Understanding How Students Develop Mathematical Models

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<ul><li><p>This article was downloaded by: [The Aga Khan University]On: 09 October 2014, At: 05:53Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK</p><p>Mathematical Thinking andLearningPublication details, including instructionsfor authors and subscription information:http://www.tandfonline.com/loi/hmtl20</p><p>Understanding HowStudents DevelopMathematical ModelsHelen M. Doerr &amp; Joseph S. TrippPublished online: 18 Nov 2009.</p><p>To cite this article: Helen M. Doerr &amp; Joseph S. Tripp (1999) UnderstandingHow Students Develop Mathematical Models, Mathematical Thinking andLearning, 1:3, 231-254, DOI: 10.1207/s15327833mtl0103_3</p><p>To link to this article: http://dx.doi.org/10.1207/s15327833mtl0103_3</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy ofall the information (the Content) contained in the publicationson our platform. However, Taylor &amp; Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to theaccuracy, completeness, or suitability for any purpose of the Content.Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed byTaylor &amp; Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources ofinformation. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly</p><p>http://www.tandfonline.com/loi/hmtl20http://www.tandfonline.com/action/showCitFormats?doi=10.1207/s15327833mtl0103_3http://dx.doi.org/10.1207/s15327833mtl0103_3</p></li><li><p>or indirectly in connection with, in relation to or arising out of the useof the Content.</p><p>This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms &amp; Conditions of accessand use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>The</p><p> Aga</p><p> Kha</p><p>n U</p><p>nive</p><p>rsity</p><p>] at</p><p> 05:</p><p>53 0</p><p>9 O</p><p>ctob</p><p>er 2</p><p>014 </p><p>http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>MATHEMATICAL THINKING AND LEARNING. 1(3) , 231-254 Copyright O 1999, Lawrence Erlbaum Associates, Inc. </p><p>Understanding How Students Develop Mathematical Models </p><p>Helen M. Doerr and Joseph S. Tripp Department of Mathematics </p><p>Syracuse University </p><p>in this article, we discuss findings from a research study designed to characterize stu- dents' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experi- enced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students en- rolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to make predictions about the situation. Our analysis of the data revealed 4 sources of mis- matches that were significant in bringing about the occurrence of shifts in student thinking: conjecturing, questioning, impasses to progress, and the use of technol- ogy-based representations. The shifts in thinking in turn led to the development of mathematical models. These results suggest that students would benefit from learning environments that provide them with ample opportunity to express their ideas, ask questions, make reasoned guesses, and work with technology while engaging in prob- lem situations that elicit the development of significant mathematical models. </p><p>Problem solving in realistic situations and the close analysis of students' reasoning continue to serve as thematic strands for research into the development of student learning. In this article, w e bring a model and modeling perspective to understand- ing how students learn and reason about problematic situations encountered in the </p><p>Requests for reprints should be sent to Helen M. Doerr, Department of Mathematics. 215 Camegie Hall, Syracuse University, Syracuse, NY 13244. E-mail: hmdoerr@syr.edu </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>The</p><p> Aga</p><p> Kha</p><p>n U</p><p>nive</p><p>rsity</p><p>] at</p><p> 05:</p><p>53 0</p><p>9 O</p><p>ctob</p><p>er 2</p><p>014 </p></li><li><p>232 DOERR AND TRlPP </p><p>setting of a mathematics classroom. This perspective brings into focus the interac- tions that the students have with the experienced situations and with the representa- tional systems they create, modify, share, and reorganize. We argue that sharing and refining ideas are essential characteristics of modeling activity. </p><p>The term model has a variety of everyday meanings as well as a variety of more technical meanings that have emerged from diverse research and practice perspec- tives. The following description of a model will likely seem familiar to research- ers, practitioners, and educators in fields such as economics, physics, chemistry, biology, mathematics, and other areas of science and engineering. A model is a system consisting of elements, relations among elements, operations that describe or explain how the elements interact, and patterns or rules that apply to the preced- ing relations and operations. However, not all systems serve as models. To be a model, a system must be used to describe, think about, interpret, explain, or make predictions about the behavior of some other phenomena or experienced system. A mathematically signQicant model must focus on the underlying structural charac- teristics of the experienced system that is being described or explained. In this arti- cle, we provide a close analysis of how learners can develop such mathematically significant models through their interactions with some phenomena or experi- enced system and with representational systems that are supported by technology and shared with other learners. We argue that, through their formation, such mod- els can provide descriptive, explanatory, and predictive power for an individual learner and for groups of learners in understanding an experienced system. </p><p>THEORETICAL FRAMEWORK </p><p>Some researchers (English &amp; Halford, 1995; Gentner &amp; Stevens, 1983; Halford, 1993; Wozny, 1992) tended to see amodel primarily as an individual's understand- ing of a system and often used the term mental model to suggest this emphasis on the individual and on the internal, psychological aspects of representations, rela- tions, and reasoning. In his work, Halford gave the following definition: "Mental models are representations that are active while solving a particular problem and that provide the work space for inference and mental operations" (p. 23). Halford went on to argue that such mental models generally reflect the experience of a given individual and are often content specific. This line of research emphasized the dis- tinction between internal (or mental) models and external (physical or otherwise outside of the mind) models. In contrast, we take the distinction between internal models and external models to lead us to an examination of the interactions and in- terdependencies of these two systems rather than their separation or independence. </p><p>To illustrate this point, consider the two-dimensional, coordinate plane. The plane could be considered an example of an external model or system. Two inter- secting perpendicular lines are commonly used to denote the plane; unit markings </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>The</p><p> Aga</p><p> Kha</p><p>n U</p><p>nive</p><p>rsity</p><p>] at</p><p> 05:</p><p>53 0</p><p>9 O</p><p>ctob</p><p>er 2</p><p>014 </p></li><li><p>DEVELOPING MATHEMATICAL MODELS 233 </p><p>on the lines provide a coordinate system. The use of two intersecting perpendicular lines to represent this external system is a model of the plane. However, as a model, this system can be seen as either an internal or an external model. If it is dis- played on a piece of paper or on a chalkboard, it is an external model. However, if it is within the head of an individual, it is an internal model. An external model may well have differences from the internal model formed in the mind of any one individual because, as Norman (1983) pointed out, internal or mental models are personal and possibly unique, reflecting the individual's experience of relevant sit- uations. </p><p>We emphasize that the model that any given individual has in his or her head may be different from the model described externally by that individual. Indeed, as was argued elsewhere (Lesh &amp; Doerr, in press), the very act of externally describ- ing the system may result in changes in the internal model of the individual who is doing the describing. Mismatches between a learner's internal model and his or her representations may lead to changes in both the internal model and in the represen- tations. External models and internal models are, in fact, interacting systems for any given individual. It is sometimes tempting to think of a model as residing in a particular set of mathematical symbols (e.g, in the symbols of a particular func- tional relation) that are external to that individual. However, the meaning of those symbols, we argue, resides partly in the head of the individual and partly in the symbols. In that sense, a model is a system whose meaning is constituted by the in- teractions of internal systems, external systems, and the representations that are distributed across these systems. </p><p>Just as a given individual can experience the interactions and interdependencies of his or her own internal and external models, groups of individuals can also experi- ence these interactions and interdependencies when the representational systems and understandings of the experienced system are shared, interpreted, and refined among individuals. As the interpretations of an external model (or experienced sys- tem) become shared, the internal models held by individuals may become modified so as to accommodate the interpretations of others. Mismatches between one learner's interpretation and another's, as well as mismatches between one learner's interpretation and some external representation, can create the need for new inter- pretations or representations. This can lead to changes or shifts in thinking by one or more learners, resulting in arefined, potentially more powerful model. In this article, we are interested precisely in how such changes or shifts in interpretations and think- ing come about for given individuals as they interact with an experienced situation, with technology-supported representations, and with other learners. </p><p>The research on mental models suggests that model-based reasoning proceeds primarily by analogy from a well-understood source system to a less well- understood target system (English &amp; Halford, 1995; Halford, 1993). In their work, Lehrer, Horvath, and Schauble (1994) suggested that analogy is at the core of model-based reasoning. Because any given model is of necessity a model of some </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>The</p><p> Aga</p><p> Kha</p><p>n U</p><p>nive</p><p>rsity</p><p>] at</p><p> 05:</p><p>53 0</p><p>9 O</p><p>ctob</p><p>er 2</p><p>014 </p></li><li><p>234 DOERR AND TRlPP </p><p>other system with its own elements, relations, operations, and rules, the idea of us- ing analogy in the development of models is to use a familiar system to understand an unfamiliar system. Lehrer et al. emphasized that models are not copies of phe- nomena, but rather, models are structural analogs between systems. Such models then "allow one to reason more clearly, both by simplifying the world and by pre- scribing rules of interaction among components that result in emergent behavior" (p. 219). This suggests that models are characterized by being explanatory, in the sense that one can reason more clearly about a system, and predictive, in the sense that the model can be used to arrive at new inferences. </p><p>Consistent with this view, our perspective is that learning mathematics involves the development of models where the emphasis is on the underlying structural characteristics of the systems and on the ability to reason with and about the sys- tems. For example, in the case of a problematic situation involving unfzmiliar mathematical concepts, learning occurs as students develop models of the con- cepts embedded in or evoked by the situation. The development of a new model is based on reasoning that draws on existing models that are related to the new prob- lem situation in some way. The reasoning that occurs in an encounter with a prob- lem situation may involve analogy from a familiar or at least partially understood system to a new system with an unfamiliar mathematical structure. </p><p>The emphasis on reasoning by analogy suggests that the mapping between sys- tems is largely one-way from the well-understood source system to the less well-understood target system. In this article, we argue that model-based reason- ing can be bidirectional, in that reasoning about one system, regardless of how well understood it is, can and does influence the reasoning about the other system. Not only is this analogical reasoning bidirectional, but often model-based reasoning is characterized by learners' efforts to resolve perceived mismatches between sys- tems as well as within a system. A modeling perspective brings to the foreground the bidirectional, developmental nature of the learner's reasoning. As mismatches within and between models are perceived and resolved, the learners change, mod- ify, and reorganize their models. The learners' internal models may change, new elements in the experienced systems may be selected for focus, and shared repre- sentations may be modified or extended. </p><p>To illustrate the bidirectional reasoning between models, consider the experi- enced physical system involving the application of a force and the motion of an ob- ject. We can identify an external model of this system by observing objects in the world around us. In particular, it is apparent that if a sufficient amount of force acts on an object, it will change position. It is often equally apparent to many students that, in their experienced world, objects do not move without force. As Clement (1983) and others have found, many students develop the idea that motion implies a force and, equivalently, that without a force there is no motion. This incomplete understanding of Newtonian mechanics can be referred to as the student's internal model of the external system. This internal model consists of the student's under- </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>The</p><p>...</p></li></ul>