This section is an annotated and commented transcription of the episode in which the students experiment with the difference graph. It is divided in two parts. In part one, the students strive to keep the difference graph on the x-axis; during the second part, they try to keep the difference graph either above or below the x-axis.
Introducing the difference graph and trying to keep it on zero
Ricardo: The computer also generates another line [turns on the difference graph] that is, em, dark blue [points at the dark blue graph; Fig. 5a] (…) so we’ll investigate what this third line is doing there, what it’s showing. So, the first thing we’ll try…
Mario: It’s called, it’s called minus because that, that purple [dark blue] line, line is, is, is pink minus blue.
Ricardo: OK, how do you know that?
Mario: It’s real, it’s quite obvious, where it says pink minus blue [points to the screen, note the area pointed at with a black arrow in Fig. 5a] at the top of the screen.
Ricardo: Aha (…) [gives the two remotes to Mario] so you move, you do whatever you want, [moves alternately right and left hands] but try to keep the dark blue on zero [points to the dark blue line], on this line [left hand runs along the x-axis]
Mario begins his first difference graph: he starts with the pink remote in his left hand and the blue one in his right hand. At the beginning of the experiment, the pink remote is kept slightly ahead of the blue one, and then the two are slowly switched in their positions. Holding the two remotes separated, he then walks forward (see the graphs in Fig. 5b).
During the last seconds of the graph production, he separates the remotes even more and says:
Mario: I’m trying as hard as possible not to make the things go opposite.
Ricardo: So here you, this piece [pointing to the dark blue graph around second 7] you had it on the line… so try to do more of that, see if you can.
While Ricardo is speaking, Mario moves both remotes back and forth, swinging the arms rhythmically. Then, he starts a new session, moving the remotes slowly in opposite directions, and produces the graphs in Fig. 6a.
During the last seconds, he says:
Mario: They’re both neutralising each other.
Mario: That’s because, because most of the time I’m, I’m, pink’s going in a straight line and blue’s going in a stripe, straight line [inaudible] [stops talking, while a new triplet of graphs starts to be created superimposed to the previous one; moves the remotes again back and forth].
Mario presses a button in the Wiimote, and a new session starts: he alternates fluid back and forth movement of the two remotes, which becomes faster and faster (Fig. 6b).
Ricardo: So here they were these lines [points to the initial part of the dark blue line in Fig. 6b] … they, oh, look! [points to the intersection of the dark blue line with the time axis, as the lines unfold] [after second 8, Mario starts to rhythmically bounce on his knees while accelerating the fluid back and forth arm movement creating the second half of the graphs in Fig. 6b]
The appearance of a third graph prompted Mario to examine the screen seeking for additional signs that could account for it. There was none with a dark blue colour. However, the sign at the top of the screen “pink minus blue,” which had been displayed from the beginning of this session but had remained unused, offered him a compelling interpretation (“it is obvious”): the dark blue line “is called minus.” The inscription above the graph included the pink and blue Wiimotes, freeing the minus to be clasped by the third graph. The dark blue graph seemed to announce its name. In Paragraph , Mario expressed an initial sense for the dark blue graph focused on its name.
Mario started the graph shown in Fig. 5b with the pink Wiimote in front of the blue one, slowly moving them towards their centre. Once they were next to each other, he continued slowly moving them along the same directions. Right before 8 s, the pink graph disappeared, possibly because the orientation of the pink Wiimote made it fall outside the field of reception. This interruption is likely to have prompted Mario to move the pink Wiimote to make its graph reappear. When it did, the dark blue graph was above the x-axis. Then his arms tensed as if trying to push the dark blue graph towards the x-axis. Mario reflected on this sense of effort (“trying as hard as possible”) as striving “not to make the things go opposite” (Paragraph ). This “going opposite” (an event, in Deleuzean sense) might have been the dark blue graph moving in a direction opposite to the desired one, such as towards the zero line. Another possibility, evoked by Mario’s use of the plural “things,” is that he saw the pink and light blue lines moving in opposite directions, instead of, perhaps, staying together. His reaction was to try to “lower” the pink and blue graphs by walking towards the monitor. However, the dark blue graph continued to inch upwards.
In all his graphical productions (Figs. 5b, 6a and b), Mario tended to move the Wiimotes in alternate directions. This is likely to have followed from tacitly adopting Ricardo’s demonstration (see Paragraph ). Ricardo gestured an alternate movement of the Wiimotes while saying “you do whatever you want” (Paragraph ). While words may leave to the interpretant a more open range of possibilities, gestures are inclined to convey unintended specificities. This tacit assumption of a wavy kinaesthetic pattern was in tension with the task of maintaining the dark blue graph on the x-axis. The graph “called minus” was not just a visual display out there but also a curve that resisted physical efforts seeming to possess a will of its own which at times led Mario to tense his movements.
In Paragraph , Mario expressed a sense for a general relationship between the pink and blue graphs: “They’re both neutralising each other.” While his ensuing account of this relationship in Paragraph  is inaudible, we hear the sense of Paragraph  as indicating the emergence of a general. Recall that dynamic relationships between components affecting each other constitute generals: “neutralizing” suggests a present continuous activity interrelating two graphs or Wiimotes.
Around second 8 of the graphs shown by Fig. 6b, Mario seemed to free himself from trying to keep the dark blue graph close to the horizontal axis, engaging in a new rhythmic kinaesthetic pattern swinging his arms back and forth and bouncing his knees. This bodily movement expressed itself visually by a wavy synchronized variation of the three graphs at once. Relieved from trying to push the dark blue line horizontally, Mario seemed to enjoy a relaxed and smooth swinging—a sense for the graphical weaving emerging on the computer screen as expressed by his wavy body motion.
Mario gives the Wiimotes to Dan, who starts a new graph. He stands still in the same position for all the session, keeping steadily the remotes at the same distance from the LED bar (Fig. 7a).
Ricardo: So that, that’s a perfect zero! [around the 8th second, laughs] [ending his graph, Dan relaxes his position, shrugs his shoulder and smiles]
Ricardo: And, can you do it while you walk?
Dan starts moving very slowly towards the LED bar with both the remotes kept steady and then backwards; he generates the graph shown in Fig. 7b:
Dan: You just have to keep the remotes in (…) one position.
Ricardo: Like, keeping [them] together?
Dan: Keeping them at the same level.
Ricardo: The same level, ok.
Dan creates then new graphs, walking again towards the LED bar, then backwards, keeping the remotes steady, next to each other.
Dan came to create a horizontal difference graph with a clear plan—stay still with the two remotes next to each other—that he had developed while observing Mario’s experimentation. He had a well-defined sense that a dark blue graph on the horizontal axis “converted” into the two Wiimotes being next to each other. Moreover, Dan easily showed in Fig. 7b that that condition was indifferent to his walking distance from the LED bar: “You just have to keep the remotes in one position” (Paragraph ). Dan articulated his sense for the relationship between the dark blue graph being on the x-axis and the range of kinaesthetic activities consistent with it in two ways: “keep the remotes in (…) one position” (Paragraph ) and “Keeping them at the same level” (Paragraph ). While the word “position” alludes to a location in space, the word “level” is customarily a term for height. So far, the children’s experimentation with the Wiimotes had not included varying the kinaesthetic quality of the Wiimotes/hands’ height, to ascertain graphical responsiveness. On the other hand, differences in height between the light blue and pink graphs had been of major significance. We surmise that Dan’s relevance of the Wiimotes being at equal levels had drifted from noticeable graphical levels to the taken-by-default levels of the Wiimotes. This “out reaching” of qualities from one signifier (e.g., graphs’ levels) to another (e.g., Wiimotes’ levels), which end up encompassing both, is an instance of what we have interpreted in Sect. 2.3 as “semiosis.”
We propose that semiosis is a key process in the formation of a general, which reflects the inherent presence of a continuum across which elements of a general communicate with each other. The graphs on the computer screen intermingle with the hand-held Wiimotes, such that “same level” and “same position” can refer to all of them and distribute qualities that remain distinct and, yet, overlapping.
Dan and Zev exchange the remotes. As soon as Zev grabs the remotes, he starts a new session and creates the graphs in Fig. 8:
Ricardo: So what do you think, how do you explain?
Zev: Well, every walk I’ve done checks, em… they have got a descending number and that’s the distance of each control on the sensor [the LED bar] and then minuses the red one from the blue one [points towards the two remotes depicted on the top of the screen]. So if they are both the same [keeps the remotes next to each other], one minus one is zero, and the same with two minus two, so when we move them back and forwards the same [moves both remotes next to each other and starts a new session, beginning Fig. 9] it stays at zero, but when we move one [moves one remote backwards while he keeps the other one still, see the region around the arrows in Fig. 9].
Ricardo: So here, this minus this distance is zero [points to two overlapped points of the blue and pink lines, then to their difference graph on the x-axis] … But here, what did it happen? [points to the two points of the pink and dark blue graphs marked by the arrows in Fig. 9].
Zev: Well, it’s, it’s different.
Ricardo: It’s different… Alright, very good.
Zev begins Paragraph 19 by articulating three propositions: (1) “they’ve got a descending number,” (2) “that’s the distance of each control on the sensor,” and (3) “then minuses the red one from the blue one.” We will comment on the sense of each one and how they concatenate:
“They” are, for the most part, the blue and pink graphs. Furthermore, since Zev had kept the two graphs going together, his saying “a descending number” (singular) might suggest that both graphs descended by the same numbers. However, semiosis allows “they” to also relate to the hand-held Wiimotes, their icons on the computer screen, or the numbers implicit in the shape of the graphs. In the event, the numbers are descending if the remotes get closer to the LED bar. While such getting closer is an action undertaken by Zev, this proposition is articulated from the point of “them,” so that they “got” a descending number. In other words, the descent is an effect passively undergone by the graph/number. The subsequent “and” signals the beginning of another proposition in the form of a juxtaposition, that is, the upcoming proposition is to be held in parallel with the previous one.
“That” brings up from the prior proposition a descending number to predicate of it that it is a specific distance between Wiimote and LED bar. This specification is inscribed in the general whose sense Zev is articulating: a general in which descending numbers, walking towards the LED bar, lowering graphs, and distances between Wiimotes and LED bar, are all mutually conditioned. The subsequent “and then” betokens an upcoming proposition that is not so much to be juxtaposed as coming after the prior ones.
After the numbers are gotten, they are “minused.” Zev states that the red one is minused from the blue one. This can be understood in opposition to the equation depicted above the graphical space that appears as if blue is to be “minused” from red. However, the object of Zev’s explanation is the case of red and blue numbers being equal, so that the result, zero, is indifferent with respect to which is minused from which.
In “one minus one is zero, and the same with two minus two,” Zev uses particular examples to illustrate a general relationship. This is an instance of what Mason and Pimm (1984) have called “seeing the general in the particular.” Zev is articulating a general that we could symbolize by: A – A = 0; however, his understanding is far from being reducible to any formal definition: it encompasses countless qualities, such as the kinaesthesia of walking with two hands next to each other, the light blue and pink graphs going at the same height, the dark blue graph staying over the horizontal axes, the vast number of numbers that can be subtracted from themselves, the nothingness that remains after taking away—minusing—what had been given, or walking ahead as a kind of “descending.” Navigating such boundless expansion of interrelated qualities is what we characterize as wayfaring a surplus of sensorimotor qualities towards grasping a general. Zev says that this general encompasses “when we move them back and forwards the same;” then he begins to point out, in words and gestures, that it excludes the case “when we move one…” and not the other one. This inclusion/exclusion criterion for a general that can be symbolized as A – B = 0 is reaffirmed by Zev (“it’s different,” Paragraph ), as he qualified the two cases pointed out by Ricardo in Paragraph .
Keeping the difference graph above or below the x-axis
Ricardo: So now we’ll try to do something similar but keep the dark blue line always above. You can walk and move your hands, but keep the black, the dark blue line above the zero.
Mario comes to the front and holds the remotes. He starts a new session (Fig. 10a):
Ricardo: So now it’s above [points to the dark blue graphs, around 10 s].
Mario creates a new graph (Fig. 10b):
Ricardo: So what do you think? In order to get the dark blue line above the zero, what do you have to do?
Mario: Well, you, you make pi, pink bigger than blue so that…
Ricardo: … the pink…
Mario: … so you keep it above but if you wanted it below you have to have blue bigger than pink.
Ricardo: Ok, so let us have it, let us have it below now.
Ricardo: Let us have it, the dark line, below.
Mario starts a new session, but very soon, he presses a button that generates again superimposed graphs. Refreshing the screen, Mario creates new graphs with a new session, starting with the pink remote in front, while the light blue one is kept farther from the LED bar (Fig. 11):
Ricardo: … [while Mario is moving the light blue remote closer to the pink one] and then slowly you get it to zero. So what did you do to, to have it under the zero line?
Mario: I had to make the blue bigger than pink.
Ricardo: So let us, eh, hand it to Dan, and so you create a pattern, you can do as many variations as you want, like walking and moving, but always keeping the blue line above the zero to start with…
Dan starts a new difference graph (Fig. 12a):
Ricardo: So what did you do, to keep it above?
Dan: Em, pink back and blue forward.
Ricardo: The pink back, further from this [pointing to the LED bar]. Em, and now keep it under this [the x-axis].
Dan: Oh! [restarts the session several times while Ricardo makes a few comments]
Dan creates the difference of Fig. 12b.
Ricardo: So what, what does it happen here, to get it below [the x-axis]?
Dan: You’ve to do the opposite… you put pink forward and the blue back.
Ricardo: Pink below, right? Ok. Great!
Ricardo: So now, Zev, do, do something like this: above, below. So first of all, but try to find out the variation, so what is it possible?
Zev is given the remotes and creates the graphs of Fig. 13a:
Ricardo: So, how did you change [the dark blue line] from below to above?
Zev: Em, by changing which controller was in front.
Ricardo: So which one was in front here? [points to the dark blue graph around the 4th second, where it is below the x-axis]
Zev: Em, [light] blue.
Ricardo: (…) And do you have a sense for why for the blue, for the dark blue line, to be below [the x-axis] then the pink has to be below [the light blue graph]?
Zev: Em, yep. Em, it’s something to do with like maths and, like, because on there, it says the [seeming to point at image of the two Wiimotes on the screen] has been taken away and then it’s hard to tell because it’s not actual numbers, but, if you have more on one side, that will be a negative number… then, then, if you have them on the other side, it’ll be a positive number, which is that [moves the dots on the screen along the dark blue line by controlling the remote].
In Sect. 4.2, Mario, Dan, and Zev characterized two complementary regions, one for the dark blue graph being above the x-axis and another for it being below. Mario separated these regions by contrasting “pink bigger than blue” and “blue bigger than pink” (Paragraphs  and ), Dan by “pink back and blue forward” or “pink forward and the blue back” (Paragraphs  and ), and Zev “by changing which controller was in front” (Paragraph ). In Sect. 2.3, we elaborated on a notion of semiosis as a drift of qualities across signifieds and signifiers. We also suggested the image of the interpretant as a continuum sustaining the “expansion, contraction and reproduction [of qualities] across signifieds and signifiers.” The events transcribed in Sect. 4.2 inspire us to visualize compositional elements for the interpretant; namely, that instead of a single all-encompassing continuum, the interpretant would be akin to a Riemann surface with various sheets (see http://mathworld.wolfram.com/RiemannSurface.html). Mario, Dan, and Zev suggested several sheets: (1) a sheet hosting the dark blue graph with a region above the x-axis, distinct from a region below the x-axis; (2) a sheet hosting the pink and light blue graphs with regions separating which one is “bigger;” (3) a sheet containing the Wiimotes distributed along regions demarcating which one is closest or farthest from the LED bar; and (4) a sheet containing the Wiimotes distributed along regions that distinguish which one is in front of which one. The regions of each sheet map out with regions on the other sheets, such that, for example, “pink bigger than blue” in one sheet maps out with the “dark blue graph above x-axis” in another sheet. Figure 14 shows an attempt to illustrate such mapping across multiple sheets: a light trace goes across several pages of a book, allowing for certain regions on different sheets-pages to mutually exchange qualities. The children collectively unfolded semiosis as hosted by an expanding interpretant with multiple sheets, allowing for the mutual and multi-layered discrimination of diverse qualities, such as above/below, bigger/smaller, back/forward, and front/behind.
In Paragraph , Zev explained how the distinction between regions in some of these sheets relates to numbers. First, he points the remotes drawn on the computer screen above the graphical region indicating that it “has been taken away,” alluding to the subtraction displayed with the Wiimotes above the graphical space. Zev thinks that “it is hard to tell” what happens because the actual numbers are not shown. This latest remark makes a fleeting allusion to an unknown: beyond estimating possible values, the position numbers per se are not given. However, Zev quickly leaves behind these unknowns to distinguish two regions by their corresponding “sides:” more on one side of the minus will obtain a negative number, more on the other side a positive number. Zev described here a fifth sheet, which included the two Wiimotes depicted on the computer screen separated by a minus, with left and right sides or regions, such that having “more” on each side maps out numbers for the dark blue graph being positive or negative.