schools, as seen in the different ways in which Russian peasants categorized colors (Luria 1976).
Whatever Vygotsky may have had in mind, or whatever development there was in his thinking, his descriptions lend themselves to the individualist and (social) constructivist readings characteristic of interpretive psychology (as distinct from scientific, physiology-oriented psychology). The living, social-material environ- ment is a condition for the adolescent to evolve the particular forms of thinking.
Piaget might have said that there are processes of accommodation and assimilation that lead to an equilibration. Vygotsky does not at all talk about the relations between participant subjects, experimenter, and other aspects of the environment making it as if the participants developed something internally. The English translation of the work – which consistently uses the adjective “mental” rather than psychological that better renders the adjective that the psychologist used in his native Russian (i.e.
psikhicheskij) – contributes to the mentalist take on development that manifests itself in the work.
again here (Fig. 7.2). The configuration is emergent and living; it is therefore appro- priate to model it as an event. This event of the emerging classification scheme includes and consists of an entire series of mini-events or phases of the larger event;
the coming to life of the configuration extends over these phases (mini-events) in which the different geometrical categories take shape. Each of these categories tagged with labels in the course of the emergent classification – e.g. cube, rectangle, cone, pyramid, and sphere – is an event realized in the coming together of different mystery objects that are said to have something in common even though they are different in size and color. Each category exists in the form of an ensemble of forms.
In terms of the analogy developed in Chap. 2, each category exists like a strand made from fibers as the trajectories of the objects come together and for a while relate because of their spatiotemporal location on the same colored sheet of heavy paper. We therefore have an emergent relation of things, which, at the end of the lesson, will have come together in groups of material things. But that coming together, that relation between things belonging or not belonging into the same group is tied up with the event of the relation between people, as shown in the sub- sections that follow with respect to different aspects important to the knowing and learning of mathematics. Just as in the example of exchange-value, which is an expression of the twofold relation of people and things, the geometrical categories will be existing, after the lesson event will have ended, as the expression of the twofold relation of people (students, teacher) and things.
In the end, all 22 mystery objects will have found a place on the floor. They mani- fest an aspect of the classical concept-learning paradigm in that each mat corre- sponds to a class of included objects standing over and against all the other objects that are elsewhere on the floor (on other mats) and thus not included. These other objects do not form some oblique class, as in the classical case where they were grouped as “non-included” (Fig. 7.1), but instead have their place in another group
Fig. 7.2 The configuration on the floor is not stable but continuously changes as more and more objects and mats are added, moved about, etc. To understand this emergence requires a transactional approach, where the minimum unit is itself an event rather than a state at a fixed point in time
of objects relative to which all others were “non-included” objects. The result is a complete (exhaustive) category system – e.g. temporally last arrangement in Fig. 7.2. The lesson shares similarities with the above-discussed paradigm used in the experiments of Vygotsky and collaborators. In those experiments, the nature of the four groups was emergent – for the research subjects – and predetermined by the names found on the back of the objects used. In those experiments, therefore, there was a hidden order, which all participants knew to be known by the experimenter but (initially) unknown to them.
The mathematics lesson provided opportunities for learning these geometrical categories before any formal rules for inclusion and exclusion were specified (e.g.
six equal faces, eight corners, and twelve edges for the cube). The concepts do not exist in the form of abstractions but in the form of groups (classes) of actual material objects. Within a specific group, each object, despite its differences with other objects on the same mat, is a manifestation of the category. By the same token, the category is known not abstractly – by means of definitions – but concretely, in the plurality of expressions available on any particular mat. The underlying method has been called the documentary method in concept learning (Roth 2017). This method originally was described as being used by investigators of cultural objects: How does the investigator of a cultural object – e.g. impressionistic painting – know and investigate the phenomenon? Subsequent studies have shown that the documentary method is used across a wide range of societal situations where people are involved in finding out about something that they do not already know (Garfinkel 1967) – e.g.
how does a coroner find out the exact cause of death? As a means for investigation, the method works like this. The investigator takes some observable fact as a con- crete manifestation of the phenomenon of interest – e.g. a painting by Claude Monet, a composition by Claude Debussy, or a poem by Arthur Rimbaud. All these cultural artifacts produced during a particular year or period of years may be taken to be concrete manifestations of the spirit of the culture. Whereas sociocultural studies recognized the role of the sensuous material in the formation of concepts, its mentalism is clear in viewing the concept as something abstracted from the con- crete. It is said that the sensuous material is perceived and transformed to give rise to a mental thing: the concept. But in the documentary method, the concept exists only in and through its material forms. It is similar to what has been described as family resemblance, where the “various resemblances between members of a fam- ily: build, features, color of eyes, gait, temperament, etc. etc. overlap and crisscross in the same way” (Wittgenstein 1953/1997, 32). Games, including that in which mystery objects are grouped on the floor, form families.3 The spirit of the times thus is known only through its concrete manifestations. Impressionism, for example, is not something abstract but known through its many concrete manifestations that are made reference to in relations with others. In the present lesson, the categories exist for the children in the same way, as concrete manifestations of ordering objects other than by color and size, an order that will be characteristic of geometry, the
3 In Wittgenstein’s families, there does not have to be a single property (characteristic) that is com- mon to all members, unlike the “families” in the classical concept formation paradigm, where there are two or more characteristics in common to all those within the family.
subject that they do not yet know but are in the process of learning. Indeed, the concrete classification on the classroom floor is the first manifestation of their learn- ing trajectory. The different categories are forming while existing individual or bundles of lifelines (moving objects currently holding a position) come to be aug- mented by an additional one (e.g. two objects on a mat that are joined by a third object after it has undergone an extensive trajectory over and above the existing configuration on the floor, as in the following sections), or a new category is formed in the conjunction of an object with an empty colored mat.
Reasoning Work
In this subsection I demonstrate in which way mathematical reasoning, in being communicated and thereby implying its intelligibility, inherently rather than acci- dently manifests itself in and as social event. The analyzed phase of the lesson begins after Connor had been invited to come forward, pull an object from the bag, and then place it. There is a description of what is to happen: that it was his turn and that he had to explain his thinking. We can hear an invitation: “Now look at the groups (Mrs. W gestures over the configuration of mats and objects), does it belong to another group or can you start a new group with that. The placement of the object (turn 1) accomplished by Connor and seen by others is the reply, that is, the second phase of the act of responding. The statement, “why does it get its own category now” (turn 3) not only presupposes and implies a norm but also, in requesting, “you have to explain your thinking,” the respondent is held accountable to the actions that preceded (i.e. the placement of the mystery object on its own mat). If a new category is formed, then there must be something different; and this difference has to be accounted for in terms of a reason. That is, what Connor has done is treated as not conforming to a norm that is yet to be enacted. The statement thus formulates what will eventually be emerging: the mathematical norm of tying a mathematical action to its account (reason).
Fragment 7.1
1 C: ((Places object on empty mat)) 2 (1.08)
3 W: NOW. befORE YOU GO ((Holds right hand out in “stop”
configuration)) you have to explain your thinking. why does it get its OWN group now.
4 C: cause this one is sort of (0.32) bigger than the other ones? ((He has walked back and now stands over his object.))
In the fragment, this tying of action and reasoning has not yet happened in its entirety: it is an event in its unfolding and it is flagged as happening. Its first phase, the categorization, has happened; and this statement, which is articulating that something is missing and thus also is issuing an invitation for the remainder to come forth, is, at this stage, part of the joint work that will have made the event happening.
That invited articulation of a reason is accepted in the forthcoming of a reply in which the size appears as the decision criterion.
The norm of tying a mathematical action to reasoning is an event that plays out in the public forum of the circle. The event and every one of its phases make sense as something that is happening. There is nothing that is not already intelligible. The event is observable as such in the sequentially organized turn taking. Indeed, the act that ties categorization and the articulation of its underlying reason exists as the relation, for the very speaking that provides the reason for an antecedent categoriza- tion also is the relation between the two people (Connor and Mrs. Winter).
Mathematical reasoning is social because it is witnessable and observable in and as that real turn-taking relation first. If it did not or could not exist for two people, it would not exist at all, as an object of consciousness. What we observe here has the twofold relation of people and things (which may be in the form of the topic of talk) described above. When at some later point in the curriculum this tying of action and reasoning was observed in the actions of individual students, it was no less social through and through – because it was intelligible as a way of proceeding in this and subsequent mathematics lessons. This was the point made in the fragmentary text
“Concrete Human Psychology” (Vygotsky 1989) from 1929, but which the author did not consequently implement, for remnants of individualistic approaches to con- cept learning appeared in the subsequently completed but posthumously published work on concept formation discussed above. To show the social nature of the work of mathematical reasoning, I draw on the framework laid out in Chap. 3, augmented here by an articulation of the structure of practical actions from the perspective of the sociologies of the visible order (Garfinkel and Sacks 1986). These sociologies focus on the public nature of the inherently joint work event by means of which the visible (social) order is accomplished such that people always are aligned as to the sense of what is currently happening.
The recurrent pattern of the practical action, categorizing by shape, may be denoted in this way: “doing [sorting by (geometrical) shape].” The “doing” refers to the actual work, which is a form of event and has to be theorized in terms of evental categories. The parenthesized statement “sorting by geometrical shape” is a gloss of the type that members (e.g. the two teachers in the classroom) use to characterize this work. Mrs. Winter had articulated such a gloss at and as the beginning of the task, in saying “when we do our any kinds of sorting activities today, we’re not going to do them by color and we’re not going to do them by size.” This gloss, here, is something like a recipe (instruction) that the children are asked to follow. The actual work of the sorting was accomplished 22 times during the task, in and as micro-events that produced this part of the lesson. The work of sorting was com- pleted in and as joint work. Sorting, as event, involves two phases: (a) the act of attributing an object to a group that in this classroom has or does not yet have a
name and (b) the act of stating a reason for the attribution. We notice in the lesson as a whole and in this fragment specifically that event “acting + reasoning” already is intelligible and makes sense. From a classical psychological perspective, the chil- dren already know how to sort and reason in an intelligible way, though the specific permanences used are not those that would be typical in a mathematical context (i.e.
color and size). Specifically mathematical is the characteristics of the classification,
“by shape,” and the tying of the placement to its specifically mathematical account (reason). In other subject areas, like the fine arts, different classificatory character- istics would have brought out the specificity of those fields, including those that sort objects by color and size. There is nothing outside the children that would not already be inside; and, conversely, there is nothing inside that would not already be outside.
In the traditional psychological literature, every phenomenon is reduced to the actions of individuals; and these actions are causally linked to supposed contents and structures of the individual mind. This literature would focus on the fact that Connor (a) did not provide the reason for his sorting action without solicitation and (b) provided a reason that is inappropriate in the mathematical context. The socioculturally oriented part of this literature might highlight that the teacher, in soliciting an explanation, “scaffolded” Connor into providing one, even though the forthcoming reason is another instance of what has repeatedly been described as inappropriate: sorting by size. The transactional approach allows us to see that the event as a whole exists for Connor, as it does for Mrs. Winter, in all of its phases though it does so asymmetrically. For both individuals, all the phases of reasoning exist (Fig. 7.3). This is so because Connor not only is categorizing the object and articulating a reason but also he is actively attending to Mrs. Winter and receiving the words; the same phases of mathematical reasoning exist for Mrs. Winter
Fig. 7.3 Mathematical reasoning is a micro-event that (asymmetrically) exists for both. It is the micro-event that intersects both lifelines, which thereby come to be immanent in one another.
Reasoning exists in a twofold manner: in and as the relation
(Fig. 7.3).4 Thus, (a) the relation of Mrs. Winter and Connor and (b) their relation to mathematical reasoning as a witnessable event are happening in and as the same micro-event in the event that brings forth the ultimate 23-object configuration as a whole (one object placed by each one of the 22 children and the first one that Mrs.
Winter placed). The upshot of this description is that there is nothing for Connor to internalize, as studies avowing to a sociocultural framework or taking a sociocul- tural lens often state in taking up from Vygotsky. In fact, the representation shows that from the very outset in the event sequence, there was nothing other that was not already Connor’s own. There is but a steady coming and going, in the event that intersects both lifelines in precisely those parts shown in Fig. 7.3. There is nothing coming from the outside that is not already on the inside, for what is outside is so precisely because we experience and comprehend it as such on the inside (Mikhailov 2001). This is why any form of mind, here taken as an event, is possible only at the interface between people, which exists in the form of an event where multiple life- lines intersect.
The lesson fragment exemplifies how the children, in their first geometry lesson, did not on their own tie classificatory actions to accounts. Like Connor, other stu- dents as well were placing their objects on separate mats and then were retreating towards their places in the circle without having provided a reason that “explained their thinking” (cf. Gina in Chaps. 1 and 3). Explaining one’s thinking is not inher- ently part of human practices. For example, in other occasions, such as the work of a fish sorting facility that I observed in a hatchery, the workers were not asked to provide verbal accounts (Roth 2005). Instead, the placement was taken as an account of their thinking; and when the placement of a specimen was error, then another worker would complete the corrective action by placing the specimen on the correct conveyor belt. The work set up for a group was accomplished by the group. There was never an issue over and about the “wrong” placement of a fish on the part of an individual worker. In other contexts analyzed in the same study, such as exploratory work in scientific research, classifications required reasoning, which was embedded and materialized in evolving operational definitions that specified group member- ship. In the same way, reasoning is specifically mathematical only in the tie of an action recognized as mathematical and of a specific intelligible verbal account that provides a justification of the action. That tying of action and justificatory account was one of the mathematical norms observed later in the unfolding geometry cur- riculum but that was a micro-event observed early on in the mathematics lesson in the relation between people. How then did that mathematical norm emerge? In that it was social relation first: it was the social relation, which was unfolding as event over the course of three turns at {talking | listening}.
Some readers might be tempted to suggest that an instance of scaffolding has occurred. But in the transactional approach it is not that Mrs. Winter, the teacher,
“scaffolds” the student Connor or provides for the conditions that create some zone
4 This description is consistent with recent neuroscientific research according to which the two events doing something and observing someone else doing it both are associated with the same neuronal events (Rizzolatti et al. 2006); and the micro-events of hearing action phrases were asso- ciated with (mirror) neuronal events observed when the individual actually does what the phrase describes (Buccino et al. 2005).