Symbolizing, Modeling and Tool Use in Mathematics Education

Introduction and overview; K. Gravemeijer, et al. Preamble: from models to modelling; K. Gravemeijer. Section I: Emergent Modeling. Introduction to Section I: Informal representations and their improvements; B.van Oers. The mathematization of young childern's language; B.van Oers. Symbolizing space into being; R. Lehrer, C. Pritchard. Mathematical representations as systems of notations-in-use; L. Meira. Student's criteria for representational adequacy; A. diSessa. Transitions in emergent modeling; N. Presmeg. Section II: The Role of Models, Symbols and Tools in Instructional Design. Introduction to Section II: the role of models, symbols and tools in instructional design; K. Gravemeijer. Emergent models as an instructional design heuristic; K. Gravemeijer, M. Stephan. Modeling, symbolizing, and tool use in statistical data analysis; P. Cobb. Didactic objects and didactic models in radical constructivism; P.W. Thompson. Taking into account different views: three brief comments on papers by Gravemeijer and Stephan, Cobb and Thompson; C. Selter. Section III: Models, Situated Practices, and Generalization. Introduction to Section II: models, situated practices, and generalization; L. Verschaffel. On guessing the essential thing; R. Nemirovsky. Everyday knowledge and mathematical modeling of school word problems; L. Verschaffel, et al. On the development of human representational competence from an evolutionary point of view: from episodic to virtual culture; J. Kaput, D. Shaffer. Modeling reasoning; D. Carraher, A. Schliemann. Index.