The Role of Communication in Learning To Model

Paul Brna, Michael Baker, Keith Stenning, Andree Tiberghien, Paul Brna, Michael Baker, Keith Stenning, Andree Tiberghien

• Publisher: Psychology Press
• Year: 2014
• Print ISBN: 9781138012714
• eBook ISBN: 9781135640026
• Topic: Psychology
• Subtopic: Cognitive Psychology & Cognition
• Index: Psychology

In this book, a number of experts from various disciplines take a look at three different strands in learning to model. They examine the activity of modeling from disparate theoretical standpoints, taking into account the individual situation of the individuals involved. The chapters seek to bridge the modeling of communication and the modeling of particular scientific domains. In so doing, they seek to throw light on the educational communication that goes on in conceptual learning.

Taken together, the chapters brought together in this volume illustrate the diversity and vivacity of research on a relatively neglected, yet crucially important aspect of education across disciplines: learning to model. A common thread across the research presented is the view that communication and interaction, as fundamental to most educational practices and as a repository of conceptual understanding and a learning mechanism in itself, is intimately linked to elaborating meaningful, coherent, and valid representations of the world.

The editors hope this volume will contribute to both the fundamental research in its field and ultimately provide results that can be of practical value in designing new situations for teaching and learning modeling, particularly those involving computers.

I

Coordinating Representations

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Chapter 1

Coordinating Mathematical With Biological Multiplication: Conceptual Learning as the Development of Heterogeneous Reasoning Systems

Keith Stenning

University of Edinburgh

James G. Greeno

Stanford University

Rogers Hall

University of California at Berkeley

Melissa Sommerfeld

Stanford University

Muffie Wiebe

Stanford University

INTRODUCTION

The goal of this chapter is to exemplify deep conceptual learning that involves the coordination of initially disparate representations into a het-erogeneous reasoning system. This coordination can be understood as the development of representational practices that occur in discourse and engage learners in social interaction. We analyze data from three perspectives with the aim of comparing and integrating approaches that are sometimes considered to be in tension. The three approaches are: a foundational semantic analysis of the heterogeneous representations encountered in the learning situation; an interactional analysis of dis-course structures that facilitate group reasoning and understanding; and an interactional analysis of how coordinated representational practices expand and stabilize within a discipline-specific domain of inquiry. Our hope is that by combining these approaches, our analysis will treat equally important aspects of representational content, discourse structure, and changes in conceptual understanding as achievements of talk-in-interaction. The chapter has it origins in a collaborative analysis reported in three separate works (Greeno, Sommerfeld, & Wiebe, 2000; Hall, 2000; Stenning & Sommerfeld, 2000). Here, we collect these anal-yses together and offer integration.

This chapter adds to a line of work in cognitive studies of mathematics education that examines how learners work at the interface between representing and represented worlds to make inferences, identify and re-cover from conceptual errors, and manage calculation (Cobb, Yackel, & McClain, 1999; Hall, 1996; Nathan, Kintsch, & Young, 1992; Nemirovsky, in press). The data we analyze are videotaped recordings of incidents of group activity drawn from a longitudinal case study of students working in a project-based, middle school mathematics curriculum unit (Goldman, Moschkovich, & The Middle-school Mathematics through Applications Project [MMAP] Team, 1995). This longitudinal case study was part of a research project comparing the use of mathematics in classrooms and adult workplaces where people work together to design things (Hall, 1999), conducted in public school classrooms and professional firms around the San Francisco Bay area. The students were concurrently learning mathematical and scientific concepts about population dynamics by learning to model populations. The MMAP curriculum was developed as a practical pedagogical response to the theoretical problem about conceptual learning that concerns us here. As psychologists and educators (e.g., Brownell, 1935; Wertheimer, 1959) have long recognized, it is one thing for students to learn the operation of a novel mathematical or scientific formalism. It is quite another for them to understand the meaning of that formalism in terms of general concepts and to master its application to new situations. MMAP’s response to this crucial educational problem is to teach mathematical formalisms (e.g., graphs and difference equations for functions) and scientific terminology and representations together in a context of their application, in the belief that this concurrent learning can provide a semantically based grasp of their application to the world. Our theoretical task is to find productive relations between analyses of heterogeneous representation systems, learning in group discourse, and studies of the interactional structure of discipline-specific representational practice to help illuminate the learning processes evidenced in this data.

This is not only a theoretical exercise or a problem that is specific to educational research. A central process in scientific or mathematical thinking involves being able simultaneously to look at and through the interface between representing and represented worlds (Gravemeijer, 1994; Latour, 1999). This is particularly true of thinking practices in which people construct and then explore models to gain access to situations that do not yet exist or that occur across scales of time and space that prevent direct observation. Although this flexible use of modeling is central to many disciplines, pedagogy has until recently focused primarily on the notational structure of formal systems of representation. This approach can trap learners in the situation of looking at complex representational systems without being able to look through them to construct and explore represented worlds (Greeno & Hall, 1997). The MMAP curriculum seeks to avoid this pitfall of the separation of formal-ism from its understanding and application by concurrently teaching mathematical formalism and scientific concepts, through the modeling of realistic situations assisted by computers. Mature mastery of a formalism does not simply replace looking at with looking through, but means that students can control the level of their attention appropriately for reasoning.

Heterogeneous Reasoning as Rules of Transformation

Linguistic inference rules turn sentences into other sentences, but in re-cent years there has been much discussion of heterogeneous reasoning systems that have representations in more than one modality (commonly diagrams and sentences) and so require rules that represent sentential information diagrammatically and diagrammatic information sententially. Theoretical interest in heterogeneous reasoning stems, in part, from the fact that much everyday reasoning is heterogeneous. We encounter information in linguistic form, but we also receive diagrammatic information such as maps, graphs, and diagrams, and even when we only encounter linguistic representations, we commonly encounter them in situations where we also have nonlinguistic perceptual input of spatial information about speakers and about their and our own embedding in the world. People generally succeed rather well in combining these different information sources, for example, by using diagrams, material models, or computer programs to simulate events in ways that support conjectures and test hypotheses (e.g., Clement, 1989; Schwartz & Black, 1996; Schwartz, Yerushalmy, & Wilson, 1993).

With this starting point, we are necessarily concerned with the semantics of formal systems—relations between the formalisms and the things they stand for. But in the kind of deep conceptual learning with which we are concerned here, target concepts are not easily differentiated from alternative interpretations by pointing at objects in the world. The classical physical concepts of weight, volume, density, and mass illustrate this point. Every object we can point to has all of these attributes, but pointing does not help differentiate the concepts. As a result, these concepts can only be differentiated by observing which physical transformations preserve which properties. Compression preserves weight and mass, but alters volume and density. Transport to the moon alters weight but not mass, and so on. These physical operations correspond to informational transformations in the representation systems we use to reason about them—informational transformations that in logic are called inference rules. Inference rules allow the transformation of representations into other representations with preservation of truth. Stenning (1999) argued for the central role of transformations in learning abstract scientific and mathematical concepts.

Practical pedagogical interest in heterogeneous reasoning stems from the conviction that students should have access to the semantics of representational systems as they learn them, both to support their under-standing of their conceptual meanings and to achieve generalization to new circumstances. Classical studies of learning the area of a parallelogram demonstrated that by having young students attend to transformations that convert parallelograms to rectangles (Wertheimer, 1959) or that slide a stack of cards, keeping area, base, and height invariant while varying the angles and perimeter of the figure (Sayeki, Ueno, & Nagasaka, 1991), the students can abstract the quantitative relation between the base, height, and area of the figure and understand and generalize the formula. Brownell (1935) studied learning of place-value addition and subtraction and showed that use of concrete models can support children’s understanding the operations of carrying and borrowing, and Resnick and Omanson (1987) found that students who benefited from experience relating procedures with numerical symbols to analogous operations on place-value blocks in reducing the “bugs” in their test performance (Brown & Burton, 1980) also talked more about the correspondence between the operations in the formal symbolic and material domains.

Barwise and Etchemendy (1994) pioneered the foundational study of heterogeneous reasoning systems through their development of Tarski’s World and Hyperproof, multimodal computer environments for learning elementary logic. Stenning and his coworkers studied the cognitive impact of such heterogeneous reasoning systems on students’ learning (e.g., Oberlander, Monaghan, Cox, Stenning, & Tobin, 1999; Stenning, Cox, & Oberlander, 1995). The upshot of their studies of undergraduate students is that conceptual learning in Hyperproof can be understood as acquisition of the strategy and tactics of using transformations for moving information between modalities. Students who learn well from the heterogeneous system acquire a deep understanding of when, during problem solving, to move information from sentences into diagrams, and when in the reverse direction. Students who fail to benefit from diagrams fail because they have not mastered these strategies. This strategic learning (in this case about the concepts of logic) can be understood as learning to coordinate the various representations used in Hyperproof into an integrated heterogeneous reasoning system containing inference rules that deal with combinations of sentential and diagrammatic information.

This chapter poses the question whether we can extend the theoretical framework developed for analyzing logic learning in Hyperproof to groups of students learning to model population dynamics. This is a substantial extension. Hyperproof is a fully formalized and implemented heterogeneous representatio...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• List of Contributors
• Preface
• Part I Coordinating Representations
• Part II Provoking More Effective Modeling
• Part III Collaboration and Language
• Author Index
• Subject Index