multiplication makes bigger; division makes smaller"

Naming the result is a separate act, and the result of splitting is, in fact, named to be consis-tent with the counting system rather than the splitting sys-tem, a practical decision, but one that has contributed signif-icantly to the confusion of the two systems of number construction.

Our pH scales, which transform the multiplicative relationships among the changes in hydrogen ion concentra-tions into our more familiar counting numbers, lull un-suspecting citizens into complicity in large scale environ-mental damage.

He advocated that, to un-derstand natural phenomena, one needed to examine their deviance from simplicity, not reduce them to their well-understood patterns.

Splitting needs to be developed as a complement to the counting worlds.

split-ting can be defined as an action of creating simultaneously multiple versions of an original, an action often represented by a tree diagram.

In particular, we believe it is a precur-sor to a more adequate concept of ratio and proportion and subsequently of multiplicative change and exponential func-tions.

Consequently, from an emergent perspective, future teachers are seen to actively construct the beliefs, supposi- tions, and assumptions that subsequently find expression in their pedagogical activity when, as students, they participated in the negotiation of classroom social and sociomathematical norms. 1; this account, a global process of appropriation from the soc

Thus far, we focused on situations in which an emergent approach might be particularly relevant. We turn now to consider situations in which a sociocultural perspective is more appropriat

The link between collective and individual processes in this approach is, theirefore, indirect because: participation enables and constrains learning but does not determine it.

However, we contend that the intqgity of rof~rrn effom is threatand if we focus it narrowly on curriculum reform and fail to locate it in a broader cultural content by considfiring the regimes of truth that sustain currant practices

Our first priority when working with the teachers at this site was to help them make aspects of their textbook-based instruction problematic. We reasoned that only then would they have reason and moti- vation to want to reform their instructional practices while waking with us.

Instead, the teachers seemed to clarify their own understanding of what should count as a mathematical difference as they interacted with their students.

One of our primary conjer;tures is, in fact, that in making these contributions, stude~w reorganize thdr individual beliefs about their own role, othm' roles, and the general nature of mathe- matical activity (Cobb et al., 1989). As a consequence, we take these beliefs to be th@ psy'Ctrolagid cbtrelates of the classroom social norms.

the renegoft'ation of classroom social norms. Examples of social norms for whole-class discussions that the teacher framed as explicit topics far negotiation included explaining and justifying solutions, attempting to make sense of explanations given by others, indicating agree- ment and disagreement, and quektioning alternatives in situ- ations in which a conflict in interpretations or solutions had become apparent.

The sod# perdfleccive is an intefactionist view of c~~f~fm~~l;al or c~llectiv~ c188sro0~~1 ~~QCRSS~SI (Baiauers- feld, Qurnmh@uer, t!k Voi@, 198$.). psychdq@ioal per- spective is a ppyrA01ogical ooastr&Miyj~t view lo$ individual students'(0r the teachw's) activity as they participate in and contribute to the development of these communal processes (von Glasersfeld, 1984, 1992).

ground theory in practice suggest a more collaborative relationship between teachers and re- searchers in which their areas of expertise are seen as com- plementary rather than as hierarchically organized

pragmatically based

We contribute to this ongoing discussion in this article by exploring possible relations between sociocultural theory and various forms of constructivism.