Understanding How Students Develop Mathematical Models

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  • 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

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    Understanding HowStudents DevelopMathematical ModelsHelen M. Doerr & Joseph S. TrippPublished online: 18 Nov 2009.

    To cite this article: Helen M. Doerr & Joseph S. Tripp (1999) UnderstandingHow Students Develop Mathematical Models, Mathematical Thinking andLearning, 1:3, 231-254, DOI: 10.1207/s15327833mtl0103_3

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  • MATHEMATICAL THINKING AND LEARNING. 1(3) , 231-254 Copyright O 1999, Lawrence Erlbaum Associates, Inc.

    Understanding How Students Develop Mathematical Models

    Helen M. Doerr and Joseph S. Tripp Department of Mathematics

    Syracuse University

    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.

    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

    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

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  • 232 DOERR AND TRlPP

    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.

    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.

    THEORETICAL FRAMEWORK

    Some researchers (English & Halford, 1995; Gentner & 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.

    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

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  • DEVELOPING MATHEMATICAL MODELS 233

    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.

    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 & 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.

    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.

    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 & 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

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  • 234 DOERR AND TRlPP

    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.

    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.

    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.

    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-

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  • DEVELOPING MATHEMATICAL MODELS 235

    standing of a physical, external system involving objects, motion, force, and some partially quantified relations among these elements (e.g., greater force implies faster motion). Moreover, the student's model has both explanatory and predictive power in the everyday world of experience, where objects in general do slow down and stop moving when a force is no longer directly applied. Changing students' in- ternal models of motion to a Newtonian model has proven to be a significant chal- lenge for science educators (Clement, Brown, & Zietsman, 1989; Hestenes, 1992; McDermott, Rosenquist, & van Zee, 1987).

    The internal models that learners have of everyday motion are well-understood systems but not a sound basis for reasoning by analogy about the Newtonian sys- tem of mechanics. It is precisely the student's well-understood model that needs to be modified and refined so as to account for the wider range of phenomena that are explained by Newton's laws. In this case, model-based reasoning proceeds from the learner's poorly understood Newtonian model of mechanics to modify the learner's well-understood model of everyday motion. In our analysis of the data, we have found several sources and types of mismatches, such as those between the student's current model of falling body motion and the reality of the collected data and those between different graphical representations of the data. The dissonance caused by model-reality mismatches and within-representational-system mis- matches leads to the continued development of the student's model.

    Modeling is seen as the interactions among three types of systems: (a) internal conceptual systems, (b) representational systems that function both as externalizations of internal conceptual systems and as internalizations of external systems, and (c) external systems that are experienced in nature or are artifacts that were constructed by others. We emphasize that we see the boundaries between these systems as fluid, shifting, and at times ambiguous. Internal conceptual sys- tems, or mental models, seem to exist mainly inside the head. Representational systems are largely embedded in the spoken language, written symbols, pictures, diagrams, and concrete objects that people use to express their internal systems and to describe external systems. External systems are those that humans experi- ence or create in the world, such as coordinate systems, mechanistic systems (e.g., Newtonian laws of motion), economic systems, and communications systems. Al- though there are useful distinctions to be made among these three systems, we see these systems as overlapping, interdependent, and interacting. It is the interdepen- dencies and interactions that are brought to the foreground here and are central to our analyses of student learning from a modeling perspective.

    Although the representational systems that students use seem to reside outside their minds and in a form that can be shared with others, it is also the case that a part of the meaning of these representational systems is in some sense within the individual. There is also a sense in which it is natural to speak of mathematically significant models as though they were pure systems or structures somehow dis- embodied from any particular representations, tools, or external artifacts. Yet, in

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  • 236 DOERR AND TRIPP

    practice, beyond the most trivial situations, it is clear that significant mathematical models seldom function without the support of powerful tools or representational systems, each of which emphasizes and deemphasizes (or ignores or distorts) somewhat different aspects of the underlying structural characteristics of the sys- tem. The models that people develop are often partly embedded in tools that may involve electronic devices but that also may involve specialized symbols, lan- guage, diagrams, organizational systems, or experience-based metaphors. Thus, in speaking of students' reasoning with models, the model is not external to the rea- soning process, and the reasoning process is not entirely internal or in the head.

    We take the position with von Glasersfeld (1990,1996) thatthe function ofcogni- tion is adaptive in the biological sense of the term, tending toward fit or viability and that cognition serves the participant's organization of the experiential world and not the discovery of an objective ontological reality. In our analysis of the data, we ex- amined how groups of learners interacted with experienced problem situations, rep- resentations, and computational tools as they developed and shared models with greaterfit and viability for theirexperience. A lackof fit or mismatch between theex- perienced system and the learner's representational system and internal system drove the development of models with greater viability for the learner. In this article, we largely focus on the kinds and sources of mismatches that occurred through the adaptive interactions within groups of learners so as to understand their develop- ment of significant mathematical models. A better understanding of the develop- ment of students' thinking and how changes and shifts in this thinking occur may help educators design learning environments that contribute to the development of useful models by students as they investigate problem situations.

    RESEARCH QUESTIONS

    This research study sought to extend our knowledge of the development of stu- dents' thinking as they engaged in modeling activities. Our particular interest was in understanding how shifts in students' thinking occur and in what ways such shifts in thinking supported thedevelopment of viablemodels. In other words, how did students change their thinking as they developed mathematical models that had descriptive, explanatory, and predictive power? What kinds of events in their inter- actions with each other, with the problem situation, and with associated representa- tions led to shifts in students' thinking? In this article, we present our findings from aclose analysis of the role played by shifts in thinking in the development of signifi- cant mathematical models by small groups of students. Our specific research ques- tions for this study are the following:

    1. What (if any) is the role played by shifts in thinking in the development of students' mathematical models during problem investigations in a college precalculus class?

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  • DEVELOPING MATHEMATICAL MODELS 237

    2. How do these shifts in thinking occur during cooperative modeling activi- ties in a college precalculus class?

    This study was designed to describe carefully how students develop significant mathematical models by closely examining the changes in their interpretations and in their representational systems as they interacted with each other and with the ex- perienced external systems. These changes in thinking were driven by the students' efforts to resolve mismatches in interpretations and relations as well as mismatches between their expectations or predictions and their representational system. In the analysis that follows, we discuss the kinds of activities that led learners to perceive and resolve these mismatches.

    METHODOLOGY AND DATA ANALYSIS

    This study was conducted in a precalculus mathematics classroom at a moderate- sized private research university. Students enrolled in the class were asked to vol- unteer for the study. A total of five observations were conducted with three differ- ent three-person groups of students engaged in modeling activities. Each observa- tion lasted for 80 min during a regular class meeting. Only one group was observed on any given day. There were several other groups in the same classroom engaged with the same problem. The group being observed did not directly interact with any other group during the class session. Three different groups were observed because the groups changed during the semester, and, on one occasion, two members of the group we planned to observe were absent. On that day, therefore, we chose adiffer- ent group to observe.

    Each session involved a small group of students working together on one of five tasks. Each student had a graphing calculator with a symbolic algebra system. The tasks covered a range of situations designed to elicit students' thinking about the underlying mathematical structure and patterns. Each task consisted of a prelab ac- tivity, which was completed by the students individually prior to their class meet- ing for the actual modeling task. The intention of this portion of the activity was twofold. First, it was intended to provide all students with an opportunity to think and reflect on the problem situation prior to the class meeting so that they could come to the class with some of their own ideas about the meaning of the problem situation. Second, it was intended that whatever representations (e.g., graphs, ta- bles, equations) or explanations the students brought to their group setting would be sufficiently different from each other's representations and explanations that this would evoke some discussion about the mismatches in their interpretations of the problem situation.

    The five tasks used in this study included two tasks about the vertical motion of a falling object, a task about the exponential growth of bacteria cultures, a task in- volving radioactive decay, and a task about the variation in the amount of sunlight

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  • 238 DOERR AND TRIPP

    over the time of 1 year. In each case, the focal point of the task was to elicit stu- dents' representations and interpretations of the underlying mathematical patterns and structure (e.g., quadratic, exponential, sinusoidal). Summary descriptions of the tasks addressed in this article appear in the Appendix.

    Data collection consisted of writing field notes during and after the observa- tions, audio taping the students' interactions as they engaged in the modeling ac- tivity, and collecting students' written work. The data were analyzed using inductive data analysis techniques. Each transcribed audio tape was coded to assist in categorizing events that led to shifts in students' thinking during the problem in- vestigations. The next step in our data analysis involved identifying recurring pat- terns and the relations among categories. According to Miles and Huberman (1994), "pattern codes are explanatory or inferential codes, ones that identify an emergent theme, configuration, or explanation" (p. 69). We sought to identify pat- terns in the coded data to assist us in identifying relations among categories as well as themes, causes, and explanations in the data. Throughout our analysis, our goals were to identify transitions or shifts in student thinking, to understand how those shifts occurred, and to describe the role played by transitions in thinking in the de- velopment of the students' models.

    A shifr or transition in thinking is a passing from one form, place, or stage to an- other in one's thinking. In previous works (Doerr, 1996, 1997; Lesh & Doerr, in press), it was argued that modeling is a nonlinear, multicycle process. As such, we expected to find evidence that learners change their thinking about a problem situa- tion, perhaps returning to earlier interpretations, as their descriptions and explana- tions are tested, modified, and extended. For example, during the ball toss-ball drop activity (see the Appendix), Diane initially stated that the rate at which the ball dropped was constant. She later asserted, however, that "as it goes down, it is, like, faster and faster." The student's initial interpretation of the problem situation de- scribed the ball as moving at aconstant speed. She changed her interpretation of the motion of the ball to one of changing speed and introduced a partial quantification ("fasterand faster") of this speed. This is an example of what we mean by a transition or shift in thinking. Shifts in thinking can be described in terms of an initial interpre- tation of the problem situation and a later interpretation that stands in opposition to the initial interpretation. Of particular interest were the events that occurred between an initial interpretation and a later, opposing interpretation. It seemed reasonable to suppose that somewhere between the two interpretations there would be evidence of what precipitated thechange in thinking. In our analysis, we focused carefully on the sequence of events between initial and later interpretations.

    One such sequence of events occurred during the trigonometric graphs activity, a prelab activity in which students were asked to find equations that matched given graphs. This activity is summarized in the Appendix. We observed the following sequence of events and transition that occurred in Mindy's thinking. Mindy began by interpreting the given graph as the graph offlx) = sin(2x).

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  • Mindy: Negative sine 2x.

    Jared stated what he thought the equation was, and Mindy questioned him.

    Jared: Negative 4x is what I got. Mindy: 4x? Diane: Because there are 4 cycles.

    The discussion continued with several incidents of conjecturing and questioning. At the same time, the students worked with their graphing calculators and used pa- per and pencil to share and test representations.

    Jared: Mindy: Diane: Jared:

    Diane: Mindy: Diane: Jared:

    Mindy: Diane: Jared:

    Mindy: Jared: Diane: Jared:

    There are a lot of cycles. Wait, 4 cycles until you get to what? IT? 1, 2, 3, ... 27c. You always go to 2n. No, it can't be. Okay, so it is 1 ,2,3,4. Is that what you are doing? So that is sine, right, negative, like it is 1, 2, 3 only. Wait. You go . . . I don't understand your counting. Like I ... No, it would go, isn't that one cycle? Down and up or . . This is one cycle. A hill and a valley. That is one cycle. So it is not cosine. No. Although it is a negative sine.

    Finally, Mindy made an interpretation that was in opposition to her initial inter- pretation.

    Mindy: Yeah, I think it is 4, too. Jared: It is negative sine.

    Mindy: Negative sine 4x.

    This sequence of events began with Mindy interpreting the given graph as the graph of the functionflx) = sin(2x). At the end of the sequence, Mindy had shifted her thinking to seeing the graph as representing the function "negative sine 4x." It is within such sequences that our analysis of the data yielded categories of events that led to the occurrence of shifts in thinking.

    Although data from five tasks were collected and analyzed, data from the ball toss-ball drop task are referred to most often, simply to provide a single context for

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  • 240 DOERR AND TRlPP

    illuminating and revealing the richness of our data. We emphasize, however, that our findings are based on an analysis of all five tasks. During the ball toss-ball drop modeling activity, three students used data collected from a motion detector to describe, explain, and predict the behavior of a falling ball (as described in the Appendix). We now discuss each of the categories of events that were found to play a role in evoking transitions in thinking as the students engaged in developing mathematical models.

    RESULTS

    The analysis of the data revealed that the use of technology-based representations, conjecturing, impasses to progress, and questioning were significant in bringing about the occurrence of shifts in thinking during these modeling activities. Each of these four categories can be thought of as evoking mismatches that occurred as the students attempted to make sense of their internal systems, the external (or experi- enced) system, and their shared representational systems. The use of technol- ogy-based representations provoked a mismatch with the students' pa- per-and-pencil representations. One type of mismatch, then, was that which occurred within or between representational systems, including words, graphs, and data tables. Incidents of conjecturing and incidents of questioning often reflected mismatches between different students' internal models as well as mismatches within a given student's way of interpreting the problem situation.

    We found that students made conjectures that were in opposition to earlier con- jectures made by either themselves or by other students. Similarly, questioning oc- curred between students as well as by students asking questions of themselves. These incidents of conjecturing and questioning often reflected a mismatch be- tween elements of the students' initial internal model and led to some new internal model or interpretation. Impasses to progress often led students to consider alter- native approaches to solving their problems; these impasses were often mis- matches between the students' internal model and the external model of the experienced system. The learners' efforts to make sense of these systems led to shifts in their thinking as they refined their models to fit more closely with the ex- perienced system and to provide greater viability for themselves.

    The Role of Technology

    We found that the use of technology served to precipitate shifts in students' think- ing by evoking mismatches within the students' representational systems and be- tween the representational systems and the students' internal models. For example, students' use of technology-based representations gave rise to an important shift in thinking that occurred early in the ball toss-ball drop activity. The students were

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  • DEVELOPING MATHEMATICAL MODELS 241

    Hilda's sketch from the preparation

    height I

    the graph on the calculator

    FIGURE 1 Comparison of a student sketch and the calculator graph.

    height

    7 6 5 " 4 " 3 " 2 1 0

    asked to compare the graphs they drew in their preparation for the class session with the graph of the data obtained from the motion detector as displayed on their calcu- lators. They were asked to comment on the similarities and differences among these graphs (see Figure 1).

    Interestingly, Hilda labeled her vertical axis 4,3,2, 1,0, beginning at the origin, rather than 0, 1, 2, 3, 4. Although this is rather unusual, it was not a problem for Hilda to switch from her scale to the more conventional scale when Diane and Grace experienced difficulty understanding Hilda's scale. This mismatch between Hilda's representation and the presumed internal models of Diane and Grace led to a change in Hilda's representational system, even though it was clear that Hilda could easily interpret both graphs as meaningful (albeit different) representations of the same phenomena.

    The students' discussion quickly became focused on the rate at which the ball fell to the ground. Hilda's initial interpretation was that the ball dropped at a con- stant speed. After viewing the graph on her calculator (see Figure I ) , however, she decided that the ball increased in speed as it fell.

    --

    -- -- . -

    time

    Hilda: The difference [between the graph on the calculator and my graph from the prelab activity] is in reality, when the ball drops, the velocity or the speed gets faster as it gets closer to the ground or whatever.

    Diane: Say that again? Hilda: I said the difference is, like when I drew the graph, I drew it that as

    soon as it starts it is the same speed all of the way down. And from the graph [on thecalculator] it shows that it starts out slower and then gets fast, like increases.

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    Hilda took the view that the ball increased in speed as it fell, apparently as a result of her work with her calculator. Thus, we attribute this shift in thinking to Hilda's inter- pretation of the graphical representation on her calculator. Whereas Hilda came to think that the ball dropped at an increasing velocity, Diane maintained her opposing view that the velocity was constant. As we analy zed the continuing discussion, how- ever, we found that neither Hilda nor Diane held on to their original thinking. They both shifted their thinking again as they engaged in interactions with the group mem- bers and, later, with the table representation of their calculator-based data.

    The Role of Conjecturing

    We found many transitions in thinking that were precipitated by incidents of conjec- turing. A conjecture refers to a potentially valid statement, opinion, or assertion ex- pressed by a student about some particular aspect of the problem situation. Multiple conjectures were often made by group members during sequences of events that led to ashift in thinking by one or more group members. The student who made theinitial interpretation (the beginning of a transition) was often involved in posing several conjectures before arriving at an interpretation that reflected a more stable shift in thinking (theend of the transition). A given conjecturecould have been supportive of another conjecture, in opposition to another conjecture, or simply a response to a question. Of particular interest wereconjectures that were in opposition to the initial interpretation that marked the beginning of a transition because anything that stood in opposition to a student's initial interpretation would have had the potential to bring about change in that student's thinking. This was a type of mismatch between one student's internal model (or interpretation) and another student's internal model. This aspect of student conjecturing was illustrated by the shifts in thinking that occurred as thestudents analyzed thebehavior of the ball at its greatest height.

    Hilda initially drew the graph on the left (see Figure 2) for the ball toss event but later changed her representation to the graph on the right. This change in how she described the relation between time and the height of the ball would suggest a shift had occurred in Hilda's thinking. Conjecturing appeared to play a significant role in precipitating this shift in thinking.

    Hilda stated that she was not sure whether to make the graph a point. Diane, however, seemed certain that the graph should have a point where it is at its great- est height. Diane made a conjecture that was in direct opposition to Hilda's more tentative interpretation of the situation.

    Hilda: This is the one I didn't understand, though. I understand it, but I didn't know if this should just be a point, you know, because when you toss a ball in the air, it kind of stays, it is in one place. I didn't know if Icould just make it a point or make it feet per seconds.

    Diane: The thing is apoint. At that point, at the highest point, it doesn't move.

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  • DEVELOPING MATHEMATICAL MODELS 243

    height height

    time time

    FIGURE 2 Hilda's sketches of the position of the tossed ball versus time

    Hilda clearly thought that the ball stays at its greatest height for some amount of time. She held on to this position in her ongoing dialogue with Diane. Diane also maintained her position, although more carefully quantifying that the time "it doesn't move" is "zero time."

    Hilda:

    Diane: Hilda:

    Diane:

    Hilda:

    Diane:

    Hilda:

    Diane: Hilda:

    Hilda:

    Diane: Hilda:

    Right. But I don't have it move any feet. I have it move per time. You know what I am saying? So it doesn't move anywhere, but it stays there for a certain period of time. But that time is too short to be like . . . But I have this like, because this is only 2 seconds I have this little time, so I just made it really small like, it is like a fraction of a second. Can it be zero at that highest point? That thing is zero, the time is zero, isn't it? Well, it has to be a certain time where it is just there. Like it can be like a millionth of a second. But I am sure that it just depends on how you graph it. I mean . . . I think there should be like zero time. [Pause] I don't know. Then, well, when it comes back down, does it come back down faster than . . . No, it is the same. It is the same speed. Okay. [Pause] And the ball dropped. I guess these two can both be down. That is just how I drew it. Isn't physics like the same? Like it is a point. Its time is zero. Yeah, that is true.

    By the end of this dialogue, Hilda changed her initial thinking that the ball "kind of stays, it is in one place" to an agreement with Diane's conjecture that the "time is zero." In addition to her final aforementioned statement, a significant indicator of this change was that Hilda changed her graph to the one on the right in Figure 2.

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  • 244 DOERR AND TRlPP

    This graph was an important element of Hilda's representational system, one that was shared with Diane and the central shared focus of the previous discussion. The opposing conjecture made by Diane appears to have been a significant factor in bringing about the shift in Hilda's thinking. We do not, however, discount other po- tentially significant factors, such as Hilda's initial uncertainty or her prior knowl- edge. Our argument is that without Diane's opposing conjecture, it is unlikely that Hilda would have changed her graph as she did. This change in the representational system is seen to be reflective of a change in Hilda's thinking about the motion of the ball. Diane's opposing conjecture served as an event that precipitated, or at least helped to precipitate, this change in Hilda's thinking by bringing into focus a mis- match between Hilda's internal model and the experienced external systems, as in- terpreted by Diane through her internal model. The resolution of this mismatch re- sulted in a change in the shared representational system.

    The Role of Impasses

    In our analysis of the data, we found that significant shifts in thinking were precipi- tated by apparent impasses in the group's work with the modeling tasks. At one such impasse, this particular group was unable to agree on the rate at which the ball fell and appeared to be at a standstill in their work. As a result, Hilda took a step back and considered the law of gravity.

    Hilda: But the only thing is that gravity, like, the force of gravity is increas- ing. So is it, I don't know. I should know from physics.

    Although considering the effect of gravity did not prove to be fruitful, it was a nota- ble shift in Hilda's thinking about the problem situation. In this case. Hilda had changed from thinking about the speed of the ball in terms of her own graph and the graph provided by her calculator to thinking about the speed as it was affected by the force of gravity. This is an example of the interactions that occurred among the student's internal model (her concept of speed as expressed externally in her graph), the shared representational system (the graph provided by the calculator), and the experienced external system of Newtonian mechanics (the force of grav- ity). The student was bringing new elements (the force of gravity) and an as yet un- quantified causal relation into her developing model. We consider this to constitute a shift in thinking, although in this case not aparticularly productive or stable shift.

    On the heels of their attempt to resolve their impasse by considering the effect of the force of gravity on the velocity of the ball, we observed another shift that was driven by an impasse in which the students found themselves. Hilda ques- tioned the degree to which the graph produced by the calculator accurately repre- sented the relation between elapsed time and the height of the ball.

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  • DEVELOPING MATHEMATICAL MODELS 245

    Hilda: But is this just, does this really exist or is it just from looking at like the graph, that is how the graph shows it. You know what I am saying? Like, just because it is in a graph doesn't mean, like, it is a calculator, you know.

    Although they had thought about the speed of the ball based on the two graphical representations (the calculator graph and their paper-and-pencil graphs), Hilda raised a question about the accuracy or even the reality of the graph on the calcula- tor. We see Hilda as having made a transition from thinking that the calculator graph provided an accurate representation of the relation between time and the height of the ball to thinking that the graph may not be an accurate representation at all. Although this shift did not prove helpful either, it was a notable shift in Hilda's thinking that occurred as an apparent result of the group being at an impasse in their modeling efforts.

    A third and more productive shift occurred immediately after the previous shift, when Grace suggested that they look numerically at the data points. Until this time, only the graph drawn in preparation for the problem-solving session and the graph of the data points drawn by the calculator had been part of their shared representa- tional system. We had observed, however, that Diane was working with the data tables in her calculator periodically throughout the activity. She even had at- tempted to draw the group's attention to the data table and provide evidence for her view. Diane had been unsuccessful in engaging the group in a discussion of the data, probably in part because the teacher interrupted the group immediately after Diane said something about the table. However, at this time, Grace's suggestion that they consider the tabular data, a suggestion that Diane was willing to receive, moved the group beyond their impasse and led the group to bring the numerical el- ements of the table and their first order differences into their developing model of the motion of the ball. The introduction of the numerical data into their representa- tional systems led to changes in how they interpreted the motion of the ball and to a more quantified measure of velocity in the experienced external system.

    The Role of Questioning

    Throughout the modeling sessions, incidents of questioning occurred in nearly all of the sequences of events where transitions in student thinking occurred. For the purposes of this article, a question refers to an expression used to request informa- tion, challenge an interpretation, or seek confirmation or clarification. Incidents of questioning played a significant role in precipitating transitions in thinking. For ex- ample, we observed that incidents of questioning played a role in driving shifts in Hilda's thinking about the velocity of the ball. Hilda initially took the position that the ball increased in speed as it fell. She did not, however, hold to this view. Follow-

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  • 246 DOERR AND TRlPP

    ing questioning by both Diane and Grace and by Hilda herself, Hilda shifted from thinking that the ball increased in speed as it fell to thinking that aftera certain point it fell at a constant speed:

    Hilda: Grace: Hilda: Diane: Hilda:

    Diane:

    Hilda: Diane: Hilda: Diane: Hilda:

    That is the ball drop, yeah. So they have the same velocity? No, they don't. Constant velocity. As it drops. Is it? I don't think. No. I think when it starts, it is slower than when it's down here. No, as it starts to drop at zero, I mean when it goes down, is it constant? When it drops? I don't think so. So it's faster as it goes down. Yeah. Faster and faster? Yeah, I think it gets faster as it goes down. [Pause]

    Grace sought confirmation and clarification from Hilda, as she explained her inter- pretation of the ball going faster and faster. Diane's question of "Is it?'appeared to challenge what Hilda was asserting. Hilda continued to explain how she was mak- ing that interpretation, referring to the graph on the calculator.

    Hilda: It starts off, like it curves instead of, if it was the same, it had the same, if it was constant it would go straight, like it would be a straight line in- stead of starting off curved and then going straight.

    Grace: Cause the graph curves when it goes down. Hilda: Right. At first it starts, it curves, it gradually goes down and then it

    goes down. [Pause]

    Hilda: You guys understand? Do you understand what I am saying? Like it should look like that [a straight line] if it was the same speed.

    Grace: Actually it does look like that, once you get to here. Hilda: Right, once you get to here, the fastest point.

    [Pause]

    Here, Hilda asked a question, seeking confirmation that her explanation and in- terpretation was understood by the other students. She then engaged in answer- ing some clarifying questions that were posed by Grace, refining her interpretation of where the position graph (see Figure 1, on the right) was a straight line and where it was curved.

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  • DEVELOPING MATHEMATICAL MODELS 247

    Grace: Hilda: Grace: Hilda: Grace: Hilda:

    Grace: Hilda:

    When it goes down. Here, draw these like little . . . Yeah. So we drew this? Or you drew this? Yeah, at first. Just go down. Just go down? Yeah. Graph. [Pause] Well, yeah it curved more. There is like, be- tween here and here this is the difference I think. Because here it is just standing still. So that could be right there. A straight line there. But I think that's the difference. It starts out slower. [Pause] It is just where it is different. Because it starts out slower. Right? Yeah. That's what I get from my graphs. I don't really know if it's true. The similarities are, eventually they are, the graphs look the same.

    Hilda changed her thinking to the notion that the graph of the ball becomes a straight line and that represented constant speed. Later in the discussion, Hilda re- stated her new interpretation of the motion of the ball as it dropped:

    Hilda: Right. I mean after a certain point I think they are the same speed.

    Hilda shifted from initially thinking that the ball fell at an increasing speed to think- ing that the speed eventually became constant. Several incidents of questioning oc- curred during the discussion among the group members. In particular, Diane and Grace asked several questions to clarify and confirm statements made during the discussion. Diane asked questions to challenge Hilda's position. Finally, Hilda asked questions to request information from Diane and Grace. We argue that the many incidents of questioning that occurred throughout this discussion played a significant role in driving the shift in Hilda's thinking. The questioning brought into focus the mismatch within Hilda's own internal model as it interacted with the shared representational system as well as the mismatch between Hilda's model of increasing velocity and Diane's model of constant velocity.

    Occurrences of Multiple Incidents

    As the categories emerged from the coding of the data, we found that most shifts in thinking were characterized by the occurrence of multiple incidents. Often two or more incidents together gave rise to a significant shift in thinking. For example, al- though Hilda had at one point expressed the view that the ball increased in speed throughout its fall, she was rather consistent in conjecturing that the velocity even- tually became constant. Just after the groups' final impasse previously described,

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  • 248 DOERR AND TRlPP

    Hilda was uncertain as to whether the velocity was constant initially or gradually increased until it reached a "fastest point." Hilda continued to struggle with this is- sue as the group examined the numerical data in the calculator table. An important shift in Hilda's thinking occurred as the group worked with the table data. Although we observed earlier that Hilda shifted to thinking the velocity may be constant from the start, statements made during the group's examination of the data table revealed that Hilda shifted her thinking to the position that the velocity gradually reached its "fastest point."

    Hilda was not the only student who reached the conclusion that the ball gradu- ally increased in velocity until it reached its fastest speed. Although Diane had held that the velocity is constant, she also made the transition to thinking that the velocity gradually reached a fastest velocity. Both of these transitions occurred as conjectures were made and as the students interpreted the numerical representa- tion of the position data on their calculators. The sequence began with Hilda as- serting her conjecture that the velocity was constant after a certain point in time, but that it increased gradually.

    Hilda: Yeah, hold on. Like I understand that after a certain point it is all con- stant. But I am talking about when it first starts to drop. Is it gradually, like, getting to that point or does it just go, you know what I am say- ing? That is what I don't understand. [Pause]

    Hilda: I guess it gradually does. I mean, it is constant. It is about . . . Grace: Well from the graph you can tell it is gradual, like, in here [the first

    part of the graph]. Diane: Yeah, I think it is gradual. Grace: But you just can't [unintelligible]. Hilda: Right. The data is more, that is just a picture. I don't know.

    [9-sec pause] Hilda: It is gradual. Because it goes 7.005 to 7.41.

    As the group continued to examine the data, they simultaneously confirmed their conjectures with both the numerical representation and the graphical representation of the data. They became increasingly convinced that the ball gradually increased in speed as it fell until it reached afastest speed. This change in thinking appeared to be a result of their conjecturing and their work with the technology. We interpret this to be an interaction of their internal models, asevidenced by their asserted conjectures, and their shared representational model, as instantiated by the calculator tables.

    The calculator table provided a data representation that was a mismatch with Hilda's earlier conjectures that the velocity was constant right from the start. The calculator representation also initiated a way of thinking about the speed, which was more quantified than their earlier interpretations. Up until this point, their de-

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  • scriptions of the speed of the ball had been semiquantitative as they described the ball as going "faster and faster" or "increasing gradually" in speed. However, at this point, Hilda expressed the speed in terms of change in position, such as from 7.005 to 7.41. This change in position was expressed without associated units of measurement. They continued to examine the position data table on their calcula- tors and eventually found what they were convinced was a measure of the fastest speed.

    Hilda:

    Hilda: Diane: Hilda:

    Hilda:

    Diane: Hilda:

    Diane: Hilda: Diane:

    Hilda:

    We can't, I can't tell the point where, see this is all level right now. [Pause] I don't know. [unintelligible] Okay, right here. Go to, like, about 64. [Pause] Okay? It goes, it says 6.7,6.6,6.5,6.4,6.3, and then it starts to go 6.1 so that is 2, it dropped off 2. Then 0, then it dropped off 2 more. I think it starts to drop off a little bit faster. Then it drops off 2 here. Then it drops off 3. It dropsoff l , 2 , 2 , ... Then it drops off almost 3 again; 3,3. I think it starts. like it drops 3 at zero, that is where they haven't dropped the ball yet. Then I think gradually it goes to 1, 1, then to 2 and then to 3. I think it stays at 3 and goes down, so I think it is gradual. So that is right. That is right? Yeah. But you might want to explain. Well, like for these distances, it is like slowly. I mean it drops really slowly. Just between here and here, these two points. You know, I just decided to pick those or whatever. Right in between here and here, it drops like . l , .2, and then all of the sudden it changes to .3 and then it stays .3 the rest of the way down.

    Hilda found what they considered to be the fastest velocity, where the velocity was measured as a unitless quantity calculated from the changes in position. This new element of change in position, with its unspecified units of measurement, became the operationalized measure of speed for the group and was part of their shared rep- resentational system. Due in part to the limited precision of the position data set, the students erroneously concluded that there was aconstant final speed. However, this conclusion was a good fit with the data that they had in their calculators. The calcu- lator representations played a role in moving the students' thinking toward an in- creasing velocity, but then the limits of the calculator-based data set supported a conjecture of a final constant speed. The use of technology-based representations,

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  • 250 DOERR AND TRIPP

    conjecturing (along with the confirming of conjectures), and questioning all played a role in this final shift to the notion of a gradually increasing and then constant speed.

    We can identify a common system shared by the students because the data showed that Diane and Hilda came to agree that a ball, when dropped from rest, will increase in speed until it reaches a maximum speed. The ball then simply con- tinues to fall at this constant speed. We take this to be the model that was devel- oped during the students' engagement with the problem during their class meeting. The students' work with the problem, however, did not end with the class session. The group was required to finish the task outside of class and submit a written re- port of their work. In addition, the problem situation was briefly discussed during subsequent class meetings. The evidence from their written report would suggest that their model was further refined and developed; however, we did not have ac- cess to the processes that led to the next stages of refinement and development. We note that in their final written work they refined their measure of speed to feet per second as opposed to simply difference in distance units. The model that they pre- sented in their written work was more closely aligned with the usual Newtonian model to describe and predict the increasing speed of falling objects.

    DISCUSSION

    We observed that multiple shifts in thinking occurred during one group's engage- ment with the ball toss-ball drop modeling activity. Although Diane maintained throughout most of the session that the rate of descent is constant, Hilda experi- enced several shifts in her thinking. In particular, Hilda initially thought the ball fell at a constant speed. She changed her thinking, however, when she observed the graph on the calculator. Hilda again changed her view when another student made a statement that was in opposition to her own view. She shifted her thinking to believ- ing that the ball may have indeed begun and continued to fall at a constant speed throughout its descent. Finally, Hilda refined her model, with speed (as measured by differences in position) as a quantity that gradually increased and then remained constant. Indeed, by the end of the class session, Diane and Hilda agreed that the ball gradually increased in speed until it reached a constant speed. They had achieved what they believed to be a measure of this final speed, based on the data recorded in their calculators. We know that their thinking shifted again because in their final written report the students expressed the speed of the ball as increasing throughout its fall.

    We see these shifts in thinking occurring as the learners' internal models inter- acted with their verbal, graphical, and technology-supported representational sys- tems, which were shared among the group members. The students' representational systems also interacted with and depended on the external (or ex-

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  • perienced) model of the ball dropping. As the learners encountered mismatches between their interpretations of the changes in the speed of the ball as it fell and the data in their calculator table, they resolved these mismatches by modifying their interpretations and by extending their representational systems to include more carefully quantified data elements (e.g., the change in position). It is in this sense, then, that we saw a bidirectional reasoning among a learner's internal model, an external (or experienced) system, and the representational systems, which are part internal and part external.

    The learners also encountered mismatches between their internal models and those of other learners. This was most evident in the ongoing conflict between Di- ane and Hilda regarding the nature of the changing speed of the falling ball. These mismatches, as mediated in part by their words, their graphs, and the table data, were brought to the foreground by instances of conjecturing, questioning, and im- passes in progress. The conjectures made by one learner, when met by an opposing conjecture by another learner or by the same learner, were followed by a shift in thinking by one or both of the learners, thus leading to the development of a more refined system for describing and explaining the experienced system. Similarly, questioning by one or more students called for information, clarification, or chal- lenges that in turn precipitated shifts in their thinking as they sought to resolve a perceived mismatch between their way of interpreting the experienced system and the interpretation of another student.

    CONCLUSIONS

    In this article, we presented findings from an analysis of student model develop- ment occurring in the context of cooperative modeling activities. We showed that shifts in thinking that occurred during engagement with modeling tasks led to the development of mathematical models of the concepts embedded in or evoked by the experienced situation. We also attempted to identify the mechanisms that tended to give rise to shifts in learners' thinking while they were engaged with mod- eling tasks.

    We found that multiple transitions in thinking did occur during the modeling tasks. We described these transitions in terms of an initial interpretation and a later opposing interpretation made by the same individual or by individuals within the same group. As we observed in the case of one learner, the development of a model involved several shifts back and forth before a more stable interpretation of the problem situation was devised. We identified four categories of events that tended to precipitate the occurrence of shifts in thinking. Incidents of conjecturing and questioning, along with the use of technology-based representations, were found to give rise to transitions in student thinking. Impasses in progress were also iden- tified as significant events that tended to drive students to consider alternative ap-

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  • 252 DOERR AND TRIPP

    proaches to understanding and solving problems. The finding that shifts in thinking were driven by incidents of conjecturing, questioning, and involvement with technology suggests that students would benefit from an environment that provides them with the opportunity to express their opinions, ask questions, make numerous conjectures, and work with technology. The interactions and interplay among the occurrences of multiple incidents is an area in need of further research.

    Our finding that impasses also play a role in the occurrence of transitions in thinking would suggest that learning environments should be created that chal- lenge students to the degree that they often find themselves unable to proceed without changing their mode of thinking. It can be productive to allow students to struggle through an impasse rather than simply help them out whenever they en- counter some difficulty. Ultimately, it is our goal that students develop an under- standing of the underlying mathematical model evoked by an experienced situation. If students are not making progress in their work with the problem, it may be helpful to encourage them to ask questions, make conjectures, challenge each other's thinking, and use technology-based representations. Learning envi- ronments characterized by such strategies will likely lead to the development of significant and useful mathematical models by students.

    REFERENCES

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    Doerr, H. M. (1996). Integrating the study of trigonometry, vectors and force through modeling. School Sclence und Mutl~emutics. 96, 407-4 18.

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    APPENDIX Ball Toss-Ball Drop Activity

    In preparation for the ball toss-ball drop activity, students were asked to sketch rea- sonable graphs for both the ball toss and ball drop events based on a written descrip- tion of the two events. The ball toss event involved holding a ball above a motion detector for afew seconds and then tossing the ball straight up into the air and catch- ing it on the way down at the ball's original height. In the ball drop event, the ball was held above a motion detector for a few seconds and then dropped toward the motion detector and caught just above the motion detector.

    The two events were demonstrated at the beginning of the class. After the dem- onstration, the students were provided with data sets for the two events on their TI-92 graphing calculators (Texas Instruments, Dallas, TX). The students were asked to compare the graphs they had sketched for homework with the graphs pro- duced using the graphing calculator. Their next task was to find a functional rela- tion that could be used to describe the position of the ball as well as to predict the ball's position at any time (t). Finally, they were to extend their model of position versus time to include a description of the rate at which the position of the ball was changing with respect to time.

    Trigonometric Graphs Activity

    In preparation for investigating the relation between the amount of sunlight and seasonal affective disorder, the students were asked to find the equations of trigo- nometric functions when given a graph with various features (e.g., amplitude, phase shift) identified. The students were given six graphs and were asked to write the functions that were represented by the graphs. Each function was a transforma- tion of one of the following trigonometric functions:At) = -sin(t),At) = -cos(t), or At) = -tan(t).

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  • 254 DOERR AND TRlPP

    In the in-class activity, students were given a table of data giving the number of hours of daylight for selected days of the year. They were asked to plot 2 years worth of this data on their graphing calculator. Their task was to find the values of A, B, C, and D in the generalized functionflx) = -A sin[B(x - C)] + D and to inter- pret the meaning of those coefficients in terms of the problem situation.

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