that tells you it is a cUBE? ((holds up the cube from the chalkboard))

Ben affi rms, but, after a pause, Joel, who has turned to Ben, negates. Mrs. Turner continues, “so why is it not a rectangular prism then?,” which Ben completes as a question-response pair, “they are all the same” (turn 19). Apparently beginning another utterance that possibly confi rms the preceding one, Mrs. Turner then produces what can be heard—because of its grammatical structure—as a question but which intonationally— because of its falling pitch toward the end—is a statement, “What’s all the same?” Overlapping her, Ben says, “I think it’s the same,” while pointing toward the chalkboard where, on the tray, there lies a cube. After a longer pause that unfolds as Mrs. Turner walks to the cube on the tray and returns with it holding it up for everyone to see. Ben continues, “because the sides are all the same” (turn 24). Mrs. Turner seeks confi rmation by restating Ben’s phrase but by modifying the “sides” into “faces,” “so all the faces are all the same” (turns 26, 28). Ben is responding in a brief pause that she leaves in the middle of the utterance. She then turns to Bavneet: “Do you agree with that?” Both Bavneet and Joel affi rm. At this point, Mrs. Turner produces a query that leads into an exchange with Bavneet (Chapter 4), who points out the vertices on the cube. Mrs. Turner then turns again in the direction of Ben and (thereby) addresses him. Mrs. Turner begins this third part of the episode (Fragment 6.2b) by describing Ben’s model, “this looks like square, square, square,” and she then continues, “how did you know to do square, square, square, square,” and repeats herself, “how did you know square, square, square, square?” (turn 57). But, after an unfolding pause, it is Joel who begins a second turn, “um square, square”; Mrs. Turner continues, however, as if she had not fi nished her previous turn, “not rectangle, rectangle,

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Figure 6.1 While Mrs. Turner is asking a question about why the mystery object in the shoebox is a cube, Ben is producing the caliper configuration with both hands, the one holding and the other measuring/gesturing.

rectangle” and then compares this to the results of another group, “like they, right Jonathan?” Across the three turns at talk, from the third cluster of repetition of “squares,” over Joel’s attempt, and to the beginning of the question about rectangles, Ben has brought his hands up from his lap and, making the caliper configurations, holds the cube fi rst up high, then brings it to eye level (Figure 6.1). Jonathan affi rms. Mrs. Turner orients back in the direction of Ben, “How do you know that?” Punctuated by pauses and interjections, Ben begins a second turn and then, as his hand moves up from his lap above the desktop forming a caliper configuration, he states, “I measured it” and then lifts his head and directs his gaze at Mrs. Turner (turn 67). Fragment 6.2b 57

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((:B)) hOW did you know; im looking at your model here. ((points to Joel’s model)) (0.50) and (0.32) this looks like square square square square how did you know t do square square square square; hOW [did you know square square square square; [((Ben brings up hands with cube, gestures, Fig. 6.1)) (1.18) ((Mrs. T shrugs shoulder, looks at Joel)) um square square =not rect]angle rectangle rectangle like <thEY> (0.30) thEY ((points to back of classroom, Ben & Joel turn around)) have rectangle rectangle rectangle or oblong oblong oblong; right jonathan? (0.24) yea. so () how did you know to go square square square square; (0.33) all along the sides and not o:blONG o:blONG o:blONG; ((moves head in same rhythm)) (1.36)

Emergence of Measurement as the Realization of Geometry 147 65 66 67

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J: T:

hOW did you know that. (0.71) um (0.32) well we (0.49) um (0.47) i measured ((caliper configuration, Fig. a)) it um ((looks up, Fig. b) um [ the ] [how di]d you measure; um because i took the (?cular) and i (0.30) made thumbs (0.45) how long it was ((caliper configuration, Fig. a)) and then i (0.43) turned it to the other side and ((caliper configuration, Fig. b)) it was the same. (0.62) so youve felt with your fINger? (0.28) and you ran along and you and you felt so you were able to measure (0.32) okay. (0.27) <now> () yOUR () group () built THIS ((picks up Joel’s “stick” model)) (0.60) and this is interesting you built a (0.83) < a cube> (0.45) a cUbe and this is how they built it. ((holds it up))

Mrs. Turner begins a second turn while Ben is still speaking, “how did you measure?” At this point, Ben produces a visible and audibly rational account that will be satisfactory for the present purposes—as per the positive evaluation that Mrs. Turner produces in repeating what Ben has said in constative form, adding an interjection of assent, “okay.” Ben provides an account in which he articulates having felt the length of the mystery object, while his hand takes on a caliper configuration placing the end of his thumb on the table: “made thumbs” (turn 69). While uttering, “then I turned it to the other side” he brings up and twists his arm, rotates the hand in caliper configuration as if he had moved it from one to another side of the cube intersecting with the fi rst at 90 degrees (turn 69). He states the result of the comparison: “and it was the same” (turn 69). Mrs. Turner then turns and begins what is going to be a recognizably different issue, and, in so doing, also affi rms that the previous topic has been completed (turn 73).

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Here, (the idea of) measurement emerges from a situation in which the children first provide a qualitative description of sides that feel the same. Mrs. Turner summarizes this initial account in terms of repeated sequences of the shape word “square, square, square, square.” She then requests an account: “How did you know that?” (turn 65). In a way similar to Melissa in the previous section, and similar to his own prior operations, Ben produces a caliper configuration together with the verb “measure” to account for what his hands have done while touching/feeling the mystery object for the purpose of building a model thereof. However, unlike the other two instances, he employs gestures independently of the object in his hands. That is, whereas Melissa in Chapter 5 and he previously formed the caliper configuration with the model in their hands, the hand movement here is entirely symbolic. In the preceding situations, the caliper configuration also has had symbolic (sign) function while the hand held the object. That is, the caliper configuration may have had its origin in the fact of holding the object—it has ergotic function, i.e., does work—and subsequently emerges to have purely symbolic function. At the transition points, there are instances where the difference between the two functions is undecidable, belonging to both functions simultaneously. In addition, rather than just using a caliper configuration, Ben “made thumbs,” which he uses to establish “how long it was.” He claims to have “made thumbs” while holding the distal phalange of the thumb against the desktop. Here, the caliper is replaced by a part of the thumb, which serves as a fixed unit of measure that no longer is derived from a preceding gesture but introduces a new element. The distance between the configurable legs (fingers) of the caliper now comes to be turned into a fixed length element that serves as the unit for comparing (at least) two different dimensions of the mystery object. Having a fixed measure may be both a fi rst in the emergence of comparison, such as when we take a stick hold it to an object, place a finger to produce a reference unit length, and hold this length to the comparison object. This in fact already is a generalization from holding two entities to be compared in length directly to one another. (Connor employs such direct comparison in Chapter 8, in an episode from the very fi rst lesson in a curricular unit that Mrs. Winter and Mrs. Turner denote by the term “geometry.”) Or it may cognitively follow the caliper, which itself might emerge from the use of the hand as an inherently modifiable “instrument” with which to compare two entities. The use of body parts as units of measurement is not surprising given the fact that such practices reach far back in human history: the foot, the (hand)span, the handbreadth, the sailor’s fathom, the mile (from Lat., mille passuum, a thousand paces), the German Elle (corresponds to the cubit, from Lat. cubitus, elbow to end of the middle fi nger), and the French pouce (thumb, one inch) all are measurement units that derive their names from body parts or bodily actions. Excavations in the Indus Valley reveal that measurement instruments have been in use since the fi fth millennium BCE, and measuring instruments have been in use during the

Emergence of Measurement as the Realization of Geometry 149 construction of many major cities during the golden era between 1750 and 2300 BCE. On the scale of human and cultural evolution, these are rather brief time scales, suggesting that measurement is quite a recent and complex cognitive feat. In the present instance, too, measurement re-emerges from the everyday experiences of the children when they are held accountable for their claims and actions.

FROM COMPARING TO COUNTING: QUANTITATIVE MEASUREMENT In the previous sections, we see the caliper configuration being used to hold an object, to feel an object, and to symbolize a measurement made between two different dimensions. The rudiments of measurement therefore have emerged. But in all instances there is the same, though adjustable standard measure that serves to compare two different items. However, true measurement comes into being if a particular measure comes to be applied repeatedly to the same dimension, that is, if a standard length comes to be associated with counting. The length of an object, thereby, becomes a multiplicity, because “the representation of a totality of given objects,” here the length unit, “is a unity in which the representations of single objects are contained as partial representation” (Husserl, 2003, p. 21). That is, true measurement is related to the application of number, which is but another term to denote multiplicity. In this section, I describe the emergence of this advanced practice, the coincidence of measure and counting. In the following lesson fragments, we can see that the suggestion of measurement emerges following a particular investigative structure that itself emerges from the exchanges between the adults in the room. The measurement arises from qualitative indications and slowly turns into an incarnate suggestion of the number of boxes needed in a stack to turn Chris’s pizza box into a cube. We have already been to this episode before. It follows the instances in which Chris has fi rst shown how there are different sides to a pizza box (Chapter 4), which then turns into an explanation of what he has to do to turn the pizza box into a cube (Chapters 1–3). Following Chris’s turn in response to Mrs. Turner’s question “what would that box have to have to make it a cube?”, I (R = Roth) utter, “how could you make a cube from pizza boxes?” (turn 24). There is a pause, an interjection, and then Chris says with low speech volume, “I don’t know.” There is another pause, and then Mrs. Winter produces an utterance grammatically structured like a question, “What would we need to do to make this pizza box into a cube?” (turn 28); but Mrs. Turner repeats what she says, before Chris completes it as a question-response pair, “take a whole bunch and stack them on top” (turn 30). He initially places his hand on the pizza box currently held by Mrs. Turner, and then moves it upward until it reaches a height approximately corresponding to the sides of the square box (turn 30).

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Fragment 6.3a 22

C:

23 24 25 26 27 28

W:

29

T:

30

C:

31 32 33

T:

34 35 36 37 38 39 40 41

42

R: C:

C: T: R:

R: J:

it would have to um ((gets a cube from chalk holder)) (0.44) that tu:rn: or this (0.21) like the:(0.62)s:: ((bends down to box)) kinda like (3.04) kinda ((hand moving along edge)) (0.23) s::quare here an here like ((rectangle)) (0.68) <but to=n=this squa> ((movement along edge of box)) it just has squares (0.45) here and here and everywhere (0.63) ´how could you make a cube from pizza boxes. (1.46) u::m ((looks at pizza box)) (1.14) <i dont know> (1.83) whose got an idea how could you (0.43) what would we need to dO: to make this pizza box into a [cube ] [in or]der to make this into a cube ((holds up pizza box)) what would we need to do to it take a whole bunch and stack them on top ((shows with hand gesture, Fig.)) (0.58) in order to do what, (0.89) in order to make it square (0.49) okay (0.66) you had a word the other day on the board; didnt you? (1.66) you stA:ck th[EM:] [yes] you stack them. ((shows with her hand above pizza box, Fig.)) (0.44)

Rather than accepting/evaluating, Mrs. Turner utters, “In order to do what?,” which Chris reifi es as a question-response pair, “In order to make it square” (turn 34), which is followed by the teacher’s evaluative turn, “Okay.” I (in the process of videotaping) offer a question, which, when there is no response, I answer myself, “you stack them,” thereby repeating a term Chris has already used and a term that Mrs. Turner had written on the chalkboard during the previous lesson. She affi rms, “Yes, you stack them” (turn 41).

Emergence of Measurement as the Realization of Geometry 151 Here, first Chris then Mrs. Turner and then Chris again (turn 44) show how stacking pizza boxes would lead to a cube. That is, the thickness of the cube is immanent in these members’ descriptions and stacking several of them leads to a new object that has about the same dimensions in all directions. Qualitatively, therefore, the repeated measurement is immanent in the description of what one has to do, but is not yet articulated as a manner of giving a precise description how many pizza boxes are actually needed to do this. Yet the extension of the cube literally is performed and felt as Chris moves his arm and hand upward. In her next utterance, Mrs. Winter provides the seed for such thought to emerge. Mrs. Winter then articulates wondering about something (“how many pizza boxes”), but Chris is talking simultaneously to Mrs. Turner, explaining what one had to do, using the same hand gesture from the box upward as before (turns 41, 44). A student calls out “ten” while Chris is still talking, and then Mrs. Turner repeats and fi nishes the issue about the number needed to make a cube. A student calls out “one,” and after a pause, Chris suggests grinning from ear to ear, “A whole bunch of pizzas” (turn 50). Mrs. Turner solicits more input, apparently without making an evaluating comment, though Daphne revises her fi rst number “four” when Mrs. Turner utters with an upward intonation “do you think four?” This is thereby heard as querying the truth of the number, with a subsequent student revision. Kendra offers a number “two thousand,” which the two teachers take with surprise interjections. But Mark makes a suggestion, which Mrs. Winter affi rms as having “it right down” (turn 68). Fragment 6.3b 43 44

45 46 47 48 49 50 51 52 53 54 55 56 57

W:

well what i was [wondering how many pizza boxes]. C: <[i think you put them here ] so its more [like] thA:t ((hand moving upward, as in Fig.)) S: [ten ] (0.56) T: well how many would you nEED until that became a cUBe. S2: ON:e (1.54) C: ((laughing)) a whole bunch of pizzas. J: daphne? D: em four? (0.36) J: do you think four? D: five ((shows five with hand)) (0.39) T: <do you need fIVe?>

152 58 59 60 61 62 63 64 65 66 67 68

Geometry as Objective Science in Elementary School Classrooms J: T: K: W: T: M: W:

(0.25) i say five> (0.43) kendra? (0.84) two thOUSand. (0.29) <whoo>= =OH: my goodness. two thousand that=ll be way longer. i think we () i () i marks got it right down the end an really long rectangular pris[m].

In this situation, the number of boxes required to make a cube from the pizza box goes unquestioned and without justification required. Some responses are qualitative, such as when Chris raises his hand from the box to some distance visually about the same as the side of the square. He also suggests “a whole bunch of pizzas,” which is associated with a smile and could have been heard/taken as a joke: You could eat a “whole bunch” of pizzas and then make a cube from the empty boxes. The suggestion no longer would have been mathematical, but would have placed the response in the gustatory domain. The numerical suggestions include 1, 4, 5, and 2,000. Mrs. Turner then calls on Jane, thereby opening what would turn out to be the final part not only of the fragment but also of the entire episode (variously discussed in Chapters 1–3, 4, and here). In her turn, Jane proposes a way of establishing how many pizza boxes there are needed suggesting a count; and Mrs. Turner will subsequently denote what Jane has done as “measuring.” At first, Jane gets up to move close to the box, gesturing along the edge facing her (Figure 6.2a). She then grabs the box between her thumb and index finger (Figure 6.2b) in the “caliper configuration and then moves the hand across the top (square) of the box (Figures 6.2c, d). She repeats the gesture, initially stating that “you go like that” and then articulates, “and count how many” (turn 70). That is, she articulates determining the number of boxes needed by taking its thickness and counting how many times this “thickness” it would take to make the same as the side of the square. Fragment 6.3c 69 70

T: J:

[j]ANe? yea um (0.39) i think if you wanted () TO make it exactly like a cube, (0.43) you have to know how ((Fig. 6.2a)) (0.45) like ((Fig. 6.2b)) (1.07) how long this is ((shows thickness of box in Jennifer’s hands)) and then ((Fig. 6.2c)) () you take that ((Fig. 6.2b, takes ‘measure’ of thickness)) and that and go like that ((makes walking movement

Emergence of Measurement as the Realization of Geometry 153

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

with fi ngers across the box, Fig. 6.2c, 6.2d)) and [count] how many.[ R: [UM ] [did everyone ´get that; S4: <

(is this the base?)> (0.48) T: jane said that we would have to measure? (0.69) how much (0.21) thIS is (1.09) and then measure, (0.61) J: ((gets up, points across the box)) you go like this and then count it. W: and count it and then what. (0.56) J: and then you can (0.36) and then you know how much to put (0.20) on <

to make a cube.> W: how many to stack on top to make a cUBe vERy interesting. (0.20) ^cheyenne. (0.40) C: um i think it will take ten boxes. I think. T: ten boxes. but, W: you know what, we will work it out. T: YES we=ll do it okay. W: we=ll work that out how many pieces to make this a cube. T: thank you very much ((gestures Chris to get back to his place))

Figure 6.2 Jane first measures the height of a pizza box using the caliper (a, b) and then shows how to count the repeated application of the measure to find out how many pizza boxes it takes to make a cube by stacking (c, d).

154 Geometry as Objective Science in Elementary School Classrooms My turn following Jane (turn 71), which addresses itself to the class as a whole (“everyone”), including its teachers, indicates that there is something remarkable that has gone on, but something which others may not have remarked: “Did everyone get that?” Mrs. Turner rearticulates, thereby affirming, what Jane has said. But she does not exactly repeat. Mrs. Turner introduces the term “measure,” which replaces the term “count” in Jane’s account. There Mrs. Turner stops, giving rise to a silence, which Jane breaks articulating her method again, “you go like this,” gesturing repeated applications of the caliper across the box, and completes: “and count it.” Mrs. Turner asks for the subsequent action, which Jane responds to by saying that then “you know how much to put on to make a cube” (turn 79). Mrs. Turner states an evaluation, which begins by reiterating/affirming the response, “how many to stack on top to make a cube” and then follows it with a more explicit evaluation, “very interesting” (turn 80). There is one more turn, in which Cheyenne offers another number, before the two teachers collaboratively bring the fragment to a close by announcing a deferral of the actual measurement to another instance in the course (“We will work it out.”).

FROM ACCOUNTABILITY TO ACCOUNTS AND COUNTING In this chapter, we see how measurement, the origin and heart of geometry, arises from the events in a second-grade classroom when the children are asked to produce accounts of what they have done or to provide accounts of what they “think.” These accounts allow them to become accountable and to make their mathematics accountable. It is precisely in and through these accounts that thinking becomes something that is not relegated to the mind, but becomes a public event visibly and hearably rational and reportable for all practical purposes of this mathematics classroom. It is not that they have to take these accounts as shared as various social constructivist mathematics educators tend to describe it: In and as of account, what is being said in this classroom is a cultural possibility and therefore inherently shared. The children participate in this collective event, their individual performances driven by the collectivity as much as by their own intentions, which inherently reflect the collective dispositions of the present field. Thus, holding the children accountable leads to thinking as a public, and therefore collective and collectively modifiable event. Participation in the event entrains and fashions the lived/living body so that any higher order function (thinking) that is the result from this participation—a form of interaction—is itself shaped by the process, that is, is itself societal through and through. That is, measurement is not just something evolving in individual bodies. It is enabled by the auto-affection of the flesh, which permits epistemic movements that are made available to others as intelligible, inherently shared and shareable expressions. Epistemic movements and counting together allow the emergence of measurement.

Emergence of Measurement as the Realization of Geometry 155 The events in this classroom also suggest a possible organization of sequences that might facilitate the emergence of measurement as a practice to be used in situations other than the particular context in which we encounter it here. Thus, we observe here a development, which begins when children handle objects and build models (with their hands). From their hand movements holding the objects they emerge hand movements that have epistemic function, such as when Melissa or Ben use them to measure the length of different sides of their mystery objects and of the models thereof. At the same time, as hand movement, it is inherently shared with others who can see in the performance of movement with intention. 2 Later on, the same hand movements may take on symbolic function, such as when Ben uses the caliper configuration to refer to the measurement and comparison has enacted before. The initial epistemic hand movement now is a symbolic hand movement with interactive function, as it demonstrates to the teacher and all the peers what has been done to ascertain the nature of the mystery object. Finally, the hand movements can be connected together into chains of movements, where each link is associated with a count of one. The total length thereby comes to be a multiplicity (grammatically expressed in plural expressions such as “20 centimeters” or “eight unit lengths”) as much as it is a unity (grammatically expressed in the singular of “the length”). In this association of repeated hand movements and a counting procedure emerges measurement. Any movement, because of the capacity of the flesh to auto-affect itself, becomes a possibility for learning and symbolization. Even machines change when they do work, and in a sense one can consider the abrasions that occur a form of memory—e.g., the memory that batteries are said to develop when they are not entirely depleted before recharging. But only the flesh is capable of transforming ergotic and epistemic movements into symbolic ones for its own purposes. Measurement tends to emerge in a culture when there is a need for it. Measurement therefore must have some utility (Davis & Hersh, 1981). In the present situation, the utility arises from the fact that it is a way of comparing and being rationally accountable. In this instance, the need emerges from the question, “How many . . .?” setting up (provoking) an answer in the form of a count or number. The method Jane articulates for producing the answer does precisely that: It shows how taking a reference length and counting how many times it fits into the side of a square. As soon as such action is used to mark off a stick, a measuring instrument such as a meter will have emerged. Measurement instruments evolved precisely to be “templates for shaping and polishing surfaces or reckoning material alignments and lengths” (Lynch, 1991, p. 80). That is, as we see in the present case, the everyday world of the children is the beginning of mathematization and is subsequently substituted by a mathematical world. This development corresponds to the historical process that led Galileo to the “surreptitious substitution of the mathematically substructed world of idealities for the only real, actually perceptually given, ever experienced and experienceable

156 Geometry as Objective Science in Elementary School Classrooms world—our everyday lifeworld” (Husserl, 1997a, p. 52). We see already in Chapter 5 how the three girls work the surfaces of their models—i.e., “polish” them—so that their artifacts come to be aligned with the ideal surfaces of the geometrical object in the shoebox. Here, they further idealize their experience, thereby contributing to the transformation of their real-lived in world comes to map onto the ideal world of the geometer. That is, mathematization and idealization occur simultaneously.

7

Doing Time in Mathematical Praxis Praxis unfolds in time and it has all the correlative characteristics, such as irreversibility, which synchronization destroys; its temporal structure, that is, its rhythm, its tempo, and above all its orientation, is constitutive of its sense. (Bourdieu, 1980, p. 137) Rhythmizing consciousness does not apprehend its object in the same way as unembellished perception does. (Abraham, 1995, p. 21)

The intellectualist approach according to which practical understanding of the world is governed by mind confuses presence in the world (Being) with the presence of the present of world, which appears in the form of things (beings), re-presentations. Thus, with the emergence of re/presentations that can be used to make something present over and over again, time has been expelled from common accounts of cognition. Thus, intellectual consciousness represents practice, synchronizes its moments, and thereby destroys the sense characteristic of praxis as it unfolds in real time.1 Practical understanding does not require formal knowledge: For example, children’s language is grammatical even prior to their encounter with formal grammar. The approach I take here, the one of mathematics in the flesh, is precisely designed to address this recurrent problem to knowing in mathematics education research, whatever the brand. Practical comprehension is comprehension in and through the flesh, enacted without the conscious mind as master and prior to any embodied image schemas. It is knowing in the flesh acquired through the flesh, which comes to be marked in participating in an inherently structured, societal and material world, from its beginning that even precedes the actual birth of a child.2 It is precisely because temporal and rhythmic features derive from the auto-affection of the flesh that the difference between the living/lived body and mind becomes indistinguishable: There is no sense possible without the auto-affection of the flesh. It makes no longer sense to operate with the Cartesian distinction maintained in the embodiment/enactivist literature, as both material body and metaphysical mind are but modalities of the flesh. The notion of praxis, which is based on knowing as performance, goes together with the notion of the living/lived body, the flesh. As soon as we move from the diachronic nature of praxis to its synchronic description in terms of knowledge, we loose all developmental aspects and necessities in the same way that synchronic linguistics loses the phenomenon of linguistic change.

158 Geometry as Objective Science in Elementary School Classrooms Time and temporality are central features of human praxis, presenting themselves in the guise of pacing, rhythms (body, voice), and vocal frequencies. These are important, not only because they structure exchanges and therefore practical consciousness (e.g., memory that is facilitated and recalled more easily with rhythmic features) but also because they serve to communicate and share emotions. Thus, “patterns of body alignment, eye gaze, speech hesitations or flow, loudness as well as overt expressions” (Collins, 2004, p. 110) communicate emotions, pride, shame, and so forth. These are precisely the features that create the alignment and mutual focus that we can observe in this second-grade classroom when following the events on a micro-scale. That is, “emotion never is a psychic and internal fact, but a variation in our relation with others and with the world that can be read from our bodily attitude” (Merleau-Ponty, 1996, p. 67). Rhythm and other periodic features structure the places we inhabit and fields that we enter: These features are written all over the fields and therefore are available to every member of the setting. In this chapter, I present a series of analyses that exhibit the interlacing of speakers and listeners. This interlacing can be observed in the complementarity of body movements, the production and reproduction of periodic features (rhythm, intonation) across interaction participants and across expressive modalities within and across the living/lived bodies of members to a setting. This interlacing underlies entrainment, the feature by means of which we come aligned with others not only on the micro-scale but also at the scales of individual development.

RHYTHM, PACE (TEMPORALITY), AND SENSE In Chapter 4, I point out and insist upon the consideration of the pauses in the exchanges between Mrs. Turner and the students. For example, her exchange with Thomas is marked by pauses and interjections that apparently do not add to the content of talk but serve interactional functions for dealing with pauses. These pauses are important not only because they give structure to and shape the temporality of the event but also because they are resources for participants and onlookers to make attributions about the knowledge of a person. For example, if there is a long pause after a teacher question to a particular student, this might be heard as evidence of lack of knowledge or as insecurity on the part of the student to proffer a response. Research conducted in the 1980s on pauses following teacher questions showed that in general, teachers tend to wait less than a second before moving on, asking another question, or pointing at a specific student to respond. That is, the teachers tend to increase the pace by breaking silences. Teachers are driving the rhythm and pacing of a lesson independently of the question whether students are actually following. Rhythm and pace are important in the constitution of society because they lead to the entrainment of individuals into collective movement and

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emotional alignment. Central to the production of rhythmic and temporal coordination is the living/lived body. Flesh is with contact, that is, in contact with the world. Contact is tact with: tact, the sense of touch. Related to the fi rst signification, tact also is the behaviorist term for an utterance evoked in contact with others and the world. Surprisingly, perhaps, tact not only refers to the sense and to a verbal response, but also to temporality: tact as stroke in beating time. We may create a new sense for the term contact, as the tact common to two or more people. Temporality, (gestural) beats, rhythm, sense, and verbal contact are some of the threads underlying this chapter. Rhythm can be understood as the production of form under the constraints of time and temporality. As form, rhythm is the product of self-reference, auto-affection. This allows a polythetic expression—i.e., produced across the different modalities of the body—to be constituted monothetically, as one unit (idea), in the auto-perception of the voice and movements of the speech organs in the organic body of the speaker. It simultaneously makes possible the monothetic constitution of the expression in the perception of another’s voice and auto-affection of the auditory organs in the organic body (flesh) of the listener. The constitution of semantic forms, sense and signification, thereby becomes indistinguishable from the auto-affection of the flesh. Affectivity then is the impossibility to distinguish the constitution of semantic forms from the sensation of one’s own flesh (Gumbrecht, 1988). In this way, rhythmic features come to constitute a “consensual domain of the fi rst order” on which language and the production of sense rests as a “consensual domain of the second order” (pp. 725–726). It is only when we can exhibit the irreducibility of these two orders that “embodiment” come to be plausible alternatives to constructivism. Because “embodiment” suggests the very division that we want to overcome, I suggest that incarnation in fact is the plausible alternative to all forms of constructivism. The upshot of the rhythmic basis of expression is that it does not make sense to listen to semantic content only. To listen only for the contents of words is like listening to psychologists or physiologists talk about hearing: They, being Cartesians par excellence, reduce this experience to recordings in the cilia and decoding mechanisms of the brain. They have already enacted the reduction that leads to the distinction of body and mind, as they concern themselves not with life but with the externalized modality of the material body. This leads us to a disembodied form of hearing. Living/ lived hearing, however, is something different, because “the spoken word— the one I proffer or hear—is pregnant with signification that can be read in the texture of the linguistic gesture itself” (Merleau-Ponty, 1960, p. 144). As a result, “a hesitation, a change of voice, the choice of a certain syntax is sufficient signification” (p. 144). Therefore, reducing a gesture to language or to some linguistically articulated schema—in the way embodiment theorists tend to do—leaves us with very little; and it leads us to accounts of knowing that are just as disembodied as the constructivist explanation. 3

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But if we were merely listening to the word-contents of the sounds that we make in communication, then teachers would hardly be able to make sense of students’ utterances; and, students, too, would hardly be able to learn. Even the simple sequence “this is a cube” could not be understood independent of other bodily expressions, which allow us to hear a question (“This is a cube?”), a constative statement of a fact (“This is a cube.”), an order (“This is a cube!”), and so on. These other bodily expressions also allow us to ascertain whether the speaker is certain, speaks with confidence, is intimidated, or tentatively proffers an answer. The temporality and rhythm of speech is integral to the expression and recognition of knowing. Hearing and speaking in real time means being attuned to and being in tune with the world across the entire spectrum of means that we produce and use to communicate, including movements, positions, prosody (speech intensity, pitch, speech rate, and many other aspects of speech), facial expressions, and so on. Moreover, it is not that we listen to the “meaning” of a word but rather, as Merleau-Ponty (1960) shows in his analyses, communicative intentions are written all over the situation. The way in which we orient corporally to the signification as speakers and audiences is implicit, “and does not suppose any thematization, no ‘representation’ of my body or the milieu” (p. 145). For example, Mrs. Turner and Mrs. Winter are listening for the pauses in the child’s speech, nervousness, expressions, thereby coming to hear what the child says not by decoding some sounds to hear words to be interpreted to find their “meaning.” In listening, they actually use their living/lived body to “take hold of a course of movements, which makes listening a course of activity itself” (Sudnow, 1979, p. 83). In listening, we do not have time to “interpret” what another person says, but we understand in hearing; we do not hear a sound and then “interpret” it as coming from a Harley motorcycle, but we hear a Harley approaching us or moving away. Rhythmic features structure the ways in which we communicate and, therefore, structure the ways in which we interact with others: They make possible interactions as the rituals on which society is based because rituals enable such material processes as entrainment, resonance, and synchrony. Thus, “successful conversational ritual is rhythmic: one speaker comes in at the end of the other’s turn with split-second timing, coming in right on the beat as if keeping up a line of music” (Collins, 2004, p. 69). Moreover, resonance does not only underlie the coordination between people, but more importantly, it underlies the constitution of sense: “sound and sense mix together and resonate in each other, or through each other” (Nancy, 2007, p. 7). A fundamental resonance therefore also is fundamental to (the constitution of) sense, which requires us to seek sense in sound, as well as look for sound and resonance in sense. Rhythmic phenomena, therefore, are not merely the characteristics of individuals but precisely of collectives: collective, ritualistic features mark interactions. This poses the question how rhythmic alignment is achieved in the face of the apparent demands that interpretation would make on cognition. In

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response to this question we may note that rhythmic phenomena are available throughout our bodies, in the beat of the heart, the circadian rhythms of sleeping and waking, diurnal hormonal changes, breathing, contractions of and along the entire digestive tract from esophagus to the colon, the vibrations of our vocal cords, and so on. Because rhythm is a carnal phenomenon, arising from the auto-affection of the flesh—the heart knows to beat in the way our hand knows and remembers to grasp after having achieved it for a first time—it is possible to produce rhythmic phenomena even though everyday situations are improvised, are not machine-like implementation of routines. It is precisely this spontaneity of communication and of the constitution of everyday life that is inaccessible to traditional psychology and metaphysical approaches to cognition. It is the experience of the rhythmic movement to itself that lies at the bottom of our experience of the constitution of sense. Rhythmic phenomena in humans and human relations cannot be thought as perceptive categories alone. Any given rhythm has to be thought as arising from the tension that exists between materially given, objective phenomena in the setting and the rhythmization of the perceived on the part of the listener: Such phenomena, as cognition, sit on the borderline between inside and outside, in fact, constituting the borderline itself. Rhythm has a performative dimension: It is actively produced as much as received and reproduced. Because of its performative nature, rhythm also implies disturbance, breaks, pauses, difference, and discontinuities—phenomena involved in syncopation. It is precisely for this reason that rhythm can be thought as the interplay of discontinuity and continuity, regularity and irregularity (Plessner, 1981); without the disturbance of rhythm, we would not be able to perceive rhythm—in the same way that we would not perceive figure without the continual change of the focal between figure and ground (see Chapter 1). Central to the present articulation of mathematics in the flesh is the intermodality of rhythm, which therefore is consistent with the phenomenological observation that my perception is not an intellectual integration, a sum of perceptions, but a holistic experience held together by tact. My living/lived body, the flesh, is rhythmically organized—it is therefore not surprising that many mundane expressions refer to rhythmic phenomena: “to be in/out of tune,” “to be in/out of sync,” “to have the groove,” “to have the beat,” or “I resonate with you.” Most if not all educators will be able to accept that communication involves more than words; and they will be able to accept that teachers use “information” from these other modes to make sense of classroom events. Similarly, educators will be able to accept that children, too, draw on these other modes in their attempts at making sense of what is happening. Especially those educators who commit to enactivist and embodiment approaches to cognition will emphasize all those other modes as being aspects of communication and cognition. However, this is not the point that I make throughout this book. In other approaches, the body

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continues to be thought separately from the mind in the sense that it provides the resources for metaphoric and metonymic extensions. These are bodies enabled by “embodied (mental) image schemas,” that is, bodies that already are viewed/thought separately from mind. That is, it is supposed that cognition is embodied in the sense that it is only through the body that mind can have developed. This, however, still allows the mind to be a separate entity, as the symbolic world now functions as if on its own. (Maturana and Varela’s machine metaphor does not help overcoming the divide between schematic plans and improvised situated actions.) The world of the body has been transcended in the metaphoric and metonymic extension and the bodily “schemas” now are no longer necessary constituents of thought or rather the sense we make in thinking. My point is different. We come to a truly performative account only if the body is a necessary condition of sense, that is, if the difference between the body of sense and the sense of the body is undecidable. In this case it makes no longer sense to speak of the embodiment of mind, because the very distinction takes us away from understanding the phenomenon in the same way that we do not understand light if we pit against each other the two forms in which it externalizes and expresses itself, as wave or as particle (corpuscle). The living/lived body is not a machine, and any patterned (schematic) account of concrete action is valid only after the fact, because human actions fundamentally are improvisations, including the acts of speaking. It is not a machine because its self-knowledge does not come from a feedback loop but from the auto-affection that leads to what Sheets-Johnston (2009) describes as the identity of thinking and movement. Moreover, it is not just a material body, as I argue throughout this book, but the flesh capable of auto-affection that constitutes a form of immemorial memory, which enables intention and symbolic memory. It is precisely because of this immemorial memory that we do not need schemas to do what we do and to do it, recognizably, over and over again. We perform the movements each time, not because of a schema that is somehow encoded in our mind, but because the movement can be repeated over and over again without requiring a representation—in the form of a mental or sensorimotor schema (plan). Just recall how my hands have recalled the telephone number, not because of a mental representation that it enacted, not because they were acting according to a schema, but because of an immemorial memory that even I was not aware of. Hand gestures employed in speaking are but vestiges of the history of thinking that also appear rather than being constitutive of the sense. Typically, we may fi nd approaches to integrating speech and gesture, where both are attributed to some cognitive model that is described by linguistic means. Thus, if those other means (modes) of communication come to be described by verbal means, then they have been reduced to language. Those other modes are subsidiary to language and linguistic means, which thereby come to be the essence of thought. Thinking is properly incarnate when all these other means are but aspects of a

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greater unit. This greater unit manifests itself in the various modes only one-sidedly. Therefore neither the whole nor the parts other than language are reducible to language. Moreover, we have to theorize thinking (moving) as a creative and spontaneous process the essence of which is not the enactment of a program (plan), the schema. Spontaneity etymologically derives from the Lat. sponte, of one’s own accord, and therefore denotes actions that spring forth entirely from their own (natural) impulse (rather than from a plan, schema). To capture the spontaneity of communicating and thinking, we need to theorize thinking and speaking in terms of performances, improvisations, which subsequently may be assessed in terms of the degree to which it corresponds to some score. It is not the schemas that cause the performance, like the musical score does not cause the musical performance, or like the recipe does not cause the particular events in a kitchen.

INTERLACING OF BODIES In Chapter 3, I articulate how in and through participation in society (corps social) we come to have socialized bodies (corps socialisé). What we do and signify is immediately endowed with sense because a socialized body produces it. This socialized body is the result of participation in social fields. By participating in social fields, we are shaped in a double way. First, our existing habitus allows/makes us see the social and material structures in particular ways, thereby shaping in which way we participate; and, second, our actions leave traces, change our bodies. We can see the continual adaptation of persons acting in this mathematics classroom to other persons and artifacts, themselves associated with cultural histories and a mutual shaping with cultural practices.4 Throughout this book, we encounter students in exchanges with other students in movement interacting over and with objects. During such exchanges, they listen, and in listening, they have to open up to be affected (to hear), which makes them exist outside and inside at the same time—and it is precisely in the capacity to be outside that their living bodies come to be fashioned. This mathematics classroom is a field—a field within the field of the school—where children are part of interaction rituals involving material structures and objects; and such interaction rituals entrain and therefore shape the children’s living/lived bodies. In fact, the physical discipline that our living/lived bodies are subjected to make them disciplined bodies, living/lived bodies that exhibit, in their practices, academic disciplines (Roth & Bowen, 2001). The children we observe in the preceding chapters of this book develop disciplined (living, organic) bodies capable of exhibiting the discipline required by the mathematical discipline they study. Accordingly, living/lived mathematical bodies are made in the process of doing mathematics in schools. The children sit on their chairs; and when they

164 Geometry as Objective Science in Elementary School Classrooms do not do so properly, they are reminded that sitting has to occur in a particular manner to be appropriate for this classroom. Thus, when Cheyenne has taken her T-shirt and moved it over the backrest—so that the latter directly comes into contact with her skin—Mrs. Winter reminds her, “Cheyenne, you need to put that shirt properly” (Fragment 4.3c, turn 75). Similarly, when Brandon’s feet “dangle” into the circle where there are geometrical objects grouped, Mrs. Winter reminds him, “Brandon, you gotta get your feet up” (Fragment 8.5, turn 33). The children sit, oriented toward the chalkboard, filled with writing or, sometimes, with posters that themselves are inscribed. Throughout this book, children interact with familiar objects, such as pizza and toothpaste boxes, Post-it blocks, tin cans, and cardboard rolls, which they learn to associate with the geometrical objects that the teachers introduce as “cubes,” “rectangular prisms,” “spheres,” “cylinders,” and the likes. In fact, what the teachers materially produce are sounds, and these sounds come to be associated with material objects that give rise to particular visual and tactile experiences. That is, sound, sight, and touch are the senses relevant to mathematics lessons; and what they produce and reproduce contributes to the constitution of mathematical sense.5 More important than the interlacing of the flesh with the material bodies of cubes, rectangular prisms, pyramids, spheres, and the like are those where the flesh, social bodies, comes to be intertwined with other social bodies. Thus, we see students engage in exchanges with other students when Jane talks to Melissa about how to model the mystery object inside the shoebox in front of them, their actions intertwine (Chapter 5). Jane pushes the shoebox to Melissa, who then reaches inside; or Jane and Melissa, taking turns, orient each other to the models they have built, sometimes working with different and sometimes with the same object. Interlacing occurs, for example, while Melissa is reaching into the shoebox while Sylvia iconically gestures the caliper configuration, thereby “directing” Melissa what she has to orient to with her hand in the shoebox. And interlacing of their bodies occurs when Jane touches the top of her model while Melissa explores it with her left hand and while exploring the mystery object in the shoebox at the same time. Such interlacing, which is an expression of coordination of living/lived bodies in movement, constitutes thinking in movement. We also see direct contact between a teacher and a student, when Mrs. Winter takes Thomas’s fi nger and moves it along the edges of the cube so that he can learn to identify the discourse of a “straight edge” with the visual and tactile impressions of a material straight edge (Chapter 4). At other times, a teacher holds an object such that a student can access it sensorily in particular ways. Thus, Mrs. Winter holds out a cube so that Thomas can see it in a particular way: Here, it is the face that he orients to and moves his fi nger across in response to the teacher’s instruction, “so which one is the straight edge?” (Fragment 4.3c, turn 66–68) or

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when she asks him “what kind of edge is that” while holding out the cube (Fragment 4.3b, turn 22–23). But the teachers’ gestures, too, leave their traces, for these are central aspects in/of the communication. In Fragment 4.1, Mrs. Turner moves her hands in a particular manner, palms facing each other and moving closer and farther away together with her talk about a rectangular prism being something like a fl attened cube (turn 13). We subsequently see students also produce similar hand gestures, such as when Ben shows what to do so that a Post-it pad would become a cube (Figure 7.1). In one instant, we observe Ben producing a hand gesture that iconically exhibits one side and the top face of a cube, and only fractions of a second later, Mrs. Winter also produces these two gestures (Fragment 6.2a, turn 08). And, to state a fi nal example, Chris and Mrs. Turner produce precisely the same hand gesture above the pizza box to show what would have to happen to turn it into a cube (Fragment 6.3, turns 41, 44). But we should not take this coordination as evidence for the fact that the audience interprets a speaker, because perceptual consciousness is very different from linguistic consciousness (Vygotsky, 1986) and therefore does not use the means and processes underlying interpretive processes that require linguistic consciousness (to produce explanation). Intellectual interpretation is unnecessary, because, according to Sheets-Johnstone (2009), such movement is thinking and no further intellectualization is required. We can understand such instances of interlacing as forms of resonance, where particular temporal patterns come to be reproduced. This reproduction sometimes is directly coordinated in exchanges between two subjects, who take complementary roles and places; sometimes this reproduction occurs later in time, as if it were the echo of a previous movement. Living/lived bodies come to be moving about as part of communication between participants that exhibit a complex interlacing of space and orientations. The living/lived body becomes an expression in this configuration. In the following fragment, Chris has turned his back to

Figure 7.1 Ben uses the same gestures as Mrs. Turner—in an episode involving Chris during a different lesson—to show how Post-it pads need to be stacked to produce a cube.

166 Geometry as Objective Science in Elementary School Classrooms

Figure 7.2 Mrs. Turner’s talk and gestures concerning a cube are coordinated with her turning toward a cube that is behind her.

the class and the teacher while placing the pizza box underneath a toothpaste box, the place from where he had taken it. Mrs. Turner makes a (summarizing) statement concerning rectangular prisms and cubes, “so that makes it a rectangular prism as opposed to a cube” (turn 17). She continues producing a clause: “so if it was a cube.” In the second part of this utterance, her upper body rotates sideways and to the right, her head turns even further so that her gaze comes to be directed to a cube currently placed on the chalk rest (Figure 7.2). That is, in this movement, her gaze moves from the pizza box to the cube and then reorients toward Chris and the pizza box. As she utters the fi nal part of this turn 17, she points in the direction of the pizza box. But Chris is still oriented to it so that he cannot see the deictic gesture (Figure 7.3a). Fragment 7.1 (from 4.1) 16

17

T:

18 19 20

C: T:

21 22

C:

(4.91) ((Chris places box, no longer looks at her, busy placing the box even after she has started again)) so <that> makes it a rectangular prism as opposed to a cUBe, because if it was a cube what would it have to have (0.58) um [thiss ] <[whawould]> that box have to have to be a cube. (0.77) it would have to um ((gets a cube from chalk holder)) (0.44) that squ:are: ((a)) or this (0.21) like the:(0.62)s:: ((b))

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Figure 7.3 Chris’s body and head/gaze movements come to be aligned with those of Mrs. Turner, as he re-orients to the pizza box after having just oriented away from it.

Chris then brings his upper body up, gazing toward the cube, which is, from his perspective, to the left and behind Mrs. Turner (Figure 7.3b). He comes up a bit further and then moves his head to the left, his gaze direction now intersecting with Mrs. Turner’s deictic gesture direction precisely at the pizza box (Figure 7.3c). Here, then, Mrs. Turner’s hand moves in a manner that can be recognized as a pointing gesture. The movement comes to be interlaced with the movement of Chris’s head and eyes. The two movements are oriented toward what may be called the focal artifact or object of joint attention. When we observe the instant in real time, it is so fast that we come to realize that any hope of modeling this fragment by an interpretive (constructive) mind, which takes in and processes information to subsequently produce an action, would fail. The constructivist mind would not have enough time to call up the embodied image schema that it then tells the body to enact. Such a model of human interaction and joint production of attention is computationally so expensive that it would take orders of magnitude longer to produce the event. “The position of another as other-than-myself is impossible if it is consciousness that has to do it” (Merleau-Ponty, 1960, p. 152). Linguistic consciousness is not only too slow but also would make contradictory demands in that it involves “constituting him as constituting, and constituting as constituting in respect to the very act through which I constitute him” (p. 152). Incarnation, Merleau-Ponty suggests, cannot be classified as simple psychological phenomenon precisely because it requires the perception of others. Chris’s gaze is directed for a while to the pizza box. Then, just as he begins a turn at talk, his head turns, gaze directed toward the cube (as

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previously) his right leg moves forward as he stretches to reach for the red cube on the chalk tray. As he comes close to Mrs. Turner, she moves her left leg backward, bringing her body around and a step away from the chalkboard, thereby creating space for Chris to get to the cube (Figure 7.4). Both are now oriented toward the cube, a cultural-historically charged object in the discipline of geometry. Their two bodies, as in a dance, have both oriented them to a geometrical object in a coordinated movement that makes space (teacher) and uses the now available space to access the object. In fact, when we look at their body positions, placements, and orientations across the sequence as a whole, we notice that as a result, Mrs. Turner ends up about one meter away from Chris, where a complex of leg movements fi rst takes her a step away from the chalkboard and then, as her right leg moves backward followed by a body rotation and a backward movement of the left leg puts her in a new position (Figure 7.5). In this new position, her body is oriented as previously in the direction of the class, with her head turned toward Chris. But the ensemble of geometrical objects on the chalk tray has now become perceptually accessible and reachable rather than being hidden behind Mrs. Turner’s body. Conversely, the poster featuring the parts of geometrical objects exemplified by a pyramid now, from the perspective of the class, now comes to be hidden behind her. In subsequent parts of this same lesson, when she makes reference to these parts, she moves again to make the visual access possible for all students. In these instances, Mrs. Turner’s and Chris’s living/lived bodies move together and in response to each other in the same way a pair of dancers move in a coordinated dance without having to intellectualize their next step or movement.

Figure 7.4 Chris’s movements are associated with complementary movements on the part of Mrs. Turner, as she turns allowing her to see where to Chris orients himself.

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Figure 7.5 Mrs. Turner, in a movement complementary to that of Chris, steps away from the position that she has held, thereby allowing Chris to take it up.

In this section, I exhibit the ways in which the movements of participants in mathematics lessons come to be interlaced, interact, produce interactions in their actions, and actions in interactions. A hand gesture on the part of one person entrains the head and body movement of another, a forward step and gaze direction toward a particular mathematical object comes to be entangled with the sideward and backward steps of another person, oriented not only to the geometrical object but to the space another person requires for accessing the object. But all these movements are so fast that they would be prohibitive in a model that is built on interpretation, a process that involves explanation and therefore is mediate rather than immediate; it would be too slow if movement were thought as enactment of embodied image schemas in the mind that order the body how to move. Such coordination of living/lived bodies in space requires, according to Merleau-Ponty, a form of consciousness very different from the linguistic one that is so important in constructivist theories. Merleau-Ponty’s findings have been confi rmed more than once in more recent neuroscientific and cognitive scientific studies. Thus, one study in cognitive science concerning the actions in the game of Tetris shows that if the player were to interpret the shape and position of the object currently visible on the screen prior to moving the falling object (rotating it, moving it sideways), the actions would be far slower than actually observed (Kirsh & Maglio, 1994). The authors offer a model in which actions have pragmatic and epistemic dimensions and are not the result of prior computations (interpretations) but rather serve to both act in and change the world and, in the process, develop a practical understanding of it. The Tetris player’s “epistemic” movement is thinking. A similar attunement to others, the setting, and to mathematical objects is observable in the events presented here, as there is a continuous

170 Geometry as Objective Science in Elementary School Classrooms reorientation between boxes understood as concrete examples of rectangular prisms and objects created to stand in for the mathematical idea of a cube. The capacity to be attuned to the changing objects of attention, both materially and ideally, means knowing to communicate mathematically, to understand mathematics in very practical ways. The different bodies that appear in my description are not equivalent. The cube on the chalk tray or the pizza and toothpaste boxes return to their places after having been used. Their part in the events leaves no traces in them. These bodies are unaffected—apart from some wear observable only over periods of time much longer than the curriculum. On the other hand, the living/lived bodies of humans are similar, their movements being coordinated, and precisely because it takes energy and because use wears them, the events come to be inscribed in them. As my long-term research in fish hatcheries showed, even the most “boring,” routine jobs—e.g., feeding fish, washing fish ponds—have observable long-term effects in the living/lived bodies of the workers, which come to exhibit increasing mastery of the work at hand: the way they move the scoop while throwing feed or what they can actually perceive in fish movement. That is, even though the effects are not immediately observable and noticeable, they bring about changes in the flesh of the workers, that is, they produce adaptation to the activity and therefore learning. Schema theory only describes the structure of the movement, not the qualitative differences in the movements that distinguish the different levels of expertise. These living/lived bodies, therefore, have to be theorized differently, they are not the same kinds of bodies; they are not bodies that come about in a modification by means of the adjectives “living” and “lived.” This is why I propose the concept of the flesh, endowed with senses, which is the source of the human body as recognized by the person inhabiting the body. The singularity of the “I” is itself the product of all the interactions, a result of the flesh that recognizes its organic body as a body among bodies. We learn because of the flesh, as our living bodies come to act in a social world, already structured in multifarious ways, including lifeless objects that are associated with particular cultural practices that themselves change in the history of the culture. Always acting in a world already full of significations, the flesh is in a social world all the while the social world is in the flesh. “The structures of the world are present in the structures . . . that agents mobilize in understanding. When the same history pervades both habitus and habitat, dispositions and position . . . history communicates in a sense to itself, reflects upon itself” (Bourdieu, 1997, p. 180). I noticed this tight interlacing of positions and dispositions (for teaching) while observing pairs of teachers work together over longer periods of time, about two to three months.6 The teachers came to resemble one another in what they were doing, from intonation, to use of specific words, to moving about and positioning in the classroom, and interactional styles. And nothing of this alignment had been present in the linguistic consciousness of the

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participating teachers until my research mapped it out for different modalities (e.g., prosody, positioning, movement, orientations, etc.). In the fragments featured here, we already have another dimension that is central to human interactions: time and temporality generally and the pacing of social events specifically. Thus, Chris’s and Mrs. Turner’s movements are not only coordinated in space but also, and more importantly, they are coordinated in time. It is as if a hidden choreography were at work, but a choreography that gives substantial amount of improvisational freedom to the performers and nobody knows future actions even seconds hence. Even the agents themselves do not know what they will be doing precisely. Mrs. Turner steps to the side, even though she did not know this only seconds before. But in stepping aside, she opens the access to the cube at precisely the instant that Chris is reaching for it. Both have already oriented to the cube, though neither has been in a position to note this orientation of the other, which was (partially) hidden because of the way in which they were currently oriented. (Chris gaze is directed toward the cube presently not visible to Mrs. Turner, and she orients to the cube while Chris is preoccupied by the attempt to get the pizza box back into its place.) Such coordination, rather than being the result of interpretations, can be better understood as resonance phenomena, thinking in movement. Resonance, a phenomenon of coordination of movement, leads us to the ritualistic aspects of interactions, which we thereby come to understand as interaction rituals. In the next section, I turn to a number of different features of interaction rituals that are all concerned with time.

INTERLACING OF PACING, PITCH, PITCH CONTOURS We are not tuned to sounds from which we extract words, but rather we hear words. However, we do not just hear words, but we hear words spoken tentatively, almost like questions; we hear questions even though the grammatical form of the utterance is a statement, and, conversely, we hear questions even though intonationally an utterances is produced like a statement; and we hear/see whether a student appears to know but cannot express herself, or whether the student does not appear to know and therefore the expression is not fluent. As teachers, we do all of this with a spontaneity that no constructivist theory can explain—even schema theory would require prior interpretation and the identification of a relevant sensorimotor schema that would then tell the body how to act. It is precisely from this spontaneity that we can learn something about knowing that we could not know otherwise. In the previous section we see how it is precisely because acting and sensing go together that people can coordinate their mutual orientations to objects of joint attention, body positions and orientations, gestures, and head movements. In this section we return to the same instant of the

172 Geometry as Objective Science in Elementary School Classrooms lesson but add a few more lines of transcript. One aspect I explicitly point out in Chapter 1 is the coordination of the hand movement and speech. Thus, in turn 14 (Fragment 7.2), the hand movements are coordinated with the verbal expressions (i.e., with movements of different components of the vocal tract, lungs, and so on); and where there might be a danger of misunderstanding, we observe the repetition of a gestural expression (i.e., movement) precisely coordinated with the speech expression (i.e., movement). Such coordination, further associated with the coordination of the living body in space and the coordination with other (living) bodies, would be a feat so complex that the computational mind processing representations would be lost in the face of the task. But this coordination is much more easily understandable if we think in terms of resonance phenomena, coordinated movements, where periodic features come to align with each other as long as the characteristic frequencies of the individual participants are not too different (in which case “synchrony” is not achieved). There is further coordination, however, which makes the interlacing of the bodies both tighter and more complex. This is why phenomenological approaches consider the entire living/lived body pervaded by tact as one expression, not merely as the expression of something thought beforehand and then made available to others. It is therefore that “my spoken words surprise me and teach me my thoughts” (Merleau-Ponty, 1960, p. 144). All of the different expressions I produce have an “immanent sense, which do not come from an ‘I think,’ but from an ‘I can’” (p. 144). This “I can,” before all thought and linguistic mediation, is a characteristic feature of the flesh. An expression therefore never is the expression of something else that went on before, thought, but the expression constitutes the very phenomenon of thinking. “For the speaking subject, expressing is to become conscious” (p. 146). This precisely is the central idea why cognitive scientists and artificial intelligence researchers interested in the role of space in cognition speak of epistemic actions: Actions not only change some aspect of the lifeworld but also constitute a form of making sense.7 Fragment 7.2 13

T:

14

C:

[it] could be like a cube thats bin (0.22) ((hand movements showing the closing of the distance between two palms)) flatten[ed] ((hands retract)) [uh] hm bu:t () ((lH moves down face of box; then holds box with lH)) s:::ohm::e are like long=an

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some are shorter than the other one[ss]

15 16

T:

17

T:

18 19 20 21 22

C: T: C:

((rH movement along right edge, then rH movement along edge pointing to Mrs. T, then again rH movement along right face)) ((looks up to her)) [uh] hm (4.91) ((Chris places box, no longer looks at her, busy placing the box even after she has started again)) so <that> makes it a rectangular prism as opposed to a cUBe, because if it was a cube whAT would it have to have (0.58) um [thiss ] <[whawould]> that box have to have to be a cube. (0.77) it would have to um ((gets a cube from chalk holder)) (0.44) that squ:are: ((a)) or this (0.21) like the:(0.62)s:: ((b))

22

Here, I orient readers to another phenomenon of central importance to the way in which social action and interaction becomes possible, and in which trouble comes to be known: features of speech, or, as it is known to specialists, prosody. Prosody includes speech intensity, speech rate, and, importantly, pitch (as well as other aspects of sound production, such as the fi rst major frequency other than pitch, generally referred to as F1). Previous research—my own and that of others—has shown that in confl ict situations, the pitch levels of two adjacent speakers tend to be very different, moving toward higher and higher pitch levels as the confl ict aggravates.8 On the other hand, in classrooms that are evaluated to exhibit a great degree of solidarity among students and with the teacher, the pitch levels of adjacent speakers tend to align. New teachers, after working together with the regular and more experienced teachers, tend

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to use the same prosodic features. New teachers generally and teachers new to a class specifically tend to become attuned to the students so that their pitch levels begin where the previous student speaker has ends; and students, in a similar manner, tend to fall into the pitch and reproduce pitch contours of the preceding teacher utterance. This is also observable in the present fragment, noticeable in the three turn changes (turns 13–14, Figure 7.8, turns 14–15, Figure 7.7, and turns 19–20, Figure 7.6). A typical example of the joining of the pitch levels between two speakers can be observed in the change over between turns 19 and 20. Mrs. Turner, who follows Chris, not only overlaps his speech but her pitch also aligns with his, leading to continuity in the melodic line of the two speakers (Figure 7.6). Another kind of “pick up” exists when the second turn employs the same pitch contour, leading not to a direct continuation of the main frequency but to the melodic line that is produced toward

Figure 7.6 Mrs. Turner’s pitch level picks up where Chris ends before returning into its normal range.

Figure 7.7 Mrs. Turner’s pitch contour repeats that previously available in Chris’s voice.

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the end of the previous utterance such as between turns 14 and 15 (Figure 7.7). The effect is that we hear the same melody produced in and by the two sound performances, which, depending on other aspects of the context, may be heard as confi rmation or as mocking of the previous speaker/utterance. Sometimes, however, the coordination becomes difficult, such as when, for example, the previous utterance continuously falls as in a statement only to suddenly rise at the end, heard as an emphasized word. That is, the intonation and the content of the utterance signal the unfolding of a constative, which allows an anticipated end point for the pitch at the end of the utterance. If the pitch, however, rises suddenly, not as the intonational flag marking a question but as emphasis, the next speaker is—unconsciously—in a quandary, having anticipated a lower ending of the pitch, which suddenly ends high. Perceptive consciousness, being of different nature than verbal consciousness, allows a rapid solution to the problem. One of the ways in which the problem may be solved is to begin halfway between the two, the anticipated endpoint of the pitch and the actual endpoint. Such a situation occurs in the change over between turns 13 and 14 (Figure 7.8). Chris comes in with his pitch about halfway between the projected endpoint of the utterance that precedes his own and the actual endpoint, by means of which Mrs. Turner emphasizes the word “flattened.” Emphases may be produced by different means, such as rapid momentary increases in pitch levels, by means of noticeable increases or decreases in speech intensity, or by means of changes in speech rate. But all of these features also are involved in the production of another structure of classroom life, rhythm, which my own research in innercity science classrooms is a way of aligning and coordinating members

Figure 7.8 The sudden and non-anticipatable jump and rise of Mrs. Turner’s pitch level creates a problem for the subsequent pitch level, here “solved” by a pitch that lies halfway between the anticipatable pitch ending and the actual pitch ending.

176 Geometry as Objective Science in Elementary School Classrooms of the same community and to both express and produce a sense of solidarity. In the next section, I turn to precisely this feature, which produces phenomena in which any conceived difference between some mind and a body becomes undecidable. In other words, there is an alignment of different forms of consciousness that cannot be reduced to each other.

RHYTHM AND PACE Musica est arithmetica nesciendi se numerare animi (Music is the arithmetic of the soul that is unaware of its own counting). —Gottfried Leibniz (quoted in Sambursky, 1973, p. 172) Rhythm is the incarnation of a formal cognition in an intuitive content. (Abraham, 1995, p. 24) Once we train our ear to hear rhythmically, we can begin to recognize the intricate complexity of communication in real time. I use but two instances from the fragment exhibited in the previous sections to illustrate how pacing and rhythm are produced together with mathematical content. The difference between the production of punctuated movements and the conceptual production (i.e., thinking) is undecidable. (The vocal production that lies at the origin of the words we hear is itself a completely corporeal phenomenon.) Rhythm is not just added to the speech: It is the rhythm of the speech and therefore cannot be perceived separately or detached from speech. Rhythm is not an epiphenomenon but it is the phenomenon without which speech would not exist. In communication there exists a “constitutive tension between the phenomenon of ‘rhythm’ and the dimension of ‘sense’” (Gumbrecht, 1988, p. 715, original emphasis). The rhythm is immanent to the production of communicative expression. But I can become aware of this rhythm, such as when I use meter notation (i.e.,–, ‿) to indicate on which syllables there are major emphases and on which there are minor emphases. In this sense, rhythm can be abstracted and represented with beat gestures, tapping, or by means of appropriate and widely used signs. In this case, the rhythm has been separated from its substrate and has become a transcendental object in its own right. The point is that in the production of speech, which is impossible to reproduce here, the rhythm is an immanent feature and integral part of the production of mathematical communication. In a strong sense, therefore, to understand how a student or teacher understands, readers have to reproduce the communication using the fragments in the same way that a musician would use a score to play a piece of music. Readers have to reproduce for themselves the phenomenon by enacting the transcriptions, including the stress patterns indicated; and thereby, in their own performances, readers experience these immanent features.

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In turn 14, Chris produces a next turn to the constative statement that the pizza box, denoted as an instance of a rectangular prism: “could be like a cube that’s been flattened” (turn 13, Fragment 7.2). I already exhibit above the coordination between the expression of geometrical content— the sides that are of different length therefore making this pizza box something different from a cube that has been flattened—and the hand gestures. In this turn 14, we also observe the production of rhythm. Fragment 7.3 14

C:

[uh] hm bu:t () ((lH moves down face of box; then holds box with lH)) s:::ohm::e are like long=an

some are shorter than the other one[ss ]

This fragment may be transcribed using meter notation, which exhibits rhythm, including the syncopation that occurs. We can see that there are two phrases, and across the phrases the da-da-dam (‿ ‿ –) rhythm typical for constative statements, where the “da” (“‿”) and the “dam” (“–”) correspond to weak and strong emphasis, respectively. | 1.11 s | 1.44 s | uh hm but () s:::ohm::e are like long=an

‿ ‿

–

‿

‿

–

‿

–

| 1.10 | 1.41 s | some are shorter than the other one[ss ]

–

‿

–

‿

‿

–

This is a three-quarter rhythm with the strong emphasis on the third beat. In questions, the strong emphasis tends to occur fi rst with the two

178 Geometry as Objective Science in Elementary School Classrooms weak emphases following. In the two phrases, Chris also produces gestures, one along each of the two orthogonal sides of the rectangular pizza box. The end of each phrase falls together with the main gesture. In the second phrase, there is a gesture that precedes the principal one, which looks the same, but it comes “off beat.” Perhaps not surprisingly, it is repeated so that it comes to fall together with the right place; but this surprise itself is surprising, as the whole performance has ended faster than even the beginning of an intellectual interpretive process would allow. Both main phrases are of almost the same length—differing by only three one-hundredths of a second. A close look at the second phrase reveals that there are more syllables, but, as the corresponding timing measure exhibits, this greater number of syllables is produced in the same amount of standard time. In a musical notation, this corresponds to two lined eighth notes (♫).9 To really appreciate the performance, readers ought not just silently read transcript and its rhythm for its cognitive content, but actually perform it, read it aloud while stressing as the meter indicates. Only then does the rhythmic consciousness, which is a performative, actually cognize what is happening: “To be able to speak about rhythm, one has to actually produce it, coproduce it in perception, in the action of the body, duplicate it, experience the process of the rhythmic in ones own living body/flesh” (Brüstle, Ghattas, Risi, & Schouten, 2005, p. 27). We also note the two syncopations preceding each main part of the respective phrases, which constitutes these parts as off the regular da-da-dam beat (i.e., ‿ ‿ –). However, overall, a new rhythm emerges, in which the syncopated and un-syncopated parts are integrated to form a new rhythm that is repeated in each of the two lines (Abraham, 1995). Together with this rhythmic moment of the performance, Chris produces a semantic-conceptual moment. Here, it is a correction of a constative statement that Mrs. Turner has made as her summary of what he has said before. Following the marker of contrast “but,” Chris describes and gesturally highlights the differences that may be observed in the sides. The two parts of the comparative statement, each pertaining to one side of a box, have the same overall rhythm. The rhythm is structured as the composition of a regular bar and one with syncopation (missing of a beat of the 3/4 rhythm, or a 2/4 beat). Aligned with the content and rhythm are the hand/arm gestures. In this performance, therefore, the difference between purely mathematical content and purely bodily performance is undecidable; speech with mathematical content and rhythmic movement of speech are but two sides of the same coin. Mathematical communication cannot be reduced to conceptual content; it is irremediably tied to the living/lived body, its rhythms and melodies. This a social body that is inherently in an interaction ritual, for which the communication is produced; but as expression, it is thought, personal and social thought simultaneously. This thought, however, generally is not cogitated beforehand but is the result of an improvised movement expression, which, as it unfolds in time, leads to the movement (i.e., development) of thought.

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Teachers, too, produce highly rhythmic mathematical communication, and, in fact, it is one aspect of the structured classroom setting that is structured by and structures the actions of the children. These structures, in turn, entrain and thereby change the living/lived bodies lastingly. This can also be seen in the repeatedly analyzed fragment, from where I extract Mrs. Turner’s turn 17. Fragment 7.4 17

T:

so <that> makes it a rectangular prism as opposed to a cUBe, because if it was a cube whAT would it have to have

Across the complex utterance—which consists of a statement, a comparison, a clause, and a question—we observe the 3/4 tact with the major stress at the end (da-da-dum) or at the beginning (dum-da-da). We see, for example, in the last part (i.e., the question), that the “what” is stressed, as this is typical for grammatically marked questions, followed by the two unstressed beats, which is the classical meter of dactyl: dum-da-da. This pattern is repeated once, falling together with the grammatical end of the sentence. The segments are of approximately the same length, though the syncopation that occurs in the fi rst verse lengthens it; this syncopation occurs as the main emphasis is missing, and we observe the beginning of the da-da-dum pattern—the classical meter of anapaest—that is constant for the remainder of the performance. (1.69) | (1.51) | so that makes it a a rectangular prism | as opposed to a cUBe, | ‿ – ‿ ‿‿‿– ‿‿ ‿‿ – | – | ‿ ‿ – | (1.45) | (1.24) bcause if it was a cu::be | what would it have to have| ‿ ‿ –‿ ‿ – ‿ ‿ – ‿ ‿ | | –

In the second line, we also observe a syncope, which occurs between the conditional clause that constitutes the fi rst half and the question that constitutes the second half of the line. Throughout this book, we can observe questions that are implemented by an emphasized interrogative that begins the phrase or utterance, with the pitch falling toward the end as it is generally observed in constative utterances. From a rhythmic perspective, the interrogative “what” would fall on a weak emphasis, but there is a strong emphasis because of the question to come. This unexpected strong beat changes the rhythm and we therefore get the syncopation where two strong emphases follow each other, enabled by a particularly elongated production of the word “cube,” which by itself takes 0.53 seconds. In both lines of the transcript, the faster phrase is heard as a clause, and this hearing

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is brought about by the faster than normal pace (allegro). As before, the conceptual aspects are inseparable from those that are purely related to the production of the utterance. Mathematical content, clauses that explicate, comment upon, or modify the main clause are recognizable in and by the different prosodic means that implement them. These features precisely exhibit the competence, which would not be detected if a speaker were to confuse or interchange main with subsidiary clauses, produce discoordination between gestural and verbal production. In fact, when gestures and speech are misaligned, researchers recognize both a lower level of cognitive development and the readiness to proceed to a higher level.10 That is, coordinated expression means that the living/lived body as a whole produces canonical or non-canonical mathematics, recognizable as such precisely in the alignment of the performative aspects of mathematical communication rather than by some form of magical appearance of inaccessible mental constructions and mental representations. Mrs. Turner, too, produces gestures while speaking even though Chris cannot see them—not surprising, perhaps, given that even congenitally blind individuals gesture when communicating with other congenitally blind individuals. She opposes the palms of her hand and produces the same gesture three times, the fi rst time with about half the amplitude as the second and third time. The transcript with the rhythms also indicates the beat part of this gesture (the closure), the reverse occurring in the non-underlined parts separating the two closing gestures. We observe that the main gestures—the ones with the large gap closing—fall together with the main emphases, occurring together with the sound heard as the word “cube.” In a way that bears striking resemblance with the performance of Chris, there is an offbeat gesture, and it is repeated in the same phrase together with the main emphasis. How is such coordination possible? Phenomenological analyses suggest that we take position in and orient to the world, and the living/lived body as a whole becomes expression. It constitutes an “I can” that is spread throughout the flesh rather than limited to this or that organ the movement of which needs to be coordinated with another organ. This is so because “I engage myself with my body among the things, they coexist with me as incarnated subject, and this life among the things has nothing in common with the construction of scientific objects” (MerleauPonty, 1945, p. 216, emphasis added). It is precisely habitus—immemorial memory—that allows the world to emerge in and from our actions by the manner “it orients to the world, brings to an attention that, like the one of the long jumper who concentrates, is an active and constructive bodily tension toward the imminent future” (Bourdieu, 1997, p. 173). In the previous fragment, we see how rhythmic features characterize the speech, as they are produced with and structure the same material that produces sounds, which we hear as words. These rhythmic features cannot be thought apart from the material that makes the sound. But, we may also observe explicit repetition at the level of language, generally structured a

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second time by the rhythmic features in the body of the sound. The same explicit features can also be found in other modes of communication, such as when the students’ (and teachers’) gestures take up and are shaped by the same recurrences in structure. Thus, for example, in Fragment 7.5 (which repeats Fragment 6.2b), Mrs. Turner, in producing the fi rst part of a question-answer sequence, utters the same word four times (turns 57, 60, 63). In turn 57, she produces three series in each of which the word “square” appears four times, “square, square, square, square.” During the fi rst sequence, her index fi nger twice moves along the four top edges of Joel’s cube model, each square circumference gesture coinciding with the performance of a “square, square” articulation. As she utters the second sequence, her index fi nger repeats the circumference gesture, including three 90° turns, making the gesture recognizably consistent with the “square” description that she utters. During the third series, the video shows Ben holding up his model, exhibiting the caliper configuration repeatedly while rotating the cube. Fragment 7.5 (Fragment 6.2b) 57

T:

58 59 60

J: T:

61 62 63

J: T:

64

((:B)) hOW did you know; im looking at your model here. ((points to Joel’s model)) (0.50) and (0.32) this looks like square square square square how did you know t do square square square square; hOW [did you know square square square square; [((Ben brings up hands with cube, gestures, Fig. 6.1)) (1.18) ((Mrs. T shrugs shoulder, looks at Joel)) um square square =not rect]angle rectangle rectangle like <thEY> (0.30) thEY ((points to back of classroom, Ben & Joel turn around)) have rectangle rectangle rectangle or oblong oblong oblong; right jonathan? (0.24) yea. so () how did you know to go square square square square square; (0.33) all along the sides and not o:blONG o:blONG o:blONG; ((moves head in same rhythm)) (1.36)

There is a longer pause. Joel then utters “square” twice, immediately succeeded by Mrs. Turner, who now produces twice a sequence consisting of three “rectangle” utterances and one sequence of three “oblong” articulations, which describe the model of Jonathan’s group. She then turns again toward Ben and repeats two sequences, one consisting of five articulations of “square” and a second sequence, which, in contrast, consists of three articulations of “oblong” (turn 63). The fi rst sequence is accompanied by

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a beat gesture where the left-hand index finger produces four beats, while apparent to all pointing to each of the four vertices of a square. Beginning with the fi fth and last “square” and into the subsequent pause, the head produces four beats. The performance that produces the three iterations of “oblong” each includes a precisely coordinated beat gesture of the head, which, with each beat, also moves further to the left as if it were inscribing the beats on an invisible sheet of music (though in the reverse direction of the normal notation). That is, the repetitions of geometry words are precisely coordinated with each repetition of a beat gesture (fi nger, head), that is, movements of the living/lived body. When we now do an analysis in terms of the strong and weak emphases, the highly rhythmic aspects of Mrs. Turner’s performance become clearly evident. As an example, I represent turn 57 using the poetic meter notation. hOW did you know; im looking at your model here. (0.50)

‿

–

‿

‿ ‿

–

‿ ‿

–

‿

–

‿

and (0.32) this looks like square square square square

‿

.

‿

‿

‿

–

‿

‿

–

how did you know t do square square square square;

–

‿ ‿

–

‿

‿

‿

‿

–

hOW [did you know square square square square;

–

‿ ‿

–

‿

‿

–

‿

We notice in the fi rst line of the following representation how the question is structured by the familiar dum-da-da rhythm. There is a break as the next part comes to be a clause (“I’m looking at your model here”). The clause is hearably produced by a substantial change in speech intensity (volume), which decreases to 1/4 of its original (from 75.7 dB as the mean intensity over the fi rst part to 69.7 dB as mean intensity of the clause part of the utterance). The clause is produced, not surprisingly given its constative nature, in the typical da-da-dum rhythm. It turns out that in this situation the rhythm is maintained across the break as the previous part ends on the strong emphasis in the word “know.” Here, then the change in speech intensity affords the perception of the ending of the clause. In the second line, the constative is produced embodying the da-da-dum pattern, the third line, which is marked by the grammatical interrogative “how,” follows the dum-da-da pattern in the same way as line four. In each case, there is a weak emphasis on the fi rst two articulations of “square,” followed by a strong emphasis on the third iteration of the word, and ends on an utterance of the word with weak emphasis.

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We note a change to three iterations in the sequences constituted by “rectangle” and “oblong.” But we observe a simultaneous shift in the rhythm, which now takes on the dum-da-da form (REC-tan-gle), certainly entrained by the normal stresses of the word in the English language (i.e., ‘rɛktaŋgl). The musical rhythm changes to the dum-da entrained by the normal stresses of “oblong” (i.e., trochee in OB-long). In the real time production of speech, there is an interaction of the musical meter with the metrical feet typical of poetry. But in poetry, especially in its written form, there are no pauses, no constraints that correspond to those associated with movements in space and the time required to complete a movement of the living body that interacts with the speaking movement. We therefore observe pauses in the speaking mode while other events occur, such as making a few steps to pick up an object. Pauses are places for syncopation to occur, where one rhythm changes into another one and where the transition cannot be anticipated and the instance of the syncopation, the turning point or bifurcation, belongs to both rhythms. On the other hand, rhythmic and other prosodic features, such as pitch and pitch contours, may be precisely timed with hand/arm or other bodily gestures (nods, pointing with chin) and other movements. Sociologists have identifi ed entrainment as an important aspect of coordination among members of a collective, with strong affective components (Collins, 2004). The term entrainment derives from the physical sciences, where it has been noticed that two pendulum clocks mounted on the same wall not too far from each other tend to align the frequencies of their pendulum swings. The two pendulums, mediated by the common mount, constitute a coupled oscillator system with the same frequencies for both; but the two clocks are in anti-phase, that is, when one pendulum is at its right-most position, the other is at its left-most position. Entrainment makes sense as soon as we accept the materiality of communication; and materiality comes with the possibility of consensual domains of different order. The primary domain is that on which the coupling (synchronization) between members to the interaction occurs, that is, the periodic phenomena (pitch, F1, rhythm), and at the second consensual level do we get the coordination at the level of sense. Resonance, as Nancy (2007) suggests, is the foundation of sense. In my own research referred to above conducted among teachers who teach together for a few months, I observed the entrainment of periodic features at different levels. For example, two teachers move in anti-phase with respect to positions they take up in the front of the classroom; their pitch levels adjust to one another, and their pitch contours take on the same shape. In this second-grade mathematics classroom, we fi nd in children’s gestures the same kind of rhythms that are characteristic features of the teachers’ talk. For example, we observe children who do not point

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to just three but four of the faces of a cube. In Chapter 1, we encounter Chris, who produces an answer to the question about other features of faces of geometrical shapes not yet mentioned in the whole-class conversation. Fragment 7.6 reproduces the turns in which Chris shows and articulates that on the cube there are squares “everywhere,” whereas the corresponding indexical gestures are reproduced in Figure 7.9. (Chris, Cheyenne) Fragment 7.6 (Fragment 1.1) 22g

23

an like ((rectangle)) (0.68) <but to=n=this squa> ((movement along edge of box)) ijst has (([a])) square (([b])) (0.45) n here= (([b])) =an=here= (([c])) =an there=(([d])) =and everywhere (([e])) (0.63)

In turn 22g, we observe a fi rst constative statement “it just has square” with a double indexical gesture consisting of a weakly and a strongly emphasized movement (beat) of the curved right hand to the face. There is then a repetition of the indexical gesture to the same face, falling together with the emphatic part of the utterance “here,” followed by the rotation of the cube and consecutive indexical gestures to other faces simultaneous with the productions of “here,” “there,” and “eve”[rywhere]. There are therefore four indexical gestures (underlined parts of words) falling together with four strong emphases, as shown in the following transcription of the rhythm.

Figure 7.9 Chris produces the same four-beat structure in his gesture while explaining the nature of a cube that Mrs. Turner enacts by prosodic means.

Doing Time in Mathematical Praxis ijst has

‿

‿

185

square (0.45) n here=an=here=an=there=and

–

‿

–

‿

–

‿

–

‿

everywhere

‿

–

–

In this representation, we note how the rhythmic features in the vocal production parallel those in the gestural movement, the strong emphases in the former falling together with the indexical gesture (underline), whereas the weak emphases fall together with the retraction of the right hand and rotational movement of the cube to exhibit the next face. The one indexical gesture produced together with a weak emphasis itself exhibits considerably smaller amplitude of the movement toward the cube face than the other iterations. The same phenomenon can be observed in Fragment 7.7, excerpted from a discussion concerning the group of objects that Ben, Cheyenne, and Ethan collected to go together with the “geometrical standard,” a cube, that they had received as the fi rst instance of their collection representing the category. Fragment 7.7 68 69 70 71 72 73 74 75

T: C: E: C:

76

T:

77 78

C:

[but its] nOT a cube because why (0.98) because its not like ((picks up red cube)) (1.14) ((holding up yellow cube)) like having sides (0.72) like thIS: ((strikes each of x, y, z faces)) (2.66) ((Cheyenne places palm on side of red cube)) doesnt have all those different sides. okay. thank you. (0.35) <[doesnt have all the sides ]>

Producing the response-part to the answer slot Mrs. Turner’s utterance (i.e., “but it’s [the Post-it pad] not a cube because why”) leaves for any one of the three students to fi ll, Cheyenne begins, “because it’s not straight,” while picking up a cube from the collection. There is a pause, which Ethan, holding another cube, brings to a close uttering “like having sides” (turn 72). There is another pause, which Cheyenne brings to an end announcing, “like this,” and then indexically denotes four faces by striking a face and rotating the cube to expose the next one to be struck (Figure 7.10). The indexical gestures are punctuated and highlighted by long retractions of the hand, producing both rhythm and particular strong emphases for each movement. That is, in this performance,

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we observe the same four repetitions of the weak-strong pattern as in Chris’s presentation of the cube—interestingly contrasting what one might expect for the three orthogonal faces of the generalized rectangular prism (those parallel to the x-y, y-z, and x-z planes in the Cartesian coordinate system). Perhaps even more surprising, as Cheyenne and Ethan both respond verbally to Mrs. Turner, they both strike faces of their respective cubes with

Figure 7.10 Cheyenne produces a four-beat structure while explaining the characteristics of a cube.

Figure 7.11 Cheyenne’s and Ethan’s performances of the four-beat structure are precisely coordinated.

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the palms of their right hands. Without gazing in the direction of the other, each directing the gaze at the cube in his/her own hand, they strike a face of the cube in perfect synchrony (Figure 7.11). That is, their respective striking of the faces enacts the same rhythm until this as occurred for the fourth time. At this point, Ethan raises his head, turns it slightly to his right, and directs his gaze at the cube that Cheyenne has in her hands. Not only do they each produce the fourfold pattern, but also they do so in synchrony (in phase). Strong rhythmic features can be observed throughout the lessons associated with situations other than indexical gestures. For example, when telling Thomas that he could not use shape words, Mrs. Winter (Chapter 4) rhythmically lists the shape words while counting with the hand, which is rhythmically beating while counting from one to three beginning with the thumb to which index and middle fi nger are added with each beat (Figure 7.12). Fragment 7.8 (Fragment 4.3) 49

W:

right; not using a sh:ape word. (0.22) so we=re not using a tRIangle CIRcle or squARE. ((rhythmically counting, beat gesture))

Figure 7.12 Cognitive content—counting of geometric concept words—is aligned with the iconic (counting) and beat (scanting) hand gestures and with the production of the rhythmic pattern.

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The transcription that includes the indication of strong and weak emphases exhibits the rhythmicity and the coordination of geometrical concept words, emphases, and counting beat gestures. We note in the second line of the transcription the coincidence of strong emphases and counting beat gestures. We also note the similarity in the structure of a strongly emphasized “not” followed by another stressed syllable (USing, GONna), which constitutes an instant of syncopation, as the anticipated rhythm changes, now beginning a familiar dum-da-da. In the fi rst line, we observe another syncopation as two consecutive syllables receive strong emphasis (shAPe, word), a sequence that we can hear as fi lled with tension. The time per syllable ratio is 0.27 seconds/syllable, which is approximately the length of the pause following the utterance. right; NOT USing a sHApe wOrd (0.22)

‿

–

–

‿ ‿

– – | <——— 1.86 s ———————> |

‿

1 2 3 so we=re NOT GONna use=a tRIAngle a CIRcle or squARE.

‿ ‿

‿ ‿ ‿ – ‿ ‿ – – ‿ – – | <————————— 2.34 s ————————————> |

In the second line, the dum-da-da pattern is maintained to the end of the utterance. In fact, the parallel in the performances of the fi rst and the second line projects the coming of a list, which its own realization both at the conceptual and at the rhythmic levels. That is, the rhythm allows anticipating the list to come, sustaining its unfolding, and hearing its ending (Selting, 2004). However, the pace is different, as the speech rate increases to 0.18 seconds/syllable. We hear this as a much faster paced speech. Grammatically, the phrase is merely an elaboration of what has been said before, at a slower pace, which gives the constative statement concerning what is not to be done greater gravity. It has been noted that practical understanding—as its correlatives of mood and attunement—temporalizes itself in anticipating the future based on past experience (Heidegger, 1977b); that is, in other words, practical understanding here expresses itself in appropriate changes of the rate of movement. We observe this at operation here, where the practical understanding underlying the performative and semantic aspects of the expression temporalizes itself, that is, not only produces itself in time but also produces temporality itself. Here we see an integral performance, where incarnate features of talk with non-cognitive content and features with cognitive content are irremediably intertwined. It no longer makes sense to speak of these as two aspects, but they may at best be one-sided expressions of the same phenomenon produced in and by the flesh. One performance produces the sound material, part of which tends to be considered in traditional research as if it were a metaphysical matter, that is,

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precisely a non-material matter. But the non-cognitive mo(ve)ment of the performance not only is inseparable from but also irreducible to the cognitive mo(ve)ment.

OF RHYTHMS AND MELODIES: MATHEMATICS IN THE FLESH AND IN COLLECTIVE BODIES [M]elody—is capable of exploiting the fascination created by the rhythm and of playing an imaginary duration on the tightened strings of consciousness. (Abraham, 1995, p. 24) In this chapter, we observe phenomena that are not explicable in most if not all currently available theories about knowing and learning. We observe events in which the difference between body and mind, the non-cognitive and the cognitive, becomes undecidable. It therefore makes no longer sense to maintain the Cartesian division inherent in the concept of the embodiment of mind; it makes no sense to maintain the idea of embodied image schemas as the source of the expressive movements in mathematical communication. In fact, the introductory quote to this section suggests that (verbal) consciousness exploits the rhythm features, a fact that is evident in the previous section (e.g., the difference between main and subsidiary clause). Non-cognitive rhythmic, prosodic, gestural, and motive phenomena are constitutive of the cognitive phenomena. The same sound material that psychologists and (mathematics) educators traditionally have reduced to “words” and their contents also are the material for non-cognitive rhythmic phenomena that require different forms of consciousness, including rhythmic consciousness and perceptive consciousness. The mathematical content—supposedly contained in or denoted by the words—actually piggybacks on the other phenomena: It cannot be thought independently from these others. It is therefore better to think cognition—the one we are familiar with in traditional psychology and (mathematics) education—as subordinate to a more comprehensive unit, for example, activity in the activity theoretic sense. There are two important ideas, therefore, that arise from the types of analyses that I exemplify here. First, the rhythmic, prosodic, gestural, and motive phenomena from which cognitive phenomena cannot be separated are consistent with the enactivist claims that knowing is performed and cannot be separated from moving about and participating in the world. However, I already point out in Chapter 3 that enactivism falls short because it does not explain the emergence of movement intention itself, which is done in the radical phenomenological approach. The immanent capacity to move—i.e., an immanent “I can move”—is the source of intentional movement. Because there is no place for theorizing the irreducible nature of (organic) body | (metaphysical) mind, traditional constructivist

190 Geometry as Objective Science in Elementary School Classrooms theories—from Kant to Piaget and (radical, social) constructivism—fall short in providing useful explanations for knowing and learning where the contents of words are subordinated to more comprehensive phenomena. Embodiment theories, too, appear to allow cognitive phenomena to transcend the body, as metaphorization and metonymization transcend the bodily schemata that are the origin of thought. This, therefore, requires a very different approach to thinking mathematical concepts and conceptions. In Chapter 10, I propose such a new way, where the mathematical sound-words stand in a synecdochical relation to the conceptual performances that they name. The forms of knowing and cognizing—those that we tend to consider independent of everything else that happens in a human body while communicating mathematical ideas—are something like the melody that exploits the fascination created by the rhythms in and of the body, not the rhythms that our linguistic consciousness points out but those that we produce and recognize with rhythmic, perceptive, and other forms of consciousness. Second, the phenomenon of entrainment shows that the evolution of mathematics in the flesh is not the result of the actions of the subject; it is not the result of an individual construction; it is not the result of an agent who changes as a function of the coupling with the world; and it is not the result of some subject’s metaphorization and metonymization of individual bodily schemata. Rather, each individual student functions in concert with others, and, importantly, with the teacher. We see how the various frequency-related phenomena—rhythms in gestures, voice, body orientations, and body positions, pitch, pitch contours—shape the collective activity; and this shaping is indissociable from affect. In being shaped, affected by others, we are literally affectively attuned to others and the field that we jointly constitute. That is, there is a coordination of the individual participation with the collective body and coordination occurs at a level not accessible or ruled by linguistic consciousness. Any bodily schema, if it actually were to emerge and exist, therefore would have an irreducible collective dimension. That is, the individuals are not free in developing just any form of knowing—by means of the traces that their participation in joint activity leaves in the individual auto-affecting flesh—but their knowing is constrained by, and takes on the aspects of collective knowing. Precisely because the body is exposed to the world—outside of itself with its senses—it can be lastingly impressed by the world. This impression occurs by means of resonance phenomena—i.e., phase synchronization—that are possible because of the periodicities that underlie the production and perceptive reproduction of communicative phenomena in both the primary and secondary consensual domains. This may become clearer with the following examples, one from the animal world, the other from a study I conducted in science classrooms. In southern Asia, observations of fi reflies conducted during the 1960s have shown that as night falls, swarms form as the fl ies gather in trees.

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Initially, they fl icker at random. But as the night goes on, the individual fl ickering comes to be coordinated with the flickering of others so that in the end, the flickering of the entire swarm is coordinated until all of them fl icker in unison. In this situation, we observe coordination across the collective at a scale much larger than the two clocks on the common wall. But such coordination does not occur between the swarms of different species of fi reflies. A similar phenomenon of coordinated rhythms has been observed in science classrooms (Roth & Tobin, 2010). The rhythms projected by a student talking from the back of the classrooms could be observed in the rocking of a leg, the beating of a pencil on the desk top, and in other body movements of students who could not see the speaker. As the latter changes the rhythm of her speech, the rhythm of the other students and in other modalities changed as well. Thus, as the rhythm accelerated while a confl ict between the student and her teacher evolved, the embodied bodily rhythms of other students accelerated as well. Conversely, as the rhythm in the student’s verbal articulation slowed down, the rhythmic performances of the other students slowed down as well. Interestingly, therefore, the synchronization of rhythms occurred across individuals and across modalities. Of course, such coordination especially across the changes could not occur if the other students had to extract the rhythmic phenomenon from the student’s performance, interpret it, and then implement their own rhythm. In the end, therefore, I observed a sort of implicit collusion of all those who have been formed and have formed the same field, that is, have been exposed to the same or similar societal and material conditions. “This collusion is the foundation of a practical intercomprehension, the paradigm of which could be the understanding that establishes itself among the partners on the same team, but also, despite the antagonism, between the players that are involved in the same match” (Bourdieu, 1997, p. 173). The practical comprehension therefore constitutes an esprit de corps, a visceral adhesion of the individual socialized body to a society, that is, a social corps (corps social). Rhythm and pitch play important roles in interaction rituals, appealing to and having sense (value) in the affective tonality that they call forth. As the term highlights, interaction constitutes a ritual; and rituals are build on cyclic phenomena, including rhythm (beat) and pitch (speech frequencies). As periodic phenomena, they enable entrainment, that is, the coordination of the frequencies among different “subsystems” that are but two sides of one whole irreducible “system.” All participants contribute to the emergence of the social phenomenon so that it would be as inappropriate to ask how any one individual contributes as it is to ask what the sound is of one hand clapping. The importance of rituals, rhythms, beats, and sound frequencies comes from the fact that they necessitate the individual human being to produce these in a manner that does not require cognitive cross-modal coordination and it provides the means for coordination within a human collective.

192 Geometry as Objective Science in Elementary School Classrooms The individual is a whole, only parts of which are cognitive. This cognitive part is but a part of the whole, subject to constraints that also determine the temporalities of the other parts (gestures, poetic rhythm). It has been noted that human beings are not cultural dopes that implement practices with machine-like routine (Garfi nkel, 1967). Rather, human beings creatively produce actions that only after the fact can be described to have been consistent or inconsistent with some rule—in the same way that we can determine only after the fact whether a piano player has played a tune correctly or incorrectly and whether she has played it in the consensually correct manner. Similarly, to draw on music as a metaphor, lessons are not enactments of scores but resemble improvisational jazz jams or improvisational dance sessions, where any individual contribution cannot be predicted with any precision. What teachers and students do, therefore, cannot be described as the enactment of a program, the transformation of a score into music. In everyday conversations, there is spontaneity at work that cannot be captured by intellectualist approaches and formal theories of knowing. We need to take this spontaneity as a model to understand the worlds of difference between the planned curriculum and the living/lived (“enacted”) curriculum. But the fact that something comes off the lessons should surprise no more than that there is enjoyable music produced in a jazz improvisation session. For understanding the intersubjectivity of the teaching-learning event we need an incarnate “description of the course of gestures instead of a cognitive attack on the program of minds” (Sudnow, 1979, p. 83). This also has consequences for the way in which we think about knowing, which always is a knowing-with rather than a knowing-of one or the other. The participants’ understanding, a phenomenon of the second-order consensual domain, is entirely based on their synchronization of mood, attunement, and practical understanding that occurs as a phenomenon of the fi rst-order consensual domain. When Mrs. Winter and Thomas communicate, each of them does not just exteriorize what they previously have thought for themselves. Even if they had done so, all consciousness—as its etymology con-, with, and sciēre, to know suggests—is consciousness for the other as it is consciousness for myself. Thomas speaks for the benefit of Mrs. Winter, using a language that he has received from the generalized other including from Mrs. Winter. For the teacher to know what Thomas knows, the latter’s knowing has to be public, available to every other participant, communicated in words, gestures, prosody, and every other available means of inherently carnal expression. What Thomas can carnally express has to be intelligible, and it therefore has to be already a possibility of expression in and for the collective as well. That is, what Thomas expresses in and through his performance are the expressions of a socialized body, lastingly impressed and shaped by participating with others in producing the world. He is part of a community, which constitutes “an affective aquifer and everybody drinks the same water from this source and

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well that he is himself” (Henry, 1990, p. 178). But we often do not know that we drink the same water and that we are not different from others who do precisely the same. Similarly, when Mrs. Winter speaks, she addresses Thomas: She talks for his benefit. What she says has to be intelligible to him so that what we fi nd expressed when she speaks, gestures, moves her body, intonates, and so on is not just her knowing but the knowing of Thomas as well. If he were not able to comprehend, then Mrs. Winter’s effort at communicating would not make sense at all. “When I speak or when I comprehend, I experience the presence of others within me or of myself in others” (Merleau-Ponty, 1960, p. 157). As a result “when I comprehend, I no longer know who speaks and who listens” (pp. 157–158). When Mrs. Winter communicates, she has to have the hope that her expression is intelligible. That is, what is expressed in and through her body also is a form of knowing-with. Much of this intercomprehension depends precisely on the ritualistic aspects of the relation in which they mutually fi nd themselves. The analyses in this chapter allow us to strengthen the argument for the material phenomenological approach to mathematical knowing and learning that I propose in this book. Rhythmic and perceptual phenomena require forms of consciousness that differ from linguistic consciousness, and because rhythms and perceptions are irreducible to the linguistic consciousness, all forms are subject to a more overarching unit.11 Moreover, rhythmic phenomena are of interest, as they are the result of auto-affection and the pervasive nature of tact, which are, as I present in Chapter 3, at the heart of the material phenomenological approach, making it distinct from that presented by the embodiment or enactivist literatures. Rhythmic phenomena particularly require bodies, as we see from the examples of fi reflies and clocks, but rhythmic phenomena that change the rhythms require the flesh, which is sensitive to changes and adapt to the changes that it produces itself. Thus, for example, “beats are not seconds or any other ‘standard’ unit of time. Instead, these are self-generated units that are used, in turn, as a kind of temporal ruler to measure the durationally varied events that are actually generating them—a nice example of self-reference” (Bamberger & diSessa, 2003, p. 128). Vocal beats, the rhythm in and of speaking, is internal to the speech sound itself, constituting a relation among speech sound events; yet it is not a conceptual phenomenon, but, instead, an entirely carnal and incarnated phenomenon. But bodies generally do not auto-affect: It is precisely the flesh that has this capacity. The conception of the flesh has the features that are required for an understanding of rhythm and perceptual consciousness as a lived—produced and perceived—phenomena irreducibly tied together by tact that is the ground of all senses that we use to make sense. Tact, shared by all senses, holds it all together because “it supposes an interior milieu, the flesh” (Chrétien, 1992, p. 147). In this chapter, I articulate different aspects of movement performances. I do so to be able to show how complex the expression of mathematical

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knowing is in and through communication. This communication is a complete, whole phenomenon, a unit that is highly aggregated, merged, and fused. There are no different performances in multiple modes that are simply put together such as if different players are put together to form a band to add another instrument to the melody. Rather, being produced in and through the same living body, the same intention, the mathematical expression cannot be thought independently of all the other aspects of this one expression. This same, then, goes for the perception of the expression as a whole. We do not just hear words or concepts, but we experience (hear, feel) a child proffer an answer tentatively, or with anxiety, or in a distraught manner. That is, as in musical hearing, “we do not piece together a hearing, putting it together out of the separate features we can name—a paste-up collage of, for instance, pitch, duration, accent, timbre, register” (Bamberger, 1996, p. 39). Rather, in perception, as apparent from the description of hearing an answer as a tentative answer, what we hear is a complex phenomenon in which cognitive and affective dimensions are irremediably intertwined. The second-order consensual domain is tied to and a function of the fi rst-order consensual domain. We hear and see one performance rather than separate properties that we have to construct and interpret before recombining them. The one performance, the performance being one in and through the one living/lived body that brings it about is the minimum unit that allows us to conceptualize what we actually hear and see. But this one living/lived body is inseparable from all the other living bodies so that there is “only one sphere of intelligibility where everything is intelligible to others and to oneself over the ground of this primordial intelligibility that is that of pathos” (Henry, 1990, p. 179). But to cognitively understand how this performance achieves what it achieves, we have to take it apart. The danger is, however, that we might think the performance consists of elements somehow and by some organ coordinated in time and content. There is a different way of thinking about the relation of the whole and its parts: They are mutually constitutive. Thus, the mathematical expression is a whole, which takes this form precisely because of the presence of all its parts. But each part is a part of the whole, not an independent element. It is a part that takes its form from the form of the whole—the part is a plural singular, constituted as such by the nature of the whole as a singular plural. Thus, we cannot and must not understand the part of an expression as an independent element (e.g., gesture) that can be reduced to, and understood in terms of, another element (e.g., language). We cannot express gesture in language, which, as gesture, only denotes the whole of practical understanding and intention in a synecdochical way.

Part C

Emergence of Geometry— An Objective Science

Introduction to Part C

Geometry is one of the oldest formalized mathematical domains. Having evolved in what came to be the earliest mathematical communities from mundane, pre-geometric experiences in a world of immediately comprehensible three-dimensional objects, it has been “handed down” and developed through the continual reproduction of its shared structures in incarnate geometrical activity. It is therefore not surprising that the Principles and Standards for School Mathematics (NCTM, 2000) identify geometry as one of the content standards for grades K–12. As well, in the more recent Curriculum Focal Points for Mathematics in PreKindergarten through Grade 8 (NCTM, 2007), emphasis is placed on K–4 students’ abilities to: identify geometrical ideas in their world; describe, model, draw, compare, and classify shapes according to their properties; investigate and analyze the composition and decomposition of two-dimensional and three-dimensional figures; and relate geometric ideas to measurement ideas. The question we may ask is how geometry as objective science comes to be reproduced in and through the actions of new generations? The psychologists’ answers to this question has changed over the years, from conditioning to information transfer to personal knowledge construction and to the personal construction of knowledge previously achieved while working in a collective. However, as I show in Chapter 2, these attempts in explaining knowing and learning have shortcomings, especially with respect to the role of cultural-historical acquisitions of humankind and their reproduction in and through the actions of new generations. Especially in Chapter 3 and again in Chapter 7, I describe—in the terms of phenomenological sociology—the shaping living/lived bodies undergo in interaction rituals. Now, the verb “shaping” sounds very agential and purposeful, as if teachers such as Mrs. Winter and Mrs. Turner intentionally acted to affect the bodies of the children, disciplining them. Both act in the very mundane ways in which teachers tend to act. The children in the second-grade classroom, too, act in the way children at this age and this grade level tend to act. They participate in “age-appropriate” tasks. What I show in Chapters 8 and 9 is, though, how this everyday way of acting entails a shaping of what students learn to do, and, therefore, how the

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very practices of a second-grade mathematics classroom shapes the living/ lived bodies of the children, “disciplining” them in a manner so that they may subsequently exhibit the discipline of mathematics. This is not the result of a continuous punitive action, but the very ways of indicating, for example, that color and shape are not allowed as task-relevant properties shapes the ways in which children view the objects at hand. Their habitus, the structured structuring dispositions that make them see and act in certain ways, change so that they learn to see similarities and differences underlying adult forms of geometry. What the children come to do, their practices, emerges from and is shaped by the interaction rituals in which they participate. That is, in a very real and mundane way, the interaction rituals constitute the foundation of the knowing that the students learn to exhibit: That is, to paraphrase Vygotsky (1978), the higher-order cognitive functions observed as a consequence of this geometry curriculum are the results of the interaction rituals that the children participated in. Vygotsky’s work—his concept of the zone of proximal development in particular—has often been used to suggest that there is some construction of knowledge in social situations, and these social constructions are subsequently constructed on the part of the individual. But in my approach, there is no individual construction of social constructions necessary. Rather, the very forms of participation are transformed in and through participation. That is, the children’s ways of experiencing truly underlie their learning, as their current participation constitutes precisely their knowing, which is transformed in its very articulation. Our living bodies are transformed and shaped in and through lived, sensuous and sense-making labor, which exhibits what we know: It is not after participation that children are changed by somehow transferring to the inside what they have experienced outside. It is precisely in the interaction ritual that they change, that is, learn. It is precisely in exchanges with others, where they listen, that they are “at the same time outside and inside” (Nancy, 2007, p. 14). The very expression of knowing in and through the living/lived body as a whole is changing knowing, an idea that—writing about the relation between thought and speech—Marxist social psychology has taken up entirely into its theories. In this third part of the book, I focus on geometrical classifications of objects as a context for investigating the role of lived experience and the living body (flesh) in the reproduction of geometry as objective science. The creation of categorization systems within mathematics has received interest among mathematics educators specifically and learning scientists more generally. I specifically use the term emergence to characterize the appearance of geometry at the cultural-historical, sociogenetic, and ontogenetic levels, because the constitutive linkage between geometrical classes (classifications) and geometrical actions occurs in a context that is or has been nongeometrical. Because the term emergence denotes an unforeseen occurrence or a state of things that arises unexpectedly, it is appropriate for situations of the type I describe where one cannot predict whether individual students

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and whole classes will achieve what the curriculum sets forth as a goal. In the lessons described and analyzed in Chapters 8 and 9, children appropriate geometrical classifications by learning to observationally distinguish between their peers’ and their own proper and improper geometrical classificatory acts. As in other work that focuses on mathematical learning in social situations, the collaborations required in making these situations the organized events that they are lead to emergent phenomena including, in Chapters 8 and 9, geometrical classifications of three-dimensional objects specifically and geometry lessons generally. Geometry, classification, and the classification of geometrical objects are integral aspects of recent curriculum documents in mathematics education. Such curriculum documents, however, leave open how the work of classifying objects according to geometrical properties can be accomplished given that the knowledge of these properties is the planned outcome of the curriculum or lesson. The fundamental question of the following chapters therefore is this: How can a lesson in which children are asked to participate in a task of classifying regular three-dimensional objects be a geometry lesson given that the participating second-grade children do not yet classify according to geometrical properties (predicates)? In my analyses, I focus on the incarnate collective work that leads to the emergence of the geometrical nature of this lesson. Thus, I articulate both the collective and the individual work by means of which the lesson outcomes—the complete classification of a set of “mystery” objects according to geometrical (shape) rather than other properties (color, size, “pointy-ness”)—are achieved. In the process, I show how second-grade children reproduce geometrical work, while operating in a division of labor with their teachers, to produce a particular set of geometrical practices (sorting three-dimensional objects according to their geometrical properties) for the fi rst time. In Chapters 8 and 9, I address three questions of how the classification of objects according to geometrical properties comes about and in what the work consists of that leads them there: “How does the ‘proper’ grouping of a collection of objects emerge from the collective task involvement of what recognizably is a second-grade classroom?” “How does this task involvement recognizably become a mathematics lesson?” and “What role do bodily experiences play in the process?” In Chapter 10, then, I propose a reconceptualization of mathematical concepts, which we simultaneously ground in philosophical considerations and empirical materials. I draw on an episode from the same lesson in which children (learn to) sort objects according to geometrical properties, here focusing on an episode involving a cylinder. In this reconceptualization, conceptions do not exist as disembodied, decontextualized, and transcendental ideas but only in concrete realizations of experiences and relations to other experiences.

8

Ethno-methods of Sorting Geometrically

How do children come to sort objects according to geometrical properties? How do children come to sort geometrically given that they tend to sort, as many studies show, according to properties such as size or color? These questions are especially important given that in the “intuitively given surrounding world . . . we experience not geometrical-ideal bodies but precisely those bodies that we actually experience, with that content that is the actual content of experience” (Husserl, 1997a, p. 23). That is, the question in this chapter will be how children come to classify objects (bodies) according to geometrical properties when they do not experience, at fi rst, geometrical-ideal bodies? And yet, if they do classify in and through classification experiences, what children bring to the classroom are the very methods, grounds, and materials upon which truly geometrical classification is built, however inadequate educators might deem these “preconceptions” to be. Educators often pay lip service to the ideas, conceptions, discourses, and so on that children bring with them to the classroom. Constructivist educators interested in the topic of conceptual change often write about “replacing” and “eradicating” misconceptions as the goal of instruction and about the difficulties to “eradicate” misconceptions.1 Piaget’s work has been read from this perspective as an attempt to show how children’s “barbaric thoughts” (Merleau-Ponty, 1945, p. 408) come to be replaced through the assimilation of and accommodation to new information, which leads to a “domestication” of children’s of untamed thoughts. Thus, much of educational thought, including the writings of Piaget, is about what children cannot do rather than what they actually do, what the world looks like to them. For example, pertaining to classification, he writes that the child “is not able to understand the relationship of class inclusion” (Piaget, 1970, p. 27). He describes children’s thinking as “primitive,” pointing out that “there is a very primitive ordering structure in children’s thinking, just as primitive as the classification structure” (p. 28). A phenomenological approach to learning recognizes that children’s forms of thought, because these reflect their experiences, are the necessary conditions for adult rationality. That is, this approach recognizes

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the everyday lifeworld as the foundation of sense, attempting to understand the experienced and experienceable that is immediately given to perception and the originarily intuitive thinking grounded in such experiences. One social science, though, does acknowledge the role of mundane thought to the constitution of everyday social situation: ethnomethodology. In this and the following chapter, I show how everyday methods underlie and are constitutive of formal geometry, that is, how the lifeworld is the actual foundation of geometrical science. I specify the nature of the lived work by means of which children produce mathematics and mathematics lessons. Ethnomethodology is homologous with its own subject matter, concerned as it is—in the present instance—with the articulation of the methods that people (ethno-) use to bring about everyday society, including everyday elementary mathematics lessons. These are considered ongoing achievements of members of society conceived as practical actors who (a) themselves are practical analysts of the world and others and (b) use whatever resources are available to constitute the sense of the ongoing everyday practical activity in and of which they are a constitutive part. This implies that the intentional and recognizable production of a lesson as a geometry lesson presupposes geometrical knowledge. In this sense, geometry as a subject and subject matter can only be, from the perspectives of children who do not yet know geometry, an emergent property. But more so than those who work within an ethnomethodological approach, I am also interested in understanding mundane classroom life in the cultural-historical nexus of the subject matter to be taught.

CATEGORIZATION, CLASSIFICATION Much of the existing (learning science) research on classification provides taxonomies that—even when taking an agent-oriented perspective—are constructions of the researcher rather than grounded in what actors in situations make available to each other. In contrast, I am interested here in the sensuous (living/lived) work that leads to the classifi cation of a set of “mystery” objects in an elementary classroom. I am interested in the production of predicates that constitute the objects in a collection as members of the same class, which, as the participating teachers’ telos, are to be characterized by geometrical properties. In this, I take an approach that differs from classical and traditional studies of children’s mathematical (geometrical) practice. I am concerned with providing a description of the “authoritative and accountable” practices in classrooms that—reflexively with the actions observable therein—are recognized by competent members (mathematicians, mathematics educators, and teachers) as mathematics lessons. I am particularly interested in the role of the organic body and other aspects of mundane everyday life that

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mediate how individuals and collectives elaborate their mathematical knowledgeability. Studies from an ethnomethodological perspective on coding and categorization in the natural sciences and in the social sciences show that to accomplish coding/categorization, coders require knowledge of the very organized ways in which the situation operates and from which their tobe-coded object (fish specimen, rock piles, hospital records) derives. That is, coders/classifiers do not code the object in, of, and for itself. Rather they code it in terms of aspects that belong to the surroundings/context from which an object is taken as a whole. This knowledge is consulted explicitly, that is, accountably, whenever problems in the classification become apparent. Relative to a geometry lesson featuring a classification task, therefore, children can be expected to learn (about) the intentions for classifying in addition to learning how to classify. An important realization of past research on classification was that the participants used “ad hoc” considerations when evaluating their questions about how to classify, which denotes the use of “‘et cetera,’ ‘unless,’ ‘let it pass,’ and ‘factum valet’ (i.e., an action that is otherwise prohibited by a rule is counted correct once it is done)” (Garfi nkel, 1967, pp. 20–21). Such ad hoc considerations also are at work when scientists evaluate coding and classification instructions such as the ones found in field guides for the identification of plants and animals in studies of classification among scientists. To understand the process by means of which geometry emerges from the incarnate, sensuous experience in the course of a lesson that Mrs. Winter, the lead teacher in this lesson, explicitly marks as a mathematics lesson (“we start a brand new unit in geo, math today”), I provide a step-by-step analysis of one episode that in many ways characterizes structural properties of the 22 sorting episodes that made for the largest part (55 min) of this 80-minute lesson. Here, as throughout this book, I denote by “episode” a clearly, participant-delimited event that begins with the teacher’s nomination of a student as having a turn (e.g., “Connor” or “who’s next?”) and by a clear transition to another student (e.g., “next one . . . Ben”). When necessary, Mrs. Winter reminds students about who currently has a turn, for example, admonishing a student who makes a contribution without having a turn (e.g., “It’s Sylvia’s turn” or “Kendra, he has to do his own thinking”). To provide a suitable context for understanding the episode analyzed, I fi rst provide an analysis of the logical structure of the task and then a description of how Mrs. Winter presents the steps and procedures she anticipates to follow.

THE LOGICAL STRUCTURE OF THE TASK In this classification task, each child gets a turn until all mystery objects have been placed (Figure 2.1). Mrs. Winter highlights at the beginning of

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the lesson, and in introducing the new topic of their mathematics work, that geometry is about shapes and the “math words about shapes” that go with them. Although she asks for and accepts from the children labels for the emerging groups (“squares,” “cube,” “tube,” etc.), and although she accepts children’s classification criteria generally, Mrs. Winter emphasizes that “color” and “size” are the two words that they cannot use to “do geometry.” Shape is a most complex phenomenon that must be dealt with in the qualitative representation of space. In Euclidean geometry, a large square and the same square with a slight nick in one side are different. In fact, geometry is not about the objects children fi nd in their everyday world; even the engineered geometrical objects that the teachers have brought to the classroom are but models that tend toward but never achieve the ideal objects that geometry is about. Shape is important in commonsense reasoning because very often the shape of an object is functional. From a logical perspective, the structure of the task requires the children to sort the objects {x, y, z . . .} such that they end up with collections within each of which the objects can be described by one or more predicates that differ from the predicates in another group. As a whole, the collection of mystery objects forms a membership categorization device to the extent that it can be used to classify further mystery objects. This device has the structure [cube/sphere/cone/. . . /pyramid] (Schegloff, 2007). The children’s task, mediated by the teacher’s input, consists precisely in developing a device (and the terms that come with it) based on their perceptual categorizations of material objects such that a set of predicates allows a mapping of the former to the latter. For a fi rst group, classification can be expressed in terms of the predicative statement ∀x(x . . . predicate[s]). Because such predication constitutes a form of generalization, it has been denoted as a stage in the conceptual development from (raw) stimulus to science. In the present situation, the task requires children to group objects according to similarities and differences: All objects within a group share one or more common predicates, which differ across the different groups (collections). But similarity is not easy to pin down. Similarity based on properties is problematic because, “[w]hen it comes to enumerating properties, we don’t know where to begin. The notion of a property, for all its seeming familiarity, is as dim a notion as that of similarity” (Quine, 1987, p. 159). Moreover, the things we encounter in the surrounding everyday world are not stable but “fluctuate, in general and in their properties” (Husserl, 1997a, p. 24). This has the consequence that identity with itself, self-sameness at any instant and across time, and sameness with other objects “are merely approximate” (p. 24). Watching the video of the lesson, we may note how children wrestle with the question whether two objects are the same or different: They both resemble a pyramid, but one has a slightly rounded top rather than a sharply pointed vertex. That is, whereas

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adults might have classified both as pyramids, the differences are salient to the children, who therefore wrestle with the very problematic that Husserl articulates. A phenomenological study of classification, therefore, has to understand how the world looks through children’s eyes and it has to view the process of classification as problematic through these very eyes.

SETTING UP THE SORTING TASK During the entire sorting task constituted by the 22 sorting episodes (student turns), the children sit in a large circle on the floor, Mrs. Winter and Mrs. Turner (dressed in black) are seated on chair s near the basket with the black plastic bag from which the mystery objects are to be drawn (Figure 8.1). The videotape of the lesson shows the children forming a large circle; all objects and categorizations therefore are perceptually accessible to all participants and everybody else can hear the current speaker and see his or her gesticulations. In her opening statements at the beginning of the task, Mrs. Winter asks the students whether they remember a conversation from the previous week about the new subject matter that they are to begin with the present lesson. Off camera, a student proffers “geometry,” which Mrs. Winter accepts by

Figure 8.1 The children sit in a circle. Mrs. Winter and Mrs. Turner (in black) sit with the children. Recognizable collections include the cubes (far left bottom), cones, and pyramids. Connor currently holds his object next to one of the rectangular solids.

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uttering the word again. This then leads to an articulation of the task, sorting, and the special conditions for this sorting task: it cannot be done by color or by size. Fragment 8.1 01 02

M: W:

03 04 05 06 07 08 09

J: W: O: W:

. . . and shape, and (serial?) and color. yea those are math words. those are the math words about the shape. good for you. anyone else want to tell us what they know about geometry? (3.59) it was a long weekend (0.29) what do you think jonathan? [what do] [size ] (0.96) it has something to do with size. anything else. oshin? color. so can we tell a shape by its color? and in fact we are going to talk about that so we may just as well talk about it right now. when we do any kind of sorting activity today, we are not going to do those by color and we are not going to do them by size. do you want me to say that again. when we ask you to sort things today, we are ´not using color as a category, and we are ´not using ´si:ze.

Quite innocuously, Mrs. Winter asks “can we tell a shape by its color?” She articulates the purpose of this lesson, to “tell shapes,” which in fact is the constant reminder to students after they have placed their object “Stop. Explain your thinking.” Fragment 8.2 10

W:

here is what you need to think about. i=m going to put a shape on this piece of paper. and you can look at this shape. you might already know something about that shape. and then i am going to ask people to come up and without–its a mystery bag—so without seeing it ((bends over to demonstrate pulling an object)) you have to decide does that shape ((points to pink paper on floor)) match or belong in the same group with the one i put out or any of the other ones out or does it need its own new colored piece of paper to sit on. so the fi rst shape that i am going to put out ((bends over to pull shape from the black plastic bag; pulls a cube and places it on the pink sheet)) there is the fi rst shape.

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S:

207

((Places another sheet of paper, brown)). okay, the next person who comes out has to decide does their shape fit in with that shape ((points to her cube)) or does it deserve its own color or its own category. its own place to be. ((She calls on the fi rst student.)) what were the two things we said we are not going to sort by. color and size

Mrs. Winter articulates the rules of the game. 2 These rules explicitly specify the predicates that cannot be used, and, therefore, the kind of grounding discourse students can or cannot use for explaining their reasons. In this, the rules specify the “language game” that is in play. In this opening episode, Mrs. Winter thereby introduces the children to the task, which includes the articulation of rules (predicates) that cannot be used in sorting the mystery objects children are going to pull from the black plastic bag. In the course of the lesson, these rules do not come to determine what students do. Rather, Mrs. Winter often encourages students to redo and rearticulate something until a point where the rules come to be descriptively adequate for what students individually and collectively have done. The point of the task is not merely to put an object with a given collection or to use it to begin a new collection. Rather, the task is to evolve category-bound predicates that allow verbal distinctions to be made for the different groups (collections, categories) of objects that are associated with different names that they come up with for the emergent groups and that Mrs. Turner records on strips of paper that she then places next to the objects. Some differences legitimately place items into different groups; other differences cannot be made part of legitimate predicates for drawing distinctions. Some differences are allowed in this “game,” evidenced by the apparently (perceptibly) different objects in the different collections that form classes; and that these objects form classes can be seen from the fact that they (legitimately) are placed on the same colored sheet.

Connor’s Turn3 During this episode, it is Connor’s turn to place his object with one of the existing collections or on an empty sheet that the teacher has placed on the floor just as Connor begins. He places his object on the empty sheet. Here I present a description and analysis of the lived work that makes these different classifications emerge from the sequential classroom talk. I show how the classification and the sequential aspects of talk are constitutive. The unit of analysis is the entire “turn” (episode) and the classification that is achieved in the end emerges from the sensuous labor completed over the course of the episode.

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Here, I use the term, “emergence” to denote the contingent nature of the present achievements, which could not be anticipated because there are other ways of classifying these same objects: Cubes are special cases of rectangular solids, which in turn are special cases of prisms (in the collection there are also triangular and hexagonal prisms); presence or absence of acute angles; number of faces (e.g., sphere = one; cone = two; cylinder = three; tetrahedron = four; . . .); and all are part of the same topological category. My analysis exhibits the children’s work of classifying, which is distributed across people and objects. It includes the walking about from collection to collection, the picking up, holding, feeling, turning, comparing, manipulating, placing, observing, etcetera of the object; and it includes attending to signals from others, such as pointing and emblematic “stop” gestures. It includes the ongoing a priori (instruction, orders, rules) and a posteriori descriptions of courses of actions that the teacher and others articulate. It includes alternative articulations, placements, and comparisons others provide. It is precisely this lived work that shapes the children’s bodies, disciplines them so that they exhibit the discipline of mathematics (geometry).

GETTING STARTED: A FIRST CLASSIFICATION At the beginning of this episode, Mrs. Winter reiterates for the class—as both teachers do throughout this lesson—the various rules for and of their “game,” sorting mystery objects (turn 01). In the present situation, it is the rule about having to articulate the thinking underlying a classification of the mystery object, which fi rst has to come from the student pulling it from the plastic garbage bag before others can articulate their own thinking. Furthermore, another rule consists of picking the fi rst object a student touches rather than feeling around to select an object. There is a pause during which Connor’s upper arm can be seen to be moving—everything from slightly above the elbow to the hand has disappeared in the bag. Members can see this as feeling around—as Mrs. Winter’s admonition to “take the fi rst one [he] feel[s]” indicates (turn 03). That is, Mrs. Winter requests Connor to act such that after the fact her admonition is an adequate description of what he has done. Adequateness always is determined after the fact because the living/lived body acts on its own based on its immanent knowing; in and through doing the relevant work repeatedly, the level of adequacy increases. Here, although a rule is rearticulated in turn 01, Connor acts in a way that Mrs. Winter (and others?) sees as not taking the fi rst one he felt (“take the fi rst one you feel”). It takes another little while before Connor pulls a mystery object, which he briefly and demonstratively holds up for others to see and which his classmates acknowledge with extended (2.18 s) “ohs” and “ahs” (turn 05).

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Fragment 8.3 01

02 03 04 05 06

07 08 09 10

11

W:

´BEfore we close this one out. remember that its connors turn first to tell us his thinking. (0.49) you have to wait until we ask for more ideas. (1.03) just take take the fi rst one you feel. (2.97) ((Connor appears to feel around in the black bag)) W: take the fi rst one you feel. (1.68) Ss: ((multiple ohs and ah’s, 2.18 s)) W: now look at the grou:ps: ((pointing around the circle of objects already there)) does it ´belong to another ´group (0.67) O:r. (0.29) ´can you start a new group with that. ((Connor facing the objects, looks at his object, looks around the circle, (0.67) K: um::: ((Kendra moves forward, points to the group of cubes [Fig. 8.2])) (0.62) W: ^kendra. (0.34) he has to do his own [thinking.] ((Connor moves to a new sheet, places his object, and walks away toward his seat)) C? [brandon.]

Just before Mrs. Winter begins to utter what we can hear as an instruction, Connor already has shifted his gaze across several collections; he directly gazes in the direction of the teacher’s hand after she has started to talk, moving her hand in a circular fashion, her index fi nger pointing in the general direction of the various collections on the floor. There already are 12 objects out on the floor in different collections distributed over six sheets, five of which are associated with labels (“ball,” “cone,” “tube,” “rectangular,” and “squares, cube”). There also is one empty sheet in the case that a student decides his or her mystery object does not fit with existing collections. Mrs. Winter now (and again) articulates for Connor what she expects him to do, “now look at the groups” (she is pointing around in a circle to the existing sheets with objects), and she continues, “does it belong to another group or can you start a new group with that?” Connor looks at his object, then—visibly to all present as he orients his head and upper body—gazes at the collections on the floor. At that time, Kendra, who is situated about two meters behind Connor is moving up, looks at his object, emits an extended “um” sound, attracts his attention (he orients towards her, Figure 8.2), and she demonstratively points to a collection containing two objects associated with the labels “squares” and “cubes” (turn 08). Some students clearly gaze in her direction; Melissa, sitting opposite to Connor in the circle waves her pointed finger toward the opposite direction. Connor briefly turns his gaze into the direction that

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Kendra points before reorienting to his right and toward beginning his own group of objects on the yellow sheet lying in the direction where Melissa has pointed. At the same time, there is a brief pause during which Kendra retreats and then the teacher, turning toward her, utters Kendra’s name with a slightly raised voice and says, with an intonation that members can hear as a teacherly admonition, “he has to do his own thinking” (turn 10). In fact, the utterance repeats, in a modified way, an instruction or rule already stated at the beginning of the episode: It is his turn fi rst to articulate his thinking, and the fact that Kendra is now pointing does not allow him to do his own thinking (fi rst). Mrs. Winter’s utterance “he has to do his own thinking,” in fact, constitutes normative work in its double orientation toward Kendra and Connor. It highlights the sensuous labor currently required of Connor, “to do his thinking,” and it asks and is understood as such by Kendra— articulated in her retreat—to keep quiet. The reprimand is recognized as such not merely because of the register in which it is delivered but also because of the way in which this turn is inserted within a (pedagogical) turn previously assigned to another student (Connor). It is recognized and exhibited as such in the retreat and silence Kendra displays. It is but one of the ways in which students’ bodies come to be affected and therefore socialized bodies (see Chapter 7). Kendra retreats so that, together, Mrs. Winter and her student establish the order properly described as “Connor’s turn.” Mrs. Winter reiterates the conditions of the task, which

Figure 8.2 Connor orients toward the collection of cubes that Kendra is pointing to.

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requires an explication of the reasons for an object’s placement, and the individual whose turn it is has to do so without help. That is, by repeating a previous instruction, Mrs. Winter marks Kendra’s previous pointing as a violation of the rules (instructions); and Kendra’s retreat is consistent with this reading. In other words, Mrs. Winter’s comment might have created a dilemma for Connor, if he wants to place the object in the category Kendra pointed to, it might mean that he has not done “his own thinking.” Both children, by responding with appropriate actions (Kendra retreating, Connor placing his object), therefore exhibit their understanding of these conditions. In this situation, therefore, the emerging participation framework between Connor and Kendra—one that children appear to choose naturally for accomplishing mathematical tasks collectively—is inhibited in favor of another one that is familiar in school situations: a student and the teacher as principal agents and the other students (as well as Mrs. Turner) as onlookers and witnesses. Although the classifications achieved in the two possible forms of division of labor may be the same, the trajectory of the lessons likely would be different, though in an unpredictable way. In this sense, therefore, this moment also constitutes a possible branching point in the lesson, which therefore will unfold as a function of the situated choice—here by the teacher—to enable the second form of division of labor rather than the first. The utterance “can you start a new group with that?” (turn 06) can be heard as an invitation to start a new group. In turn 10 Connor does this despite the possible tension that might be perceived between his action and Kendra’s pointing to the “square, cube” group. She does not just point but repeatedly jettisons her hand forward, in an apparent attempt—she does not just stick out the index fi nger—to attract Connor’s attention to the pointing. Kendra’s actions also constitute a form of sensuous, living/lived labor that we need to take into account to understand the episode, and in fact, it constitutes a form of instruction that might be glossed thus: “Put your object to the group toward which I am pointing,” which, as it will turn out and unbeknownst to the participants at this instant, is the one that the teacher will be satisfi ed with at the end of this episode. But at this instant, Connor does act consistent with the condition that he has to do his own, unaided classifi cation and thinking.

FEEDBACK AND FURTHER INSTRUCTIONS While Mrs. Winter is turned toward and talking to Kendra, Connor gets up, walks toward the empty sheet, picks up his mystery object and produces, as he is placing the object, a new collection. He then moves toward his place in the circle of students, thereby indicating his turn to

212 Geometry as Objective Science in Elementary School Classrooms be completed, when Mrs. Winter interpellates him, “Now, before you go, you have to explain your thinking” (turn 13), holding her hand out in a way that knowledgeable individuals recognize as stop (turn 13). In this, Mrs. Winter makes it known that from her point of view more needs to be done before Connor can sit down. She continues by articulating, while pointing to the mystery object, that he has to explain why the mystery object is placed in its own group (turn 13). She does not merely ask Connor to articulate his thinking but more precisely, she asks him to explain why the object gets its own group, emphasizing the word “own” by pronouncing it with signifi cantly greater speech intensity (volume) than the other surrounding words. Without hesitating, Connor, who has walked back to his object and has pointed to it, suggests, “’cause this one is sort of bigger than the other ones.” He picks up the object and begins to walk when Mrs. Winter interpellates him again, “Connor, just a minute, stop for a sec.” She then asks him to “remember” (the rules): “we are not counting size . . . and we are not counting color” (turn 16). Fragment 8.4 12 13

W:

14

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15 16

X: W:

17 18 19 20

C: W:

(1.08) NOW. befORE YOU GO ((Holds right hand out in “stop” configuration [Fig. ])) you have to explain your thinking.´why does it get its ´OWN group now. cause this ↑ `one is ↑ sort of (0.32) bigger than the other ones? ((He has walked back and now stands over his object, looking down on it.)) that s[ize. ] [connor] ((Connor picks up his object)) just a minute stop for a sec. ((Hand held out in “stop” position)) (0.53) remember WE:R not telli counting ´si::ze (0.62) ((pulls up left-hand index fi nger with right index)) ↓ <so thats bigger or smaller, and we are not> cou:nting ^co:lor. ((pulls middle fi nger, as in counting 1 then 2)) ((in “giving a lecture tone”, pulling on his fi nger for 1 and 2)) ((Connor is looking at her until she is done, with his object in his hand.)) (0.50) <this> (0.21) <

`um>. (0.99) SSo ´how is it this? (0.24) different now.

Ethno-methods of Sorting Geometrically 21 22 23

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(0.83) or the same as other shapes. (0.23)

In turn 13, Mrs. Winter requests Connor to explain his thinking before he can go, that is, return to his place. She brings forward her hand with palm open in the way drivers are signaled to come to a stop (turn 13)—one of the many ways in which the children’s living/lived bodies (flesh) come to be affected because of the structures of the field. Connor, who has already stopped—sign that the injunction has worked, that his body has come to be fashioned a bit—comes to face his teacher at this very instant. This appears to be a form of gesture found in classrooms more generally, whereby a teacher, like a crossing guard delaying oncoming traffic, directs the attention of the class/group as a whole to a particular issue (Koschmann, Glenn, & Conlee, 1997). In uttering, Connor completes what we can hear as a request-response pair; he thereby may learn to articulate the grounds for acting in this rather than another way. He may learn to be accountable for his actions, even if the account is not delivered each and every time. Here specifically Connor is put in the situation, as Mrs. Winter articulates not wanting to have the result of the classification or any answer from Kendra (“he has to do his own thinking”). Mrs. Winter particularly asks Connor to articulate the grounds for his decision to place the object on its own. Rather than reworking his explanation, which Mrs. Winter has evaluated negatively (“we are not counting size”), Connor enacts what is called a preference for self-repair. That is, he repairs the choice rather than the explanation. Connor hesitates (turns 17–19). In asking how his shape is similar to (turn 22) or different from (turn 20) existing shapes, Mrs. Winter provides a specification of the task and therefore, a specification for the lesson to become a geometry lesson rather than some other lesson—for example, an art lesson where perspective and apparent size are of importance, or where students are to learn about color and the illusion of depth. Connor provides a predicate but Mrs. Winter stops him before articulating that he has to remember not to count size or color. There are pauses and some stuttered utterances on Connor’s part, which the teacher follows by asking how “it” and “this” is different “now.” It is precisely here that the teacher asks the student to make a distinction between, on the one hand, his earlier explanation grounded in his familiar experience with the world and his everyday competencies and, on the other hand, what is to become the specifically mathematical explanation fostered in and by means of this lesson. The “now” follows the articulation that size and color are not counted—i.e., the predicates in terms of these two categories though possible are disallowed—so that others, consistent with mathematics, can subsequently emerge, after this singular point, and with them the nature of the practices that make this observably a mathematics lesson.

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Again, there is a pause (turn 21), providing the teacher with an opportunity to articulate the task in a new, alternate way: specify the similarities between this and the other (collections of) objects. In turns 20–22, Mrs. Winter in effect asks Connor to classify an object that he already has classified. Since the fi rst time, however, she has noted color and size as illegitimate predicates—“improper” (knowledge) categories—the latter having been used by Connor to put his object into a group of its own. By asking him to classify again rather than to further clarify and elaborate his rationale—as she does in those instances where the object ultimately remains in the collection where students have placed them (i.e., the “correct” one)—she is inviting Connor to do (try) the sorting again. She does not invite him to classify or rethink, but in fact asks him how “it” is “different now.” Saying “now” distinguishes this moment from an earlier one, the one prior to having made the selection of the group on its own. Since then, she has rearticulated the rules of the sorting game: Neither color nor size is to be used. Just prior to her turn, Connor articulates size as the predicate that distinguishes his object from all the others, thereby deserving its own group; that is, Mrs. Winter has articulated a rule that is inconsistent with the one Connor used. “Now” means that Connor is asked to classify his object again given that his previously stated predicate is inconsistent with the rules of the game, which therefore requires another predicate for making a decision. In fact, it is not the repeated statements that cause Connor to make the right classification, a way in which some researchers might want to characterize this instant. He is asked repeatedly to try (a different) classification until such a point that the rule is descriptively adequate of what he has done before and until Mrs. Winter appears to be satisfied. At this point he would be allowed to return to his seat. Moreover, at this point his actions could have been described as consistent with the rules. If the utterance of a rule were able to cause student actions, then it would be surprising why Mrs. Winter has had to reiterate repeatedly these corrections that cumulatively shape the living/lived bodies of the children in the course of the lessons—whenever a student uses color and size, or whenever someone else provides hints for classification before the student whose turn it is has the opportunity to explain his/her thinking. In the course of the lesson, therefore, the restatements of instructions and rules become resources for understanding that the classification has to be redone; and when these restatements do not occur, the foregoing classificatory action can be considered appropriate and consistent with the rules of and instructions for the sorting game.

A SECOND ATTEMPT The episode continues with a pause before Connor utters two interjections, followed by another pause. During this time, he walks away from the colored

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sheet having picked up his object, and in a long curving trajectory moves around the other collections, gazing at them in passing and toward the sheet labeled “rectangular.” He holds his object directly against one of the two larger objects (turn 24), which we (members of the culture) can see as a comparison: He brings the object with his right hand to the others and passes his index finger of the left hand over from the object on the floor to his own. We then hear him make the comparison that we have seen him enact physically: “tiny bit, I say it is a tiny bit different” while shaking his head twice sideways as if in negation and looking the teacher squarely in the face (turn 24). As his fingers run over the surfaces, he is in a position to sense the flatness and smoothness each of the two rectangular (square) and painted surfaces presents, that is, he can sense the bodies in the way and with the content that they are really experienced rather than the geometrical-ideal bodies he is to learn about. The lived difference between the objects is therefore made available to the others in the room, most importantly for Mrs. Winter, to whom he is orienting himself, as shown in his gaze direction. At least it appears to be that way. For, after a brief pause, Mrs. Winter questions him again, “but how?” (turn 26). There is another brief pause, and, with a subdued voice, Connor utters that he “can’t really say how.” Fragment 8.5 24

C:

25 26

W:

27 28

C:

29 30 31

C: W:

32 33

W:

km ↑ this ((walks in a curved path toward the collection of “rectangulars,” Fig)) one::s:sorta ((holds his object against two others)) (0.91) <

ti>ny bit, (0.83) i say it is a tiny bit <

different>. (0.60) but ^how::. ((head moves down in “teacherly” manner)) (0.72) ((shrugs shoulders)) <

i cant really say> <how>. (1.33) <

itll be (big[ger?)]> [I WA]NT you look at that block ((points to block)) and i want you to take it to each group ((points around circle)) and i want you to see:: (0.56) whether it looks the same as any of the other groups ((Connor holds his object directly next to one of the rectangular prisms)) or if it is different from all. (0.32) brandon you gotta get your ´feet up.

216 Geometry as Objective Science in Elementary School Classrooms 34 35

C:

(5.68) ((Connor holds his object next to the cubes, quickly next to the cone, then the pyramid)) <

um. a. um.> i ↑ thinki::t probably: go (0.98) THIS one. ((places it with the two cubes))

Here, Connor has provided proof, visible and audible to everyone: the reason for his previous classification. Connor says that his object is “a tiny bit different” while holding his object next to the collection of rectangular solids. And this tiny difference may—but does not have tomake all the difference, for in the sphere of the merely typical, the likeness of one thing with other things always is approximate. Mrs. Winter requests Connor to state how they differ and Connor responds that he “can’t really say” and then tentatively—see the rising pitch toward the end of the utterance marking questions—offers a possible answer (“It’ll be bigger?” [turn 30]). She begins her turn before Connor has ended, requesting the student to look at his object and to walk around and compare it with all the collections so that he may determine “whether it looks the same as any of the other groups or if it is different from all” (turn 31). We can hear the “I can’t really say” as an indication of his experienced inability to provide what the teacher is asking for and, simultaneously, as a request to be instructed in how to arrive at the expected response. Mrs. Winter does precisely that in providing instructions for how to go about comparing the mystery object to all others and to decide whether “it is different from all.” It is only in this latter situation that Connor’s original choice of beginning a new collection is legitimate. In fact, Mrs. Winter is asking Connor to do what he already has done—and observably so, as he moves his fi nger repeatedly across the ridge between the two objects and thereby enacts the comparison. He has scanned all of the collections and has decided that his object should be placed on a separate sheet of paper, being in a category of its own. He has passed by all the other collections and has gone to the one with the largest objects. He directly holds his object to the two largest ones in the collection, passing his fi nger from one to the other thereby showing the relation in juxtaposing his mystery objects with two others, making visible his test of the size comparison, and verbally articulating that his actual, lived experience of different heights. He can feel it: The difference is real. Asking him to do this again does not make sense unless the outcome of the fi rst time has yielded an inappropriate result. Thus, by stating the condition after Connor has started a new collection, Mrs. Winter implicitly calls his action and decision into question, which differs from her previous evaluation that only reminded Connor to justify his solution. Without taking up the intervening student’s suggestion (not completely recoverable from either of the two soundtracks), the teacher begins another instruction. Pointing around to the six collections, Mrs. Winter says “I

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want you to look at that block and I want you to take it to each group and I want you to see whether it looks the same as any of the other groups or if it is different from all” (turn 31). In this, she rearticulates instructions already provided in turn 06, using pointing gestures in the same way marking each collection on the floor with a beat (rhythmic) gesture while slightly pausing. At the same time as Mrs. Winter talks, Connor has compared again his object to the two larger rectangular ones. In fact, she asks Connor to do what he has done already twice, first perceptually, then while walking to the collection of rectangular objects. In giving the instructions again, we can see that which has occurred between the two situations as inconsistent with the description that the instruction provides prospectively for what is anticipated to happen. It is a way of bringing about another sequence of actions that then can be compared again against its prospectively provided description (i.e., the “instruction”). And engaging in this sequence again leaves a trace in Connor’s body, shaping what and how Connor will act at some future instant. This and the previous section can be viewed as the organization of classroom talk and environments that has been termed I-R-E-C sequence, which involve a situations where a student reply does not lead to a positive (accepting) assessment by the teacher (Macbeth, 2004). Thus, following an initiation of the turn (I), a student replies (R) with the statement of trouble all the while having the opportunity to initiate or correct in the same turn. In the next turn (E), the teacher then evaluates and initiates a correction. Finally, the student or teacher corrects (C) in the fourth turn of the sequence. Although it is possible that the I-R-E-C sequence can sometimes be found in its simplest form, it “can on any actual occasion show expansion to an ‘nth’ turn” (p. 710). Mrs. Winter’s utterance in turn 16 thereby sets up the sequence, in providing Connor with the opportunity to locate trouble in the previous turn 14. This is (partially) completed once Connor has placed his object in a collection where it will be once he sits down and the next student gets a turn. Mrs. Winter continues to evaluate (turns 20, 22), and Connor self-corrects in turn 24. The continuation of the evaluation, and therefore the assessment that this is a sequence, further receives support from the fact of the repeated utterance of “now,” which both mark points in time and generate a parallel structure. In each case, the “now” marks relevant prior actions. This line of analysis, therefore, suggests that repair might be one of the most important aspects of producing intended learning outcomes. In this instant, Connor does not merely participate in the I-R-E-C sequence. Rather, his active participation produces and reproduces this interactional form, shaping and reshaping the ways in which he participates now and in the future. But if the patterns of his actions change (i.e., his practices), then his living body has changed. It is through expansions of the basic pattern that a student comes to respond (act) in the “proper” way, which therefore is a clear expression of the shaping of his living/

218 Geometry as Objective Science in Elementary School Classrooms lived body in interaction ritual. The student’s living/lived body comes to be socialized, but the social body (society) itself is changed because in the intuitively given world, self-sameness and self-identity are never given, always only approximate.

CLASSIFICATION: REQUESTING AND PRODUCING AN ACCOUNT Connor turns around, holds his object right next to each of the two objects in the “square, cube” collection, then to the taller of the two objects in the “tube” collection, then gets up and, in passing, holds his object to the object on the “cone”-labeled sheet. He stops short of the “pyramid” and begins to talk, “I think it probably go this one” (turn 35) while placing his object right next to the larger cubic object (of about the same size). Here, rather than making defi nitive statement, he uses the adverb “probably,” which expresses a high degree of likelihood but not certainty. In fact, it is a device that allows members subsequently to change a statement more easily than when they have committed themselves fi rmly to one or another choice. Although this is the place where the teachers ultimately want the object to be placed Mrs. Winter requests a reason, “and can you tell us why you think that?” Fragment 8.6 35 36 37 38 39 40 41 42 43 44 45

C:

W: C: W:

C: X:

<

um. a. um.> i ↑ thinki::t probably: go (0.98) THIS one. ((places it with the two cubes)) (0.46) and can you ´tell us why <

you think that>. (1.29) cauz:: these are more squares. (0.40) < ((Connor looks up to the teacher, picks up his cube)), (1.06) they are all squares i think (0.57) ah ((Connor looks up toward the student.))

There is a pause and Connor, who has gotten up, now gets back on his knees, waves his right hand over the entire collection, and utters “cause these are more squares” (turn 39). Mrs. Winter produces interjections that both acknowledge receipt of the previous turn and provide the previous speaker with the possibility to continue and elaborate. Connor takes this opportunity to elaborate, “they are all squares, I think” (turn 39). In this utterance, he not only specifies that there are “more squares” but more

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importantly that there “are all squares” (turn 43). However, he follows this qualification of the earlier predicate by saying, “I think,” which is a way of marking the utterance as a subjective and revisable description. In uttering “they are all squares,” Connor refers his audience to the faces of a particular object and thereby constitutes a predicate that is bound to a particular class of objects: cubes.

REQUESTING AND PRODUCING A NAME AND PREDICATES FOR THE COLLECTION A central part of learning geometrical classification is the naming of objects, because the category names ultimately become metonymic devices for referring to and denoting geometrical classes of objects. Requesting and producing the names and predicates for the emerging collections of objects also is an important aspect of this lesson generally, exemplified in this episode specifically. These names thereby become integral parts of situations and, in fact, they come to index these situations and particulars thereof. As I show in Chapter 10, these names thereby come to bear a synecdochical relation to the situation as a whole and to the phenomenon that mathematics educators tend to denote by the term conception. Here, the classifi cation episode continues as the teacher asks Connor what the indicated group has been said to be about (turn 46). After a one second pause, Connor, however, asks her what she means. By querying “what do you mean like?” he not only queries her about the sense of what she has said but also he (a) marks what she has said as not meaningful and (b) requests that she state more clearly what she has meant to say. Rather than bringing the sequence to a closure, he extends and opens it up further by rendering problematic the signification of what preceded. Here, he questions the nature of the question, that is, what she is asking him about really but not yet evident. Mrs. Winter clearly replies using a category of an acceptable answer (“the name of that group”). But rather than asking “what . . . that group was about?” she now requests producing the name of the group that they earlier wrote on a sheet of paper and placed right next to the colored sheet with the cubes. When there is a no response, she exhibits preference for self-correction in saying, “what’s written on the card.” Connor utters “squares,” which the teacher acknowledges in repeating. But she also denotes his response as incomplete “squares and . . .” This type of turn at talk is a designedly incomplete utterance (Koshik, 2002), because it is designed to allow students to self-correct by offering a slot (“. . .”) that they can complete. As Connor does not answer, Cheyenne proffers the called for second term “cube” (turn 56). In both instances, Mrs. Winter repeats and thereby positively evaluates the utterance of the previous speaker. She then asks whether “it” “meets the criteria of having the square or the cube?”

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Fragment 8.7 46

W:

47 48

C:

49

W:

50 51 52 53 54 55 56 57 58

59 60 61 62 63 64 65 66

67 68 69 70 71 72 73

C: W: c: W:

X: W: C: W:

X: X: C: W:

em an ↑ what did we say that group was about ((points to the group of cubes)) (1.00) <

what do you mean li[ke?>] ((looks up to her, still pointing)) [WHAt] was the (0.15) WHAt did we put for the name of that group. ((still pointing)) (1.51) ((still pointing, then pulls hand back)) whats written on the card. (0.83) <

squares> ˇsquare an::d? (0.20) ((has moved forward, points to the label)) cube (0.25) cube. does it meet the criteria of having the square or the cube? ((looks at Connor sternly, nods while talking)) (0.25) <

no> (0.25) do you think it does? (0.84) like what do you mean? (1.10) ´does it match. We said THAT this group ((points)) was ´squa::re (0.31) or cube (0.49) ((looks at Connor, nods)) does it match that? (0.41) <yes.> (0.48) or::. ((gazes at teacher)) yes. (0.70) <

o>kay. () ben you wanna add? ((nods in direction of Connor)) () thanks connor.

Connor does not respond. Someone else off camera almost inaudibly says “no” (turn 60). Oriented toward Connor and thereby clearly addressing him, the teacher reiterates her question in a different form, “do you think it does?” A pause unfolds (turn 63), and then Connor asks Mrs. Winter to articulate what she means (turn 64). Another pause unfolds and Mrs. Winter responds by asking whether there is a match, and she rearticulates the antecedent condition for the other objects in the targeted collection, “We said that this group was square or cube” (turn 66). Mrs. Winter then asks again whether “it” matches “that.” Initially, there is no response; then

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someone in the class says “yes” with a very low, almost inaudible voice. There is another pause followed by an interjection (“or”) that some student utters, before Connor produces an affi rmative “yes” while gazing at the teacher (turn 71). Following a brief pause, Mrs. Winter utters “okay” and thereby accepts Connor’s response. By turning to invite another student (Ben), she makes available to everyone present that Connor’s turn—expressly marked as such at the beginning of the turn—is coming to an end. There is nothing more to be added, especially after Mrs. Winter explicitly thanks Connor (turn 73). In the course of this sorting episode, the object that Connor initially has placed on a sheet of its own now has ended on a sheet where there are already some objects. In response to repeated teacher requests, Connor eventually affi rms that his mystery object matches the criteria of the second collection in which he has placed it. We can hear the repeated requests on the part of Mrs. Winter (turns 58, 62, 66) as insisting that Connor produce a specific answer, here agreement with the previously stated criteria. There are utterances that allow us to doubt an agreed-upon classification, as we can hear not only “yes” but also a negation (“no” [turn 60]) and a conjunctive “or,” which ordinarily is used to coordinate two or more sentence elements that are not of the same type and therefore constitute alternatives. In this situation, although Connor clearly has classified the object, the unfolding events make evident to everybody present that in this situation it is not an appropriate classification in the normative framework hidden from the students but apparently attended to by the teacher. This classification is not the result of some abstract action but of real living/lived work requiring energy and resources. Repeatedly, Connor exhibits bewilderment, a form of affect, with Mrs. Winter’s questions about what the predicates are of the group where his object should be placed (“What do you mean like?” [turn 48] and “like what do you mean?” [turn 64]). It is as if the bewilderment on his part is produced by the suppression of his mundane classification and the predicate he uses while retaining a sense of what this game is about. These utterances also are instructions for Mrs. Winter to teach him. Here, he asks the teacher to explicate what she means to say by her previous utterances. That is, he explicitly instructs Mrs. Winter what is required for him to make the lesson contribution that will allow them to go on because they have arrived at an endpoint for which the accomplishment of the instruction “explain your thinking” is an adequate description. He thereby contributes not only to the living/lived work of learning but also to the lived/living work of teaching by instructing the teacher on how to do it to meet his current needs. Preceding this part of the episode, Connor has placed his mystery object with the collection that it ultimately (at the end of the lesson) ends up in. But at this point, there has not been an explicit articulation of the predicate or predicates that make this object a consistent member of the collection. What the present part of the episode achieves in, and as result of, its

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sensuous work is this: the production of an answer that allows the previously articulated predicates of the collection also to be predicates of the object at hand. The object turns from a mystery object into an object that belongs to a known group or collection that already has known predicates and two names. Or, to put it in theoretically more edgy terms, a category already existed out there among them and became perceptually embodied in the collection labeled “squares/cube.” This category therefore “acquired” another one of the mystery object. It will continue to “acquire a certain proportion of [the category] as long as it is given life in the organization of tasks, skills, and evaluations” (McDermott, 1993, p. 271) in the current lesson. In the present situation the agreement between existing class and the mystery object arises from a series of sequentially uttered alternatives and in the course of sequentially uttered repetitions of (a) requesting stating the name(s) of the collection and then (b) ascertaining that the names/criteria also are suitable for the new object. It is only when the mystery object is identified as extending the existing collection of the squares/cubes that it turns from a mystery object to a knowable and known object; and, by becoming a member of an existing collection and adopting its name, it also becomes a geometrical object from the perspective of the knowledgeable observer, analyst, curriculum planer, and teacher.

ACCEPTING THE CLASSIFICATION AND CLOSING OUT THE EPISODE Once a student has had his or her turn and has produced a satisfactory articulation of the predicates on the basis of which the sorting has been done, Mrs. Winter calls on others whether they have something to add. In the present situation, only one other student seeks and gets a turn. This is also the case in this sorting episode. Fragment 8.8 73

W:

74 75

B:

76 77 78 79

W:

<

o>kay. () ben you wanna add? ((nods to Connor)) () thanks connor. (0.66) the um the ah the square that connor ´sai:d (0.46) it loo::ks=similar to the squares and cubes. (0.82) okay. (0.52) anybody else wants to add to it? (2.15) no? okay, next ONE ben.

Here, after being invited by Mrs. Winter to contribute, Ben articulates that “it” looks similar to the squares and cubes, that is, “it” looks similar to the objects that already are on a sheet of paper associated with another

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piece of paper that has printed on it, the words “square” and “cube” (turn 75). Mrs. Winter acknowledges the statement, asks whether someone else has anything to add, and then invites the next student, who happens to be Ben, to draw an object from the bag. By moving on in this way, the events signal that whatever has been the result, it is now appropriate for the projected lesson outcome, which the students do not yet know but which they experience and witness as something unfolding. All members of this group know—explicitly or implicitly—that going to school and participating in a lesson means that teachers teach something that students are to learn; whatever is achieved in the face of repeated changes in classification within specific episodes and across all episodes taken together therefore can be assumed to be the desired end result. Whatever the projected criteria for the sorting, the present sorting episode can be taken as consistent with what is supposed to be the outcome, which is realized in the fi nal set of collections. The projected outcome has been achieved and the mystery object is now associated with and constitutive of an existing collection, as its predicates have been extended to the new object.

COLLECTIVE ACHIEVEMENT OF GEOMETRY DURING A SORTING TASK The classification lesson consisted of 22 student “turns,” of which I closely analyze one in the preceding sections (another one is at the heart of Chapter 10). In this section, I articulate the processes and products of a sorting lesson at the macro-, that is, classroom-level across the sorting task in its entirety. I begin by providing a descriptive analysis of (a) the pragmatic structure of the task, that is, the structure as it emerged from the labor participants do and (b) the outcomes that this labor as a whole achieves.

The Pragmatic Structure of the Task How does this lesson emerge as a geometry lesson from the sequentially ordered talk? In other words, how can this lesson be a geometry lesson given that the children do not yet know geometry (it is their fi rst lesson)? The short answer to be elaborated in the following is this: As soon as formal geometry is understood as being grounded in the mundane, everyday (non-, pre-geometrical) competencies of the students, then their present knowing—as given by the ways in which they participate in classroom interaction rituals—is a condition of their future knowing—as given by their subsequent ways of participating in classroom interaction rituals. What their living/lived bodies do is the work of learning. The transformation occurs in and as a consequence of the lived, sensuous labor (i.e., “doing a lesson”) that they participate in. This perspective radically changes how we think about the relationship of teaching and learning.

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Traditionally, mathematics (as science) curricula are designed based on the logical structure of the field. However, whereas the logical structure provides us with a disciplinary perspective on a task, it remains an empirical matter whether this structure bears relevance to the living/lived labor children do and whether children actually do the task. This logical structure therefore does not determine the unfolding events that it is said to undergird, as Piaget tended to assume. Thus, returning to the episode, there existed a contradiction of which Connor was not necessarily aware: some of the rectangular objects are distinguished from some of the cubes only by “size,” given that the pair of squared sides of the former are precisely the same size as the sides of the cube. Structurally and logically, this can be expressed as: Group 1: Group 1: ∀x(x is cube), Group 2: ∀y(y is rectangular prism with same base as x) Comparison 1: ∃x∃y(y is taller than x) Furthermore, the collection of “rectangular solids” contains objects with one pair of squares and two pairs of equal “oblongs” and objects with three pairs of unequal oblongs (slabs) whereas the “cube” collection contains objects with three pairs of squares (rectangles of equal sides). This situation is symptomatic for some of the troubles that have emerged in the course of this lesson where the children wrestle with the classification of an object that looks like a cone but has a rounded rather than the pointed top that the others had. That is, the children are confronted with the fact that the actual contents of their experience, the rounded cone top, somehow have to be brought into alignment, in and through their living/lived work, with the nature of the object as a cone, which, geometrically defi ned, has a vertex. (In fact, from a calculus perspective, there is a difference in the sense that one is differentiable and has a continuous derivative whereas the other objects have a discontinuity at the point.) Thus, the structure of a task is an empirical matter even in situations where the researchers/teachers have done everything possible to precipitate the emergence of task and logical structure, for example, by stating and restating the task goals and providing other forms of instruction. I therefore turn to the actual structure of the sorting task that pragmatically emerges from the labor of this lesson in which I have participated and which I subsequently have observed repeatedly during my analyses. In all 22 cases (one student was absent), the students place the object either on an empty sheet or group it with an existing collection without saying a word; in most instances, the student begins to walk back to his/ her seat when Mrs. Winter then invites the individual to articulate his/her thinking, that is, to articulate predicates on the basis of which s/he has

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placed their objects where s/he did rather than somewhere else (“Now can you tell us what you are thinking?” “Explain your thinking!” “You have to tell us why!” “Before going on, explain your thinking!” “Don’t go away, tell us why you put it there?”). As part of articulating similarities and differences, the student holds the mystery objects to other objects and in various collections, sometimes uttering “this is the closest object to it” (Oshin), sometimes running the fi ngers across the mystery and comparison objects, sometimes running fi ngers along edges and corners and naming them as such, sometimes articulating roundness (e.g., tube, cone, sphere) or noting the absence of sides (sphere). Throughout the lesson, Mrs. Winter also reiterates other utterances that members can hear as instructions: “Just take the fi rst one you feel!” or “The first one you feel!” Although these “instructions” do not cause students to act differently, they shape what they actually do and, therefore, shape their living/lived bodies in the same way as other instructions and interdictions, about how to sit, how to wear clothing, where to place legs and feet, and so on. All of this is part of a pedagogy that shapes the living/lived bodies of the children, and, therefore, the forms of expressions (knowing) they exhibit in and through participation. The sorting task begins with the Mrs. Winter’s articulation of the rules relating to color and size. Throughout the lesson, however, color and size continue to be prevalent predicates in the classification of the objects pulled from the bag. Many children use size (e.g., “this is bigger than the other ones” [Connor], “that square one [points to cube] is a bit shorter and this one [points to cylinder closer to her] is a bit shorter” [Geena], and “this one [points to rectangular prism] is smaller than that one [points to cube]” [Cheyenne]) to articulate similarities and differences between objects, followed, when applicable, by reiterations of the rule (“we are not counting color, size”) or describing what has happened in a tone that what is said can be heard as a complaint (“So you are looking at size again”). As a result of Mrs. Winter’s contributions to the collective sensuous labor, the sorting session comes to be orchestrated, both teachers being in the conductor position, stating, restating, and reiterating instructions and rules; but it is the orchestration of a partially improvised session, because only Mrs. Winter and Mrs. Turner know the score and the children attempt to play such that it comes to be consistent with the score they do not know at the time. If no teacher restates and reiterates a rule or instruction, the ongoing or immediately past act of classification and predication can be seen and heard as consistent with the (from the children hidden, yet to-be-discovered) rules and instructions. It is seen and heard as a successful action so that now the rules and instructions are descriptions of rather than prescriptions for what has happened.4 After the fact, the rules and instructions may be said to have led to the successful classificatory act. In this lesson, the children use many verbal predicates and descriptions about what they feel when touching the object. “Pointed,” “round” while circling or placing the hand around the object, “smooth,” “pointy,” “flat”

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while gliding the hand along the side. The videotapes show children holding their mystery objects immediately next to other objects already in collections, sometimes gliding a finger from one to the other, sometimes pointing out perceptual similarities, sometimes using iconic gestures to show how we (onlookers) have to look to see similarities and differences (e.g., how two cones of different size have different angles at the tip). These actions affect their senses so that what they say expresses and is consistent with their living/lived experience of the objects at hand. They talk about lines, points (“corners”), and sides (“round sides” “flat top” “no sides”) and about shapes (“all squares,” “triangles,” “rectangles,” and “circle”). Following repeated solicitations by the teachers to name a category, children propose these same terms, which Mrs. Turner duly records on prepared sheets of paper and places next to the relevant group. Eventually, children also use number—counting the number of edges to distinguish triangular, hexagonal, and octagonal prisms. Many of these actions—which are produced by individuals but for an audience that is presupposed to already comprehend their sense—are tactile rather than visual, as the sense of touch frequently is drawn upon. And this sense is not independent from motor acts by means of which characteristics of the mystery objects are explored and exhibited. In this lesson, formal geometry emerges from children’s sensuous actions and talk precisely when they cease to use color and size as predicates. At that point, the lesson has achieved this: (a) students act/talk according to formal geometry in the sense that they use proper rather than improper predicates; (b) they participate in geometrical practices as handed down and developed by generations of practitioners and students of the discipline; and (c) they have had the opportunity to experience geometry sensuously, in action, both their own and vicariously in those of their peers. 5 The children do not begin the classification tabula rasa, but bring their mundane competencies of classifying and of participating in lessons generally to this class. It is in and through their participation in the sensuous labor of classifying that their mundane classifying practices become consistent with geometrical classifying practices. It is not that the students have overcome, or teachers eradicated classification conceptions, but, rather, children’s own participation in classifying has changed their sensuous classifying actions.

Lesson Accomplishments Classification depends on an aim and the inclination of those who classify. In the present situation, however, the students cannot know at the outset what the end result will or should be, because the correct classification is the object/motive of the activity, that is, the intended learning outcome. One important accomplishment of the lesson, therefore, is for students to have come to know after the fact the structure that underlies their task, which is the precise structure that they realize in the existing division of labor with their teachers. At the end of the sorting task, this

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accomplishment is publicly available on the floor before them for everyone to see—i.e., for scrutiny if necessary. But this structure is only one of many different classifications that can be produced all consistent with the canon of classical geometry. The accomplishment of this classification therefore is the contingent result of the constitutive relation between the emergent and continuously changing configuration and the randomly drawn next object. Because the emerging configuration becomes a referent for future classifications, the sequential order of talk comes to fold back on itself and thereby integrates itself over (like a mathematical convolution of a function with itself). At the outset of the lesson any predicate is as good as any other. The ordered collections emerge in the course of selecting some predicates over others and setting (reiterating) rules for excluding some predicates; and with the collections emerges a geometrical arrangement that constitutes the intent of the task discoverable by the children only after the fact. But this knowledge of the intent is, according to Wittgenstein and others, a condition for the proper classification of the mystery objects. The children therefore are in a contradictory situation whereby they have to classify without knowing what the classification presupposes and is built on. How can this be? In my perspective, this is so because of the ordered (classified) objects and the socially ordered mathematical practices stand in a mutually constitutive relation so that proper classification practices and classes emerge simultaneously. Although objects and ordered mathematical practices stand in a constitutive relation, the objects do not explain the order of the unfolding, sequentially ordered talk. They are not evidence of the order but stand in a reflexive relationship to the unfolding ordered and orderly labor. Once a person in the process of categorizing can see the ordered system, she or he can provide a legitimate account of the instructions. Thus, geometry as a subject matter does not begin with the announcement that a new mathematics unit in geometry begins or that this is a geometry lesson, but with the children’s sensuous classifying according to rules that exclude color and size as predicates. But this beginning is grounded in and made possible by actions that are not geometrical at all: These non-geometrical actions become the very possibility and condition of geometry. It is in the sensuous labor that labor is transformed. Moreover, to hear teacher instructions as consistent with geometry—that is, not just as arbitrary rules but as constitutive of geometry and the ordered properties of collections qua geometrical objects and properties—it takes an understanding of geometry that this and the following three weeks of daily lessons is intended to develop. The children are an integral part of the sensuous production of these lessons; and in their productions, they make available resources from which, reflexively, they may extract what is geometrical about geometry lessons. From the teachers’ perspective and within their rationality, the task has been accomplished when the different collections are consistent with

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an ordering according to geometrical properties, that is, when the collections and children’s actions legitimately can be labeled using names from a geometrical discourse. At the end of the day, 22 objects are grouped precisely into 10 different collections that are consistent with a classification according to geometrical properties (predicates of the objects). This classification is inconsistent with the predicates of size and color, as differently sized and colored objects can be found in the same groups. When all is said and done, the proper classification has emerged as a collective achievement, irreducible to teacher or student, as even the teachers are held to the task of specifying more clearly what they really mean by their questions and instructions. To make (i.e., produce) sense, sensuous labor has to make sense (i.e., be intelligible); and the various parties hold each other accountable for the intelligibility of their utterances. The fi nal result of the classifi cation task then provides students with concrete materials that can be used to substantiate the sense of “no color” and “no size” in geometrical classifi cation. Each utterance in the unfolding, sequentially ordered conversation contributes to the contingencies in and of the living/lived situation. In this chapter specifically and in this book as a whole, I am concerned precisely with the nature of the sensual labor—as available to participants and analysts—that achieves such a set of collections. A classical paper on the formal structures of practical actions—a paper that established the foundations of ethnomethodological and conversation analytic studies—specifi es the investigators’ task as one of identifying the work actually done that is described or glossed in natural language (Garfi nkel & Sacks, 1986). The theoretical formulation these authors propose for the work, “doing [sorting according to shapes],” brackets the description of what is being done, recognizing it as a gloss of the real work “doing” that realizes the contents of the gloss. This formulation is consistent with the distinction between a rule (instruction) and the actual work (practice) of following the rule. The participants in this situation are heard to be speaking a natural language used for the natural and “objective production and objective display of commonsense knowledge of everyday activities as observable and reportable phenomena” (p. 163). Here, in accordance with developing an ethnographically adequate account of concerted activities and their contexts, I exhibit the sensuous labor of classification by focusing on one initially problematic categorization of an object because participants (teachers and students) are holding each other accountable for clarifying what they really meant to say and do when there is trouble. In the course of the lesson recorded on the videotapes, we observe the emergence of category families as children begin to make comparisons and distinctions with other more similar collections but leave out those that evidently do not belong in a collection. That membership categories form families or collections pertains to the fact that some can be heard or

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seen as “going together” whilst others cannot be so seen, used, or heard in this way. For example, the cone and pyramid come to be alike (both are “pointy”) and their slight differences are articulated for grounding their belonging to different categories; they could be in a group on the basis of their acute angles, which some other mystery objects do not have. At the same time, neither of these “pointy” objects is compared to the sphere or cylinder groups—though in one instance there was a comparison with the corners of a cube (“pointy”). In another instance, a child holds an uncertain object next to and compares it with the rectangular solids and cubes but not against cylinders, spheres, or cones. The sphere then comes to be compared to the cylinder, based on predicate of “roundness,” but not to the cubes, rectangular solids, or cones and pyramids. The result of the lesson therefore is not the classification and categorization of the three-dimensional objects as such but an articulation and delimitation of permissible predicates, which are co-constitutive of the objects as geometrical objects and the lesson as a geometry (rather than some other) lesson. This is so because discourse (logos) has an apophantic function, that is, the function of allowing a phenomenon to be visible in the way it shows itself from itself (Heidegger, 1977b). Naming lets something be seen, but only something that shows itself from itself can be named. However, rather than constituting a chicken-and-egg situation, the two can be seen as emerging together when Being and presence come to be understood in terms of beings and the re-presentation of the present. In other words, together with the dehiscence between Being and beings emerges the signifier (language)—signified (thing) relation, which involves sections of the material continuum that come to mutually refer to each other. In the course of this analysis we fi nd the members of this classroom (teachers, students) to continually perform common sense analyses of typical entities in this classroom—e.g., Connor’s queries concerning the signification of Mrs. Winter’s utterances presuppose that he could not fi nd one, i.e., an analysis of intelligibility—and the typical objects for a lesson on three-dimensional geography are no exception. Children’s common sense understanding of classification, categories, and predicates are mobilized to make this a geometry (rather than some other) lesson by selecting, supporting, and transforming those aspects that become constitutive of geometry while also deemphasizing others that are not suitable for allowing the geometrical nature of the objects to appear. Predicates that are suitable for talking about things do not necessarily “go together” as invariant components of geometry lessons. The labor accomplished by Mrs. Winter’s repeated comments articulating the nonpermissibility of size and color as predicates establishes topic appropriateness, and with it, the clarifi cation of what a topic typical for a geometry lesson can be. To sum up: Throughout the lesson, we can see students (a) articulating common sense knowledge for themselves and others, and, in the process,

230 Geometry as Objective Science in Elementary School Classrooms (b) making common sense knowledge available for an explicit transformation into a more formal structure (membership categories, membership categorization devices). Geometry (context) therefore is allowed to emerge from children’s non-geometrical lifeworlds in and through its constitutive relation with actions (category-bound devices) that are recognized as geometrical. The objects pulled from the black plastic bag are collected on the different pieces of colored paper, and the appearances of these collections provide a form of perceptual “gestalt.” The collections are evidence of geometrical structure, pointing to and are documents of the geometrical nature of these objects. The structure is for this lesson, occasioned as a geometry lesson, and the geometrical properties emerge as an achievement of the lesson, which becomes mathematical with the geometrical nature of the objects. There is a recursive and constitutive relation between what the sensuous labor participant members do and what the nature of the societal activity that they participate in is and becomes.

9

Reproducing Geometry as Objective Science

In the preceding chapters I focus on the performative aspects of geometry– geometry as a form of cultural-historically motivated, sensuous labor—and its individual and collective dimensions. As every other science, geometry exists in and as of an open chain of known and unknown researchers as the constitutive subjectivities that subjectively enable the science as a whole (Husserl, 1939). That is, we cannot just think of geometry as an individual, personal performance and experience, but it is precisely in, through, and out of the subjective experience of every person doing geometry that the totality of the science comes to be constituted. In this chapter, I am precisely concerned with the reproduction of geometry as an objective science from the experience of performing and experiencing geometry of my second graders and their teachers. Fundamentally, geometry is objective because it is discoverable—by each member of society in his/her sensuous actions, and in each generation—rather than discoverable because it is objective. It is precisely for this reason, too, that there is a continuous origin of geometry and a long cultural history of it as well. My study was originally designed to investigate how three-dimensional geometry comes (or does not come) into being in and through the incarnate, sensuous, and sequential labor of second-grade students and their teachers. While I was participating during the fi rst lesson, when students were asked to classify a series of mystery objects I was especially interested in fi nding out how students could classify three-dimensional objects according to geometrical properties given that recognizing the latter was the curricular goal. I asked myself, “How could children classify geometrically given they currently attended their fi rst lesson of geometry?” And watching the lesson unfold, I asked myself, “How can it be that children behave in ways that would allow an unexpecting visitor to recognize this lesson as a geometry lesson even though the children do not yet know or are at the beginning of their knowing geometry?” In this chapter, I further discuss the fi ndings of the preceding one with respect to three pairs of terms that cannot be reduced to each other or, in other words, each pair exhibits a contradiction. Much like the philosopher Edmund Husserl, who—well-prepared with a PhD in mathematics

232 Geometry as Objective Science in Elementary School Classrooms and postdoctoral training in psychology—used geometry as an example for illustrating general points about the “crisis of the European sciences,” I understand the present analyses as speaking to issues that go beyond geometry and mathematics. The pairs of terms therefore include (a) the sequential order of classroom talk and the ordered field of objects (category system) that evolves, (b) the following of instructions and the collective achievements, and (c) the emergence of disciplinary knowledge (disciplinary practices) from pre-disciplinary knowledge that frequently contradicts the former (as in “misconceptions”). In speaking to these issues, I contribute to rethinking how cognition might be examined consistent with the production of an ethnographically adequate description of concerted activities and their context.

EVOLVING CATEGORIES AND SEQUENTIAL CLASSROOM TALK Two of the questions I set out to answer are “How does the ‘proper’ grouping of a collection of objects emerge from the collective task involvement of what recognizably is a second-grade classroom?” and “How does this task involvement recognizably constitute a mathematics lesson?” In search of answers, I present in Chapter 2 Piaget’s ideas about development for two reasons. First, it characterizes an approach still taken by many educators who attempt to locate cognition exclusively in the human mind. Second, it is a teleological approach where children’s competencies are analyzed in terms of their differences with adult scientific reasoning that constitutes their telos. I also chose the van Hiele scheme as a referent because it suggests that second-grade students should not be ready (able) to classify objects using properties of shapes. In contrast to the predictions of both theories, the analyses in Chapter 8 show that the children already have developed tremendous classificatory competencies prior to their coming to this particular lesson, the fi rst one of a sequence labeled “geometry” or “doing geometry.” What the children learn in this geometry curriculum generally and in the classifying task in Chapter 8 specifically is not the classification of objects: This, they already do competently, for example, when they distinguish objects by means of color and size. They children also are keen observers, noting aspects in the objects that in fact distinguish them from proper geometric objects, that is, their non-ideality. In fact, the classification lesson as planned presupposes sensual classificatory competencies. The question for the teacher is how to allow a differentiation of these competencies into properly geometrical, on the one hand, and different, worldly competencies, on the other hand. The difficulty is that worldly objects are very different from geometrical objects, and even the model objects that Mrs. Turner and Mrs. Winter have brought to the classroom are worldly rather than

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geometrical ideal—though perhaps further along in the asymptotic refi nement of real toward never-achievable ideal objects that geometry deals in. However, what children do become competent in is this: how to contribute to a collective sorting activity such that their intelligible actions themselves come to constitute a two-category class of proper and improper actions. Together, therefore, the children and their teachers bring about a lesson in and through their sensuous labor such that it can be appropriately named (here) a geometry lesson. Together, in and through a division of labor that itself is an achievement of the sequentially ordered classroom talk, students and their teachers do it by eliminating from the produced accounts of sensuous action all those that involve predicates based on the empirical categories size and color and that disregard certain properties of the real sensible objects that are insufficient to move them out of the class surrounding the (yet to be disclosed) geometrical ideal object. They also do it by using only those that are invariant in spatiotemporal terms with respect to the gestalt (shapes) of objects. That is, these are precisely those predicates that from the perspective of formal geometry are consistent with discipline as a science based on transcendental properties. The function of the sequentially ordered whole-class conversation is clear: the identification of improper classification actions and their replacement by the proper forms. The necessary revisions are based on self-correction, which is the preferred form of action. The second important part of the living/lived instructional process lies in assisting the child currently classifying (and all those who witness) to distinguish among those incarnate (classificatory) actions that are proper for the task at hand (Ap) from those that are improper (Ai).1 Because all Ai are publicly produced and available, children now learn, after the fact, the relevance of teacher instructions (“no color, no size”) in their own, immediately preceding actions. That is, the children act first prior to learning whether and to what extent their actions are consistent with some rule. This relationship between sensuous acting and rules exists not only for children in the process of learning but also for any form of sensuous action, even those of accomplished professionals such as engineers, scientists, or cooks. Whether they have followed “the procedure” or “the recipe” can, in any practical case, be established only after the fact. In the organization of this sorting task, students and teacher reproduce the conditions underlying classical concept learning experiments, where participants were asked to derive categorization rules from sets of instances and non-instances of a concept. More specifically, in the sequential organization of the lesson we can find the same organizational structure used in the classical rule induction experiments. That is, given an initial example, participants are asked to discover a hidden order as they proffer their own examples followed by feedback as to whether the sample belongs to the hidden category and as to the explanations for the periodically announced hypotheses about the rule. The difference of such studies with my own

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lies in the fact that here the entire set of rules underlying the ultimately achieved categorization has been achieved collectively, contingently, and emergently in the sensuous labor that makes the lesson. As a result of the collective labor that brings about this lesson, new categories (classes) emerge for the children. These categories emerge not in the sense of Piaget and van Hiele on an intra-personal level, that is, as the necessary stages through which the individual has to go prior to accessing the next level. Nor do they emerge because the teacher has told students what the classes and families of classes are. Rather, it is in, through, and from their individual, collective, and always incarnate sensuous labor that not only the preferred categorizations but also the predicates emerge. An essential aspect of this labor is the sequentially organized, organizing classroom talk—that is, the interaction rituals. From these rituals emerges the geometrical nature of the produced arrangement on the floor together with the geometrical nature of all objects themselves. That is, what children do eventually know, their “higher-order cognitive functions,” is achieved in and the consequence of resonance phenomena that ritualize their interactions. This involves, on the part of the students sensuous actions: perceiving, pointing, touching, feeling, walking about, measuring, comparing, orienting, giving instructions, and requesting accounts and explications. In this, my analyses provide an answer to the third question that oriented my research, “What role do incarnate experiences play in the progress [of sequentially organized classroom talk]?” My analyses therefore provide an articulation of the microgenetic processes by means of which cultural artifacts such as cubes, cones, and cylinders—which do not have intrinsic sense—come to be associated with intelligible, recognizably mathematical practice. In the sorting lesson, a configuration with the structure [cones/ rectangles/cube and square/ . . . /] emerges. Within this structure, the mystery objects become part of a classification where the truly ideal geometrical objects, unavailable in practice, constitute their transcendental exemplar. The sensible objects themselves become geometrical in and through the types of predicates that are used to put them into collections rather than because they are mathematical objects in and of themselves. 2 The objects are referred to as “cones,” “rectangles,” “cubes,” or “squares” in the same way that any appropriate specific thing (token) comes to be denoted by the category name “tree” ([ideal] type). My analyses also makes apparent the two other dimensions of Saxe’s (2002) analytic framework—sociogenesis, concerned with appropriation and spread of cultural forms, and ontogenesis, the shifting relations between form and function of cultural artifacts—though it does so in a different way. Despite ample evidence that children make reference to each other’s contributions (e.g., “the square that Connor said”), more important appears to be the emergence of the classification as a cultural artifact. It serves, already during its emergence and in incomplete form, as a resource in all subsequent classifications. That is, the classification as available in

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the physical, public space at the core of their circle is like a snowball affecting anything coming thereafter. That is, the subsequent states of the structure have a cultural history, here constituted in and through the collective, sensuous labor. This has the result that sequentially ordered classroom talk folds over itself, where previous choices, crystallized in the emerging material order in the center of the circle, afford and constrain subsequent choices. In the present study, for example, Connor compares his mystery object to the objects already classified and therefore to the emergent order as a whole. This order is the result of collective, equally sensuous labor and, therefore, a representation of the collective norms for classification that are in the process of establishing themselves. In the process, the function of individual forms changes. Forms initially exist as intentional objects of the task at hand but, as soon as they are acceptably placed, they become resources in the classification of subsequent mystery objects. Moreover, they are not only resources that serve individuals but also exist as an emergent perceptual order (di-vision) that organizes collective perception (vision) of additional objects. That is, the emergent order embodies and develops a particular habitus, a way of sensing and sensuously acting toward the world. This habitus can only observed in action, it “is constituted in praxis and is oriented toward practical functions” (Bourdieu, 1980, p. 87). In and through the sensuous labor of this lesson, which requires the mobilization of habitus, the latter not only comes to be exhibited but also, simultaneously, undergoes transformation. There are many situations in everyday life where it is crucially important to make (perceptual) distinctions, even if these distinctions cannot be articulated in so many words. Some of these, such as the workplaces that I studied in my previous research, do not require verbal articulation of reasons as long as others can observe the classification to be adequate in and for the task at hand. Thus, for example, the workers in a fish hatchery sort juvenile salmon without being able to provide a description of their classification criteria (Roth, 2005). Yet they do classify and correct others when the latter have misclassified a fish in the process. When I asked to be taught how to classify juvenile salmon, the workers invited me to “just look.” But mathematics, as science, is based on that which can be verbally articulated, so that all those things we know but cannot articulate in one or another way do not constitute mathematics and cannot be conceptualized as such. That is, these sciences require accountability, which is produced in verbal accounts (see Chapter 6 and the relation of accounts, accountability, and the emergence of the objective nature of geometry). Verbal accounts use language, Gr. logos, which cultural-historically has come to be equivalent to logic (Gr. logos) and rationality. The specific objective nature of mathematics generally and, here, geometry specifically arises from the fact that key elements can be and are brought into accounts, that is, into discourse that re/presents the present whenever necessary. It is, therefore, re/presentation, part of accounting practices, that makes mathematical reasoning timeless.

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Thus, when Connor says that he cannot say how his object (cube) differs from the rectangular prisms (Fragment 8.5, turn 28), though he has held the former next to the latter while saying that it is “a tiny bit different,” his predicate falls outside of what is properly formal mathematics. Only when his object is different from all others present may he start—following the rules of the game—a new group. Using color and size in predicates are improper actions (Ai) in this classroom, and the disjunction of color and size from geometrical properties lies at the heart of the nature of mathematics. Of course, the mystery objects could be classified in terms of size and color; and these would or might be probable categories in other situations.3 But here, the rules of the (categorization) game articulated repeatedly in the unfolding, sequentially ordered conversation explicitly discourage such predicates. The geometry of threedimensional objects generally and that of classification of three-dimensional objects specifically has to be disconnected from size and color for geometry to emerge as the transcendental science that it is. This is the point Kant (1956) makes when he suggests that color is an empirically grounded category, whereas that of triangle can be derived as transcendental category based on the division of (Euclidean) a priori given space based on pure intuition (“by means of three lines, of which any two taken together are longer than the third, one can draw a triangle,” p. 206). Many adults would fi nd themselves in a situation similar to the children if an instructor asked them to classify objects according to “topological” principles before knowing topology. The task would then require all of the children’s mystery objects to be placed on the same sheet and, for example, coffee mugs, teacups, and toruses to be put on another sheet. The sequential order of classroom talk provides the resources for discouraging the use of color and shape. Mrs. Winter, in going next after a student, may affiliate (making A an Ap) or disaffi liate (making A an Ai) with the predicate designed to establish the category. She does not say that the predicate is wrong but she says that for the present purposes, color and size are not proper. This means that the capacity of a category to explicate something on an occasion of use depends on the sequential organization of the talk in which it is used, including the capacity of the categories developed in this lesson to explicate a geometrical property. The sequential talk realizes, mobilizes, and evolves a particular language game. As in any game, it is the sense of the game and real mastery while playing that matters rather than the abstract knowledge and symbolic mastery one may have about it.4 What children develop by participating in the sorting game, sensuous as any other game, is precisely this sense of the game that distinguishes the geometer with know-how from the geometer with knowingthat. The relation of the sensuousness of activity and the sense we develop engaging is not fortuitous—there is a necessary, founding relation between the sensuousness of doing something and the sense we make in and of it. This sensuousness is not found in bodies, it is found precisely in the flesh.

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The talk-in-interaction that I present here therefore constituted and presupposed the lesson as a geometry lesson, which in turn is a resource for the quiddity (essence) of the talk as talk of a mathematics lesson. Hence, geometry as a sub-field of mathematics emerges from the particular turn-taking routines of the sequentially ordered (sensuous and sensible) classroom talk that leads to the production, evaluation, solicitation, and so on, differentially located with different members.

FOLLOWING INSTRUCTIONS AND THE EMERGENCE OF DISCIPLINARY PRACTICES Instructions constitute the second dimension of my answer to the question “How does the ‘proper’ grouping of a collection of objects emerge from the collective task involvement of what recognizably is a secondgrade classroom?” Instructions and sensuous instruction following are more complex issues than normally assumed. For example, during the months while writing this book, I happened to a road construction on the right part of the road during a cycling tour. Even before the last oncoming car had passed through the one-way stretch, the signal person suggested, “Go! Outside the crown.” I did, staying to the right of the plastic posts separating the construction from the usable road. I hear a voice shouting behind me “Outside the crown!” and immediately move to the left of the posts. There is no more shouting. Here, then, I act even though I have never heard the word “crown” in the context of road construction. I follow the instruction “Outside the crown,” by staying to the right of what “Outside the crown” teaches me to be a crown, on the side away from the usable road as marked by the plastic posts. The shouting voice teaches me that the outside is on the usable side, that is, for me the inside. In this manner, I have learned to associate sensuous actions and verbal instructions so that after the fact I can be said to have acted fi rst inconsistent then consistent with the instructions. I have learned to recognize a crown in the process, and also where to fi nd—or how to know whether I am at—the inside and the outside of this crown. I learn this categorization and classification in and through the sensuous praxis of classifying; and I learn to follow instructions by following instructions. I can know whether I follow an instruction only when my living/lived body (flesh) acts in a manner that after the fact turns out to be consistent with possible accounts. It is precisely when I no longer hear shouting behind me that I know that what I do and have been doing for a while is consistent with the description (account): I have stayed outside the crown. It is in sensuous praxis that I come to face the practical contingencies that the actions themselves create and I can become aware that I have followed the instructions I received once the outcomes of my actions have been evaluated and sanctioned (“On the outside!”).

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My analyses presented in Chapter 8 speak to the role instructions play in classrooms generally. In this lesson, which leads to the emergence of a classification of 22 mystery objects, instructions are an integral part so that the end result—the ordered field of objects—cannot be understood without taking the instructions into account. This relationship between instructions and contingent, incarnate action is not straightforward. We therefore have to take instructions into account here, but not in the way instructions frequently are portrayed, that is, as causal antecedents of those sensuous actions that are said to implement the former. Particular instructions—e.g., “No color!” and “No size!”—do not prevent students to continue drawing on color and size in the predicates they articulate. We should not take this as an observation to think of children as having a deficit of understanding instructions for there is evidence that instruction following requires understandings that the instructions presuppose. The fact that the presence of color and size in children’s predicates decreases over time shows that children learn what it means to sensuously act so that the instructions are descriptions of the adequacy of which can be established a posteriori— just as this has been in my own experience of following the instruction “Outside the crown!” Thus, my analyses are consistent with a theoretically akin study of third-grade children doing science experiments (Amerine & Bilmes, 1990). It found that children do have a well-developed sense of accountability for actions, which, in my situation, is a sense of accountability for actions in terms of predicates that the children use for classifying their mystery objects. A second dimension of instruction emerges from my analysis: Students themselves provide instructions, here concerning the manner in which they are to be instructed. In Chapter 8, I repeatedly show how Connor tells the teacher what she needs to do to assist him in (a) accomplishing the desired task of placing his mystery object in proper rather than improper fashion and (b) providing a proper rather than an improper account. Thus, for example, when Connor utters “What do you mean?” he achieves quadruple work: (a) by asking for the signification of what has been said before it marks trouble; (b) by connoting that the signification of a preceding utterance (teacher instruction) is unclear; (c) by initiating a repair sequence; and (d) by employing a fi rst-part of the turn such that the teacher’s response (in explicating what she really meant) repairs the trouble that has emerged. When we listen to a teacher articulating instructions, it is easy to see why educators see actions as following and “being caused” by the a priori instructions that describe them. The link between instructions and actions derives from a maxim (“apparatus”) that describes everyday, mundane, and commonsense competence. This sense makes members see observations that appear in sequence not only as related but the second one as engendered by the fi rst. In Chapter 8 specifically and throughout this book more generally, I show that successful courses of actions are the conditions for the a posteriori determination that instructions have been followed. It is

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intuitively evident that a course of action determines its outcome in a physical sense; but a course of action, in the way it comes to be articulated and formulated in a posteriori accounts, makes certain aspects of the outcome relevant, noticeable, and “discourse-able.” Here, these aspects precisely are the geometric properties, which the teachers note in the construction of group labels such as “squares” and “cubes.” Therefore, successful instruction itself is an outcome of the sequentially ordered classroom talk and, in being as successful as it is, allows the emergence of (three-dimensional) geometry as a recognizable field of inquiry. In the same way as I have discovered the relevance of the instruction “Go! Outside the crown,” the children discover for themselves and collectively with others the relevance of the instructions “does some shape match and fit in the groups already on the floor or does it deserve a new group on its own” with the constraint, reiterated repeatedly, “not to go by color and not to go by size.” They do so in the course of their incarnate, sensuous engagement with the objects (perceiving, holding, picking up, placing, replacing, comparing, touching, turning). It is in and through their concrete, sensuous labor of “doing [sorting mystery objects],” where the bracketed part is a description that signifies the actual work done. In the same way as I do while cycling into the construction zone, it is in the (lived) praxis of classification that Connor comes to face the practical contingencies that the actions themselves create; and he can become aware that he has or has not followed the instructions he received once the outcomes of his actions have been sanctioned. For one, he is not only asked to provide an account of what he has done and, implicitly, why he has engaged in the particular course of actions. He also is asked to reconsider his classification under a set of constraints rearticulated for the purpose of guiding or constraining his revision.

EMERGENCE OF A DISCIPLINE IN AND FROM (PRE-DISCIPLINARY) KNOWLEDGE My analyses speak to a third aspect hardly ever addressed in learning research: How do we learn mathematics (science, history, etc.) if all that we have available are our pre-mathematical (scientific, historical) and often non-mathematical (scientific, historical [e.g., “misconceptions”]) ideas and practices? Readers may already anticipate this answer: In the same way as I learned construction-site discourse (“outside the crown”) based on my everyday non-construction-site material practices. First, the word “crown” makes sense as soon as I see one, even when I have never seen something as a crown before. It is not that I had not seen such separations involving plastic posts before, but it is precisely at that instant that the word “crown” takes on significance in the context of road construction. I know a crown when I see one, but the crown became a crown only because I heard

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the word before. But when I saw the crown, it had clearly demonstrable aspects, even though it took another instruction before I could differentiate its inside from its outside. Explicitly grounding the teaching of mathematical concepts in everyday experiences may increase rather than decrease the complexity of the learning task. From a historical perspective, for anything like present-day geometry to emerge, there have had to be “apodictic” (clearly demonstrable) aspects in the pre-geometrical world of the founders of the discipline—such as those making the crown a crown and distinguishing its inside from its outside. These apodictic aspects provide the materials for idealizations that are characteristic of the field of geometry. These idealizations themselves are projected but never achievable outcomes of continued refinements to make, for example, dies that have six equal squares as their faces adjoining pairs of which stand at right angles to each other. Similarly, the everyday world of children provides them with the materials and grounds on and from which their first geometrical understandings emerge, not as something radically strange and different, but as something that has the potential to make sense from the fi rst moment. This pre-geometrical world not only is a world of objects but also is the necessary condition for fully fledged geometry to arise—just as my non-road-construction world is a pre-condition for the emergence of a road-construction world in which the sound transcribed as “Go! Outside the crown” entails (should entail) a particular set of actions (i.e., become “the flesh of words,” Rancière, 2004). However, as an analysis of Husserl’s “The Origin of Geometry” shows, these early geometrical objects are not mere objects, because the coexisting human beings and the coexistent cultural objects cannot be reduced to pure (mere) thinghood (Derrida, 1989). These pure bodies only exist in and through real material bodies, which also have, besides their spatiotemporal characteristics, sensible “material” qualities including color, heat, heaviness, hardness, and size. These sensible qualities rather than the geometrical properties are the first facts given to children in their sensuous experience of a real, lived-in world. The teachers’ challenges to children’s classifications therefore confronted the latter with the factual nature of the objects of their perception and their lifeworlds. Because such challenges concern what appears to be obviously right and out there, the children’s frequent consternation with teacher demands is quite understandable. At the same time, for formal geometry as we know it today to emerge from children’s carnal engagement and from the sequentially ordered talk, the spatial shapes, temporal shapes, and shapes of motion have to become available to the children from the totality of perceived objects. But the objects’ independent and concrete everyday reality, which comprises all the material, sensuously sensible qualities and the totality of their predicates have to be disregarded in the process. The pedagogical function of the classification episodes is to assist children in sifting through the totality of predicates for their sensuous experiences to

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arrive at a selection that meets with the approval of the teachers. Alternatively, the pedagogical function is to provoke new predicates to become available to the children, which they then can use in their classificatory considerations—just as I learn to use the predicates “inside” and “outside” in the context of a road construction crown. The members of the group, collectively, realize this pedagogical function in a situated and contingent way, however “successful,” “imperfect,” “partial,” or “unsuccessful” others may characterize this to have happened. It is precisely in their collective sensuous labor organized in and by means of the sequentially ordered classroom talk that students and teacher achieve “ordinary discursive organizations [that] yield practical mathematical objects” (Macbeth, 2003, p. 269). Surfaces, edges, lines, corners, and points initially constitute the distinguishable features of objects. As we see in Chapter 4, Mrs. Winter and Mrs. Turner posted a diagram that had precisely those distinguishable features as their topic. But, as we see in the actions of Thomas and Bavneet, such public postings in and of themselves do not lead to the isolation of the properly geometric habitus from one that characterizes everyday perception, with its very different way of vision and di-vision. In the sensuous praxis of working with such materials, preferential properties are continually refi ned and related (for example, an object’s faces that meet and are bordered by lines, points, and corners). The emerging preferential properties are ideal and idealized constructions that are valid across historical time, space, and cultures. Because of this validity, idealizations are reproducible with identical intersubjective sense—whereas the objects of the everyday lived-in world are subject to continuous change. Such an ideal and idealized construction can arise from sensuous activity only under the condition that “the apodictically general content, invariant across all conceivable variations of the spatiotemporal sphere of gestalts is taken into account in the idealization” (Husserl, 1939, p. 224). This conception of the nature of geometry also is relevant and useful in the present study of the emergence of geometry in children’s classifications and related tasks. The objective of the sorting task precisely is to arrive at a classification where objects of similar gestalts (forms, shapes) are placed together so that collections emerge that materialize ideal objects with invariant spatiotemporal properties (as articulated in the predicates). These collections differ from other collections in the spatiotemporal properties, while permitting other properties to differ and be the same within and across collections, respectively (Figure 2.1). That is, there might be red and green objects in the same class of cubes, or red cubes and red cylinders. But all objects of cubic gestalt, for example, have to be in the same collection, for otherwise geometry would not properly emerge. In this, some possible classifications, though geometrical through and through, remain undesired because contradictory to the planned curriculum. In Chapter 5, Mrs. Turner wants Melissa specifically to distinguish the cube from other parallelepipeds (rectangular prisms), even though it can be considered a

242 Geometry as Objective Science in Elementary School Classrooms special case of such idealized objects. Similarly, from the perspective of topology, a cone, a pyramid, and a cylinder could be placed in the same category with a cylinder. This classification could be done based on the explanation that a cone over a polygon P is a pyramid with base P, a cone over a circle constitutes a classical cone, and a cone is nothing but a generalized cylinder (round or with n rectangular surfaces over each side of P) in which the meridian is concentrated in a point. But classifying cones, pyramids, and cylinder together would not have been what Mrs. Turner and Mrs. Winter wanted, that is, the separation of the shapes into the three now-classical and preferential categories (types). In the present lessons, a formal discipline (geometry as praxis rather than as body of knowledge) begins to emerge from a sensuous sorting activity that children are asked to accomplish without prior formal disciplinary knowledge. How is such emergence possible? From pragmatic and methodological perspectives, members to organized arrangements such as mathematics classrooms are continually “engaged in having to decide, recognize, persuade, or make evident the rational, i.e., the coherent, or consistent, or chosen, or planful, or effective, or methodical, or knowledgeable character” (Garfi nkel, 1967, p. 32) of their inquiries. In this classroom, these inquiries pertain to sorting, classifying, providing reasons, counting, graphing, sampling, reporting, and so on. The objective nature of the discipline (here geometry) therefore exists in and through the emergence of the discipline in the practical actions of real, incarnate, sensual human beings. Therefore, it is only in their contingent productions that the objective sciences (here formal geometry) are reproduced across generations. It is not therefore that geometry lessons cause students to do certain things; nor is it that in their actions, students produce geometry anew. Rather, the transcendental and material nature of the objective sciences emerges each time when a member (here students) produces and thereby reproduces practices that are recognizably and therefore collectively scientific and objective in nature. Because transcendental concepts are taken up by new transcendental concepts, newcomers to a science do not have to reproduce the phylogeny of it in the way Piaget proposed (e.g., Piaget & Garcia, 1989). As members of culture, however young we are, we take on cultural accomplishments and tend to forget the sensuous work that has led to their emergence; we forget because there is no requirement or personal need to do so. We can therefore observe in the present lessons how the objects of geometry are “taken over with the sort of naiveté of a priori self-evidence that keeps every normal geometrical labor in motion” (Husserl, 1997a, p. 29). A corollary is that earlier concepts are not necessarily recoverable at some later time so that ontogenesis does not repeat or reproduce phylogeny in form or content. My analyses therefore provide a new perspective on the relationship of interest to (mathematics) educators between cultural-historically achieved sciences and their production and reproduction in present-day classrooms.

10 Rethinking Mathematical Conceptions

Concepts and conceptions are of central interest to mathematics educators, and there appears to be hardly an article in the literature that does not contain multiple uses of one or the other term.1 There are also continued efforts to come to grips with the nature of mathematical conceptions, which have been theorized, for example, in the form of a (complementarity) duality of mathematical processes and products; a three-world hypothesis including embodied, proceptual (procedural and conceptual), and formals worlds; and the dialectics of concept and concept defi nition. Consistent with the reigning Kantian/Piagetian underpinnings of mathematics education research, conceptions are supposedly derived from experience by means of abstractive processes. Abstraction, which is understood as occurring at several levels, is a process whereby some property comes to be regarded irrespective of the contextual particulars (see Chapter 2, section on Kant and Piaget). An abstraction always is an abstraction from something. Associated with abstraction is interiorization, whereby experiences come to be abstracted from and transcend their original sensual context to a metaphysical realm. Here, whatever is abstracted can now be freely deployed, operated on, or projected into some other material or situation. Typical for such a transformation is Piaget’s theory of the development from concrete operations to formal operations. Interiorization and abstraction are linked, leading, in most general form, “to the isolation of structure (form), pattern (coordination), and operations (actions) from experiential things and activities” (Steffe & Cobb, 1988, p. 337, emphasis added). Thus, whereas some view children’s actions of enumerating stacked pizza boxes—as Chris enacts this in Chapter 1 or Jane in Chapter 6—in terms of mental models, which are thought to have been abstracted and exist independently of concrete situations, others, attending to the deployment of gestures and other bodily aspects, show that students deploy during this task embodied knowledge, that is, knowledge that exists in the sensorimotor transactions between children and their lifeworlds. Both phenomenological studies and recent neuroscientific research on the role of mirror neurons suggest that such abstraction does not occur.

244 Geometry as Objective Science in Elementary School Classrooms One problem with existing conceptions of mathematical conceptions is the fact that they attribute agency to mathematical learning in situations that appear logically impossible. Consider, for example, the situation we observe in Chapter 8. Here, two teachers ask their second graders to classify geometric objects (including cylinders, cubes, pyramids, rectangular solids, and spheres), which they have designed for the students to develop the mathematical concepts of cylinders, cubes, pyramids, rectangular solids, and spheres. The children are faced with the task of classifying these objects, which requires them to have one or more conceptual schemes; this conceptual scheme, however, presupposes that the children already know how to classify the object. Children are asked to act in a situation where they do not yet have knowledge of the goal state that their actions are supposed to reach. This agential (intentional) approach to mathematical learning is apparent in many recent publications that sketch the situated learning of mathematical concepts. In these and similar situations, students are said to engage in the “construction” of mathematical conceptions prior to knowing what the object is that they are to construct. Agency, however, requires an intention and, as cultural-historical approaches to activity teach us (see Chapter 3), the objects/ goals of actions require a vision of the ultimate outcome. But students cannot have such an image of the mathematical concept because, confronted with the unknown concept, they have no conceptual order with which to think and imagine what it is that they are asked to learn. Moreover, to be able to classify examples and non-examples of a concept, students need to have the concept; and, to have the concept they need to be able to classify examples and nonexamples. Students have as little to intend the mathematical concept as Christopher Columbus has had for intending the discovery of the Americas. If he had the knowledge required for intending to go to the Americas, he would not have been in the position to discover it and he would have called the people he encountered “Americans” rather than “Indians.” That is, mathematical learning conceived as the active construction of concepts fails to theorize the radical passivity underlying learning events when we are confronted with the unknown, foreign, and strange (Waldenfels, 2006). But how do I “construct” a mathematical concept when I do not have the competency to distinguish examples from non-examples; and how do I distinguish examples from nonexamples if I do not already have the concept? In contrast to the going theories concerning mathematical conceptions as abstractions from bodily and embodied experiences, I present in this chapter an alternative theory in which conceptions do not exist other than through their concrete realizations in incarnate experience (i.e., movement). As the following analyses and images from the classroom show, hands, which I thematize throughout this book, again play an important part in the constitution of a conception. As before, the hand serves me as a synecdoche for tact, the sense of touch spread throughout the body, central to the constitution not only what we sense but also the sense of the sensibly sensed. But tact is impossible in rest; feeling an object and feeling oneself feeling both require movement, for otherwise, as I show in Chapter 1, any sense impression

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vanishes within seconds. This proposal therefore radically breaks with Kant and Piaget; and it breaks even embodiment and enactivist theories (see Chapter 2), because conception and singular, incarnate experiences can no longer be thought independently. Understanding a concept therefore means that any one experience—e.g., hearing the concept word “cube” rather than a mere sound that the International Phonetics Association transcribes as “kju:b”—potentially activates any one of the different living/lived experiences that make up the conception as a whole—e.g., cube. Individual experiences, including the sound words, therefore synecdochically stand for the conception as a whole—for example, the sound “kju:b,” which belongs to the conception of cube. “Having a conception” essentially means “being in movement,” because without it, there is no sensing, sense, or sense-making. In the next section, I provide an exemplary episode from the same classification lesson that figured in Chapters 8 and 9, where children are asked to pull three-dimensional geometrical “mystery” objects from a bag and then either add them to existing groups or start new groups. In the process, they come to engage in a variety of experiences with the objects. I provide a first analysis in terms of the phenomenology of perception.

AN EXEMPLARY EPISODE Prior to each child’s turn, Mrs. Winter ascertains that there is an empty colored sheet on which a new group of solids can be started (see Chapter 8). The children are instructed to either place the solid they pulled from the bag with an existing solid or group of solids or to begin a new group. The present episode begins with the selection of Kendra as the next student, which is collaboratively achieved between the teacher and a student (Justice) who actually names Kendra. The teacher then reiterates for Kendra the rule of the task, reaching into the bag and grabbing the fi rst thing she touches (turn 04). Jonathan and Joel, oriented toward and pointing to some group of objects, talk at very low volume until the teacher reminds them that it now is Kendra’s turn to explain her thinking (turns 07, 09). 2 Kendra is digging for a solid. Fragment 10.1 01 02 03 04

05 06 07

W: J: W:

j: W:

um=whose next (0.14) ((turns to his right, points)) kendra kendra:: ((Kendra begins to approach)) (1.08) want you take the [first thing (0.28) that you touch ] [((Kendra reaches into the bag)) ] <its open.> <

yea.> (0.86) now remember it is kendras turn.

246 Geometry as Objective Science in Elementary School Classrooms 08 09 10

j: W:

11 12 13

X: Y: S:

<(??)> ((points to some objects; side talk)) (1.03) jonathan? () to explain her thinking. (3.26) ((Kendra digs for an object in the black garbage bag)) OO::::[ ::: ] ((Fig. 10.1a,b)) [yea] ((Fig. 10.1c)) aoo that. ((Fig. 10.1d))

Kendra fi nally pulls out a solid, which the others greet and acknowledge with a variety of sounds and words: “oo::::,” “yea,” and “aoo that” (turns 11–13). Kendra, who has looked toward her classmates while her hand is inside the black bag, now reorients to look at the object, fi rst from the side (Figure 10.1a,b); then she rotates it to get a look at the circular end (Figure 10.1c,d). In this, the object presents itself under different perspectives (visual and tactile) that differ as the child continues to interact with it—much as the cube is presenting itself differently as Chris rotates it in his hands (see Chapter 1). This rotating of the cubes in the hands may be of a similar kind as the saccades of the eyes that move without our knowing, but that in this movement allow some figure to appear and remain a constant sensuous impression (see Chapter 1). Appealing to different senses, these perspectives also appeal to different forms of consciousness, especially different from linguistic consciousness. Already Kendra has had the kinds of experiences that lie at the basis of the conception of a cylinder, for, as the cube Merleau-Ponty (1945) discusses, its different “sides are not only invisible but inconceivable” (p. 237). To be conceivable, the cylinder would exist for itself, which it cannot do as it is an object. The sides of the object Kendra holds are not projections but precisely sides of a sensuous object. When these are perceived “successively, with the appearance they present in different perspectives, I do not construct the idea of the geometrized projection which accounts for these perspectives” (p. 237). This is so because the object is in front of the person and reveals itself through the perspectives. All the experiences that follow have to be seen in the same light. Just before Mrs. Winter continues to instruct Kendra to look around and to figure out whether the solid she has gets its own spot or whether it is like the solids in another group (turn 14), Mrs. Winter checks whether she has an empty slot (colored paper), articulating what she is doing under her breath at very low speech volume, “how many weren’t out there?” (turn 14). At the same time, Kendra lifts her gaze briefl y toward the other object groupings on the ground (Figure 10.1e), then clearly orients her gaze in such a manner as for everyone to see that she is looking as instructed. Simultaneously, her right hand grabs hold of the object as if in pick-up confi guration (Figure 10.1f,g), whereas the left hand appears to touch the other side, gliding along the surface (Figure 10.1f–h).

Figure 10.1 Kendra gazes at her object and then looks around at the existing category systems.

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248

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Fragment 10.2 14

T:

15

K:

kay; ((Fig. 10.1e)) look around, ((Fig. 10.1f)) does it get its own (1.16) <

how many werent out there?> [((Fig. 10.1g)) (0.49) ] [((places new purple sheet))] ´does it get its own spot? (0.51) or is it like another one. [(5.39) ] [((Fig. 10.1h-j))]

After scanning for more than five seconds, while turning the object in her hand, getting to hold it in different fi nger–hand configurations (Figure 10.1i), Kendra nods while now holding the object at the tips of her two hands at the curved surface (Figure 10.1j). She then nods and demonstratively (i.e., with a loud “clack”) places the object with two other, much larger cylindrical objects on a sheet labeled “tube.” Until now, Kendra has not spoken, though her orientations and actions show that she has been attending to the task. It is not surprising then that the teacher invites Kendra in Fragment 10.3 to articulate her thinking (turn 16). Kendra begins, “it’s because o hm,” while moving back toward the group of cylindrical objects (turn 18). The conjunctive “because,” which indicates that something has been done for a reason, announces that she is in the process of providing a description of the grounds for her placement (classification) that she has made on perceptual grounds. She fi rst leans on the cylinder with the entire weight of her upper body on the object, then, while uttering “it’s shorter,” she moves her object next to the taller of the two previously placed cylinder (turn 18). She grabs the cylinder in a pickup-circular-object configuration and holds it above in such a way that the two circular faces are only centimeters from each other (turn 18, c). Fragment 10.3 16

T:

17 18

K:

and tell us your thinking. (0.33) its because o hm; ((Fig. a)) (1.13) its shorter ((Fig. b)) bu um ((Fig. c)) (1.18)

There is a pause, during which Kendra erects her body while turning her right hand so that the cylinder comes to lie length-wise in her open palm, fingers wrapped around (Figure 10.2a). As she produces expression and thinking, the left hand, which she heretofore has used to lean on, moves up in front of her body to grab hold of the cylinder, her gaze clearly oriented toward it,

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rolling the cylinder while uttering “round” (Figure 10.2b). Kendra continues turning the object in her hands, rolling it between her open palms (Figure 10.2c,d), orients her gaze toward the existing group of objects, and then points to the two cylinders on the floor by touching their top surfaces (Figure 10.2e). Fragment 10.4 19

((Fig. 10.2a)) theyr sort of ((Fig. 10.2b)) (0.43) round; ((Fig. 10.2c)) (0.49) ((Fig. 10.2d)) like this ((Fig. 10.2e)) one and this one

Figure 10.2 Kendra wraps her fingers around the cylinder.

In Fragment 10.5, Kendra engages further with the existing objects by picking up the larger one with her left hand in a pick-up-circular-object configuration, and, while repeatedly uttering in part and whole the causative conjunctive “because,” brings the tall cylinder next to her own, holding them side-by-side (Figure 10.3b). She then turns the large cylinder so that its circular end becomes exposed to her gaze (Figure 10.3c) only to return it to its place on the “tube”-labeled sheet on the floor. A long pause follows. Fragment 10.5 20

((Fig. 10.3a)) becau Fig. 10.3b)) (0.95) because um ((Fig. 10.3c)) (4.07)

Figure 10.3 “Comparing” two cylinders.

250 Geometry as Objective Science in Elementary School Classrooms Fragment 10.6 begins with Kendra roll her cylinder in her two hands (Figure 10.4a). She continues to manipulate and hold in different ways the cylinder while haltingly articulating what it felt when she fi rst grabbed it (“when I got it”): It “felt like smooth” (turn 21). Before uttering the word smooth, her hands had been wrapped around the object, like a person cuddling a cup of coffee to warm up the hands that hold it. The gesture suggests both roundness and smoothness. Kendra is fully oriented toward this object rather than to the others, both in her physical orientation and in the topic of her talk. Fragment 10.6 21

((Fig. 10.4a)) when i ((Fig. 10.4b)) was- (0.99) when i got it ((Fig. 10.4c)) felt (0.52) like (Fig. 10.4d)) (0.96) smooth (0.52)

Figure 10.4 The cylinder—which in Greek is kylindros, the roller—affords “rolling” actions of Kendra’s hands.

But as she utters “smooth,” Kendra turns fi rst her head, then her entire upper body toward the existing group. Leaning onto her left hand, she bends forward to hold her cylinder next to the smaller one on the floor (Figure 10.5a), then withdraws the object again, turning it in her hand so that she comes to look at one of the circular sides (Figure 10.5b). All the while, she utters “and um” and pauses. Fragment 10.7 22

and=um ((Fig. 10.5a)) (2.33) ((Fig. 10.5b))

Figure 10.5 More comparison actions.

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At the end of the previous fragment, Kendra has begun to reorient toward the object itself, which she now in Fragment 10.8 describes as having “round sides,” which contrast “but” and ends with “fl at top.” In the course of the utterance and especially during the longer pause, Kendra changes the way in which she holds and manipulates the object, seizing it between but a few fi ngers (Figure 10.6a–c), then having it lay in her fl at palm (Figure 10.6d,e) fi ngers curling around. While uttering “round sides,” Kendra’s two hands bend around the object, rolling it in her palms before turning it so that one of its flat ends comes to rest in her right palm, the other hands curling around the sides (Figure 10.6g). The fragment ends with the articulation of the end as “fl at top,” at which time she clearly is oriented toward the now exposed top, the other circular end fi rmly resting in her right palm (Figure 10.6h). Fragment 10.8 23

24

((Fig. 10.6a)) it had [ (1.76) ] round [((Fig. 10.6b,c,d,e))] ((Fig. 10.6f)) sides ((Fig. 10.6g (0.53) but flat ((Fig. 10.6h)) top (0.59)

Figure 10.6 The cylinder affords many different ways of holding it.

There is a pause (turn 24), which provides the teacher with the space to enter into an exchange that is captured in the fi nal Fragment 10.9. Kendra places her object on the brown paper and orients away from the assembly of objects on the floor, all the while pushing the object closer to the other cylinders. But as the teacher’s question unfolds, Kendra picks up the object again clearly gazing at its circular end (turn 25, a) and, as she answers the teacher’s question about the fit with the existing group, she hits the circular top with her right hand while her eyes are directed toward the teacher (turn 27, b).

252 Geometry as Objective Science in Elementary School Classrooms Fragment 10.9 25

T:

26 27

K:

okay, (0.37) we called that group the ^tube. do you think that fits in with (0.26) the ((Fig. a)) ^tube? (0.44) ^yep. ((Fig. b))

A PHENOMENOLOGICAL READING The entire episode has lasted but a little more than one minute. And yet, Kendra, the student who is to learn about three-dimensional geometrical objects, has not only seen an object but also more importantly manipulated and therefore felt it in a variety of ways. More so, after having been asked to explicate her classification, she also produces a description in a very hesitating way—much as we see Chris produce his description of a cube (Chapter 1). In words, gestures, and orientation (toward classmates and teacher), Kendra articulates and expresses thinking in movement for others while drawing on the same gestures and manipulation that she has enacted during the initial part of the episode without simultaneously talking. This thinking is produced in, by, and through her living/lived body, which simultaneously serves as an expressive means that make her thinking and classifying available to others. Her thinking exists in and as of the movement, inseparable from it, as movement is the very matter of thinking. We have no evidence of any mental concept behind what she does, because the conception is supposed to be the outcome of this lesson. The capacities of this living/lived body, however, derive from the self-affection of the flesh as outlined in Chapter 3. It is only when a “selfreflexive sensory modality exists” that the physical body can play any role in and be the foreground of experience (Sheets-Johnstone, 2009, p. 83). For example, the memory of a cylinder underlying its recognition—our eyes that move in such a way that we see a cylinder despite the partial way in which it exhibits itself and our fi ngers that curl around the object even without thinking “tube” or “cylinder”—arises from and exists in the form of the self-affected flesh. Our bodies know what to do so that we interact with some object as a cylinder rather than as a cube, and this knowing is totally immanent and does not require mediation by an intellectual consciousness. If recognition required intellectual consciousness, then Kendra could just spill its tokens, that is, the words that constitute

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intellectual consciousness. She does not, but she moves. All her thinking is made available in and as movement. In classical theories of knowing it is out of and from such living/lived experiences that human beings are said to abstract conceptions such as that of a cylinder, where the conception acts as genus that categorizes the different experiences (applications) that act as species. That is, children—once they can be said to have learned a concept—have abstracted something that is common to all the cylinders on the paper sheet. All these share common properties, which distinguish them from other conceptions based on different properties. Among the many ways in which Kendra experiences her object is that of hearing the sound that linguists using the IPA conventions transcribe as “tju:b” and that everyone can hear as an instance of the sound word (concept) “tube.” This word is taken to be naming and therefore taking the place of the concept. It is not surprising, therefore, when some educators come to suggest that a student may “know” a concept (i.e., remember, recall, produce the concept word) or have a conception but not know how to apply it. Clearly, such a conception of mathematical conceptions separates different forms of experience, taking the use of a concept word as the conception itself, which therefore can be separated (after all, it is supposed to be an abstraction) and think it separately from other instances that go with a particular object. From a phenomenological perspective on knowing, however, conceptions and experiences cannot be theorized separately: Sounds and the words they realize are but an integral and inseparable aspect of the experienced world. We do not require the mediation of intellectual consciousness and its tokens, words, to recognize a cylinder as a cylinder; our moving eyes make the object appear as a particular one that also affords the curling of fi ngers around it. From the perspective developed throughout this book, Kendra expresses/ experiences/ thinks her object by multiple means associated with the different ways in which she grabs hold of it (fingers around the circular perimeter, around the curved surface) or acts upon the cylinder (rolling it in her two hands, holding it against other cylinders facing ends or with axes parallel). It is in and through her movements that the object comes to be what it is. She orients both toward the material ob-ject by itself and toward the ob-jects in the reference group; she, the sub-ject, also orients herself—body, gaze— toward the teacher. In this, the living/lived expression is not only for herself but also for the others. Kendra manipulates the cylindrical object and thereby lives in different ways, which provides the ground for her subsequent intellectual understanding of the cylinder concept. She knows her object as that what it is when she knows what happens to her sense impressions when she moves (eyes, bodies, head) with respect to this object (Merleau-Ponty, 1945). This is so because the “experience of my own movement conditions the position of an object . . . that I am able to interpret perceptual appearance and construct the [object] as it truly is” (p. 236). That is, Kendra knows her object as a cylinder because she

254 Geometry as Objective Science in Elementary School Classrooms knows what will happen to her perception as soon as she moves eyes, head, hand, or body with respect to the object. But this knowing is not in terms of intellectual consciousness; it is an immanent knowing, the knowing of a selfaffected flesh. Similarly, she touches an object and knows it to be a cylinder when she knows what will happen as soon as she moves the palms of her two hands that hold it: It begins to roll with respect to her two palm surfaces. As a consequence, to conceive a geometrical object requires her to take up a position in space that sets the object in perspective. Despite the brevity of this episode, Kendra comes to enact a range of experiences all of which constitute ways in which a cylinder can be experienced by means of her senses. Knowing the concept of a cylinder will mean that she can activate the incarnate traces of these different experiences when needed—such as a subsequent task that asks her to build a model of a geometrical object that she can only touch but not see (as described in Chapter 5). Whereas Merleau-Ponty was concerned with perception alone, subsequent philosophers extended his reasoning to other senses as well. The important role of touch, smell, sight, hearing, and taste in conceptual understanding should not come as a surprise given that the senses are the very ground from which sense emerges. The transcendental of sense (or what is ontological in it) is touch, smell, sight, hearing, and taste. Sensibility, sensuousness, and sense are irremediably enfolded into one when it comes to true master—symbolic master being related to knowing words without knowing to enact the movements to which they pertain (e.g., sports journalists, who talk about athletes and athletics that they have not the slightest ability for, have nothing but a symbolic mastery of the sport). The episode exposes the embedded nature of so-called abstract concepts in concrete experience: At the precise moment that Kendra is asked to explain her classification, she draws on different forms of singular experiences of movement (holding, rolling, feeling) of the cylinder. That is, her articulation of the abstraction (categorization) really is the bringing together of the general and a series of particular movements (perspectives). The general here is inseparable from and dialectically tied to the particular because, in fact, the general always realizes itself in the particular (singular). Kendra’s conception is not different from the (concrete) individual experiences, floating somehow over and above them, but is passage of the concrete into the abstract and the abstract into the concrete.

A REFORMULATION OF MATHEMATICAL CONCEPTIONS Sensibility and ideality are one through the other, one for the other, and one in the other. (Nancy, 2002, p. 46) Received theories of conceptual knowledge are contradictory in that they allow for mathematical concepts to be empirical and transcendental.

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Sometimes authors write about “conceptual framework” and “children’s concepts.” The conceptual is then contrasted with, for example, the figural (perceptual, sensorimotor). The disjunction at the heart of the idea of conceptions theories is evident also when authors refer us to application errors, which are said to occur when students know some concept but cannot apply it in concrete situations. In contrast to such theories, philosophers recently have suggested (as in the quote that opens this section) that sense impressions and sense cannot be thought independently: The body of sense is the sense of the body (Nancy, 2006). Knowing is in movement; and knowing a language no longer is distinguishable from knowing one’s way around the world (again: movement). Here I offer a way of thinking mathematical conceptions that is not caught in the (logical) contradiction of earlier approaches. If all the expressions examined above are taken into account, then mathematical conceptions are multiplicities that cannot be reduced to any one of their singular instances (forms of experience, capacities), whether these be in verbal (hearing, uttering “tju:b”), gestural, or perceptual form. In fact, I suggest that a conception is a unitary capacity constituted by the various capacities for movement as a singular plural. Thus, all of the different gestures, manipulations, and utterances Kendra produces are integral and constitutive part of her conception. And it is precisely when the different instances are linked, so that any single one can evoke in her some or all of the others, that she instantiates the conception. There is no longer a difference between knowing a concept and applying a concept, for being able to utter a concept word does not constitute the conception. Each singular past experience, remembered by the flesh that auto-affects itself during an initial movement, is a competency that activates the others so that transitions between them are part of incarnate knowing of the concept just in the way phenomenological philosophers and neuroscientists have described the organization of conceptual knowledge.3 The relationship between individual capacities and the conception of which it is a part can be thought of in terms of irreducible webs of relations. Such webs can be thought of in terms of networks in which the intensity of a node is mediated by—directly or indirectly—the intensities of all other nodes. All capacities come to be associated with and defi ne a particular set of capacities (nodes), where these and other possible forms of capacities defi ne the set (Figure 10.7a). This set is open—has no boundaries—so that any new capacity can become part of it. One possible capacity is constituted by recurrent sounds (heard as “tube” or “cylinder”), that is, words and sentences that denote or describe the object at hand. In this model children have formed a conception when two or more different capacities come to be related in the form of a network (Figure 10.7b). Knowing a concept then means that any one of these nodes (experiences) can activate any or all other possible capacities (Figure 10.7c), that any movement may entail all other movements that they are associated with. Thus, when a tube or a cube

256 Geometry as Objective Science in Elementary School Classrooms

Figure 10.7

A conception of conceptions.

appears under a particular visual projection, other possible projections are activated as well—e.g., one of the ways in which the cylinder feels when held in the hands—so that the person immanently knows what will happen to perception if the projection is changed. Seeing a particular projection also activates the node associated with the sound tju:b or kju:b. Thus, one may know a sound (e.g., tju:b) but if it does not yet (as children taught color words to name a set of objects but who do not group same-colored objects) or no longer (as in amnesia) activate other movement capacities, it is not part of the conception. In other words, the person is able to “apply” the concept and brings it to bear even when the activation has been by means other than the sound word (Figure 10.7c). In the way formulated here, the approach differs from all others discussed in this book, because a conception is defined in terms of the actual capacities arising from the auto-affection of the flesh rather than of one or more properties that they have in common. The conception exists only in and of the set of capacities and their relations to one another—the entire set (singular entity) constitutes a plurality of singularities. Generalization here means that any one (singular) movement (possible form of experience) has come to synecdochically stand for the set (plurality) of movements (possible forms of experience) that have arisen from auto-affections of the flesh and contributes to the activation of any other singular capacity as all other capacities contribute to it. In the context of a concept, therefore, the belongingness of any singular capacity to the set is defi ned by its context, which consists in all the other capacities. As such, a conception no longer is metaphysical, beyond the senses and the flesh: As result of a generalization it is as concrete as all the specific competencies of the flesh that constitute it. The whole (conception) only exists in and through its parts (individual capacities), and the individual capacities as these exist in and through the whole. The sound tju:b is but one aspect of the experiential manifold (Figure 10.7c) the function of which it is to activate the (possibility for) manifold as a whole. The sound functions as a synecdoche, that is, a part of a manifold that comes to stand for the manifold. Thus, the sounds tju:b or 'silindǝ are specific auditory (sensory) capacities associated with a network of capacities that one can have with a material cylindrical object: These sounds are part of, and come to stand for, the totality of these capacities that have arisen

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from auto-affections of the flesh. This, however, cannot be the whole story, as material signs generally and sound-words specifically have a dimension that is not bound to the singularity that exists in the auto-affected flesh. The other part of the story derives from the development described in Chapter 6, whereby some movements seem to be “detached” from the particulars of situation and take on symbolic functions. Now, to produce the symbolic gesture or sound word, the same movements are required. But the material sign also refers not only to this situation but also to all situations of this type and it does so for all competent individuals. That is, when the movement no longer is solely doing something in the world or attempting to find out about it, then a dehiscence occurs between the being of the movement to the movement as standing for something else. It is precisely this replacement of something present by a re-presentation that we go from simple movements to symbolic movements. It is the form of these movements that are recognized by others as generalizations of their own movement forms. Specifically, therefore, the sounds that go with particular experiences are not mere sounds—e.g., in the way a bird might produce it as a predator approaches, leading the entire flock to depart. These sounds are tied to situations with predators. But in human beings, as developmental psychologists have pointed out long ago, a sound, such as “'teibl,” which initially is identical to the presence of a table, comes to be detached and function independently as the name of a particular kind of object (e.g., Piaget, 1997). My point here is that even when the names of objects are detached and become categories, they still activate all the nodes that are associated with the sound. In this way, knowing the concept (name) always means knowing when to use (speak, hear) it, under which conditions it applies, and so forth.

CONSEQUENCES A fi rst consequence of this way of understanding conceptions lies in the inseparability of a conception and its application. Recognizing the sound tj:ub does not constitute a conception; nor does placing a word in an empty slot designed for eliciting a verbal or written response. Rather, a conception always pertains to the activation of the capacities of the flesh, and therefore, reflexively, is always an incarnate conception. The tighter the links between the different nodes in Figure 10.7, the more readily one particular incarnate capacity activates all other experience-based nodes. What evolve are not the capacities arising from self-affection but their organization. As the conception forms, individual capacities come to be moments, which no longer can be thought as independent entities because they depend on all the other capacities linked to them. One might therefore expect that the cross-situational use of conceptions is associated with the strength of these links as well. The mentioned neuroscientific research on the role of mirror

258 Geometry as Objective Science in Elementary School Classrooms neurons in spatial and social cognition predicts that especially observing others touch and manipulate objects provides opportunities for the learning from others. A second consequence is the mutually constitutive nature of individual experiential nodes and a network of movement possibilities as a whole (i.e., conception). It is a dialectical relation characterized by a situation where the whole is present in each of its part: “The two sides of the relation entirely interiorize the other in a way that from now on the two are identical in their difference” (Sève, 2005, pp. 131–132). As experiments with computer models of such networks show, strong (experience-based) inputs will draw the entire network to a state corresponding to one conception (category) rather than to other possible conceptions. The state of each node influences the states of all other nodes; in turn, the state of each node is the result of the influences of all other nodal states: It is an organic whole where the whole forms the parts, while at the same time it is formed by these through an embryological or historical process. That is, when the connections between different incarnate movements is high, any single movement may have a strong influence on all other movements consistent with the same conceptual object and therefore to activate the conception (network) as a whole. Stronger links between nodes (movements) that belong to the same entity mean that the conception is realized more easily and rapidly than when the links are more tenuous. Thus, for example, one might expect that if Kendra had been a (classical) Greek child she might have more easily learned the mathematical concept corresponding to the sound ‘silindǝ (i.e., cylinder) because in her language the term would have meant “roller,” which corresponds to Kendra’s experience of rolling her object between the two hands. Similarly, as a Greek child, she might have more easily learned the concept of a cube, which, in her native language, would have denoted a die to play with. A third consequence of these considerations is that there are no simple concepts. Every concept has components and is defi ned by them. The conception Kendra forms through her movements that lead to incarnate capacities for movements therefore is a combination. It is a multiplicity, although not every multiplicity she forms is conceptual. “What is proper to the concept is that it renders components inseparable within itself: distinct, heterogeneous, and yet not separable, this is the status of the components or what defines the consistency of the concept, its endo-consistency” (Deleuze & Guattari, 1991/2005, p. 25). A conception such as Kendra’s therefore can be understood as a point of coincidence, condensation, or accumulation of its components. Each component is an intensive feature, which must be understood not as general or particular but as a pure and simple singularity—“a” possible world, “a” face, “some” words—that is particularized or generalized depending upon whether it is given variable values or a constant function. A mathematical concept thus is not found in its genus or species but in the composition of its various material and

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experiential appearances. A conception arises from a heterogenesis—that is to say, an ordering of its ever-new movement nodes by zones of neighborhood. “The concept is in a state of a god’s eye in relation to its components. It is immediately co-present, without distance, to its composites or variations, endlessly traversing them” (p. 26). But despite its constitution in and through movements and the traces movements leave in our bodies (because of auto-affection), the conception is neither denotation of states of affairs nor signification of the lived; it is the event as pure sense that immediately runs through the components. A concept therefore is “an incorporeal, even though it incarnates or effectuates itself in bodies” (p. 26). A fourth consequence is that conceptions evolve from primary experience as these give rise to auto-affection and movement capacities that are connected and mutually mediate and stabilize each other. The question with which I begin Chapter 8—classification of objects when a conception of the object does not yet exist—no longer poses itself because in the way articulated here, conceptions emerge in and through sensuous movements (i.e., labor), never consisting in anything else but activated prior incarnate capacities for movement. The problem does not pose itself because conceptions exist only in and through the immemorial memory that comes from auto-affection of the flesh. The term “sense” both designates the organs of immediate apprehension of some thing and the universal (concept, conception) underlying the thing. Therefore, “the two senses of the word must then have, in their distinction and in the opposition that this distinction presents, the same sense” (Nancy, 2002, p. 46). As a consequence, “the sense of the word sense is thus in the passage of each one of the two significations into the other” (p. 46). Because our understanding of a thing as a kind of thing exists in our (tacit) understanding of how the sense impressions change if we were to change our (spatial) relation to it and in the changing relation of different sense impressions, conceptions can be understood as networks that emerge from lived (rather than intellectual) reorganizations of incarnate movements. In the course of the lessons that I participated in and observed, the secondgrade students developed conceptual understandings of geometrical objects through their extended, incarnate labor involving them. In this chapter I provide a conceptualization of conceptions that does not separate these from the living/lived incarnate labor but makes conceptions a dialectical complement of all incarnate capacities for movement that—arising from the auto-affection of the flesh—are used synecdochically to denote conceptions and associated concrete experiences.

Epilogue From the Flesh to Society in the Mind

Embodiment has become one of the fashionable terms in the educational literature. A problem that this literature presents is that it lacks the provision of mechanisms that make plausible the fact that the living/lived body—or, rather, the flesh—is always involved in “cognition,” not merely a stepping-stone to abstract intellectualism. As Sheets-Johnstone (2009) suggests while citing the likes of Lakoff, Johnson, and Varela, the embodiment and enactivist literatures continue to evoke “the possibility of a disembodied relationship” and, with it, “the spectre of Cartesianism” (p. 215). In this book, based on close, phenomenological analyses of mathematics lessons in a second-grade classroom, I not only describe the incarnate, sensuous labor of doing geometry but also articulate an epistemology that differs from those that currently dominate the discourse in mathematics education, mathematical cognition, and philosophy of mathematics. Throughout this text, I emphasize a distinction that is never made in the mathematics education literature: that between material bodies, including the material human body and the living/lived body of our experience, a distinction that phenomenological philosophers make between the body (Ger. Körper, Fr. corps) and something sometimes translated as the fl esh (Ger. Leib, Fr. chair). The specific contribution to cognition is made by the flesh (Leib, chair) rather than by the (material) body, which is that aspect of the flesh that it shares with and makes it indistinguishable from other bodies. But knowing is due to the flesh, not due to the body: There are many bodies that populate this world that do not have the special feature that we refer to as knowledge. The hand, as an integral and constitutive part of the living body, serves me throughout this book as a synecdoche for tact, which is the foundational sense that founds other senses and the sense we make of the sensibly sensed. Throughout this book, we see the hand at work: changing, sensing, and symbolizing the world filled with cultural-historically charged objects. That is, the hand, in its movement, underlies sensibility and sensuousness and, therefore, sense. The work of the hand is sensuous, and without recognition of this sensuous nature, we fail to appreciate knowing as foundationally the consequence of and located in the living/lived body. The hand

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is the synecdoche for touch, which, “as a self-reflexive modality appears to be a uniquely human character” (Sheets-Johnstone, 2009, p. 83). The hand is of special importance because it is not only at the origin of the sensibly sensed that allows us to make sense, but also it is the organ through which the word becomes flesh, such as when, for example, Cheyenne moves her T-shirt following the Mrs. Winter’s admonition or when Melissa begins to fashion a cube from the plasticine ball in her hand after having announced “I’m making it a cube.”

SENSUOUS LABOR AND INCARNATION In this book, I focus on the experience, the relation of the objects and the nonverbal phenomena of touch and gesture. The question remains, however, how something like a cube comes to be integrally related to the verbal articulation, verbal thought, and the word. Particularly in Chapter 10 I show how what initially are movements and material phenomena among other movements and material phenomena become symbols. The becoming of signs occurs in an instance of dehiscence that leads from pure presence—the unmediated presence of the present in presentation—to the presence of the non-present—the re-presentation. That is, pure present does not require representation, in other words, movement does not require intellectual consciousness. Only that which is absent, non-present, requires representation to be present in the present. But movement made present again in the form of re-presentation is dead—no wonder that philosophers such as G.W.F. Hegel, and Derrida (1972) discussing him, refer to the verbal sign as a tomb (for the movement I might add), “a tomb that one cannot even make resound” (p. 4). In Chapter 1, too, we see that Chris’s hands are moving expressively without and before speech. These expressive movements constitute a presence of the present—without representation. It is only when the movements stand for other movements, movements that have occurred in the past, that they take on representational function. The sequence from ergotic movements to epistemic and symbolic movements of the hand that precede language was already a result of my research in science education. Phenomenologically, therefore, intuition “owes its phenomenological power . . . to intentionality” (Henry, 2004, p. 328). It is precisely when “appearing is accomplished as the making-see of intentionality [that] the subordination of language to phenomenality . . . opens the way to the intentional conception of language” (p. 328). A way to the answer of the question about the relation between movements of the flesh and language also has been sketched in the phenomenological literature. In his discussion of apophansis, Heidegger (1977b) points out that language (Gr. logos) discloses—makes manifest, lets be seen (Gr. phainesthai)—what shows itself from itself: speech (logos as apophansis) makes the visible accessible. Neither speech nor the sensually accessible are independent of the movements of the living/lived body. For Marx and

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Engels (1969), material activity and material intercourse with others constitute the real language of life; material activity and material intercourse have spoken language and ideas as their product. Writing and speaking, therefore, are two incarnated modes of language that are closely related and in fact interlaced with material life generally and the hand specifically, which constitutes the essential connection between humans and their world, and therefore also embodies intentionality. Thus, the gestures of the hand could never transpierce language unless the word and the hand already had something in common (Franck, 1986). “There is no voice except a carnal voice, and our whole body is what it presupposes to be what it is, and what, in return, the voice makes into its mouthpiece and therefore the highest manifestation of mind” (Chrétien, 1992, p. 101). They have in common that they are expressions of capacities anchored by an “I can” held together that precedes the coming of the ego, intentionality, and self. In this book, I offer a (material, sociological) phenomenological epistemology of knowing and learning. This approach has the advantage that it highlights knowing as it is actually expressed and therefore publicly exhibited by students and teachers alike. This approach has at least two advantages over other approaches. First, the phenomenological approach orients us to what students and teachers really attend to in their sensible/sensual labor. Second, what they attend to are the expressions of others, which we, researchers, fi nd exhibited in the same way that students and teachers observe it as part of their everyday labor of schooling. Third, because students and teachers have to make their thinking available to one another, thoughts are not something that are theorized to be subjective and properties of the individual but something that only comes into being in and through public expression, which is inherently movement because stasis does not allow figures to stand against ground. Fourth, because thought comes to be only in material expression (for the person communication as much as for the intended audience), we are enabled to study how students are transformed in expressing themselves and in attending to the expressions of others (which leave their impressions in them). Mathematics educators often pay lip service to grounding mathematics teaching and learning in what and how students actually know and experience. Despite all the rhetoric about the importance of students’ experiences in the learning of mathematics, most mathematics educators do not provide evidence that they are actually concerned with understanding the world through the eyes of the learner. But if we want to understand learning, especially, what students can rationally intend given that they do not know the object of their learning, we need a radical phenomenology, one that begins by investigating how the world looks like to students and teachers, and what kinds of world they currently inhabit. From the gestures that the children in this study deploy, we cannot infer that they have some “mathematical understanding” in the traditional sense. The children do make explicit aspects of the world salient in and relevant to their sensuous

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sensible labor. For example, Chris moves his hand along the two sides of the pizza box, the two sides that we, teachers, see not being square; and we hear Chris utter the word “square,” which follows the teacher’s question, “What does this box have to have to be a cube?” The position taken here differs from that in the enactivist literature. Varela et al. (1991) propose to look for knowledge at the “interface between body, society, and culture” (p. 179). In the position articulated here there is no interface: Mind is in society and culture as much as society and culture are in the mind. Similar positions can be found in activity theory from Vygotsky to Leontjew and to the present day. Maturana and Varela (1980) take societies merely to be “systems of coupled human beings” (p. 118), whereas the position here is the converse: The specifically human being is a result of society rather than preceding coupling. In terms of phenomenological sociology (also in activity theoretic terms), there is mind because there is society. For Maturana and Varela, “the linguistic domain and the domain of autopoiesis are different domains” that “do not intersect,” “although one generates the elements of the other” (p. 120). In phenomenological sociology, however, the two are parts of the same phenomenon: Habitus and the field, language and the person are but different, one-sided aspects of the sensuous, incarnate labor that we may observe. Mind is possible precisely at the borderline, between inside and outside, between the sensibly sensed and sense. In Chapter 3, I articulate how in material phenomenology, auto-affection and affectivity are theorized as underlying the existence of community. In Chapter 7 particularly, I empirically show how periodic phenomena—intonation, rhythm, and complementarity of movements in space—are present throughout this second-grade mathematics classroom. In opposition to the social constructivist theories to mathematics education, therefore, I conceive of community as the condition for observing the emergence of mathematics rather than the results of negotiations that individual Selves engage to produce rules and “sociomathematical norms.” Material intercourse of humans, as Marx and Engels say, is the source of ideas, representations, and consciousness. The intersubjective community exists because of the auto-affection of life and affect that singularize each individual rather than through a shared “interpretation” of a putatively common (material) world. Verbal consciousness and everything intelligible to others and oneself exists on the basis of a primary intelligibility that exists in and as of pathos of material life, affectivity. The phenomenological matter in which this intelligibility appears therefore is precisely the flesh, in which life auto-affects itself.

IMPLICATIONS FOR STUDYING MATHEMATICAL KNOWING The aim of this book is, in part, to respecify the real, sensuous work of “doing geometry” generally and, in these lesson, “doing [classification

264

Geometry as Objective Science in Elementary School Classrooms

according to geometrical principles]” more specifically. I began the research in this second-grade classroom with the assumption that whatever children did could be counted correct in some classification (and therefore language) “game” and the associated rationality instead of with the assumption that the children were in error, to varying amounts. The real question that has emerged for me in the course of my inquiry then is this: “What is or are the games and the rationality that we are dealing with? That is, I take the game and children’s pre-geometrical rationality each as a phenomenon sui generis, not as some faulty and diminutive forms of adult games and rationality. The sense of the game of geometry generally and of the classification of three-dimensional objects specifically could not be a predicate of children’s learning intentions. Children therefore are not in a position to construct the categories, because intentional objects such as cube, prism, and so on, as those “of ‘triangle’ . . . can be fulfi lled only by an indefinite intuition, for which each conscious lived experience offers an expected but insufficient moment” (Marion, 2002, p. 126). Nevertheless, both historically (at the beginning of a science or during one of its “revolutions”) and ontogenetically, a discipline as it exists on a given day has evolved from conditions and rationalities that essentially involve material and thinking practices that have evolved through our participation in the intuitive everyday worlds of our experience, which are often inconsistent with and even opposed to their (“scientific,” “mathematical”) successors. I end this book with three brief but far-reaching comments that have implications for theory and method in the learning sciences more broadly. My fi rst comment concerns the ways in which researchers construct their descriptions and explanations. I did not begin my research program with the assumption that I participated in and observed on the videotapes are geometry lesson a priori. If the events in this classroom can be categorized as constituting a geometry lessons, then I take this to be an achievement that emerges from the ongoing transactions, in which I also played my (reflexive) part. This allows me to highlight and understand the order that makes these lessons recognizably geometry lessons rather than something else—a science lesson, for example, given that the children used science words when the teacher solicited what they understood when hearing the term geometry. These have been lesson for which Mrs. Winter and Mrs. Turner have had the intent to occasion geometry and geometrical practices. They have announced it as such. Students participate without knowing (being able to know) what this implies, because they have no precedent to which they could compare what is happening or that they could use as resource for their actions. Chris, Thomas, Bavneet, Connor, Melissa, and all the others who act in ways inconsistent with Mrs. Winter’s or Mrs. Turner’s conception of geometry fi nd out precisely after their actions that these have been inappropriate. The motive of activity (Tätigkeit, deyatel’nost’), which emerges in and as the field of geometrical objects, still is hidden from them and, in fact, is one of the outcomes of these lessons. Thus, like the

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emergence of intention from the fi rst carnal movements that auto-affect the flesh, the public and accountable intention of playing games—including the one of sorting mystery objects, building models of mystery objects, or pronouncing oneself about edges and faces—are outcomes of playing the games for a fi rst time. As the sensible sensuous labor unfolds, more and more children sort and classify the objets trouvé in their classroom in ways consistent with formal geometry. It is that instant when this consistency becomes apparent to them that geometry as objective science is discovered again, has a new origin. This discoverability lies at the origin of the objectivity of geometry rather than the other way around. Formal geometry therefore is an emergent property of these lessons, which recognizably become lessons in geometry even though the children do not yet know the discipline. For children the lesson becomes a resource for finding geometry in their own actions. In this way, the lessons I analyze throughout this book become object lessons in geometry in two ways. First, in the older, chiefly historical sense, these are object lessons because the students examine material objects as the basis of their instruction in basic geometrical concepts. Second, these are object lessons in the sense of a striking practical example of how particular lessons emergence from the ongoing sensual and therefore sense-making behavior of teachers and students. My second comment concerns the relationship between scientific culture conceived broadly and its production and reproduction in school classrooms. It has been noted that we truly understand a science only when we grasp its inner history, which is an epistemological problem, of the continual production and reproduction of the invariances that characterize spatiotemporal objects (Husserl, 1939). The sciences are transcendental because they are continuously produced and reproduced in intersubjectively intelligible—that is, public and accountable—ways in and by the sensuous labor of concrete individuals. The nature of a transcendental science does not lie in the fact that it can be handed down, as external historicist conceptions would put it, but in the fact that its history is contingent upon the reproduction and transformation in the sensible sensuous labor of real concrete individuals in fl esh and blood. The nature of a science, therefore, is an epistemological problem. My work illustrates the case of one science, geometry, and how its transcendental nature is produced in the ongoing, incarnate, and mundane work of an elementary classroom. My analyses highlight three quite distinct yet intersecting developments at any instance of a situation: (a) the second-by-second unfolding of the ongoing tasks (microgenesis), (b) the dialectical individual/collective developments in the community at hand (ontogenesis, sociogenesis), and (c) the historical developments of a field within a particular classroom. These dimensions are mutually constitutive and therefore cannot be reduced one to the other. This is a main reason why all theories that focus on individuals and their “constructions”—that is, all the ones I present in Chapter 2 including the social constructivist one, which has individuals construct their knowledge

266 Geometry as Objective Science in Elementary School Classrooms after they participated in the social construction—are inappropriate for theorizing the emergence of a micro-culture of geometry that reproduces the historically evolved, transcendental nature of the domain. My fi nal comment concerns the method of doing and writing research that aims at providing an ethnographically adequate description of concerted activities and their context. As other analyses, mine show that “classroom talk is a highly complex matter not easily summarized, quantified, listed in terms of ‘rules’ or separated into ‘roles’” (McHoul & Watson, 1984, p. 299). This complexity derives from the fact that even a simple utterance (“e.g., “Go! Outside the crown.”) is polyvalent with respect to sense and function, as do the different kinds of words used—including making a statement about the lack of understanding and being an instruction for how to remediate the problem. Furthermore, my analyses throughout this book show that much of what happens in classrooms presupposes an enormous amount of practical understanding exhibited in the mundane and routine practices children exhibit in response to the tasks that their teachers require them to complete. But these mundane and routine practices students bring to school are “glossed over [by researchers] in favor of the ‘remarkable’ precisely because it is routine—thereby ignoring its essential complexity and artfulness” (p. 299). I chose my analytic method with the explicit purpose of bringing this frequently neglected complexity back into educational and learning sciences discourses.

Appendix Transcription Conventions

Following Selting, et al. (1998) I use the transcription conventions exemplified and described below. The only exception occurs in Chapter 5, where I indicate the pause lengths in seconds rather than one hundredths of a second.

Notation

Description

Example

(0.14)

Time without talk, in seconds

more ideas. (1.03) just

()

Pause of less than 0.10 seconds; in Chapter 5, less than 0.5 seconds

kay. () bert

((turns))

Verbs and descriptions in double parentheses are transcriber’s comments

((nods to Connor))

::

Colons indicate lengthening of phoneme, about 1/01 of a second per colon

si::ze

[]

Square brackets in consecutive lines indicate overlap

S: s[ize ] T: [colby]

this one

Underlined part coordinates with a gesture described; lH and rH indicating left and right hand, respectively

this ones:? ((rH moves down, up, down right face, Fig 4.1b))

((:B))

Colon prior to letter in double parentheses: The speaker directly addresses another person “B”

57

<

>

Piano, words are uttered with less than normal speech volume

<

um>

< >

Pianissimo, words are uttered with very low, almost inaudible volume

<this>

< >

Forte, words are uttered with greater than normal speech volume

<that> makes

< >

Fortissimo, much louder than normal speech volume

<hU:::::::ge.>

< >

Allegro, faster than normal speech rate

<[whawould]> that

T:

((:B)) hOW did

(continued)

268

Appendix

Notation

Description

Example

< >

Lento, slower than normal speech rate

<but () its like a flA:Tcube.>

< >

Ethnographic description of speech that is enclosed in brackets

<because its like a sort of (0.60) vertex>

ONE bert

Capital letters indicate louder than normal talk indicated in small letters.

no? okay, next ONE bert.

.h, hh

Period before “h” indicates in-breath; “h” without period is out-breath

.hhī, hh hh

(?cular)

Question mark with whole or part word in parentheses indicate possible hearings of words or missing sound

(serial?), (?cular)

(??)

Question mark(s) in parentheses: Inaudible i (??) word(s), the approximate number given by number of marks

,?;.

Punctuation is used to mark movement of pitch toward end of utterance, flat, slightly and strongly upward, and slightly and strongly downward, respectively

T:

=

Phonemes of different words are not clearly separated

loo::ks=similar

↑↓

Arrow up, down: significant jump in pitch up or down

is ↑sort, ↓<so thats

`´^ˇ

Diacritics indicate movement of pitch within the word that follows— down, up, up-down, and down-up, respectively

`um; ´sai:d; ^Cheyenne; ˇsquare

T:

so can we tell a shape by its color? does it ´belong to another ´group (0.67) O:r.

Notes

NOTES TO THE PREFACE 1. While doing the very last edit of this manuscript, I happened to come across a book by Maxine Sheets-Johnstone (2009). I noticed not only that she is very much in agreement with my own approach, but also that she is one of the few Anglo-Saxon scholars who does not hesitate using the term flesh in very much the same sense that I use it throughout this book. 2. Roth, W.-M., & Thom, J. (2009a). Bodily experience and mathematical conceptions: From classical views to a phenomenological reconceptualization. Educational Studies in Mathematics, 70, 175–189. Reprinted with permission from Springer. 3. Roth, W.-M., & Thom, J. (2009b). The emergence of 3d geometry from children’s (teacher-guided) classification tasks. Journal of the Learning Sciences, 18, 45–99. Taylor & Francis. 4. Roth, W.-M. (2010). Incarnation: Radicalizing the embodiment of mathematics. For the Learning of Mathematics, 30 (2), 2–9.

NOTES TO THE INTRODUCTION 1. Synecdoche is a figure of speech whereby the part of something, here a soundword, is used to denote the whole, here all the experiences and competencies relating to a conception (as per Chapter 10), or a whole is used to denote one of its parts (e.g., the class name “tree” for a specific specimen of a tree). 2. My hunch is that it is precisely because of this phenomenon that there is a gap between plans and situated action (Suchman, 1987), between instructions and the instructed action (Amerine & Bilmes, 1990). 3. The italicized words constitute the Latin and Greek versions of “The word was made (became) flesh,” a passage from the New Testament according to John 1:14 (Biblos, 2011, http://bible.cc/john/1-14.htm). 4. The Kantian schemata are already of conceptual nature: the schema “is the pure image (the schema is a non-sensible image) by which, in general, an image is possible” (Nancy, 2003, p. 149). See also Sheets-Johnstone (2009), who, coming from a phenomenology of movement in dance, critiques the embodiment (Johnson, Lakoff) and enactivist approaches (Varela et al.) whose very notions evoke “the spectre of Cartesianism” (p. 215). The term embodied is a lexical band-aid,” she writes, that barely covers a problem created “by a schizoid metaphysics” (p. 215).

270 Notes 5. Rorty’s (1979) Philosophy and the Mirror of Nature is a direct, tongue-incheek reference to such a conception of the relation between knowing and the world. 6. In cultural-historical activity theory, a form of social psychology that has arisen from Karl Marx’s ideas, consciousness is the consequence of practical activity in the world, the transformation of which is accomplished by the hand. Thus, the working of the hand precedes the working of mind. 7. This point has been developed most notably by Heidegger (e.g. 1977b) and subsequently by hermeneutic philosophers (e.g., Ricœur 1991). 8. A parallelepiped is a geometrical three-dimensional figure consisting of six parallelograms. A special case is the cuboid, where the six faces are rectangles; and the cube is a special cuboid with all faces being square.

NOTES TO CHAPTER 1 1. There are forms of consciousness that differ from the intellectual-linguistic one, including the visual perceptive consciousness (Vygotsky, 1986), rhythm (Abraham, 1995), and groove (Roholt, 2010). These forms of consciousness exist only in the act. They cannot be translated into language and denoted by words. If you do not follow the exercise, you fail to experience the forms of consciousness that go with—and are required to understand—what I say/write. 2. On the philosophical use and distinction of types and tokens see, for example, the entry “Type versus Token” in Quine, 1987, pp. 216–219. 3. Logodaedalus is the subtitle of Nancy’s (2008) Discourse of the Syncope, in which the author analyses Kant’s futile attempt to separate philosophy and literature. Daedalus is the master artisan in and of Greek mythology. He built the castle of Minos including an intricate maze called Labyrinth. He also built wings for his son Icarus and himself to escape from Crete. With the term “Logodaedalus,” Nancy denotes Kant as the master artisan of logos, the mind. 4. Here, I am not going down the discursive road of and surrounding “meaning,” because it is part of the problem where knowing is reduced to words, when in fact part of knowing is not verbal at all (Roth, 2004). Word-meanings are used to denote aspects of knowledge by means of a term that is precisely not of the kind, just as “sensorimotor schemas” reduce to language forms of knowing of a radically different kind. 5. The problem was articulated in a widely cited paper by Harnad (1990). Among designers of embodied robots, which are allowed to learn through interacting with the world, the problem is said to have been solved (e.g., Steels, 2008). 6. Etymologically, subject is the correlate to the ob-ject, the thing thrown before. The word derives from the Latin subjicere, a composite of to throw, cast (jacere) and sub, under. The subject of activity is under the dominion or rule of the object. The two, subject and object are mutually constitutive, precisely as it is theorized in cultural-historical activity theory (Leontjew, 1982). Activity sublates, that is, keeps and overcomes the opposition between the two terms. 7. See, for example, the special issues on gesture, Educational Studies in Mathematics, 70 (2), 2009 (Radford, Edwards, Arzarello, 2009), and on bodily activity and imagination, Educational Studies in Mathematics, 57 (3), 2004 (Nemirovsky & Borba, 2004). 8. Details on the study and curriculum may be obtained from the author.

Notes

271

9. His full name is Marie François Pierre Gauthier Maine de Biran, but he published under the nom de plume Maine de Biran (without any given name). 10. Michel Henry develops material phenomenology throughout his oeuvre, but most notably in Philosophie et phénoménologie du corps (Henry, 1965) and Incarnation: une philosophie de la chair (Henry, 1990). 11. In conversational terms, pauses in speaking that are longer than one second are substantial and longer than the norm. Ample research on wait time in the late 1970s and early 1980s showed that teachers generally do not wait that long prior to intervening. 12. For the transcription conventions used throughout this book, see Appendix. 13. Bourdieu (1997) devotes a long section on the intellectualist, scholastic attitude in the social sciences, which reduces knowing to mental representation. 14. There is an inherent limit to our language, which forces us to speak of Chris’s hand, fi ngers, and body, when in fact hand, fi ngers, and body are Chris— otherwise we would be introducing into our conceptualization the very situation we want to overcome: split of body and mind (Self). 15. I am thinking here of Kant’s a priori schemata and of Chomsky’s innate universal grammar. 16. For example, the psycholinguist McNeill (2002) takes the position that there is one underlying conception to speaking and gesturing, whereas his colleagues Alibali and Goldin-Meadow (1993) suggest that in periods of growth, there may be multiple conceptions underlying communication. 17. Phenomenology is not concerned with what individual people experience and even less how they feel. Phenomenology is the science of phenomena, that is, it investigates the conditions under which we have this versus that experience and how these experiences come to be. 18. The Necker cube is named after the 19th century crystallographer Louis Albert Necker (de Saussure), who fi rst described the phenomenon investigated in this section. Psychologists tend to refer to this phenomenon as an illusion (e.g., Gregory, 1994). 19. According to a theory initially articulated in the 1940s, retinal receptors respond to fluctuations of illumination and little if at all to steady illumination. Thus, to see an object as an object, the eye has to move continually to and away from it. 20. A self-portrait featuring him using this device can be seen by visiting URL www.makart.com/resources/artclass/fi nder.html 21. I did this exercise using my laptop while traveling on a plane. When I returned home, I found that the standard explanation that Gregory (1994) provides is actually disputed today. The problem with standard psychological explanations is that they draw on categories such as “interpretation,” when in fact no interpretation occurs in the sense that this concept has received in philosophy. The “perspective explanation” featured in Wikipedia is but a possible explanation, because the process itself has not been studied by means of phenomenological reduction that allows us to investigate how the different experiences we have actually come about. 22. There exists now an increasing body of work. For an older review see, e.g., Gallese and Goldman, 1999. 23. The emergence and development of mirror neurons has been explained using Hebbian learning principles according to which neurons that fi re together are wired together. Thus, neurons responsible for hand movement are wired together with neurons responsible for observing hand movement (e.g., Keysers & Perrett, 2004).

272

Notes

NOTES TO CHAPTER 2 1. This is the case both for spatial cognition (Rizzolatti, Fadiga, Fogassi, & Gallese, 1997) and for social cognition (e.g., Gallese, 2003). 2. Numerous philosophers interested in the nature of everyday knowing have noted this, notably among them Heidegger (1951) in his lectures on Kant. 3. The English intuition translates Kant’s term Anschauung, the noun version of the verb anschauen, “to look at.” It does not surprise, therefore, that the image and imagination (Ger. Einbildung, literally, to form an image inside) come to constitute the fi rst modern figures of a faculty that produces its object and, in this manner, becomes a purveyor of knowledge (Nancy, 2003). Nancy deconstructs Kant’s approach. Both Anschauung and Einbildung constitute vision as an analogy of knowing—in the same way that “I see” can be used to indicate understanding. Rorty (1979) criticizes the conception of mind as mirror of nature. 4. Historically, Galileo suggested that scientific knowledge requires the elimination of all subjective elements (i.e., sensible qualities, sensations, affect, impressions), and, therefore, the separation of known (nature) and knower. Descartes formalized the objective elements in terms of geometry applicable to res extensa (extended things) that are separate from res cogitans (thinking things). Because the world is a sensible one, Kant is forced to ask about the conditions of any experience, which he has to postulate in terms of an a priori existing space (and time) that had been expelled earlier from knowledge by Galileo and Descartes (e.g., Henry, 2004). 5. We know pain because of the experience of pain, because only the experience of pain allows us to know what pain is (Henry, 2003). 6. As noted, both Anschauung (intuition) and Einbildung (imagination) draw on vision as the metaphor for knowing. 7. One of the criticisms comes from van Hiele (1986), who notes that Piaget “repeatedly interprets from an adult perspective, and that his interpretations . . . are unacceptable in many respects” (p. 94). Van Hiele then spends his entire Chapter 15 to show in which way Piaget has erred. 8. Felix Klein, a German mathematician of the 19th and early 20th century, synthesized non-Euclidean and Euclidean geometries by proposing the study of space that is invariant under any given group of transformations. In Chapter 8 of Didactical Phenomenology of Mathematical Structures, the Dutch mathematician and mathematics educator Freudenthal (1983) critiques Piaget’s work concerning geometrical objects and especially Piaget’s uptake of Klein’s Erlangen program. 9. In contrast, Husserl and Heidegger show that it is because of a dehiscence of time, a dehiscence in which pure presence breaks into presence and representation, the presence of the present, that we come to know conceptually anything at all. 10. Vorstellung and the corresponding verb vorstellen are used to speak about thought and thinking. Literally, however, the word means “to place” (stellen) “before” (vor-), where philosophers have extended Kant to read the vor(“before”) both spatially—as Heidegger in his notion of Gestell (Heidegger, 1977a)—and temporally—as Nancy (2003), who ponders about how for Kant thought precedes experience. 11. Figures, images (Ger. Bild) in the mind are central to Kant’s approach (Nancy, 2003), and intuition and thought are theorized in terms of the verb einbilden, to make for oneself an image of. In his book on Kant’s metaphysics, Heidegger (1951) conjugates Bild with various prepositions: Abbild (copy), Nachbild (imitation, “after-image”), and Vorbild (model, “fore-image”).

Notes 273 12. This position of the subject that has to wonder about the relation between its own thinking and that of others also characterized the work of those who subscribe to “social constructivism” (e.g., Cobb, Yackel, & Wood, 1992). The best the subject can do is to think of its own practices, interpretations, conceptual schemes, or chains of significations as “taken-to-be-shared” with those of other members of the community. This is radically different from cultural-historical activity theory, which takes that the world and “all these things are experienced by the child as the jointly sensed and jointly taught attention of adults to himself” (Mikhailov, 2001, p. 27). 13. There are other possible theories to be discussed (J. Bruner on concept learning), but culturally and historically they do not take the same central role in the fi eld of mathematics education as the two reviewed here. This is likely due to the fact that these theories are generic and not specifi c to mathematics. 14. This model has also been employed to characterize development in other mathematical domains such as algebra or differential equations. 15. Some translators of Husserl’s work choose to render Leib as Body (capital B) and leiblich as Bodily (capital B) to distinguish the living body (Body) from any other (material) body. The embodiment/enactivist literature does not (sufficiently) note or insist on the difference. 16. This analysis is also consistent with the results of research in the cognitive sciences, where actual rotations are much more economical than mental rotations from a computational perspective on the mind (Kirsh & Maglio, 1994). Expert Tetris players rotate objects in the world rather than representations thereof in their minds. 17. Physicists are well aware of the need for different descriptions of a “system” when the conditions change. Thus, for example, the description of the (random) motion of water molecules no longer is valid when two plates at narrow distance are heated to different temperatures. This gives rise to Benard cells. That is, rather than moving randomly as they should, water molecules move in circles within hexagonal (Benard) cells arranged like a honeycomb. Conversely, when physicists move from macroscopic scale to atomic scales, their models radically change from classical physics to quantum mechanics. Maturana and Varela attempt to explain everything human in terms of biology, when in fact the conditions have changed and a different kind of description is required. 18. Biology, as a science, studies manifestations of life rather than life. Manifestations are externalities, which do not allow us to understand the underlying phenomenon. Hegel (1979) was possibly the fi rst to point to the impossibility to understand a phenomenon by studying its manifestations; Marx and Engels (1973) show how the British economists of their times completely misunderstand capital markets because they study externalities of exchange-value and use-value, which are only the outer manifestations of value as such. Henry (2003) similarly critiques biology for missing an understanding of life. 19. Maturana and Varela think cognition bottom up, taken the way in which life realizes itself in a concrete manner, the organism, as their starting point. But if the concrete organism only realizes the possibility of life, then theorizing has to take a different trajectory. It needs to identify in the concrete that which is general, not to take the concrete wholesale as the starting point of theory (see, e.g., Bourdieu, 1992). 20. In French, Bourdieu (1980) speaks of the sense (sens) rather than of logic (logique) of practice, where sense is that inherent in the sense of the game rather than underlying the (explicit, defi ned) sense of a word.

274

Notes

21. Formulations such as in the book title How the Body Shapes the Mind (Gallagher, 2005) do not help, because the two one-sided perspectives on human life are reified rather than seen as two ways in which a more encompassing unit manifests itself (see, e.g., Mikhailov, 2001). The very project that opposes body and mind returns us to thinking life and movement in terms of externalities and the Galilean/Cartesian division of body and mind. 22. The very idea of meanings that are denoted by words is a metaphysical project that places primacy on the radical autonomy of the constructive mind, because the relation between the word and its meaning has to be posited by me qua consciousness (Merleau-Ponty, 1960). 23. Henry (2005), Marion (2004), and Nancy (2003) discuss visual perception with respect to the arts, where painters allow us to see new phenomena, and, because they are new, these could not have been anticipated. The spectator is passive with respect to the new, has to receive it as a gift. Similar concepts are pursuit in the phenomenology of the foreign/strange (e.g., Waldenfels, 2006). 24. Such descriptions require differential equations. When a separation of variables is possible, such as for a moving physical body, the two subsystems can be described independently. In the modeling of dynamic cognitive processes, the separation of variables for solving the differential equations is not possible (van Gelder & Port, 1995). 25. The mathematician René Thom (e.g., 1981) developed catastrophe theory, which allows models that have both qualitative and quantitative aspects. Thus, for example, this theory allows to model the evolution of new biological or cognitive structures yielding qualitative differences when certain parameters are changed quantitatively.

NOTES TO CHAPTER 3 1. The most productive research group in this area is situated in Italy; its work is found in the leading science journals, including Science. An example of the research referred to is Kohler et al., 2002. 2. Levinas (1978), Marion (2002), and Henry (2003) are some of the philosophers who are concerned with the fundamental role of passivity in life and being. 3. He apparently died suddenly of coronary thrombosis at the age of 53, leaving a number of his works unfi nished. 4. Neuroscientifically, intention, intention recognition, and the ability to learn by imitation all are related to the presence of mirror neurons, that is, neurons that mirror the activity of motor neurons. An easy introduction to this field is available online at http://www.scholarpedia.org/article/Mirror_neurons 5. Some neuroscientists model this auto-affection using Hebbian learning, which also can be used to account for the association between neurons in the pre-motor cortex and those responsible for visual and auditory capacities (e.g., Keysers & Perrett, 2004). The authors describe how their Hebbian learning model accounts for the way in which human or infants learn to grasp objects in particular (and, perhaps, particularly efficient) ways beginning with “initially cumbersome movements” (p. 504). 6. See Merleau-Ponty and the touching-touched, seeing-seen relationships that leads to the self-reflexivity that comes with and is constitutive of thought. 7. I tend to think this relation in terms of the concept of singular plural. Thus, the collective only exists in and through its members, but the members are members only because of the collective they form. 8. Recent neuroscientific research shows that emotions and empathy have their origin in the same mirror neurons that allow the individual to recognize actions and intentions that it is capable of in others.

Notes

275

9. Topologically, a human body is a torus, with the digestive tract constituting the hole. The contents of the stomach, therefore, are outside the body proper, in the hole surrounded by the body.

NOTES TO THE INTRODUCTION TO PART B 1. “Soul,” which translates the Latin animus in Aristotle’s De Anima, itself a translation of the Greek title Peri Psykhès, could also have been translated by the words “psyche” or “mind.” 2. This is not the same as sensorimotor abilities or schema, for intentional sensory capacities, as intentional motor capacities, are a consequence of selfaffection rather than their constitutents.

NOTES TO CHAPTER 4 1. This sentence simultaneously alludes to, without referencing something specifi c, various works by Jacques Derrida (e.g., 1982, 1996, 2003) that articulate the inherent contradiction of language, which simultaneously is mine and not mine, understood and not understood, or translatable and untranslatable. 2. Both of these instances are extreme cases, in fact, deceiving the human being involved. But both cases beautifully exhibit how we make sense generally by attributing rationality and intentionality to the conversational contributions of others. 3. In Canada, both English and French are recognized as national language. All documents that the federal government produces exist in the two languages. Because speakers in the Parliament may speak either language, broadcasts from the sessions will involve translators who begin as soon as a speaker switches into the language different from the one in which the channel/radio normally broadcasts. 4. Riœur (1991), for example, a well-known theorist of interpretation, conceives it as a process that consists of a dialectic of practical understanding and explanation. Practical understanding precedes, envelopes (accompanies), and concludes explanation, but explanation elaborates and develops practical understanding. “Explanation” is not part of perception, so that perception does not interpret. 5. I insist on the presence of the pauses, because these contribute to the temporality of the event, that is, the way in which the participants experience the event. For example, teachers may hear pauses as indications of lack of knowing, uncertainty, or timidity and then act accordingly. This temporality is central to my argument about mathematics in the flesh, especially as elaborated in Chapter 7. 6. In neuroscientific parlance, we would say that only someone who has developed the mirror neurons for a particular action is able to see the movement of another as intentional action (Gallese et al., 2004).

NOTES TO CHAPTER 5 1. Some mathematics educators legitimately ask the question how embodiment and enactivist theories inform instruction. 2. The term model is polysemous, denoting both the object copied—if children were to make geometric figures following a model—and the object made to

276

3. 4. 5.

6.

7. 8.

9.

Notes depict the structure of something else—as is the case in this chapter, where the children build models of a mystery object that they cannot see. In geometry, an affi ne transformation can also be defi ned in vector notation, as a linear transformation of a vector x by an operator A followed by a translation b: x → Ax + b. In this chapter, the pauses have been measured to the closest second, that is, pauses of 0.50 seconds and larger are rounded up to one second, and pauses below are indicated by empty parentheses “().” We do not know what Melissa does inside the shoebox, as we cannot see it. Yet research in which subjects observe sets of lights in the dark recognize those that are affi xed to a human body as representing intentional movement whereas other lights, moving about in random fashion, are not attributed to human movements. This suggests that movement and its intentions are recognized even if the information is only partial. This is one of those instances where it becomes evident that mathematics is different than poetry or literature, where different readings and hearings are allowed concerning the same object. In mathematics and science, however, the language is unitary (Bakhtin, 1981). Here, in the interaction only one “interpretation” of the mystery object is supported. Because social interactions are the source of higher mental functions, the unitary nature of mathematics comes to be reproduced. The D 2h symmetry group includes identity, 180° rotation, two reflections for each of the pair of faces, whereas the Oh symmetry group includes identity, 90°, 120°, and 180° rotations (around two- and four-fold axes). The verb to formulate is a technical term in ethnomethodology, denoting parts of speech act where an agent not only does (says) something but also articulates what s/he is doing. Here, Melissa not only changes her model but also says that what she is doing is “making it else.” Imitation in terms of an image is consistent with Kant’s concept of Einbilden, whereby an image (Bild) comes to be formed inside the mind; the verb einbilden also is used in the sense to imagine something that does not exist.

NOTES TO CHAPTER 6 1. It is not surprising, therefore, that Marx and Engels (1969) make consciousness a function of practical activity: “The production of ideas, representations, of consciousness is primarily and immediately interwoven with the material activity and material intercourse of men, language of real life” (p. 26). 2. This is a feat that is possible because of the mirror neurons that allow us to re/cognize any action with its intention (e.g., Gallese et al., 2004).

NOTES TO CHAPTER 7 1. The connection between sense and practice—constitutive of Le sens pratique (The Sense of Practice), the title of Bourdieu’s (1980) book—is lost in English, where the French word sens is translated as “meaning” (a word that does not exist in French) rather than as “sense.” For Bourdieu, who emphasizes the knowing of real people, in flesh and blood, it is a non-intellectual sense of practice rather than the intellectual meaning of practice that characterizes what we do in the everyday world, including teaching and learning mathematics in second-grade classroom. Not surprising, therefore, the English translation of the book title is Logic of Practice, which emphasizes language, Gr. logos, whereas

Notes 277

2. 3. 4. 5. 6. 7. 8.

9.

10. 11.

Bourdieu points out that a very different logic underlies practice, namely one similar to the sense of the game a player has—as opposed to knowledge of the game characteristic of most commentators, spectators, and scientists. There now exists a large body of work on the effect of pre-natal sound exposure on post-natal experiences of the child (for an extensive review of all dimensions of these phenomena see Lecanuet, 1996). My hunch is that precisely because of this feature, the reduction to linguistic structures and schema renders the embodiment approach unconvincing to many mathematics educators, especially those of the constructivist kind. Sheets-Johnstone (2009) points out that such changes and adaptations also constitute forms of thinking in movement. There are other senses, but these are not mobilized in this or, for this purpose, any other mathematics classroom: smell and taste. Two relevant studies are Roth, Tobin, Carambo, and Dalland, 2004, 2005. Artificial intelligence researchers have not been interested in situated cognition; but some have, including Agre and Horswill, 1997. One study was conducted in an urban (inner-city) school, where there were many confl icts between students and teachers (Roth & Tobin, 2010). This study focused on a confl ict arising over a conceptual issue, that is, how to best fi nd the number of valence electrons for a particular element. Here I only present “simple” cases that do not address the differences between musical and poetic rhythms. Natural language is marked by pauses, which we also fi nd in music, whereas the material of the poetic work is the phoneme, so that the poetic rhythm—as present in the written poem—comes to be structured differently than the played musical work, which has an explicit notation for rhythm, including the meter, pauses, and so on. Moreover, everyday language is not composed: It is continuously improvised (MerleauPonty, 1960), in contrast to composed music or poetry. In everyday language, rhythm is a continuous achievement. The analysis of rhythms characteristic of metric phonology therefore is rather limited for analyzing the form and function of rhythm in everyday speech (Auer & Couper-Kuhlen, 1994). That is, unlike in poetry, speech does contain such a phenomenon as silent beat. Much of the work that Susan Goldin-Meadows has published, together with her students and co-workers, makes this suggestion. See, for example, Church and Goldin-Meadow, 1986. In dialectical terms, this overarching unit sublates the oppositional moments, which means, it both overcomes/obliterates them and keeps them, each being but a one-sided way in which the overarching unit realizes itself.

NOTES TO CHAPTER 8 1. A quick Google search using the search “eradicate AND ‘conceptual change’” will list many examples of texts, more so in science than in mathematics education, in which such statements can be found. 2. In recalling the kinds of cases where a game is said to be played according to a defi nite rule, Wittgenstein (1958) lists, among others, an example highly relevant here: “[There] is a variety of cases in which we should say that a sign in the game was the name of a square of such-and-such a colour. The rule may be an aid in teaching the game. The learner is told it and given practice in applying it.—Or it is an instrument of the game itself” (¶53, ¶54). 3. In this chapter, I use the term “turn” in two senses. On the one hand, it is an “emic” (insider) term, a part of the lesson that our participants are turned to. The analytical (outsider, “etic”) equivalent would be “episode.” On the other

278 Notes hand, consistent with the conversation analytic tradition, I use the analytic term “turn,” which is short for “turn at talk.” Thus, in Connor’s “turn” there are many turns at talk, as shown in and by the analyses presented here. 4. Because of the choice of the episode, intended to highlight the lived, sensuous labor of classifying (rule following) that members of a group exhibit for one another to deal with some breakdown, “successful” rule following is not shown in the episode analyzed. Of course, “success” and “no teacher comment” are mutually constitutive. In not making a comment or request, the teachers exhibit to and for students their evaluation of the action as correct. 5. See Gallese, Keysers, and Rizzolatti, 2004, for how perceiving an action— given the recent neuroscientific research on mirror neurons and social cognition—presupposes the ability to perform it. That is, observing means actively participating as the same neurons fi re that would fi re if the child were to act.

NOTES TO CHAPTER 9 1. I borrow this notation from Schegloff (2007), who used it to describe the R (relationship) and K (knowledge) categories in Sacks (1974), each of which may be proper (Rp, Kp) or improper (Ri, Ki). 2. The mystery figures are not inherently geometrical objects but becomes such in and through the practices of the group members. This situation is similar to the one involving a baseball umpire (Bill Klem), who stated, “It is nothing until I call it,” it is only in and through the call that a throw comes to be “a ball” or “a strike.” 3. At college, for example, I had chosen size, color, and spatial relation of objects as the theme for one of my art courses; I showed, among others, how color mediates the perception of space. 4. Sports commentators often appear to know the games they comment better than the multi-million dollar players who have made a fortune based on their competence in playing.

NOTES TO CHAPTER 10 1. In this chapter, I begin by following earlier distinctions in using the term mathematical concept whenever a mathematical idea in its official version is involved (i.e., at the collective level) and reserve the use of mathematical conception when dealing with the realization of a mathematical concept in and by the individual. 2. As previously articulated (Chapters 7 and 8), the utterances “I want you to take the first thing that you touch” and “now remember it is Kendra’s turn” are part of the labor that comes to shape the children’s living/lived bodies—forms of discipline that have material (in bodies) and ideal (in minds) consequences. 3. In fact, neuroscientists explicitly refer to the phenomenological studies Merleau-Ponty conducted concerning the human understanding of concepts, according to which, for example, knowing that something we see is a sphere because we know the consequences our movements with respect to an object (eyes, head, body) will have on our perceptions of the object.

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Index

Note: Italics indicate names of individuals.

A Abstract, 37, 79, 254; concept, 39, 45, 54, 253, 254; knowledge, 45, 61, 236; mathematics, 57; mind, 20, 84 Abstraction, 21, 44, 45, 46, 48, 54, 55, 87, 112, 143, 243, 244, 253, 254 Affectivity, 70, 71, 76, 159, 263 Apodictic, 36, 240, 241 Apophansis, 229, 261 A posteriori, 208, 238, 239 A priori, 15, 27, 36–39, 44–45, 69, 208, 236, 238, 242, 264, 271, 272 Archetype, 54 Aristotle, 26, 55, 275 Attunement, 74, 134, 160, 161, 169, 170, 174, 188, 190, 192 Auto-affection, 4, 28, 33, 60, 64–71, 75, 84, 127, 134, 135, 140, 154, 155, 157, 159, 161, 162, 190, 193, 255, 256, 257, 259, 263, 265, 274 Auto-perception, 159 Autopoiesis, 5, 53, 263

B Bakhtin, M. M., 77, 79, 276 Being, 8, 63, 157, 229 Being-there, 71 Being-with, 71 Biology, 52, 58, 273 Bourdieu, P., 16, 35, 72, 73, 75, 76, 92, 157, 170, 180, 191, 235, 271, 273, 276, 277

C Cage, J., 58

Catastrophe theory, 58, 274 Catchment, 79 Chair (Fr.), 1, 16, 260, 271 Classification, 11, 40, 128, 167, 197; abstract, 45; classifying, 46–48, 201–230, 231–242, 243–259; impoverished, 43 Cognition, 57, 109, 148, 149, 189, 190, 192 Cognitive science, 1, 16, 20, 46, 50, 60, 139, 169, 172, 273 Conceptual change, 87, 201, 277 Congruence, 43, 114, 136 Consciousness, 4, 20, 50, 61, 62, 63, 68, 69, 75, 78, 167, 263, 270, 274, 276; affective, 59; bodily, 80; emergence of, 57, 276; form of, 1, 3, 5, 9, 10, 22, 54, 59, 65, 66, 79, 169, 176, 189, 190, 246, 270; for myself, 192; for the other, 192; intellectual, 10, 26, 28, 30, 33, 56, 59, 93, 157, 252, 253, 254, 261; intellectualist, 59; intellectualizing, 5, 28, 29; knowing, 2, 63, 136; linguistic, 5, 9, 28, 54, 78, 85, 93, 165, 167, 170, 190, 193, 246; linguistically mediated, 78; loss of, 56; mode of, 3, 9; musical, 8; perceptive, 26, 54, 175, 189, 193, 270; perceptual, 193; practical, 158; reflection, 77; refraction in, 77; representational, 38; rhythmic, 54, 59, 66, 78, 85, 178, 189; rhythmizing, 157; strings of, 189; verbal, 43, 175, 189, 263 Constructivism, 10, 35, 36, 41–44, 60, 107; alternatives to, 159;

288

Index

approach, 21, 36, 43, 44, 74, 84; constructivists, 36, 44, 50, 52, 112, 137; educators, 52, 55, 63, 154, 201; explanation, 159; idealist, 73, 75; literature, 6, 110; mind, 4, 15, 20, 21, 27, 63, 167; monad, 15, 58; organism, 61; Piagetian, 8, 43, 46, 48, 53, 83, 243; position, 69; project, 21; radical, 21, 36, 44–46; social, 10, 11, 36, 154, 190, 263, 265, 273; theory, 6, 8, 11, 15, 169, 171, 263; view, 28, 52; way, 63 Contact, 5, 7, 15, 21, 26, 27, 31, 64, 73, 76, 85, 108, 11, 134, 135, 139, 140, 159, 164 Content, cognitive, 178, 187; noncognitive, 188, 189 Contingency, 7, 15, 21, 31, 32, 64, 208, 227, 228, 234, 237, 238, 239, 241, 242, 265 Corporeality, 54, 55, 59, 73, 76, 176, 259 Corporeity, 69, 70 Crick, F., 51 Cultural-historical activity theory, 7, 9, 10, 16, 17, 21, 32, 52, 61, 62, 76–80, 112, 168, 197, 198, 202, 231, 235, 242, 244, 260, 270, 273

D Dance, 5, 15, 54, 59, 168, 192, 269 Deaf-blind, 63, 80, 106, 109, 111, 135 Deixis, gesture, 60, 79, 89, 166, 167, 217; sign, 2 Derrida, J., 1, 16, 26, 36, 79, 240, 261, 275 Descartes, R., 41, 59, 157, 159, 186, 269, 272, 274 Development, cognitive, 35, 36, 48, 52, 180 Dialectical (materialism), 254, 259; approach, 16, 80, 243, 275, 277; movement, 69; mutual constitution, 55, 56, 6173, 77, 194, 227, 258, 265, 270, 278; psychology, 16, 63; relation, 77; theory, 16, 77, 78 Disposition, 68, 72, 73, 75, 76, 92, 154, 170, 198 Domain, cognitive, 54 Dualism, 35, 52, 62, 74 Dürer, A., 32

E Ego, 71, 262; alter ego, 17 Embodiment, 1, 4, 6, 7, 10, 11, 16, 17, 19, 21, 27, 34, 66, 68, 159, 277; hypothesis, 55; mathematics, 269; mind, 162; literature, 113, 157, 193, 260, 273; theories, 36–59, 70, 71, 75, 83, 110, 112, 137, 140, 159, 161, 190, 245, 275 Emotion, 6, 61, 76, 77, 158, 159, 274 Enactivism, 9ff, 35, 52–59, 74, 84, 189; account, 135; approach, 17, 68, 74, 84, 161, 269; body in, 85; claim, 189 literature, 6, 16, 66, 85, 110, 157, 260, 263, 273; perspective, 83; research, 68; system, 61; theory, 6, 8, 17, 31, 66, 70, 71, 75, 84, 137, 140, 245, 275 Entrainment, 6, 11, 61, 66, 71, 85, 86, 154, 158, 160, 163, 169, 179, 183, 190, 191 Epistemic (movement), 40, 85, 140, 142, 143, 155, 169, 172, 261 Epistemology, 21, 35–59, 71, 260, 262, 265 Ergotic gesture, 139, 140, 143, 148, 155, 261 Erlangen program, 41, 42, 121, 272 Ethnography, 28, 228, 232, 266, 268 Ethnomethodology, 202, 203, 228, 276 Euclidean, 41, 42, 45, 136, 204, 272; non-Euclidean, 272; pre-Euclidean, 72 Exteriority, 71, 73, 74 Externalization, 92, 159, 162

F Field, 73–75, 76, 107, 154, 163, 191, 263; cultural, 108; ordered, 232, 238; perceptual, 93; social 163; structures, 213 Frequency, 173, 174, 190 Freud, S., 15, 22 Freudenthal, H., 10, 19, 20, 35, 43, 111, 112, 114, 115, 136, 272 Function, cognitive, 73, 198, 234

G Galileo, G., 41, 59, 155, 272 Game, 3, 9, 19, 207, 236, 264, 278; adult, 264; categorizing, 52; geometry, 264; mathematical,

Index 289 54; playing, 265; rule of, 207, 214, 236; sense of, 68, 74, 221, 273, 277; soccer, 9, 54; sorting, 214, 217, 236; tennis, 19; Tetris, 169 Greece (Greek), 34, 48, 72, 83, 138, 140, 141, 250, 258, 269, 270, 275 Growth point, 79

H Habitat, 170; inhabit, 5, 8, 47, 73, 75, 107, 108, 110, 158, 170, 262 Habitus, 73; 74–76, 163, 170, 180, 190, 235, 241, 263; emergence of, 68–70 Hegel, G. W. F., 261, 273 Heidegger, M., 1, 16, 73, 188, 229, 261, 270, 272 Henry, M., 1, 4, 5, 16, 44, 57, 59, 60, 66, 67, 70, 72, 76, 83, 92, 194, 261, 271, 272, 273, 274 Husserl, E., 1, 3, 16, 19, 34, 49, 51, 52, 62, 114, 134, 136, 138, 149, 156, 201, 204, 205, 231, 240, 241, 242, 265, 272, 273

I Iconicity, 121, 165, 187; gesture, 55, 60, 78, 89, 119, 120, 121, 122, 226; representation, 136 Imagination, 3, 37, 39, 42, 114, 135, 270, 272 Immanence, 26, 28, 31, 33, 60, 62, 64, 66, 69, 70, 76, 80, 92, 108, 151, 172, 176, 189, 208, 252, 254, 256 Improvisation, 4, 22, 161ff, 225, 277; dance, 54, 192; freedom, 171; jazz, 5, 192; movement, 178; music, 5; situated action, 162 Incarnation, 4, 21, 28–29, 64, 159, 167, 176, 261–263; activity, 197; action, 233, 238; agent, 10, 68; being, 19, 242; capacity, 257ff; carnal expression, 192; carnal presence, 51; conception, 257, 259; description, 192; engagement, 239, 240; experience, 46, 203, 234, 244, 245, 254; features, 188; knowledge, 21, 55, 255; labor, 231, 234, 259, 260, 263; mode of, 262; movement, 258, 259, 265;

phenomenon, 193; presence subject, 180; suggestion, 149; thinking, 162; voice, 262; work, 199, 265 Innatism, 80, 271 Intellectualism, 5, 25, 28, 29, 35–59, 73, 74, 75, 157, 165, 168, 192, 260, 271 Intelligence 1, 4, 65, 68, 91, 172; artificial, 16, 277 Intelligibility, 43, 46, 47, 50, 93, 154, 192–194, 228, 229, 233, 234, 263, 265 Intention, 1, 2, 6, 10, 20, 29, 55, 57, 58, 63, 65ff, 69, 71, 80, 88, 106, 108, 133–137, 140, 162, 194, 244, 265, 274, 276; actionrelated, 60; activity-related, 127, 135; classificatory, 203; collective, 127; communicative, 160; conscious, 26; development of, 57; explicit, 27; investigative, 109; learning, 264; movement, 79, 84, 127, 155, 189; neuroscience, 274, 276; operative, 92; sensorimotor, 66, 135; speaking, 94, 106, 109; teacher, 106, 110 Intentionality, 2, 5, 10, 21, 26, 43, 62, 68ff, 71, 74, 108, 244, 261, 262, 275; act, action, 57, 73, 75, 197; aim, 20; bodily, 5; of disclosure, 5; learning, 43; motive, 135, 275, movement, 49, 140, 189; nonintentional, 6, 21, 44; objectrelated, 20, 27, 92, 235, 264; operative, 2; potential, 55; production, 202; representative, 68; seeing, 63; sense impression, 65; sensorimotor, 21, 275; speaking, 78; thinking, 78; touching, 63 Interaction ritual, 16, 61, 67, 71, 72, 73, 75, 76, 163, 171, 178, 191, 197, 198, 218, 223, 234 Intersubjectivity, 15, 17, 52, 136, 138, 192, 241, 263, 265 Intonation, 94, 96, 100, 105, 109, 120, 144, 145, 151, 158, 170, 171, 175, 210, 263 Intuition, 4, 10, 34, 37–39, 41, 50, 83, 261, 272; early, 45; empirical, 38, 39; immediate, 34; indefinite, 264; metric, 41; first, 41; pure, 38, 236; topological, 41, 45 Invisible, 5, 27–28, 50, 59, 113, 182, 246

290 Index Irreducibility, 5, 54, 70, 78, 84, 159, 189, 190, 191, 193, 228, 255

J Jazz, 5, 83, 87, 192 Johnson, M., 5, 53, 54, 269

K Kant, I., 7, 8, 10, 15, 17, 20, 21, 35, 36–38, 38–41, 42–46, 49, 50, 52, 53, 60, 136, 190, 236, 243, 245, 269, 270, 271, 272, 276 Klee, P., 58 Klein, F., 41, 121, 272

L Lakoff, G., 5, 54, 55, 57, 260, 269 Language, English, 183; everyday, 43, 277; game, 207, 236, 264; national, 92, 275; native, 258; natural, 228, 277 Learning paradox, 44 Leib (Ger.) 1, 16, 49, 260, 273 Leontjew, A. N., 16, 77, 127, 263, 270 Lifeworld, 44, 47, 93, 94, 107, 156, 172, 202, 230, 240, 243 Linguistic, 157, 163, 270; consciousness, 9, 28, 54, 78, 85, 93, 165, 167, 169, 170, 190, 193, 246; form, 78, 83, 107, 108, 112; gesture, 159; means, 162; mediation, 172; mind, 59; schema, 159; structure, 5, 54, 277 Logodaedalus, 36–38, 270 Logos, 4, 54, 85, 229, 235, 261, 270, 276

M Maine de Biran, P., 1, 2, 3, 16, 21, 29, 57, 60, 65–69, 75, 271 Manifold, 37, 256 Marx, K., 16, 76, 77, 78, 83, 198, 261, 263, 270, 273, 276 Maturana, H., 5, 10, 17, 52, 53, 55, 58, 59, 61, 162, 263, 273 Mediation, 33, 39, 54, 60, 61, 66, 67, 78, 112, 133, 134, 136, 138, 169, 172, 183, 203, 204, 252, 253, 255, 259; immediate, 29, 34, 38, 68, 72, 74, 75, 77, 92, 93, 105, 133, 163, 169, 197, 202, 259, 276; unmediated, 2, 4, 54, 112, 261

Melody, 2–5, 7, 8, 67, 174, 175, 178, 189–194 Membership categorization, 204, 228, 230 Memory, 2–4, 10, 26, 28, 60, 65, 67–68, 69, 79, 91, 140, 155, 158, 252; immemorial, 2–4, 33, 66, 69, 75, 76, 162, 180, 259; movement, 26, 33, 65; pad, 76; symbolic, 162 Merleau-Ponty, M., 1, 3, 10, 16, 21, 36, 42, 49–52, 55, 57, 58, 62–64, 70, 78, 80, 91, 92, 106, 158, 159, 160, 167, 169, 172, 180, 193, 201, 246, 253, 254, 274, 277, 278 Metaphor, 1, 5, 6, 8, 35, 54, 61, 83, 110, 162, 192, 272; conceptual, 54, 56, 57; metaphorization, 53, 54, 56, 61, 69, 110, 112, 190 Metaphysics, 7, 15, 20, 35–59, 60, 62, 64, 83, 84, 112, 157, 161, 188, 189, 243, 256, 269, 272, 274 Metonymy, 1, 5, 83, 110, 162, 219; metonymization, 190 Misconception, 113, 201, 232, 239 Model, cognitive, 162 Mood, 71, 188, 192 Müller-Lyer Illusion, 32, 33 Multiplicity, 50, 149, 155, 255, 258 Mundanity, 197, 198; action, 197; classification, 226; competency, 226; experience, 113, 134, 197; expression, 161; practice, 266; work, 265 Music, 9, 58, 83, 176, 182; composition, 277; context, 8; jazz, 83, 192; making, 4, 9, 110, 277; metaphor, 192; meter, 183; musician, 176; notation, 178; perception, 8, 194; performance, 7, 163, 192, 277; phrase, 143; rhythm, 183, 277; rock, 143score, 7, 163; sheet, 5, 7; tact, 15; teacher, 83; theory, 8

N Nancy, J.-L., 1, 4, 15, 31, 62, 160, 183, 198, 254, 255, 259, 269, 270, 272, 274 Necker cube, 29, 30, 34, 67, 271 Nuance (musical), 8, 143 Núñez, R., 5, 10, 54, 55, 56, 57, 112

Index 291 O Objectivity, 4, 136, 265 Object/motive, 16, 124–127, 226 Ontogenesis, 29, 41, 114, 198, 234, 242, 264, 265 Ontology, 254 Organic body, 6, 27, 28, 38, 49, 51, 57, 59, 65, 77, 87, 159, 163, 170, 189, 202, 258 Other, 52, 98

P Pace, 60, 90, 136, 137, 158–163, 176–189 Passion, 21, 28, , 57, 84, 85 Passivity, 3, 6, 17, 44, 58, 62, 64, 65, 68, 71, 80, 84, 87, 244, 274 Pathos, 71, 194, 263 Pedagogy, 76, 87, 97, 210, 225, 240, 241 Performance, 5, 26, 49, 78, 84, 85, 103, 176, 178–182, 188, 192, 194; conceptual, 190; expressive, 6; individual, 154, 231; live, 5, 8; mathematical, 8; movement, 155, 193; musical, 7, 163; praxis, 157; rhythm, 191; sound, 175 speaking, 163; thinking, 163 Periodicity, 66–67, 71, 73, 75, 158, 172, 183, 190, 191, 233, 263 Phenomenology, analysis, 22, 33, 49, 57, 68, 92, 180, 260; approach, 16, 43, 61, 65, 172, 201, 262; epistemology, 262; inquiry, 114; investigation, 33, 111; material, 7, 10, 21, 28, 53, 61, 63, 64–71, 74, 76, 77, 80, 84, 193, 262, 263, 271; nonintentional, 44; observation, 161; origin, 121; perspective, 51, 253; philosophers, 15, 35, 52, 65, 255, 260; primitives, 26, 113; radical, 189, 262; reduction, 271; sociological, 16, 71–74, 84, 262; studies, 32, 35205, 243, 278; terms, 50; thought, 21 Phylogeny, 29, 242 Piaget, J., 8, 10, 35, 41–44, 44–46, 46–49, 50, 51, 52, 53, 60, 61, 83, 87, 112–115, 121, 131, 135, 136, 190, 201, 224, 232, 234, 242, 243, 245, 257, 272 Piano, 6, 7, 9, 58, 60, 192, 267

Pitch, 8, 67, 73, 75, 130, 145, 160, 171–176, 179, 183, 190, 191, 194, 216, 268 Poetics, 16, 277; meter, 182; poetry, 183; rhythm, 192 Practice, 30, 44, 73, 127, 148, 155, 157, 163, 192, 198, 213, 217, 228, 235, 242, 273, 276, 277, 278; accountable, 202, 235; advanced, 149; classifying, 226, 227; conceptual, 55; cultural, 163, 170; disciplinary, 232, 237–239; discursive, 127; geometrical, 139, 199, 202, 226, 264; material, 47; mathematical, 227, 234; mundane, 226; practical action, 80, 106, 202, 228, 242, 270, 276; practical comprehension, 73, 74–76, 157, 191; practical sense, 73, 74, 76; practical understanding, 8, 75, 76, 157, 169, 188, 192, 194, 266, 275; routine, 266; social, 74; thinking, 264 Pragmatism, 169, 223–224, 242 Praxis, 16, 77, 235, 239, 242; empirical, 136 mathematical, 157–194; methodological, 114; phenomenological, 29–34; sensuous, 237, 241 Presence, 4, 26, 261; in the classroom, 87, 109; in the present, 26, 92, 157, 261; in the world, 1, 15, 157; representation, 26, 272 Process, 69; abstractive, 45, 46, 243; autopoietic, 53; biological, 55; categorizing, 227; classificatory, 205, 235; cognitive, 274; creative, 17; descriptive, 36, 248; differentiation, 70; explanatory, 271; emergence, 203; historical, 155, 258; instructional, 233; interactional, 107; interpretive, 165, 172, 275; learning, 48, 53, 134, 233, 237; life, 78; material, 160; mathematical, 243 measurement, 139; metaphoric, 5; microgenetic, 234; self-organizing, 53; sense-forming, 114, 134; socialization, 72; sorting, 223; spontaneous, 160; systemic, 56; technological, 34; transformation, 139 Psyche, 15, 22, 275

292 Index Psychology, 22, 30, 35, 41–44, 76, 91, 106, 159, 161, 167, 189, 197, 232, 271; cognitive, 16; developmental, 41–44, 257; dialectical materialist, 16, 63, 78; experiment, 29, 32; geometry, 41; Marxist, 76, 77; naturalistic, 52; neuropsychology, 61; perception, 29; Russian, 78, 80; social, 198, 270; societal, 16 Reductionism, 76 Representation, 15, 16, 21, 25, 33, 35, 37–39, 42, 43, 52, 63, 67, 68, 70, 71, 79 133–137, 149, 160, 162, 172, 182, 185, 235, 261, 263, 271, 273, 276; ephemeral, 16; gestural, 135; iconic, 136; image schematic, 140; mental 20, 21, 25, 162, 180, 271; partial, 149; qualitative, 204; subjective, 63; symbolic, 136 Research, cognitive, 36 Resonance, 71, 160, 165, 171, 172, 183, 190, 234 Rhythm, 158–163, 193; alignment, 160; beating, 103, 187; communication, 179; consciousness, 54, 59, 66, 78, 85, 178; coordination, 159; counting, 102, 187; features, 157ff, 183, 185, 187; movement, 178; noise, 66; pattern, 187; performance, 7, 178; phenomenon, 11, 84, 85, 189, 190, 193, 217; rhythmicity, 5, 66, 188

S Schema, 4, 5, 6, 8, 42, 45, 53, 54, 66, 69, 159, 162, 163, 170, 171, 189, 269, 271, 275, 277; bodily, 110, 162, 190; container, 54; embodied, 54; image, 53, 54, 55, 56, 79, 140, 175, 162, 167, 169, 189; sensorimotor, 7, 61, 66, 85, 162, 171, 270; source-path-goal, 112 Self-affection, 11, 16, 54, 65, 69, 79, 80, 84, 93, 106, 108, 252, 257 Self-correction, 217, 219, 233 Self-donation, 70 Self-knowledge, 162 Self-movement, 66, 70, 79 Self-organization, 53 Self-reference, 159, 193

Self-reflexivity, 80, 261, 274 Self-sameness, 34, 136, 204, 218 Self-understanding, 61 Signification, 1, 20, 46, 56, 72, 84, 114, 135, 159, 160, 163, 170, 212, 219, 229, 238, 239, 259, 268, 273 Situatedness, 56, 64, 69, 73, 162, 209, 211, 241, 244, 269, 274, 277 Socialization, 6, 10, 21, 58, 59, 64, 72, 73, 75, 85, 108, 109, 127, 163, 191, 192, 210, 218 Sociogenesis, 198, 234, 265 Sociology, 16, 73, 183; phenomenological, 6, 10, 16, 17, 52, 67, 71ff, 84, 197, 262, 263 Sociology, phenomenological, 6, 10, 11, 16, 17, 52, 58, 67, 71, 197, 263 Soma, 76 Sound-word, 79, 190, 257 Spontaneity, 47, 59, 76, 161, 163, 171, 192 Structure, beat, 164, 166; cognitive, 42, 44, 274; dispositions, 72, 73; formal, 6, 228, 230; grammatical, 145, 149; linguistic, 5, 54, 112, 277; logical, 46, 203, 224; material, 75, 163; mathematical, 62, 113, 272; noncognitive, 1; social, 72; temporal, 157–194 Subjectivity, 4, 44, 47, 70, 107 Suffering, 62, 63, 66, 70, 76 Synecdoche, 1, 7, 9, 65, 190, 194, 219, 244, 245, 256, 259, 260, 261, 269 Synchrony, 67, 157, 172, 187; synchronization, 157, 183, 190, 191, 192 Syncope, 38, 140, 142, 143, 178, 270; syncopation, 161, 177, 178, 179, 183, 188 Synthesis, 37, 42, 50, 69, 75, 113

T Temporality, 31, 55, 109, 136, 137, 157, 158–163, 165, 171, 188, 192, 193, 233, 240, 241, 265, 272, 275; conception, 42; coordination, 159; duration, 41 Timbre, 194 Thinking in movement, 54, 113, 164, 171, 252, 277 Token, 19, 234, 252, 253, 270 Topology, 41, 45, 83, 114, 121, 236, 242, 275; category, 208; intuition,

Index 293 41, 45; primitive notion, 46; principle, 236 Transcendence, 37, 39, 45, 70, 176, 199, 233, 234, 236, 242, 254, 265, 266

U Undecidability, 3, 5, 38, 62, 84, 140, 142, 143, 148, 162, 176, 178, 189

V van Hiele, P. M., 10, 46–49, 232, 234, 272 Varela, F. J., 5, 10, 17, 42, 52, 53, 54, 55, 57, 58, 59, 61, 162, 263, 269, 273

Vision, 31, 35, 43, 50, 63, 64, 85, 112, 135, 235, 241, 244, 272 Voice, 2, 6, 23, 92, 108, 109, 158, 159, 174, 190, 210, 237; carnal, 262; inner, 92 von Glasersfeld, E., 10, 21, 44, 46 Vygotsky, L., 16, 61, 73, 78, 79, 165, 198, 263, 270

W Watson, J., 51 Wittgenstein, L., 227, 277 Western, cognitive science, 16; culture, 42; non-Western culture, 19; scholarship, 16 Word meaning, 27, 78, 270