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Ontogenetic Origin of Mathematical Reasoning

In the preceding section we show how the children in the class were beginning to reason mathematically (Sylvia did, Melissa did not in the episode). At the beginning of this curriculum on geometry, the children did not tie their actions to reasons. In this example, we enter the video recording in the middle of the task that had taken up almost the entire first lesson by means of which the teachers had introduced the three-week geometry unit. The task began like this. The teacher pulled an object from a black plastic bag and explained that the task consisted in either putting the object to a group of like objects or to start a new group. Because hers was the first object, the teacher put a colored mat on the floor and placed the object on it. Then she added another, empty mat to the floor. The teacher called a child by name, who then had a turn at pulling an object (without feeling around) and at placing it with other objects or on a mat of its own. The children were asked to state their thinking, but neither color nor size was to be used as reason for classifying an object. During that entire lesson, only one child in that class of 24 placed an object and provided an account (reason) to explain why s/he had categorized his/her object in that way.

We enter the lesson during Connor’s turn, who already had placed his object (a cube) on a mat of its own—even though the geometrically savvy could see that there already existed a group of that type of objects. Prior to turn 01 in Fragment 5.2, the teacher has asked Connor to take a look at his “block,” take it to each of the existing groups, and decide whether it belongs to any one of these. The offprint in turn 01 features Connor holding his object next to a cone, which occurs after he has held it next to a group of objects on a mat labeled “cubes, squares,” and following which he will be holding it to a pyramid. He then completes a {question | reply} in turn 02. The fragment exemplifies how the children, in their first geometry lesson, initially do not tie classificatory actions to accounts, exemplified here in Connor’s placement of his mystery object on a mat and then remaining silent without having provided a reason. Indeed, most children immediately began to retreat to their seat so that the teacher, in each case, said something of the kind that she does in turn 04 in the fragment above. In Connor’s case, there was little time between his action and the reply, which came as a request to state a reason for thinking in this way.

Fragment 5.2

Failing to provide a reason is not inherently a fault; and there is nothing unusual about not providing reasons for classifications. Thus, for example, in a fish sorting facility that we observed in the course of five-year ethnographic study, the workers were not asked to provide verbal accounts (Roth et al. 2008). Instead, the placement of the fish was taken as an account of the workers’ thinking; and when the placement of a specimen was “erroneous,” then another worker engaged in corrective action by placing the specimen on the appropriate conveyor belt (Roth 2005). Similarly, as observed in a scientific laboratory, when the actions of a scientist were questioned, it was always because other members were seeing something relevant that the actions had not acknowledged (e.g. moving the slide under a microscope when others said they had seen a research relevant object). The results of the action are treated as accounts of the work.

In this classroom, however, reasoning is counted only when it is tied to a specific verbal account that makes it mathematical rather than something else. In the fine arts, students might indeed legitimately classify by color (but not by size), whereas in the science class, they may legitimately classify by size (but not by color and shape). That tying of action and justificatory account so that it is recognizably mathematical is one of the “socio-mathematical norms,” one of the higher functions observed later in the lessons (as described in the preceding section). It did not exist early on in the unit. How then do such social norms emerge? They do so in and as of the transactional order (social relation). In the present situation, the norm is the social relation anchored on two turn pairs, one that Mrs. Winter completes and the other one that she initiates; and she does so with the same turn.

{C: ((Places object with the two cubes))

W: and can you tell us why you think that д C: coz these are more squares.

The second turn begins with a conjunctive “and” rather than with some other way of continuing, which could be a “but” or “are you sure?” Instead, the “and” accepts what has been done (i.e. positively evaluates it, at least for the moment) and simultaneously asks for something additional (provides an opportunity for adding something) that apparently is missing from what has preceded. The turn also provides a description, an instruction for what is missing to come both as process (“can you tell us”) and content (“why you think that”). As such, it also is the first turn of a {query I reply}, which is constituted as such by the second turn, “coz there are more squares.” Importantly, then, Connor has produced a classification and he has provided an account. That is, he has completed the two parts that we characterize above as the structure of practical (social) action: He has done the work of a classification by placing his object with others on a mat labeled “cubes, squares” and he has provided an account (reason) for doing so: doing + “coz theses are more squares.” That “and” in the middle turn invites both the next part and its connection to the first. Importantly, that invitation cannot be simply attributed to Mrs. Winter. As we see in Chap. 4 and as we repeat above, an invitation is an invitation only because there is an acceptance. Whatever Mrs. Winter says also rings in Connor’s ears. What Connor first produces as a reason for Mrs. Winter eventually becomes a reason for himself. His action, in taking up the invitation and by providing a reason accepting it, is an integral part of the joint (social) action. There already exists a communica- tional competence that is generative of the higher order function that we observe.

Connor may not have produced the two parts on his own, doing so only after having been invited to add the second part. But he is already doing the two parts and, therefore, does not have to somehow internalize them, as is so frequently suggested in the literature. There is nothing “socially constructed” visible in their circle or between Connor and Mrs. Winter that the former can now “construct in his mind.” Moreover, if Mrs. Winter were said to be scaffolding, she would not be constructing but only providing a context for it.11 Instead, for the traditional educational psychologist to state that Connor reasons mathematically, the boy only has to make the second part follow the first without Mrs. Winter having a turn in between.

We note above that the turn “and can you tell us why you think that” is understood as part of a joint (social) action. In this sequence, there are two turn pairs (I, II) hinging on one turn that “glues” the other two turns together. It is a contextual modification that allows the joining of the two turns. That middle turn completes Pair I by evaluating what has preceded, both in affirming it and constituting its shortcoming, which precisely is a statement of the reasoning that accounts for the classificatory act. This middle turn also initiates another pair (II) that invites the recipient to state his thinking. The next turn—following, as we see in Fragment 5.2, a considerable pause—accepts the invitation by stating that there are more squares. The middle turn thereby offers up an invitation not only to add something but in fact to connect the second, discursive action to the first. It is as the sequence of classification, solicitation of thinking, providing reason—a sequential order of distributed work in which the student takes the first and third, and the teacher takes the middle slot—that we first observe the social praxis of classifying (sorting) three-dimensional objects. There is an order at work that is witnessable by those present: classificatory action followed by provision of reason. It first exists in the sequential turn-taking order of student (S) and teacher (T) in the form S-T-S and later is observed in a single turn S (e.g., Melissa, Sylvia). That is, there is a reduction of a transactional order S-T-S ^ S that parallels the process of pairing from {classifying | requesting accounting | accounting} ^ {classifying | accounting}.

So far we have achieved only the first part of showing that a higher psychological function, here tying action and reason, first is a social relation. The second part has not yet been shown: the sociality of mathematics (geometry). Any suggestion that mathematics is a social practice has to show the essentially mathematical dimension in an observed action. We see part of this additional work in Fragment 5.3, which derives from the earlier part of Connor’s turn when, after having placed his object on a mat of his own, Mrs. Winter asks him to state his thinking, especially inviting him to address “why does it get its own group?” The reply (turn 01) states that the object is bigger than the other objects (already on the floor). The reply to the reply (turn 03) requests to stop and to remember what previously has been stated as a rule (norm): size and colors do not count in this (language-) game.

Fragment 5.3

“When a house is painted, scaffolds are used. But the scaffold has nothing to do with painting. It

supports the painter in doing what she has to do.

Here, then, we have an evaluation of the turn that provides a reason. The evaluation exists in the statement that size is not counted, and, as if attempting to make sure that the recipient produces a statement that does not include size. Color, too, is discounted (“we are not counting”). In that game that they are playing, which earlier has been named geometry, some accounts (reasons) are counted in, and others are counted out. The tie of action and account, as we show above, is social. A distinction is made between (a) those inherently (social) accounts that are in and applicable to the game in play and (b) those that are out. Those in play, therefore, are typical in the game of mathematics (geometry), whereas others, though these are indeed possible accounts and therefore social, lie outside the domain of mathematics.

When the account is provided, the classificatory work is completed. The children have the opportunity to learn, by actively attending to the evaluations of what they have said, which accounts are valid and included and which are not and therefore excluded from the game in play. We may also say that the children learn a rule. They do so not by listening to it being stated in words—in the course of this lesson, Mrs. Winter finds herself in the position of having to restate the rule eight times in the course of the lesson—but by constituting a living part of the transactional order that requests them to select an alternative reason whenever their first option had been color or size. It is by (affectively as much as intellectually) attending to the pairing in {turn 01 | turn 03}—which here discounts the contents of turn 01—and by acting in appropriate ways that the children come to know the rules for participating in the game of mathematics rather than some other game.[1] Accounts that address shape are in, but accounts in terms of size or color are out. As a consequence, those accounts that remain in the game, the (social) rules that describe what is happening in this classroom, are social and mathematical through and through. They are so not because some negotiation has led to them but because they are both a transactional order (event) and the content thereof (product).

  • [1] It should be apparent that children’s turns are “appropriate” as long as the equivalent to turn 03does not discount what they have said or done.
 
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