making their own connections: students' understanding of multiple models in basic electricity

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Making Their Own Connections:Students' Understandingof Multiple Models in BasicElectricityJoshua P. Gutwill , John R. Frederiksen & Barbara Y.WhitePublished online: 07 Jun 2010.

To cite this article: Joshua P. Gutwill , John R. Frederiksen & Barbara Y. White (1999)Making Their Own Connections: Students' Understanding of Multiple Models in BasicElectricity, Cognition and Instruction, 17:3, 249-282, DOI: 10.1207/S1532690XCI1703_2

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Making Their Own Connections:Students’ Understanding of Multiple

Models in Basic Electricity

Joshua P. GutwillDepartment of Education in Math, Science, and Technology

University of California, Berkeley

John R. FrederiksenCognitive Science Research Center

Educational Testing Service

Barbara Y. WhiteDepartment of Education in Math, Science, and Technology

University of California, Berkeley

In this study we explore the educational impact of teaching high school students sev-eral models of the same phenomenon. In particular, we create different sets of modelsof static electricity (each set containing 1 model of particles and 1 model of aggregatesof particles) that are connected in specific ways and measure the effects of these linkson further learning of circuit behavior. Four groups were run through a 2-week curric-ulum on direct current electricity. One group received “coordinated” models that areconnected together by their mechanisms and representations. Another group received“noncoordinated” models that do not connect via mechanisms and representations.The other two groups served as control groups for the Coordinated andNoncoordinated conditions. Contrary to the hypothesis that Coordinated modelswould be easier to learn and apply, posttest results show that students in theNoncoordinated condition outperformed those in their control group; meanwhile, theCoordinated group did not outperform its control group on these tests. An analysis ofprocess data gathered while students were working through the curriculum suggests

COGNITION AND INSTRUCTION, 17(3), 249–282Copyright © 1999, Lawrence Erlbaum Associates, Inc.

Requests for reprints should be sent to Joshua P. Gutwill, Department of Visitor Research andEvaluation, The Exploratorium, 3601 Lyon Street, San Francisco, CA 94123. E-mail:[email protected]

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that students in the Noncoordinated group were actively combining the differentmodels presented in that treatment condition. We conjecture that the effort of trying toachieve coherence among models may have led to a better understanding as measuredby the posttests.

In high school physics classes, teachers often provide students with multiple waysof thinking about a phenomenon. For example, students learn to determine the ve-locity of a dropped ball in at least two ways: (a) the velocity is the result of an accel-eration caused by a gravitational force, or (b) the velocity is due to the conversion ofpotential energy to kinetic energy as the ball moves through a gravitational field.Although tightly connected in the domain of mechanics, these two ways of findingthe velocity of a dropped ball are disparate and employ different models (Hestenes,1992). The first has more of a causal flavor: A gravitational force between the earthand ball accelerates the ball, which causes a change in its velocity. The second ismore of a constraint system: A certain amount of energy, constrained by the initialheight of the ball, changes form during flight. How do students make sense of thesedifferent models? How do they create connections between them? Does it help stu-dents if instructors use models that have clear links between them?

The research reported here investigates the cognitive and educational impact ofteaching multiple models of a particular phenomenon. Specifically, we connecteddifferent models of electricity by linking the mechanisms and representationsacross the models; then we measured the effects of these links on further learningof circuit behavior. To take a concrete example, let us suppose that we teach stu-dents two models of electricity, one at the “particle” level of electrons and anotherat the “flow” level of current. Connecting the models means that each one woulduse a version of the same intuitive mechanism. In this case, push would explainboth why electrons move in the particle model and why charge moves in the flowmodel: Crowded electrons push each other apart, and current flows because com-pressed charge pushes outward to expand. In addition, the representations in themodels would be linked by a mental zooming process: Imagine looking at a multi-tude of dark particles on a white background and then zooming out your mentalview; eventually, you would see a single grayish region (see Figure 1). The moreparticles, the darker the region would become.

Because the models employ linked mechanisms and representations, we call thisa coordinated set of models. In a noncoordinated set, individual electrons wouldpush, but current would not. Instead, it might flow from high levels of charge to low

250 GUTWILL, FREDERIKSEN, WHITE

FIGURE 1 Zooming out from an area con-taining many particles yields a single gray re-gion.

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levels of charge, thereby employing a kind of fall mechanism. The representationsalsowouldbenoncoordinated,becausezoomingout fromparticlesmovinghorizon-tally does not lead to charge levels moving vertically (see Figure 2).

To define our terms more carefully, a model is a simplified, runnable system thatrepresentsaprocessor interaction (White&Frederiksen,1990). It consistsofasetofrules and representations that enable one to step through time to predict and explainphenomena. Instructional models may be instantiated in several types of media, in-cluding computer simulations, static drawings, mathematical equations, and words.Learning a model means constructing a mental model that shares critical featureswith the instructional model (Hestenes, 1992). In this study, we have created modelsthat represent and explain the behavior of electric circuits. The mechanisms withinthe models are intuitive causal processes, such as pushing or falling, that can be usedto explain a variety of phenomena (Clement, Brown, & Zietsman, 1989; diSessa,1983, 1988, 1993). Representations in the models are the diagrams used to displayvarious objects, such as electrons, charge, batteries, and so on. Links among modelsare created through simple “transformations” of the mechanisms and representa-tions used in the models (Lakoff, 1987). We describe the details of our curricularmodels and the coordination process later in this article.

The main hypothesis was that Coordinated models—ones that use linked mech-anisms and representations—would be more understandable and effective thannoncoordinated models.1 Applied to the domain of direct current electricity, thehypothesis was that students who learn coordinated models for charge flow wouldbe better able to solve circuit problems, because their knowledge of circuit rela-tions would be well-grounded in linked mechanisms and representations.

STUDENT DIFFICULTIES IN ELECTRICITY

Electricity provides an excellent context for testing the effects of Coordinated mod-els, primarily because integrating ideas about static electricity (electrostatics) andcircuit theory (electrodynamics) appears to be enormously difficult for students(Eylon & Ganiel, 1990). Perhaps because the causality is straightforward in

UNDERSTANDING MULTIPLE MODELS IN ELECTRICITY 251

1This research comes from Joshua P. Gutwill’s doctoral thesis (Gutwill, 1996). Consequently, the co-ordination hypothesis for connecting multiple models was his. The other authors had competing hypoth-eses for how to link models, which will be explored in the discussion section.

FIGURE 2 Zooming out from an area con-taining many particles does not produce a“fluid level” that moves up and down.

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electrostatics, physics students more readily understand the essence of it: Likecharges repel, and unlike charges attract. Unfortunately, they are much less suc-cessful at comprehending ideas in electrodynamics. Most distressing is the findingthat students are unable to explain the origins of particular voltage distributions incircuits (Eylon & Ganiel, 1990; Frederiksen & White, 1992; Millar & Beh, 1993;White & Frederiksen, 1990).

In a study of advanced high school seniors’ ideas about electricity after in-struction, Eylon and Ganiel (1990) found that most students did not understandthe relation between behavior of circuit devices at the macro level and interac-tions of electric charge at the micro level. Although students performed reason-ably well on a test involving capacitor circuits, a number of responses revealedinconsistent micro–macro reasoning. A minority of the students believed thatthere would be no current in at least one wire leading to a capacitor, but that bothcapacitor plates would become charged. This is impossible because in order forboth plates to become charged, current would have to flow through both wires.The students’ answer indicates a divide between their macro models of circuitbehavior and micro models of electron motion.

In depth interviews with eight other advanced high school students indicatedthat this problem is more widespread than shown by the written test used in Eylonand Ganiel’s study. Seven of the eight students either made no connections be-tween micro and macro at all, or made connections that still divorced certain as-pects of the two models. Most common was a discrepancy between voltage(macro) and potential (micro).

In our own research, we have noted a similar conceptual gap betweenelectrostatics and electrodynamics. Taking a case study approach, we (Frederiksen& White, 1992) interviewed a 14-year-old high school student just as she finisheda physical science course. The student was quite capable of explaining the electro-static processes of charging by induction and was competent at solving quantita-tive circuit problems. When asked for a connection between them, however, shesaid, “I have absolutely no concept of the relation between the two” (p. 216).Pressed to define the concept of voltage, she responded, “I don’t know. Theytaught them as facts, formulas. I never knew what they were” (p. 216).

Although traditional instruction often fails to facilitate a link betweenelectrostatics and electrodynamics, students sometimes try to construct their ownconnections. In a study (Gutwill, Frederiksen, & Ranney, 1996) of students’ un-derstanding of electricity in a standard high school course, we found that most stu-dents spontaneously shifted among three different perspectives when explainingtheir answers to qualitative circuit problems. These perspectives included particle(e.g., electrons, protons, neutrons), aggregate (e.g., current, energy, power), andcircuit topology (e.g., open, series, parallel). However, only the highimprovers—students whose scores increased significantly from pretest toposttest—shifted perspectives in a “coherent” manner. In the context of the study,


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shifting coherently meant that the student seemed to be using descriptions at oneperspective to explain behavior at another. This suggests that being able to connectmultiple models or perspectives is an important part of expertise in this domain.

According to this and other research on learning electricity, the problem stu-dents face in making a connection between micro and macro probably stems fromtheir difficulty in grasping the abstract concept of voltage and relating it to micro-scopic phenomena (Dupin & Johsua, 1987; Eylon & Ganiel, 1990; Frederiksen &White, 1992; Millar & Beh, 1993). Without a deep understanding of what voltageis, students can neither solve circuit problems effectively nor relate circuit con-cepts to ideas in electrostatics, such as electric repulsion. This study investigateshow to design multiple models so that students may overcome these difficulties.

The Coordinated models hypothesis is that Coordinated models will help stu-dents comprehend the meaning of voltage and its role as causal agent in the circuit.Because Coordinated models concretize the concept of voltage with similar intu-itive mechanisms (versions of push) in multiple contexts, students should have agreater chance of learning what voltage is and of applying it in learning to solvecircuit problems. In short, students who learn Coordinated models should have amore stable understanding of voltage, and thereby be better able to use it at boththe micro and macro perspectives.

Although this hypothesis may seem compelling, several alternative hypothesesare plausible. For example, one might believe that the most important factor inmodel design is choosing the best mechanisms and representations; how the modelsare connected is less critical. Hence, Coordinated models may not necessarily yieldthe deepest understanding in a domain. Yet another viewpoint might argue for teach-ing Noncoordinated models, because they provide students with explicitly differentways of thinking about electricity. Armed with models that use different mecha-nisms and representations (like push and fall), students would be prepared to tackle awider range of problems than students who learn Coordinated models.

DESIGNING COORDINATED MODELS

We have suggested the need for connecting models in the domain of electricity. Butwhat exactly do connected models look like? What kinds of design features do theyhave?

Built on Simple Mechanisms

The mechanisms used in the models are simple and intuitively understandable, akinto diSessa’s (1983, 1988, 1993) phenomenological primitives, or p-prims for short.diSessa (1983) argued that p-prims, small knowledge elements that are abstracted


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from life experience, form the basis for people’s naive explanations of physicalphenomena. For example, if asked to explain how a pool stick can cause a cue ballto move, a person would typically invoke the force as mover p-prim (“a directed im-petus acts in a burst on an object,” resulting in “displacement and/or speed in thesame direction”; diSessa, 1993, p. 217). The cue ball moves because the pool stickexerted a force on it. For most people, this is simply understood and requires no fur-ther explanation. By building models on p-prims that are consistent with formal sci-ence concepts, instructors may help students overcome preconceptions while con-structing new, more scientifically valid knowledge.

In related research, Clement and his colleagues (Brown & Clement, 1989;Clement, 1993; Clement et al., 1989) have employed such intuitive ideas in“bridging analogies” to help students understand various difficult concepts in me-chanics. For example, many physics-naive students have difficulty imagining anupwards force when considering why a book rests on a table. To help students un-derstand the origins of this force, Clement intentionally drew on students’ springi-ness p-prim. Through a set of analogies among springs, flexible boards, and rigidtables, students are encouraged to construct a model that represents all objects, in-cluding the table, as slightly springy. After thinking about the analogies, studentscorrectly predict an upward force on a book by a table.

In this study, we provide students with models of electricity that directly buildon p-prims like push, carry, and fall. Our approach is distinctly different from thebridging analogies work, because we do not make analogies between, say, chargemovement in a wire and water draining (falling) out of a tank. In our models, fall orpush simply are the mechanisms for charge movement; the model is presented as asimplified description of actual phenomena, not as an analogy between actual andrelated phenomena.

The Zoom Transformation

The essential feature of coordinated models is that each model utilizes mechanismsand representations that are linked to those in the other models via a simple mentaltransformation, such as zooming (Lakoff, 1987). The zooming process is rooted inthe physical experience of moving closer to or further from a set of objects.2 Imag-ine moving further away from someone’s front yard, for example. At a certainpoint, the blades of grass blend together, becoming a single lawn. Such transforma-tions do not have to be literal in the sense of requiring visual perception. Accordingto Kosslyn (1980), the experience of zooming in on a mental image is similar to theexperience of visually perceiving an object grow larger as one walks toward it. Thissuggests that when students encounter a transformation of mechanisms and dia-


2Lakoff (1987) labels this a “Mass to Multiplex” transformation because the object in one’s mentalview changes from a single entity to a multiplex of objects, or vice versa.

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grams in a curriculum, they should be able to represent that transformation men-tally.

In Noncoordinated models, the models also would be at different perspectives(Particle and Aggregate), but the mechanisms and representations would not belinked by this zooming process.

A General Strategy of Model Progression

The main goal of this educational approach is to help students learn models of elec-trodynamics by connecting them to models of electrostatics. To do this, we followthe lead of our previous research (Frederiksen & White, 1992; White, 1993a;White, Frederiksen, & Spoehr, 1993) in two respects.3 First, we insert an “Aggre-gate” Model in between the Particle Model and the Circuit Model. This model ex-plains the behavior of charge, an aggregated substance consisting of many elec-trons. Hence, it is intermediate in perspective—neither zoomed all the way in toparticles, nor zoomed all the way out to lightbulbs, wires, and batteries.

Second, we design the set of models to progress by a process of emergent be-haviors. This means that the behaviors that evolve out of running one model be-come the fundamental behaviors of the successive model (cf. Frederiksen, White,& Gutwill, 1999; White & Frederiksen, 1990; White et al., 1993). We can illustratethis emergence of model behavior in a computer simulation. As electrons pushapart in the Particle Model simulation, the electrons spread from a crowded regionto an emptier one. This emergent behavior, namely moving from a region withmore to one with less, becomes one of the fundamental behaviors of charge in theAggregate Model (see Figure 3).

Similarly, the movement of charge leads to properties of the Circuit Model.When charge moves through a resistor in a closed circuit in the simulation, for ex-ample, a difference forms in the amount of charge on either side of the resistor.This charge difference is referred to as a voltage difference in our simplifiedmodel.

In past research, the Aggregate Model was more abstract than the ParticleModel, inheriting its mechanisms from the Particle Model (cf. Frederiksen et al.,1999; White, 1993a; White et al., 1993). The representations in the AggregateModel were also more abstract. For example, the model used abstract represen-tations of the amount of charge (through a bar graph) rather than illustrations ofthe charge itself. One new feature of our study is that here the Aggregate Modelitself employs explicit mechanisms and representations just as the ParticleModel does, so they are at a similar level of abstraction. In other words, the rep-


3The previous research was performed by the Qualitative Understanding of Electrical Systems andTroubleshooting group at Bolt, Beranek, & Newman, Inc. Laboratories, the University of California,Berkeley, and the Educational Testing Service.

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resentations in the Aggregate Models show actual charge, either through densityof the charge or height of a charge pile. Our study investigates the educationalimpact of using models that all employ specific mechanisms and more concreterepresentations.

We created two sets of models, one coordinated and one noncoordinated. Eachset contained three qualitative models: one at the particle level (electrostatics), oneat the circuit level (electrodynamics), and one at the aggregate level. See Figure 4for the model progressions. The Aggregate Models in the two sets determinewhether or not each set is coordinated. In one of them, the mechanisms and repre-sentations link with the Particle Model by a zoom transformation, thereby formingthe coordinated set. The other Aggregate Model does not have mechanism or rep-resentational links, providing us with a noncoordinated set.

The models in both sets advance from particles to aggregate to circuit behavior.There are empirical reasons for taking such a bottom-up approach. First, startingwith a Particle Model probably builds best on what students already know and un-derstand. As previously mentioned, the mechanism of electrons pushing eachother apart is quite easy for students to understand (Frederiksen & White, 1992;White et al., 1993). Second, research in the field of economics indicates that start-ing at the micro level and moving to the macro promotes greater understandingthan moving in the opposite direction (Fizel & Johnson, 1986).


FIGURE 3 The simulation’s behavior for the Aggregate Model (charge moves from more toless) emerges from the Particle Model simulation (electrons push apart).

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Each set consists of three models. The next sections provide a portrait of each ofthe models needed for the two complete model sets. All of the models in the studywere instantiated in computer simulations. For clarification, we point out that ourmodels of electricity need not have employed computer simulations. In fact, stu-dents often learn scientific models in other ways, including talking with an instruc-tor, drawing pictures, manipulating formulas, or observing physical processes. Wechose computers as our primary medium because they are particularly conduciveto representing runnable systems like electric circuits.

Local Models of Electricity

The Particle and Aggregate Models are called local models because they deal withlocal interactions, either between electrons or between adjacent sections of a con-


FIGURE 4 The two model progressions, illustrated with diagrams from the computer simula-tions. On the left, the Coordinated Aggregate Model is connected to the Particle Model via azooming transformation. No such connection exists in the Noncoordinated Aggregate Model onthe right.

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ductor or resistor. In addition, these models are electrostatic in that they do not in-clude whole circuit behavior. Examples of diagrams that were used to representconductors, resistors, and batteries in each of the models is shown in Figure 5.

Particle Model. In the Particle Model, a multitude of electrons interact witheach other, with the material they inhabit, and with batteries. The model should pro-vide students with the basic sense of mechanism for why electricity moves. The fol-lowing is a brief sketch of the Particle Model.

Electrons move because they push on each other to spread apart. When moreelectrons are in one place than another, those in the crowded area push harder thanthose in the emptier area, so electrons move from the former to the latter. Resis-tance is modeled as a blocking process in which “imperfections” in the material actas obstacles in the electrons’ paths.4 As shown in Figure 5, the difference between


FIGURE 5 Representations of conductors, resistors, and batteries used in the simulations forthe three local models. (Note: The size of each has been reduced to create this figure.)

4In the interest of simplicity, we avoided modeling resistance as an attractive interaction betweennegatively charged electrons and positively charged nuclei. In all of our models, there is only one type ofcharge. The “imperfection” metaphor is akin to the notion of doping in semiconductor materials.

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a resistor and a conductor in the Particle Model is that the former has more imper-fections (rectangles) that block the movement of the electrons (circles).

Batteries act to carry electrons from one terminal to another, to produce a differ-ence in the number of electrons across the terminals. This difference is called avoltage difference. The battery in Figure 5 has a difference of 4.0 (6.0 – 2.0 = 4.0).

The Particle Model concretizes the concept of voltage in the form of a pushfrom crowded electrons. The push (voltage) is distinct from the spreading (move-ment). To ensure that the model describes only electrostatic phenomena, studentsnever see a complete circuit with particle representations.

Coordinated Aggregate Model. To create a model at the aggregate per-spective that connects to the Particle Model, we chose mechanisms that could resultfrom a zoom transformation. In other words, the mechanisms in the CoordinatedAggregate Model (CA Model) are virtually the same as those in the Particle, exceptthat they act on a single entity (charge) instead of a multitude of individual objects(electrons). A brief sketch of the CA Model is as follows.

In the CA Model, a single substance composed of many electrons, calledcharge, pushes on the walls of its container to spread out or expand. When morecharge is in one place than another, the charge in the more concentrated areapushes harder than the charge in the diluted area, so charge moves from the formerto the latter. Resistance is modeled as a blocking process—“imperfections” in thematerial act as obstructions around which the charge must flow. Figure 5 showsthe representations for resistors and conductors in the CA Model. The differencebetween them is that the resistor has more imperfections (dark rectangles) betweenits sections of charge (shown in shades of gray) than the conductor. These imper-fections slow the movement of the charge. This is similar to the blocking processin the Particle Model’s resistors.

Batteries act to carry charge from one terminal to another to produce a differ-ence in the amount of charge across the terminals. This difference in amount iscalled a voltage difference. Because the battery moves charge in the direction op-posite to natural charge movement, it creates the potential for charge to move. Ifeach terminal of a battery were connected to either side of a neutral conductor, forinstance, the charge on the positive terminal would push to spread out into the con-ductor. Meanwhile, charge on the other end of the conductor would push into thenegative battery terminal because that terminal has less than the normal amount ofcharge.

Like its particle counterpart, this model makes voltage concrete by employing apush mechanism. Hopefully, the distinction between pushing and moving helpsstudents distinguish between voltage and current. This model lies in between parti-cle and circuit behavior in that it describes electrostatic phenomena but also is used


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to derive whole-circuit relations. We will revisit the notion of model derivationwhen we describe the Circuit Behavior Model (CB Model).

Noncoordinated Aggregate Model. The Noncoordinated AggregateModel (NCA Model) also takes an Aggregate perspective, but it uses differentmechanisms from those in the Particle Model and the CA Model. Although theNCA Model is internally consistent, its mechanisms and representations cannot begenerated by a zooming transformation from the Particle Model. A brief sketch ofthe NCA Model is as follows.

In this model, an aggregated substance composed of many electrons, calledcharge, drains down to seek its own level. When one place has a higher amount ofcharge than another, the higher charge moves down to the lower charge. Resis-tance is modeled as a constricting process—imperfections in the material constrictthe path through which charge may flow. As shown in Figure 5, resistors differfrom conductors in the size of the imperfections (length of the vertical line) be-tween piles of charge. A longer imperfection in resistors constricts the amount ofcharge that can flow in a given time interval. The constricting process of the NCAModel stands in contrast to the blocking process of the Particle Model, adding tothe lack of coordination between the two models.

Batteries in the NCA Model raise charge from one terminal to another, to pro-duce a difference in the height of the charge piles across its terminals. This differ-ence in height is called a voltage difference.

Unlike the other two models, the NCA Model concretizes voltage via a fallmechanism. Although students should be able to understand this mechanism forcharge movement, it is not linked to the push mechanism in the Particle Model by asimple transformation. Hence, students should have more difficulty integrating themodels. Like the CA Model, the NCA Model describes electrostatic phenomenaand also leads to whole-circuit relations.

Global Model of Electricity

Circuit Behavior Model. The CB Model is quite different from the previousthree models. The Particle Model, CA Model, and NCA Model all describe local,mechanistic behaviors of electricity, and each one has its own set of mechanismsand representations. In contrast, the CB Model explains global behaviors, and ituses the representations from the Aggregate Models.

This model primarily describes the qualitative relations one might find in anygood introductory electricity text. For example, students learn that more resistancein a circuit leads to less current, that series resistors act like voltage dividers which“share” the battery’s voltage, and that parallel resistors offer less resistance than asingle resistor. The model focuses on the global behaviors of entire circuits.


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The relations between resistance, charge flows, and voltage differences in theCB Model emerge from the behavior of charge found in the Aggregate Models(Frederiksen et al., 1999; White et al., 1993). For instance, in the local models, re-sistance is something that slows down the flow of charge. However, once a resistoris connected to a battery in a complete circuit, a new property of resistanceemerges. In addition to slowing charge flow, resistive boundaries now actuallycreate charge differences. They accomplish this by causing charge to accumulateon one side of a section boundary as it moves off the other side. Because thecharges on either side of the boundary cannot equalize instantly, a charge differ-ence develops (see Figure 6).5 This only happens in a complete circuit and, hence,is an emergent property of the Aggregate Model.

Model Coordination

The Particle and CA Models are coordinated because the mechanisms and repre-sentations they employ for charge movement, resistance, and battery action aretransformable via a mental zooming process. The Particle and NCA Models arenot coordinated by this process. Figure 4 shows the links or lack of links amongrepresentations from these models. Table 1 lists the mechanisms for each of themodels.

EXPERIMENTAL METHOD

The purpose of this investigation was to compare the performance of two groups: aCoordinated group and a Noncoordinated group. The Coordinated group learned aset of models that all employ the push mechanism for why charge moves. TheNoncoordinated group learned a model set with both push and fall as the mecha-nisms that cause charge movement.

Four Groups (Two Treatmentand Two Control) Were Needed

Comparing a Coordinated group to a Noncoordinated group would confoundmodel coordination with the type of Aggregate Model used in the study. For exam-ple, suppose the CA Model was simply a better model than the NCA Model. We


5This figure uses representations from the noncoordinated model, but could have used diagrams fromthe coordinated model instead. The CB Model employs representations from either of the aggregatemodels, depending on the treatment group.

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might erroneously conclude that the coordination between Particle and CA Modelwere responsible for the Coordinated group’s superior performance. To separatethe effects of model coordination from the particulars of each Aggregate Model, weran two control groups. Participants in the control groups did not learn the ParticleModel, but only the Aggregate and CB Models. The full, four-group design isshown in Figure 7.

The control group for the coordinated condition, labeled C-Control, learnedAggregate and CB Models that use the push representations—charge density, ob-stacles at resistive boundaries, and lateral movement in batteries. The other controlgroup, NC-Control, also learned Aggregate and CB Models, but with mechanismsand representations for fall—piles of charge, constrictions at resistive boundaries,and vertical movement in batteries. These control groups allowed us to investigate


FIGURE 6 The formation of a charge difference across a resistor in a complete circuit. Theability of resistor boundaries to create charge differences is new to the Circuit Behavior Modeland emerges from both of the Aggregate Models.

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whether there were any direct effects of the push and fall mechanisms and repre-sentations.

Comparing Treatments to Controls

The intention was to design both the push and fall mechanisms and representationsto be understandable for students, thereby allowing the two control groups to per-form equally well. With this experimental design, however, equivalent perfor-mance of the control groups was not required. Instead, the analysis compares eachtreatment group (Coordinated and Noncoordinated) to its control (C-control andNC-control, respectively). Previous research in electricity education found that stu-dents who learn only an Aggregate Model perform worse than students with an Ag-gregate and a Particle Model (White et al., 1993). This suggests that the two treat-ment groups should both outperform their respective control groups. In fact, such acomparison in this study could be viewed as a replication of previous experimentalresults.6 The question unique to this study is, “Will the two treatment groups out-perform their respective control groups by different amounts?”

Participants

The participants in the study were 62 high school students, ages 15 to 17 years, fromtwo ethnically diverse schools in the San Francisco Bay area. All students in thestudy had recently completed either Grade 10 or 11. Because the material requiredsome knowledge of atoms and electrons, only students who had taken a high schoolchemistry course, but had not yet studied physics, were allowed to participate. Ade-quate algebra skills were also needed for the curriculum, so students were accepted


6It is not a perfect replication, however. As previously mentioned, the Aggregate Model in the Whiteet al. (1993) study was more abstract than the Aggregate Models in this study.

TABLE 1Electrical Phenomena and the Mechanisms Used by Each Model to Describe Them

ElectricalPhenomenon

ParticleMechanism

CoordinatedAggregate Mechanism

NoncoordinatedAggregate Mechanism

Voltage Push Push FallCurrent Spread apart Spread out/expand Drain downResistance Block Block/obstruct ConstrictBattery action Carry Carry Raise

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only if they had completed a high school algebra course. Students worked insame-gender pairs during the study and were paid for their time.

Students who chose to participate agreed to attend a 1-hr “prestudy” session inwhich they took two pretests. The pretests were used for the purpose of matchingparticipants across groups. Sixty-nine students attended the pretest session. Ofthose, 64 were chosen as participants in the study, with the intent of placing 16 stu-dents in each of the four conditions. Unfortunately, midway through the study, onepair of male students dropped out and could not be replaced. This left three of thegroups with 16 students and the fourth (the C-control group) with only 14 students.

Matched Groups

During the prestudy session, students took one test that assessed their algebraicskills and another that measured their understanding of simple circuit relations. Theformer, called the Algebra pretest, included some arithmetic problems, but focusedprimarily on proportional and algebraic reasoning. In previous research, algebraskills correlated highly with ability to solve circuit problems, even qualitative ones(Frederiksen et al., 1999). The latter test, called the Batteries and Bulbs pretest (BBpretest), measured whether students realize the need for a complete circuit, and


FIGURE 7 Four-group (two treatment and two control) design to test the effects of model co-ordination on learning.

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tested their ability to predict the relative brightnesses of lightbulbs connected in aseries and a parallel circuit. In order to be understandable to students as a pretest, theBB pretest employed realistic-looking circuit diagrams (i.e., the diagrams resemblephotographs of circuits). See the Appendix for a sample problem from each of thetwo pretests.

Because students worked in pairs, the groups were created by matching pairsrather than individuals. The pairs were matched across groups based on three crite-ria: gender, Algebra pretest score, and BB pretest score. The test score means andgender breakdowns for each group are shown in Table 2. To further control for dif-ferences among participants, the Algebra pretest scores were included as acovariate in all analyses.

Curriculum Format

During the 2-week curriculum, participants worked in isolated pairs to learn quali-tative and quantitative models of electrostatics and simple circuit theory. The cur-ricular materials consisted of a student workbook and an instructional videotape,which, together, led students through a series of computer simulations and interpre-tations regarding how electricity works.

The curriculum centered on an “Inquiry Cycle” (White, 1993b) in which stu-dents thought about circuits, made concrete predictions, ran simulations, inter-preted the results, and applied their new knowledge to real electrical devices. Thiskind of cycle, a simplification of the scientific method, has been successful in help-ing students construct models in the domains of electricity (e.g., Gunstone, 1991)and mechanics (e.g., White, 1993b; White & Frederiksen, 1998).

Guided by a written workbook, students followed the Inquiry Cycle each timethey investigated a different topic or type of circuit. During the Question stage, avideotaped instructor motivated a new topic by introducing an interesting


TABLE 2Pretest Means and Gender Breakdown for Matching Across Groups

Algebra Battery and Bulb Students

Group M SD M SD Female Male

Coordinated 84.7 13.9 51.9 25.4 10 6C-control 82.7 15.6 50.0 28.3 8 6Noncoordinated 84.4 15.5 55.6 21.3 10 6NC-control 84.9 17.1 51.9 24.3 8 8

Note. Students worked in same-gender pairs. C-control = control group for the coordinatedcondition; NC-control = control group for the noncoordinated condition.

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real-world phenomenon, like lightning. Then, students were faced with a narrowerquestion about electricity, such as “How does electrical charge behave in a con-ductor?” The narrow questions were directly addressed by computer simulationand discussion. The instructor’s introduction also included information necessaryfor making predictions in the first experiment, such as the idea that electrons pushaway from each other.

The Prediction stage required students to make predictions about a particularexperimental setup (e.g., how much charge will move across a given resistiveboundary?). Making predictions required students to remain on task and may havemotivated them to think more deeply about each experiment. Predicting outcomesalso may have increased students’ personal investment in each exercise (Hatano,1986).

Just after making predictions, the students ran a simulation of the circuit or ofthe circuit components on a Macintosh computer. This was the Simulation phase.

When each simulation was complete, students returned to their workbooks tointerpret the results. In this Interpretation stage, students used what they alreadyknew (or what they learned from the videotaped instructor’s introduction) to makesense of the new data. In addition, they compared the results with their originalpredictions and explained any discrepancies. This method of promoting cognitiveconflict has shown promise for building more accurate models in electricity(Closset, 1985; Frederiksen et al., 1999; Steinberg & Wainwright, 1993; White &Frederiksen, 1990; White et al., 1993) and in mechanics (Minstrell, 1989; White,1993b). Once students had discussed their interpretations of the results with eachother, they played the videotape again to hear the instructor’s interpretation. Theintention of the video feedback was to prevent students from straying too far fromthe curriculum’s target “expert model.” Moreover, having students compare theirinterpretation with that of the instructor may again have engendered productivecognitive conflicts.

The three stages of Prediction, Simulation, and Interpretation were revisitedseveral times within a workbook chapter to give students experience with differentaspects of a certain topic. After a topic had been fully explored, students turned tothe Application phase.

During Application, the videotaped instructor applied the new informationfrom the chapter to real-world electrical phenomena and devices. For instance, atthe end of the chapter on electrons in conductors, the instructor explained the phe-nomenon of static discharge that often occurs after one walks across a carpetedfloor. The workbook employed a scaffolding and fading method (Collins, Brown,& Newman, 1989; Palincsar & Brown, 1984) for the Application phase by askingstudents to make their own applications in later chapters.

In summary, the curriculum consisted of workbooks, computer simulations, andan instructional videotape. It employed an Inquiry Cycle to take students throughsets of experiments about electricity. By predicting, simulating, and interpreting,


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students constructed mental causal models of the electric circuit under the guid-ance of a videotaped instructor.

To measure student learning gains as a result of working through this curricu-lum, we employed six posttests, which will be described in the next section.

RESULTS

The data reported in this article came from two sources: written assessments andvideotape of students working through the curriculum.

Data From Written Assessments

The written assessments, given before and after the curriculum, include forcedchoice items and short answer questions.

Overall pretest to posttest differences. We already described the purposeof two of the pretests, Algebra and BB. A third pretest, called the Relative Bright-ness test (RB test), was given after the students had been assigned to groups but be-fore instruction had begun. It consisted of multiple choice questions about the rela-tive brightnesses of lightbulbs in various circuit configurations (see the Appendixfor a sample problem). Both electricity pretests (BB and RB) were given again asposttests, thereby providing a measure of the change in students’ understanding ofsimple circuits. Overall, the students in the study significantly improved their per-formance on both of the electricity tests from pretest to posttest; BB test: t(61) =4.67, p < .001; RB test: t(61) = 5.86, p < .001). The effect size of the increase in testscores on both tests was 0.8σ.7 This improvement indicates that students learnedabout the behavior of circuits over the course of the study. The group differences onthese and other tests are described in the next section on posttest results.

Group differences in the posttests. The six posttests in the study were de-signed to measure students’ understanding of the three types of models in the study:Particle, Aggregate, and CB. For all posttests, we performed a two-way analysis ofcovariance (ANCOVA), using Group (treatment–control) as one factor, Coordina-tion (coordinated–noncoordinated) as another, and Algebra pretest score as a


7Throughout the analyses, effect sizes were calculated using the square root of the error variance.When an assessment was given as both a pretest and a posttest, as in the case of these two tests, the errorvariance was that of the pretest.

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covariate. To reveal differences among the four groups, we employed threeplanned comparisons of the following:

1. Mean scores of the two control groups (two-tailed test). This comparison al-lowed us to determine whether any performance differences were due to thedifferent mechanisms and representations in the two Aggregate Models.

2. Means of the Coordinated treatment group and its control (one-tailed test).This comparison permitted us to investigate whether learning a ParticleModel that was coordinated with the Aggregate Model led to better perfor-mance.

3. Mean scores of the Noncoordinated treatment group and its control(one-tailed test).8 This third comparison allowed us to determine whetherlearning a Particle Model that was not coordinated with the AggregateModel led to better performance.

Taken together, these three comparisons provided evidence as to whether a coordi-nated model set is superior to a noncoordinated set.

Table 3 shows the adjusted least square means from each group on the posttests.For a sample problem from each of the posttests shown in the table, see the Appen-dix.

In the Particle Model test, students explained the behavior of electrons in vari-ous devices (conductor, resistor, battery) and in whole circuit situations. The for-mat of the test included short answer, multiple choice, and diagrammatic questions(see the Appendix for a sample problem from this test as well as the other tests dis-cussed following this). The results from Table 3 indicate that the two treatmentgroups, Coordinated and Noncoordinated, reliably outperformed their controlgroups on this test; Coordinated versus C-control, t(57) = 1.92, p = .03;Noncoordinated versus NC-control, t(57) = 1.78, p = .04. The effect sizes forlearning a Particle Model (i.e., the Coordinated and Noncoordinated outperform-ing their controls) were both 0.7σ. A planned comparison revealed no statisticaldifference between the two control groups, t(57) = 0.05, p = .96. This result is per-fectly understandable: On Particle Model problems, the students who studied aParticle Model scored higher than those who did not.

In the Aggregate Model test, the object of interest was charge rather than elec-trons. Students were asked to determine how charge behaves in primarily electro-static situations (i.e., without circuits). See the Appendix for a sample problem.Planned comparisons revealed no reliable differences among the groups;Coordinated versus C-control, t(57) = –0.28, p = .61; Noncoordinated versusNC-control, t(57) = –0.41, p = .66; C-control versus NC-control, t(57) = 1.00, p =


8The one-tailed tests were appropriate because previous research led us to expect that groups wholearned a Particle Model would outperform those who did not (White et al., 1993).

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.32. The effect sizes for both the Coordinated and Noncoordinated groups were0.1σ. This suggests that the two Aggregate Models were equally understandable tostudents in all groups.

Several tests dealt with students’ understanding of circuits (i.e., understanding ofthe CB Model). Already mentioned were the BB and RB tests, both of which servedas posttests. Two other tests focused on students’ grasp of the CB Model: the Rela-tive Voltages and Currents test (RVI test) and the New Diagrams test (ND test).

In the RVI test, students predicted qualitatively the voltage distributions andcurrents in various circuits. This multiple choice test employed representationsthat were identical to those found in the curriculum and was designed to revealwhether certain mechanisms and representations have an impact on students’ rea-soning about voltages and currents in circuits.

The ND test introduced students to the standard, scientific circuit diagram. Thetest required students to judge qualitatively the relative voltages and currents invarious circuits. Except for the new representations, questions in this test weresimilar to those in the RVI test. The ND test measured transfer of problem-solvingskills to representations that did not support the mechanisms in the models. Allfour of these posttests assessed students’ understanding of the CB Model—under-standing woefully lacking in students in many traditional classrooms (Dupin &Johsua, 1987; Millar & Beh, 1993).

ANOVAs revealed a consistent pattern across three of these posttests: theNoncoordinated treatment group consistently outperformed its control group,whereas the Coordinated group did not. Figure 8 illustrates the pattern.

Starting with the RB test, we found a group difference between Noncoordinatedand NC-control, t(57) = 2.44, p < .01; effect size: 1σ; no difference between Coor-dinated and C-control, t(57) = 0.05, p = .48; effect size: 0.02σ; and no differencebetween the two control groups, t(57) = 0.56, p = .58.


TABLE 3Group Adjusted Least Square Means and Standard Errors on Each Posttest

Test Score

Coordinated C-control Noncoordinated NC-control

Posttest M SD M SD M SD M SD

Particle Model 61.0 4.4 48.3 4.7 60.7 4.4 49.2 4.4Aggregate Model 64.2 3.2 65.6 3.4 68.5 3.2 70.3 3.2Relative Brightnessa 69.3 2.7 69.1 2.9 76.3 2.7 66.9 2.7Relative Voltages and Currentsa 57.4 4.8 59.1 5.1 65.5 4.8 53.6 4.8New Diagramsa 51.2 6.4 57.3 6.8 61.4 6.4 47.3 6.4Batteries and Bulbsa 78.7 4.0 70.1 4.3 66.9 4.0 70.5 4.0

aMeans are from multiple choice responses.

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The next test of circuit behavior, the RVI posttest, showed a similar pattern ofresults: The Noncoordinated group outperformed its control group, t(57) = 1.76, p= .04; effect size: 0.9σ; whereas the Coordinated group did not, t(57) = –0.25, p =.60; effect size: 0.1σ. There was again no difference between the two controlgroups, t(57) = 0.80, p = .43.

The third test to show this same pattern was the ND posttest (multiple choicequestions only). The planned comparisons revealed a marginally significant dif-ference in mean score between the Noncoordinated group and its control, t(57) =1.57, p = .06; effect size: 0.6σ; but no difference between the Coordinated groupand its control, t(57) = –0.65, p = .74; effect size: 0.1σ. Once again, there was nodifference between the control groups, t(57) = 1.07, p = .29. This pattern, appear-ing across tests that used very different representations, suggests a robust result.

In the BB test, which employed realistic-looking diagrams of circuit ele-ments, however, the planned comparisons revealed no differences across thegroups. There was no difference between the Noncoordinated group (M = 75.9)and its control (M = 80.4), with t(57) = 0.90, p = .19; effect size: 0.2σ, nor be-tween the Coordinated group (M = 87.4) and its control (M = 81.3), with t(57) =0.69, p = .25; effect size: 0.2σ. There was also no significant difference betweenthe control groups; C-control versus NC-control, t(57) = 0.14, p = .89. Perhapsthere were no group differences on this test because the diagrams in the test por-trayed circuit elements in a realistic way. This may have cued memories and ex-periences of actually connecting such physical devices—experiences that arecommonplace in elementary and middle-school science curricula. Students’


FIGURE 8 Consistent pattern of mean multiple choice scores across three posttests.

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problem solving could have been based on recall of these experiences ratherthan on the models in the study.

From the results, we are left with the question: Why did the Noncoordinatedgroup do so well on three of the tests? The results disconfirmed the hypothesis thatCoordinated models should be easier for students to learn and should facilitate abetter connection between electrostatics and electrodynamics. Instead, the resultsshowed that having Noncoordinated models led to greater performance on electro-dynamics (circuit) problems.

Process Data

To better understand the posttest results, we turn to data gathered while studentswere working with the curriculum. An analysis of the process data revealed someevidence that the Noncoordinated group was combining mechanisms and objectsfrom the Particle and Aggregate Models. This suggests that those students made aconcerted effort to bring together disparate mechanisms. Such work may have hada payoff, and may explain the Noncoordinated group’s greater problem-solvingskills in the circuit behavior posttests.

Data from daily videotape recordings of eight matched pairs of participants(two pairs from each group) may shed light on students’ thinking as they studiedthe curriculum. Because our resources allowed us to videotape only two pairs pergroup, the analysis of the process data is suggestive at best and cannot be taken asstatistically reliable.

Perhaps the best place to look for reasoning patterns that may elucidate ourposttest results is where students who had worked with the Particle Model first en-countered the Aggregate Model. Since the control groups never studied the Parti-cle Model, they were excluded from this analysis.

The most striking finding was that students in the Noncoordinated groupseemed to combine elements of the Particle and Aggregate Models in their expla-nations for charge behavior. Although we might expect such integrated reasoningfrom the Coordinated group, it was somewhat surprising to find it in theNoncoordinated group. This finding suggests that the Noncoordinated studentsconstructed their own connections among disparate models and thereby achievedsome measure of coherence in their understanding.

In this section, we present examples of ways in which they attempted to com-bine models. By studying how the participants talked with each other about thebehavior of charge, we found that they combined models in three ways: (a) theyused Particle Model objects with Aggregate Model mechanisms, (b) they usedParticle Model mechanisms with Aggregate Model objects, and (c) they com-bined aspects of Particle Model objects and Aggregate Model objects. For each


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of these types of model combinations, we provide one representative examplefrom each treatment group.

Combining Particle objects and Aggregate mechanisms. The first typeof model combination involved students using objects from the Particle Model withmechanisms from the NCA Model. An example of this is that students talked about“electrons draining down.” We contend that this sort of reasoning suggests a com-bined model that incorporates aspects of both the Particle Model and AggregateModel because electrons are particles and draining is a mechanism from the Aggre-gate Model.

During the first exercise in the Aggregate Model section of the curriculum, thenoncoordinated students were faced with the problem presented in Figure 9.

One pair of students, Don and Craig,9 responded to this problem by jokingaround a bit. Don drew two downward curving lines that connected the right andleft conductor and two horizontal lines that indicated new charge levels, as illus-trated in Figure 10.


FIGURE 9 Workbook questions in Exercise 1 for students in the Noncoordinated andNC-control conditions.

9We use pseudonyms for students’ names.

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Don then playfully considered the possibility that multiple objects will “ride thewaterslide”:

D: Well, I think that 5 points of this will ride the waterslide—C: God, they think we’re dumb or something.D: Ride the waterslide, then see, they’ll have fun, and then they’ll fill up right

here.C: Yup, I agree.D: But it’ll hurt when they run into each other at the end of the waterslide.C: [Laughing] It’ll hurt. Like when you hit the water, you’re going real fast.

Kapow!D: [Writing] Waterslides are fun.

This segment shows that Don was able to combine the idea of multiple objects fromthe Particle Model with the mechanism of falling or draining from the NCA Model.His references to “they” and Craig’s personified “you” indicate discrete objects,but Don’s drawing, his mention of a “waterslide,” and his explanation that the ob-jects “fill up” all suggest a fall mechanism.

Although this analysis revealed model combinations among the students inthe Noncoordinated group, such an analysis was nearly impossible to performwith the Coordinated group. In the Coordinated condition, the Particle and Ag-gregate Models were designed to use the same mechanism for charge move-ment, namely push. For students to say that “the electrons push each other” tellsus nothing about whether they were combining models or simply reasoning withthe Particle Model. This limitation made it difficult to compare the reasoningpatterns of the Noncoordinated and Coordinated groups. Nonetheless, we be-lieve it is an interesting result that students in the Noncoordinated group werecombining models. In the next section we describe another way in which thesestudents blended the models.

Combining Particle mechanisms with Aggregate objects. The secondtype of model combination entailed bringing mechanisms (such as push) from theParticle Model into the Aggregate Model. In this scenario, the students reasonedabout NCA Model objects and Particle Model mechanisms. For example, studentssaid that the “charge levels push each other.”


FIGURE 10 Don’s drawing in his workbook to show a“waterslide” through which charge may move.

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Another pair from the Noncoordinated group, Trista and Jenny, ran the com-puter simulation in Exercise 1 and saw that the charge drained until it becameequal. At that point, they imported the mechanism of push from the ParticleModel. This came on the heels of a brief discussion about charge conservation:

T: That’s the property of conservation of initial energy.J: It only moves around, but it doesn’t, it’s never destroyed.T: It’s never what?J: It’s never destroyed. The charges are never destroyed.10

T: Uh hmm. They’re gonna reach an equal level but they have, both pushingon each other [gestures by bringing her two vertical palms together hori-zontally] and an equal amount of pressure and stuff. Energy.

J: So it reaches an equilibrium and the charge won’t be destroyed. Somethinglike that.

T: Uh hmm. They did it and [inaudible] reach an equal level right.J: Yeah.

In her statement that the charges will “reach an equal level,” Trista seemed to bethinking about charge levels moving up and down until they were equal. However,she then immediately stated that when the levels are equal, the charges will be“pushing on each other” equally. The push mechanism was found only in the Parti-cle Model, not the NCA Model. Hence, it appears that Trista has combined a Parti-cle Model mechanism with an NCA Model object.

As before, it was difficult to apply this analysis method to the Coordinatedgroup, because their mechanisms were already similar across models.

Combining Particle objects and Aggregate objects. So far, we have seenthe Noncoordinated participants combine objects from one model with mechanismsfrom another model. In a third type of model combination, students tried to recon-cile the differences between a model of discrete objects on the one hand and a modelof continuous objects on the other. This type of model combination should be visi-ble in both the Noncoordinated and Coordinated groups because both of the groupslearned one model with discrete objects (Particle) and another model with continu-ous objects (Aggregate).

At the end of the first exercise, Trista and Jenny from the Noncoordinated groupbegan to wonder what would happen if the objects had an odd amount of totalcharge. In other words, what would happen if the final amounts of charge on each


10Remarks like “the charges are never destroyed” were ambiguous with respect to model combina-tion because they could have meant either “the [aggregate] charge in the two conductors is never de-stroyed,” or “the [individual electron] charges are never destroyed.”

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conductor were not whole numbers, but were fractions? Posing such a questionsuggests thinking of charge as indivisible, more similar to a collection of discreteobjects than to a continuous substance:

J: What if there’s an unequal charge? [laughs] I’m stuck on that. If there’s anunequal charge, does charge pass back and forth?

T: Does charge what? Balance?J: Like, unequal charge.T: Uh huh.J: It just happens that one is 5 and one is 15, then they come together at 10 and

10.T: There is no such thing as an—J: An odd charge. I mean like an odd thing.T: No, in reality there isn’t. They always have to be equal. I guess. Oh well.

Jenny believed that charge may pass back and forth instead of becoming com-pletely equal in the two conductors. This indicates that she conceptualized chargeas fundamentally indivisible and discrete, an idea more in line with electrons in theParticle Model than with charge in the NCA Model. On the other hand, she alsospoke of an “unequal” or “odd” charge, which may indicate her viewing it as a sub-stance.

It turns out that Jenny was quite prescient because the very next problem, Exer-cise 2, involved so-called unequal charges (see Figure 11).11

Trista immediately recognized that this was exactly the situation Jenny had justmentioned:

T: Oh, there we are! All [inaudible] for you.J: Yay!

When they reached the question about charge stopping its movement, they becameconfused:

J: How will the charge stop moving? You can’t like split a charge, can you?T: I don’t know.J: What does it? Well, that’s a funny question. If there’s like one extra, it will

just keep bouncing back and forth, won’t it?T: There’ll be a difference of one right?J: Yeah, so it wouldn’t, really, never stop?


11When the simulation ran in Exercise 2, the charge drained from left to right until each conductor hada charge pile of 12.5 units.

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T: I don’t get it.J: We could write a guess and then say I don’t know. So, if it would try to bal-

ance, it would just keep on going back and forth, back and forth, back andforth [gestures back and forth horizontally with her pencil]. Unless theysplit it up, and I don’t think you can do that. And it can’t go anywhere, be-cause then it’ll be destroyed.

T: So when, so when will it stop moving then?J: I don’t know. Never? Maybe like 25 divided by 2 and then there’s like

always one left. [Laughs] [Reading] It’ll stop moving when it reachesan equality. But it will never reach an equ—. OK. [Writes then turns toT] But in this case it might not end, or it might not reach—

T: So what are you writing?J: I write that the charge stops moving when it reaches an equality, but in this

case it might not reach an equality.T: Yeah, I guess that’s the only explanation.

Again, they seemed to be conceptualizing charge both as a collection of discrete ob-jects and as a single, continuous object. Only an odd amount of a continuous sub-stance could balance or reach an equality, yet only a single object could bounceback and forth and be impossible to split.

One of the two student pairs from the Coordinated group also showed evi-dence of combining the objects of these two models. The representations in Ex-


FIGURE 11 Workbook questions in Exercise 2 for students in the Noncoordinated andNC-control conditions.

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ercise 2 were different for the Coordinated group, but the questions wereidentical (see Figure 12).

Tania and Alicia responded to the questions in Exercise 2 by talking about split-ting an electron:

T: OK. This is kind of tricky because we know there’s no such thing as a halfelectron, right? So, it’s like forcing it again, there’s like not really a way tobalance it. So my theory is that they’re gonna try and connect, but there’sgonna be an electron that’s gonna ricochet itself back and forth. You knowwhat I mean? Because it’s not gonna have anywhere to go.

A: Yeah. So, what, there’s gonna be like, 12 from each side, but there’s gonnabe an electron that’s gonna go [motions back and forth with her hand.]

T: Yeah, back and forth.A: Sharing.T: Yeah.

This pair, although looking at the “substance” representation for charge in theworkbook, was still operating from within the Particle Model. They had directlyimported the discrete objects (electrons) into this situation involving continuousobjects.

In the next section, we discuss the meaning and implications of these results.


FIGURE 12 Workbook questions in Exercise 2 for students in the Coordinated and C-controlconditions.

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DISCUSSION

The purpose of our study was to investigate how people make sense of multiplemodels of a particular phenomenon in science. The main hypothesis was that stu-dents would construct the greatest understanding if the models were coordinated.Coordinated models are connected to one another via a simple mental process, suchas zooming in or out on the mechanisms and representations in the models.

The results appear to have disconfirmed this hypothesis. Although theCoordinated and Noncoordinated treatment groups performed equally well ontests involving local models (Particle and Aggregate), differences appeared intheir understanding of the global model (CB). On three different tests of the CBModel, students in the Noncoordinated condition outperformed students in theNC-control group. In contrast, the Coordinated group did not score any higher thanthe C-control group. This suggests that learning the Particle Model had a greaterbenefit for the Noncoordinated group than for the Coordinated group.

One possible explanation for these results is that the students in theNoncoordinated group simply had more intuitive mechanisms at their disposal.Because they could employ both push and fall mechanisms, they might have beenable to tackle a wider array of problems, thereby achieving greater success on theposttests. Such an interpretation, however, ignores the evidence that students in theNoncoordinated condition appeared to be making their own connections betweenthe push and fall models. An analysis of the process data revealed that students inthe Noncoordinated condition were actively combining aspects of the Particle andAggregate Models. We found three types of model combination: importing objectsfrom the Particle Model to the Aggregate Model, importing mechanisms from theParticle Model to the Aggregate Model, and combining objects from both the Par-ticle Model and the Aggregate Model. If the important factor had merely been thatthe Noncoordinated models provided students with more mechanisms, modelcombination would not have been so prevalent in the students’ reasoning.

We believe that the mental work of actively constructing combined models it-self may have been responsible for the Noncoordinated group’s greater benefitfrom learning the Particle Model. In this regard, we note that the Coordinated andNoncoordinated situations provided students with differential opportunities to cre-ate connections across the models. In the case of the Noncoordinated group, thedifference in mechanisms in the models challenged students to think about howone model’s objects or mechanisms might be incorporated within or mapped to an-other model. This accounts for the first two types of model integration we found. Incontrast, because the Coordinated models shared the same mechanisms, therewere fewer occasions for students to build their own connections. Both instruc-tional treatments included a shift from models of discrete objects to those of con-tinuous objects and, thus, also provided an opportunity to build connections acrossthe models—connections which we discovered in both groups during our analysis.


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The shift from discrete to continuous objects provided the Coordinated group’sonly opportunity for building their own connections across the models. In sum-mary, more frequent opportunities for active connection-building could explainwhy the Noncoordinated group gained more from learning the Particle Model thanthe Coordinated group did.

Besides the possibility of increased engagement, what benefits are there for stu-dents who build their own connections across models? One important side effect ofproviding models that use different mechanisms and representations is that studentsmay better realize that the models are only models and not “true pictures” of an un-seen world. According to Hestenes (1992), successful physics students must learn todistinguish between the conceptual world, which contains models produced by thescientific community, and the physical world, which consists of real things and pro-cesses. Both pairs of students in the Noncoordinated group questioned whether frac-tional charges exist “in reality.” In fact, one of the pairs even conjectured that thedecimal in “12.5” might simply mean that some charge is still actually moving eventhough the computer simulation depicted a static state. Such statements suggest thatthe studentsweredistinguishingbetween the realworldofelectricalphenomenaandthe conceptual world of the electricity model and its representations.

Learning to think about science as a process of model building, rather than as asearch for “facts,” can affect students’ attitudes and behavior (Hammer, 1995;Linn & Songer, 1993; White & Frederiksen, 1998). Research has shown that whenstudents have a constructivist view of science, they work hard to understand scien-tific concepts. In contrast, students with a positivistic outlook typically adopt astrategy of memorization. By providing students with slightly inconsistent modelsof the same phenomenon, we may encourage them to realize that models are con-structed entities and not discovered truths.

In light of our findings, should educators simply provide students with disparatemodels of a phenomenon and hope they build their own connections? We think not.Instead, we advocate designing model sets that are sensible and coherent but thatalso challenge students to think deeply about the phenomena being modeled. In thisstudy, it appears that providing models with unlinked mechanisms and representa-tions engendered deep thinking: Students tried to integrate the different models.This may have been responsible for the success of the Noncoordinated group.

Our previous research (White et al., 1993) on students’ understanding of multi-ple models of electricity suggests yet another way to challenge students. We taughtstudents a progression of models, from Particle to Aggregate to CB, but designedthe Aggregate Model to be more abstract than the Particle Model. For students tomake sense of the behavior of charge in the Aggregate Model, they had to use themechanisms from the Particle Model. Encouraging students to engage in this kindof connection-building across models seemed to be beneficial. Students wholearned all three models outperformed students who learned only the (more ab-stract) Aggregate Model and the CB Model, even though learning more models


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imposed a greater cognitive load. As in the current study, we found that challeng-ing students to engage with the models led to better problem-solving performance.

In future research, we hope to determine which method of challenging studentsis more effective: teaching noncoordinated mechanisms or helping students moveto a higher level of abstraction. Our hypothesis is that the latter would be superiorbecause more abstract models can lead to greater transfer across problems and do-mains (White, 1993a).

In conclusion, this study disconfirms the notion that multiple models should becoordinated by having similar mechanisms and transformable representations ex-plicitly presented within each model. Instead, the results suggest that students canprofit most from learning multiple models when those models fit together well butalso contain cognitive challenges that lead students to build their own linkagesamong the models.

ACKNOWLEDGMENTS

Joshua P. Gutwill is now at the Department of Visitor Research and Evaluation atThe Exploratorium in San Francisco, CA.

This study was supported by funding from the Educational Testing Service. Wegratefully acknowledge Sue Allen, Ming Chiu, Andrea diSessa, George Lakoff,Michael Ranney, Christina Schwarz, and two reviewers for their helpful com-ments on a previous version of this article.

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APPENDIXSample Test Questions and Accompanying Diagrams

Test Sample Question

Sample Diagram forCoordinated and

C-control

Sample Diagram forNoncoordinated and

NC-control

Algebra 12 – 5x = 2. Solve forx.

N/A (text only) N/A (text only)

Batteries andBulbs

Is the lightbulb on oroff?

RelativeBrightness

Compare thebrightnesses of thetwo lightbulbs.

Particle Model What happens to theelectrons when theconductors areconnected?

Aggregate Model What happens to thecharge over time?

Relative Voltageand Current

Compare the voltagesand currents ofResistors 1 and 2.

New Diagrams Compare the voltagesand currents ofResistors 1 and 2.

Note. C-control = control group for the coordinated condition; NC-control = control group for thenoncoordinated condition; N/A = not applicable.

Bulb 1

Bulb

2

Battery

+_

Bulb 1

Bulb

2

Battery

+_

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making their own connections: students' understanding of multiple models in basic electricity

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