Koeno Gravemeijer

Participating in Classroom Mathematical Practices

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students’ diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.

Learning to reason about distribution

The purpose of this chapter is to explore how informal reasoning about distribution can be developed in a technological learning environment. The development of reasoning about distribution in seventh-grade classes is described in three stages as students reason about different representations. It is shown how specially designed software tools, students’ created graphs, and prediction tasks supported the learning of different aspects of distribution. In this process, several students came to reason about the shape of a distribution using the term bump along with statistical notions such as outliers and sample size. This type of research, referred to as “design research,” was inspired by that of Cobb, Gravemeijer, McClain, and colleagues (see Chapter 16). After exploratory interviews and a small field test, we conducted teaching experiments of 12 to 15 lessons in 4 seventh-grade classes in the Netherlands. The design research cycles consisted of three main phases: design of instructional materials, classroom-based teaching experiments, and retrospective analyses. For the retrospective analysis of the data, we used a constant comparative method similar to the methods of Glaser and Strauss (Strauss & Corbin, 1998) and Cobb and Whitenack (1996) to continually generate and test conjectures about students’ learning processes.

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 childerns language B.van Oers. Symbolizing space into being R. Lehrer, C. Pritchard. Mathematical representations as systems of notations-in-use L. Meira. Students 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.

Emergent Models as an Instructional Design Heuristic

In this chapter, the design of an instructional sequence dealing with flexible mental computation strategies for addition and subtraction up to one hundred, is taken as an instance for elaborating on the role of ‘emergent models’ as an RME design heuristic. It is explicated how the label ‘emergent’ refers both to the character of the process by which models emerge within RME, and to the process by which these models support the emergence of formal mathematical ways of knowing. According to the emergent-models design heuristic, the model first comes to the fore as a model of the students’ situated informal strategies. Then, over time the model gradually takes on a life of its own. The model becomes an entity in its own right and starts to serve as a model for more formal, yet personally meaningful, arithmetical reasoning. The analysis of the exemplary instructional sequence, is used to show that the transition from model-of to model-for involves the constitution of a new mathematical reality that can be denoted ‘formal’ in relation to the original starting points of the students. Furthermore, attention is given to the dynamical character of the emergent model; there is not one model, but the model is actually shaped as a cascade of inscriptions. The latter observation forms the starting point for a more detailed discussion of the role of these individual inscriptions in the learning process.

Computer Algebra as an Instrument: Examples of Algebraic Schemes

In this chapter, we investigate the relationship between computer algebra use and algebraic thinking from the perspective of the instrumental approach to learning mathematics in a technological environment, which was addressed in the previous chapter.

Preamble: From Models to Modeling

The objective of this chapter is to situate this book by giving a global overview of the history of the change in perspectives on symbolizing and modeling in the mathematics education community. This history describes a shift from the use of symbols and models as embodiments of mathematical concepts and objects in instructional practice, design and theory, to explorations in semiotics as a central field of interest. Underlying this shift is a shift from correspondence theories of truth to contextualist theories of truth. The latter category encompasses constructivism and socio-cultural theory, which constitute the main background theories that are currently adopted in the mathematics education community. The chapter starts with a discussion of two instruction theories that have incorporated the classical use of manipulative materials and visual models. These concem the so-called ‘mapping theory’, which has emerged within the context of information-processing theory, and Gal’perin’s theory of the stepwise formation of mental actions. Next follows a sketch of the constructivist critique. This is followed by a discussion of the role of (cultural) tools from a socio-cultural perspective. Finally the change in ways of describing and conceptualizing symbolizations that has emerged recently is addressed. In relation to this, the semiotic notion of a sign as an integrated signifier/signified pair is discussed. This is complemented with a discussion of the notion of an inscription as the material correlate of a sign, and of the instrumentation of ict tools.

Emergent Modelling as a Precursor to Mathematical Modelling

This chapter discusses the relation between ‘emergent modelling’ and ‘mathematical modelling’. The former that has its roots in RME theory constitutes the main theme of this chapter. It is argued that mathematical modelling requires a preceding learning process, since it requires abstract mathematical knowledge to construe a mathematical model. The emergent-modelling design heuristic offers a means for shaping a series of modelling tasks that may foster the development of that abstract mathematical knowledge. The emergent-modelling heuristic is illustrated with an instructional sequence on data analysis.

Using multimedia cases for educating the primary school mathematics teacher educator: a design study

The overarching goal of this chapter is to better understand how multimedia video case studies can support the professionalization of primary-school-mathematics teacher educators. We investigate the use of multimedia cases to support teacher educators in learning to mathematize and didactize and to learn how to use multimedia cases with their student teachers. The research study has an exploratory character; we will present a framework for the use of multimedia cases as a tentative answer, grounded in the researchers’ experiments and design activities.

Transforming Mathematics Education: The Role of Textbooks and Teachers

In this chapter, we discuss the question of how we can encourage mathematics education to shift towards more inquiry-oriented practices in schools and what role textbooks and teachers play in such a reform. The stage is set by an exposition on the need for curriculum innovation in light of the demands of the twenty-first century. This points to a need to address goals in the area of critical thinking, problem solving, collaborating, and communicating. However, previous efforts to effectuate a change in mathematics education in that direction have not been very successful. This is illustrated by experiences in the Netherlands. In relation to this, the limitations of transforming education using textbooks and problems with up-scaling are discussed. To find ways to address these problems, an inventory is made of what can be learned from decades of experimenting with reform mathematics education while trying to achieve the very goals that are discerned as crucial for the twenty-first century. On the basis of this inventory, suggestions are made for shaping textbooks in such a manner that they may better support this kind of transformation. At the same time it is pointed out that the latter requires a complementary effort in teacher professionalization and a well-considered alignment of both efforts.

Promoting science and technology in primary education: a review of integrated curricula

Integrated curricula seem promising for the increase of attention on science and technology in primary education. A clear picture of the advantages and disadvantages of integration efforts could help curriculum innovation. This review has focused on integrated curricula in primary education from 1994 to 2011. The integrated curricula were categorised according to a taxonomy of integration types synthesised from the literature. The characteristics that we deemed important were related to learning outcomes and success/fail factors. A focus group was formed to facilitate the process of analysis and to test tentative conclusions. We concluded that the levels in our taxonomy were linked to (a) student knowledge and skills, the enthusiasm generated among students and teachers, and the teacher commitment that was generated; and (b) the teacher commitment needed, the duration of the innovation effort, the volume and comprehensiveness of required teacher professional development, the necessary teacher support and the effort needed to overcome tensions with standard curricula. Almost all projects were effective in increasing the time spent on science at school. Our model resolves Czerniac’s definition problem of integrating curricula in a productive manner, and it forms a practical basis for decision-making by making clear what is needed and what output can be expected when plans are being formulated to implement integrated education.