The Emergence of 3D Geometry From Children's (Teacher-Guided) Classification Tasks

Abstract

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 present study therefore is this: How can a lesson in which children are asked to participate in a task of classifying regular 3-dimensional objects be a geometry lesson, given that the participating 2nd-grade children do not yet classify according to geometrical properties (predicates)? In our analyses, which are inspired by ethnomethodological studies of work, we focus on the embodied and collective work that leads to the emergence of the geometrical nature of this lesson. Thus, we report 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 (color, size, “pointy-ness”) properties—are achieved. In the process, our study shows how geometrical work is reproduced by 2nd-grade children who, in a division of labor with their teachers, produce a particular set of geometrical practices (sorting three-dimensional objects according to their geometrical properties) for the 1st time.

Notes

¹We use the adjectives sensual and sensuous in their dictionary meanings of “pertaining to the senses,” in the way these terms have been used by philosophers such as G. W. F. Hegel and K. Marx.

²There are other possible theories to be discussed (Bruner on concept learning), but culturally and historically they do not take the same central role in the field of mathematics education as the two reviewed here, likely because they are not specific to mathematics.

³In recalling the kinds of cases where a game is said to be played according to a definite rule, Wittgenstein (1958) listed, 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” (p. 26e, 26–27e)

⁴In this article, we 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 hand, consistent with the conversation analytic tradition, we use the analytic term turn, which is short for “turn at talk.” Thus, in Colby's “turn” there are many turns at talk, as shown in and by the analyses presented here.

⁵In conversational terms, pauses in speaking that are longer than 1 s are substantial and longer than the norm (Jefferson, 1989); ample research on wait time in the late 1970s and early 1980s showed that teachers generally do not wait that long prior to intervening (Tobin, 1987).

⁶We thank Tim Koschmann for pointing this and the following points out to us.

⁷Because of the choice of the episode, intended to highlight the work of classifying (rule following) that members of a group exhibit for one another to deal with some breakdown, “successful” rule following is not shown. 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.

⁸We 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).

⁹The mystery figures are not inherently geometrical objects but become such in and through the practices of the group members. This situation is similar to the one involving a baseball umpire 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.”