Richard Lehrer

Design Experiments in Educational Research

In this article, the authors first indicate the range of purposes and the variety of settings in which design experiments have been conducted and then delineate five crosscutting features that collectively differentiate design experiments from other methodologies. Design experiments have both a pragmatic bent—“engineering” particular forms of learning—and a theoretical orientation—developing domain-specific theories by systematically studying those forms of learning and the means of supporting them. The authors clarify what is involved in preparing for and carrying out a design experiment, and in conducting a retrospective analysis of the extensive, longitudinal data sets generated during an experiment. Logistical issues, issues of measure, the importance of working through the data systematically, and the need to be explicit about the criteria for making inferences are discussed.

Building functional models: Designing an elbow

Previous work suggests that children largely perceive science as the passive observation and recording of events. In contrast, studies of practicing scientists show a concern with building and testing models. In this study we investigated the role of a design context for developing childrens understanding of science as the construction and revision of models. Grade 1-2 children were given the task of building a model that works like a human elbow. Via discussion, model building, evaluation, and revision, children came to understand that not only motion but also constraints on motion were important qualities to include in their models. Moreover, review of classroom activity and analysis of the postmodeling interview sug- gest that as early as first grade, childrens model-evaluation skills may be quite amenable to development: In comparison to a nonmodeling peer group, modelers were largely able to ignore perceptual qualities when asked to judge the functional qualities of models. Further, they showed an understanding of the modeling process in general that was similar to that of children 3-4 years older. J Res Sci Teach 34: 125-143, 1997. Recent studies suggest that children perceive science largely as a passive process of ob- serving and recording events. In their view, good scientists are those who attend acutely and make accurate and complete records of all that they observe (Carey, Evans, Honda, Jay, & Unger, 1989; Songer & Linn, 1991). Yet, accounts of the work of professional scientists paint a far different picture, one dominated by building and testing models (e.g., Giere, 1988; Hestenes, 1992). In these accounts, model building and testing are essential to the development of theory—models both channel observations and drive the resulting interpretations (Hestenes, 1992). In contrast, childrens views of models, like their views of scientific practice in general, reflect a belief that observation is theory-free and unproblematic. Thus, children tend to view models primarily as physical copies of phenomena, rather than as tools in the service of theory construction and testing (Grosslight, Unger, Jay, & Smith, 1991). In this study, we investigated the role of a design context for developing childrens appreci- ation of science as a model-construction and model-revision process. We chose the design focus because the process of design affords the potential for children to construct, apply, debate, and

From Physical Models to Biomechanics: A Design-Based Modeling Approach

In this study, we used a design context for developing childrens understanding of the natural world via the designing, building, testing, and evaluation of models. In this instance, we asked children to design models of the human elbow. Childrens models were then used as the basis for an exploration of the biomechanics of the human arm. The investigation of biomechanical principles is a major extension of our earlier research. By building on childrens design-based models, we were able to engage students in an investigation of the relation between force and the location of the attachment point of the biceps. In so doing, we were able to provide children with opportunities to develop their understanding of the relations between mathematics and science through the construction and interpretation of data tables and graphs. Design is a form of problem solving in which thinking, tool manipulation, and materials are reflected in the construction of an artifact (Bucciarelli, 1994; Roth, 1996; Simon, 1981). In turn, artifacts become objects that facilitate the sharing of knowledge. Moreover, because many design problems are, at least initially, inherently unspecified, the problem solver must often first define the problem and then explicitly state and justify the relation between the problem and the proposed solution. This promotes a view of knowledge as purposeful, fluid, and conditional (Lehrer, 1993). In this study, we used a design context for developing childrens understanding of the natural world via the designing, building, testing, and evaluation of models. We chose a design focus because the process of design affords the potential for children to construct, apply, debate, and evaluate models, rather than to simply

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.

Models and Modeling Perspectives on the Development of Students and Teachers

This special issue of Mathematical Thinking and Learning describes models and modeling perspectives toward mathematics problem solving, learning, and teaching (Lesh & Doerr, 2003). The term “models” here refers to purposeful mathematical descriptions of situations, embedded within particular systems of practice that feature an epistemology of model fit and revision. That is, “modeling” is a process of developing representational descriptions for specific purposes in specific situations. It usually involves a series iterative testing and revision cycles in which competing interpretations are gradually sorted out or integrated or both—and in which promising trial descriptions and explanations are gradually revised, refined, or rejected. The latter emphasis on the “fitness” of models is critical because it suggests that models are inherently provisional, and it emphasizes that they are developed for specific purposes in specific situations—even though they may endure for longer periods of time, and even though they generally are intended to be sharable and reuseable in a variety of structurally similar situations. The distinction between model and world is not merely a matter of identifying the right symbol-referent matches; rather, it depends intimately on the accumulation of experience and its symbolic representations over time. Models bootstrap MATHEMATICAL THINKING AND LEARNING, 5(2&3), 109–129 Copyright

Reasoning about Structure and Function: Children's Conceptions of Gears

Twenty-three second graders and 20 fifth graders were interviewed about how gears move on a gearboard and work in commonplace machines. Questions focused on transmission of motion; direc- tion, plane, and speed of turning; and mechanical advantage. Several children believed that meshed gears turn in the same direction and at the same speed. Many second graders provided very incomplete expla- nations of transmission of motion. Most children confused mechanical advantage with speed. Yet as the interview proceeded, several fifth graders generalized conceptions about transmission of motion into a rule about turning direction. They increasingly justified their ideas about gear speed by referring to ratio. Chil- drens reasoning became more general, formal, and mathematical as problem complexity increased, sug- gesting that mathematical forms of reasoning may develop when they provide a clear advantage over sim- ple causal generalizations.

Developing Model-Based Reasoning in Mathematics and Science

Abstract It is essential to base instruction on a foundation of understanding of childrens thinking, but it is equally important to adopt the longer-term view that is needed to stretch these early competencies into forms of thinking that are complex, multifaceted, and subject to development over years, rather than weeks or months. We pursue this topic through our studies of model-based reasoning. We have identified four forms of models and related modeling practices that show promise for developing model-based reasoning. Models have the fortuitous feature of making forms of student reasoning public and inspectable—not only among the community of modelers, but also to teachers. Modeling provides feedback about student thinking that can guide teaching decisions, an important dividend for improving professional practice.

Supporting the Development of Conceptions of Statistics by Engaging Students in Measuring and Modeling Variability

New capabilities in TinkerPlots 2.0 supported the conceptual development of fifth- and sixth-grade students as they pursued several weeks of instruction that emphasized data modeling. The instruction highlighted links between data analysis, chance, and modeling in the context of describing and explaining the distributions of measures that result from repeatedly measuring multiple objects (i.e., the height of the school’s flagpole, a teacher’s head circumference, the arm-span of a peer). We describe the variety of data representations, statistics, and models that students invented and how these inscriptions were grounded both in their personal experience as measurers and in the affordances of TinkerPlots, which assisted them in quantifying what they could readily display with the computer tool. By inventing statistics, students explored the relation between qualities of distribution and methods for expressing these qualities as a quantity. Attention to different aspects of distribution resulted in the invention of different statistics. Variable invention invited attention to the qualities of “good” measures (statistics), thus meshing conceptual and procedural knowledge. Students used chance simulations, built into TinkerPlots, to generate models that explained variability in a sample of measurements as a composition of true value and chance error. Error was, in turn, decomposed into a variety of sources and associated magnitudes—a form of analysis of variance for children. The dynamic notations of TinkerPlots altered the conceptual landscape of modeling, placing simulation and world on more equal footing, as first suggested by Kaput (Journal of Mathematical Behavior, 17(2), 265–281, 1998).

What Kind of Explanation is a Model

We describe modeling as a form of explanation that is particular to science and, based on a research program conducted over the last 15 years, identify the conceptual resources and practices that must be developed for school students to become initiated into this kind of reasoning. We point out that modeling is difficult for novices to grasp but is treated by school science as self-evident, which may account for the fact that it is widely misunderstood by learners and educators alike. We close by considering components of instruction, especially classroom norms and tasks, that best support the long-term development of modeling.

Developing Model‐Based Reasoning

Key elements of the structure and function of models in mathematics and science are identified. These elements are used as a basis for discussing the development of model‐based reasoning. A microgenetic study examines the beginnings of model‐based reasoning in a pair of fourth‐ and fifth‐grade children who solved several problems about chance and probability. Results are reported in the form of a cognitive model of childrens problem‐solving performance. The cognitive model explains a transition in childrens reasoning from tacit reliance on empirical regularity to a form of model‐based reasoning. Several factors fostering change in childrens thinking are identified, including the role of notations, peer interaction, and teacher assistance. We suggest that model‐based reasoning is a slowly‐developing capability that emerges only with proper contextual and social support and that future study should be carried out in classrooms, where these forms of assistance can also be part of the object of study. Mode...