Cobb starts with a short overview of the relation between symbolizing and mathematizing. Symbols that students use have influence on math course and results (Kaput, Thompson). Mathematical reality is symbolizing into being (Sfard, Dorfler). Mathematics is an activity that is mediated by the use of symbols and tools (Meira, van Oers). People's activity with symbols should be analyzed as an integral aspect of their mathematical reasoning (Cobb, Meira). The purpose of this paper is to explore the interplay between the development of ways of symbolizing and the development of mathematical meaning from a social constructivist perspective. This research is motivated by an interest in instructional design (as part of developmental research). Instructional sequences go with conjectures about (1) possible learning trajectories, and (2) the means that might be used to support and organize learning. The social constructivist perspective appears from Cobb's interest in collective mathematical development and the fact that this involves modifications in taken-as-shared ways of symbolizing.
RME & symbolizing: use of emergent models, elaborated by identifying four types of activities (in task setting, referential, general and formal) tied to specific task-settings and involving situations specific imagery.
Inscriptions are symbolizations that have physical form.
This introduction is followed by the description of the teaching experiment in the area of statistical data analysis in the seventh grade. For the experiment minitools were designed to fit with taken-as-shared ways of reasoning by the students and supporting the reorganization of that reasoning.
Cobb describes the emergence of the mathematical practices and illustrates them with a few protocols. The practices consisted of: (1) the exploration of qualitative characteristics of collections of data points, and (2) the exploration of qualitative characteristics of distributions.
The use of the tools influenced the process of the students mathematical development and its products. Cobb specifies this observation by specifying the interplay between the development of ways of symbolizing and of mathematical meaning in terms of chains of signification and types of activities. In relation to the minitools he mentions the notion of affordances, characterized as social accomplishments and refers to Meira: "instructional devices should be thought of in connection to some task, system of activities, and cultural context in relation to which they make sense". This implies that for instructional design the focus should be on how students might reason with inscriptions as they participate in an evolving sequence of (classroom) mathematical practices.
In the end Cobb clarifies his use of the term model (derived from the RME theory): "a model is defined in terms of signifying relations established in activity for some purpose. Further, models are seen as originating not from situations but from activity in and reasoning about situations." The model and the situation being modeled co-evolve. Modeling in this view is a process of reorganizing both activity in and about a situation, the situation becomes to be structured in terms of mathematical relationships.
Modeling in the traditional way, where models are abstractions from situations, serves to clarify an overall goal of instruction by delineating a perspective we want students to be able to take on their modeling activity. The above sketched role of modeling serves mathematical development. Dus aan de ene kant modelleren ten dienste van het leren van wiskunde (daar gaat het in dit artikel om), en aan de andere kant heeft het onderwijs als doel om leerlingen bewust te maken van het modelleren als toepassing van de wiskunde. In dat laatste verband wordt verwezen naar de traditionele rol en definitie van modelleren.
Zie See Confrey, Jere and Shelley Costa en See Roth, Wolff-Michael, Michelle K. McGinn voor min of meer vergelijkbare aandacht voor tool-use.