Conducting Teaching Experiments in Collaboration With Teachers

ion so that the results of activity can be anticipated and activity itself becomes an entity that can be manipulated conceptually (Sfard, 1991; P. W. Thompson, 1994b; von Glasersfeld, 1991 a). One of the central tenets of RME is that the starting points of instructional sequences should be experientially real to students in the sense that they can engage immediately in personally meaningful mathematical activity (Streefland, 1991). In this regard, P. W. Thompson (1992) noted from the emergent perspective that: If students do not become engaged imaginistically in the ways that relate mathematical reasoning to principled experience, then we have little reason to believe that they will come to see their worlds outside of school as in any way mathematical. (p. 10) As a point of clarification, it should be stressed that the term experientially real means only that the starting points should be experienced as real by the students, not that they have to involve realistic situations. Thus, arithmetic tasks presented by using conventional notation might be experientially real for students for whom the whole numbers mathematical objects. In general, conjectures about the possible nature of students' experiential realities derived from psychological analyses. It also can be noted that even when everyday scenarios used as starting points, they necessarily differ from the situations as students might experience them out of school (Lave, 1993; Walkerdine, 1988). To account for students' learning, therefore, it is essential to delineate the scenario as it is constituted interactively in the classroom with the teacher's guidance. A second tenet of RME is that in addition to taking account of students' current mathematical ways of knowing, the starting points should be justifiable in terms of the potential endpoints of the learning sequence. This implies that students' initially informal mathematical activity should constitute .a basis from which they can abstract and construct increasingly sophisticated mathematical conceptions as they participate in classroom mathematical practices. At the same time, the situations that serve as starting points should continue to function as paradigm cases that involve rich imagery and thus anchor students' increasingly abstract mathematical activity. This latter requirement is consistent with analyses that emphasize the important role that analogies (Clement Presmeg, 1992; Sfard, 1994), prototypes (Dorfler, 1995), intuitions (Fischbein, 1987), and generic organizers (Tall, 1989) play in mathematical activity. 12. CONDUCTING TEACHING EXPERIMENTS 319 In dealing with the starting points and potential endpoints, the first two tenets of RME hint at the tension that is endemic to mathematics teaching. Thus, Ball (l993b) observed that recent American proposals for educational reform are replete with notions of 'understanding' and 'community'-about building bridges between the experiences of the child and the knowledge of the expert (p. 374). She then inquired: How do I as a mathematics teacher create experiences for my students that connect with what they now know and care about but that also transcend the present? How do I value their interests and also connect them to ideas arxi traditions growing out of centuries of mathematical exploration arxi invention? (p. 375.) RME's attempt to cope with this tension is embodied in a third tenet wherein it is argued that instructional sequences should contain activities in which students create and elaborate symbolic models of their informal mathematical activity. This modeling activity might entail making drawings, diagrams, or tables, or it could entail developing informal notations or using conventional mathematical notations. This third tenet is based on the psychological conjecture that, with the teacher's guidance, students' models of their informal mathematical activity can evolve into models for increasingly abstract mathematical reasoning (Gravemeijer, 1995). Dorfler (1989) made a similar proposal when he discussed the role of symbolic protocols of action in enabling reflection on and analysis of mathematical activity. Similarly, Kaput (in press) suggested that such records and notations might support the taking of activities or processes as entities that can be compared and can come to possess general properties. To the extent that this occurs, students' models would provide eventually what Kaput ( 1991) tenned semantic guidance rather than syntactic guidance. In social terms, this third tenet implies a shift in classroom mathematical practices such that ways of symbolizing developed to initially express informal mathematical activity take on a life of their own arxi used subsequently to support more formal mathematical activity in a range of situations. A discussion of RME as it relates to the design of instructional sequences involving technology-intensive learning environments was provided by Bowers (1995). It is apparent that the interpretation of RME I have given implies conjectures about both individual arxi collective development. For example, the speculation that students' models of their informal mathematical activity might take on a life of their own and become models for increasingly abstract mathematical reasoning implies conjectures about the ways in which they might reorganize their mathematical activity. Further, this conjecture about individual development