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Constructing QuantitativeProblem Representationson the Basis of QualitativeReasoningRolf Ploetzner a & Hans Spadaa University of Freiburg, Department of Psychology ,Niemensstr. 10, D‐79085, Freiburg, Germany Phone:049 / (0) 761 / 203‐2484 Fax: 049 / (0) 761 /203‐2484 E-mail:Published online: 28 Jul 2006.

To cite this article: Rolf Ploetzner & Hans Spada (1998) Constructing QuantitativeProblem Representations on the Basis of Qualitative Reasoning, Interactive LearningEnvironments, 5:1, 95-107, DOI: 10.1080/1049482980050107

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Constructing Quantitative Problem Representationson the Basis of Qualitative Reasoning

ROLF PLOETZNERHANS SPADA

We describe the instructionally and psychologically motivated cognitive simulation modelSepia in which qualitative and quantitative problem representations in physics are concep-tualized as complementary representations. The main emphasis is put on how qualitativeproblem representations can be taken advantage of to facilitate the construction of quanti-tative problem representations. The application domain is classical mechanics. Sepia hasbeen implemented as a knowledge-based system by means of an equation-based repre-sentation language. Two different mechanisms are employed in the model to coordinatequalitative and quantitative problem representations. On the basis of these mechanisms notonly can it be demonstrated how qualitative and quantitative problem representations canbe coordinated to solve physics problems successfully and efficiently, but it can also bepredicted how misconceptions come into play if problems are to be solved which ask for aprecise quantitative solution. These predictions are tested empirically.

INTRODUCTION

Research on learning and problem solving informal sciences such as physics has undergonea major shift with the discovery that the empha-sis of quantitative reasoning is very often mis-leading. With respect to many problems, plainquantitative reasoning is not sufficient for suc-cessful and efficient problem solving perform-

ance. Consequently, artificial intelligence re-

search, educational research as well as, psycho-

logical research direct increasingly more

attention towards qualitative reasoning; and the

role it plays in learning and solving problems.

In this article, we describe the cognitive simu-

lation model Sepia (see also Ploetzner, 1995;

Ploetzner & Spada, 1993). Its development

Interactive Learning EnvironmentsVolume 5, pages 95-107Copyright © 1998 by Swets & ZeitlingerAll rights of reproduction in any form reserved.ISSN: 1049-4820

Rolf Ploetzner, University of Freiburg, Department ofPsychology, Niemensstr. 10, D-79085 Freiburg, Germany;Tel: 049 / (0) 761 / 203-2484; Fax: 049 / (0) 761 /203-2496; e-mail: ploetz@psychologie.uni-fteiburg.de.

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96 INTERACTIVE LEARNING ENVIRONMENTS Vol. 5

started from the assumption that the ability toconstruct and coordinate qualitative and quan-titative problem representations is a precondi-tion for successful and efficient problem solvingin physics. Four main goals guided the develop-ment of Sepia. The first goal was to examine thevarious knowledge structures which underliequalitative and quantitative problem repre-sentations in physics. The second goal was toidentify reasoning mechanisms which allow oneto construct and coordinate qualitative andquantitative problem representations on the ba-sis of these knowledge structures. The third goalwas to investigate how incorrect qualitative phys-ics knowledge or so-called misconceptions affectthe construction and coordination of qualitativeand quantitative problem representations. Thefourth goal was to explore how a learning environ-ment could benefit from the insights gained onthe basis of Sepia.

Sepia's application domain is classical me-chanics. Under scrutiny are textbook problemswhich involve motion in one dimension. Theknowledge investigated with respect to theseproblems is qualitative and quantitative physicsknowledge about various concepts in dynamics(e.g., mass and force) and kinematics (e.g., dis-placement, velocity and acceleration).

In Sepia, qualitative and quantitative problemrepresentations are conceptualized as comple-mentary representations. Each representationis based on qualitative and quantitative physicsknowledge, respectively. Though Sepia solvescertain classes of problems exclusively on thebasis of qualitative reasoning, in this paper themain emphasis is put on how qualitative prob-lem representations can be taken advantage ofto make subsequent quantitative problem solv-ing successful and efficient. Qualitative problemrepresentations facilitate the construction ofquantitative problem representations in Sepiain two different ways. First, qualitative problemrepresentations make possible the derivation ofadditionally required quantitative information,not available to the model beforehand. Secondly,qualitative problem representations provideconstraints to be met by quantitative problemrepresentations.

Sepia reconstructs how correct qualitativeand quantitative problem representations canbe constructed and coordinated during problemsolving. It also allows one to predict where miscon-ceptions might come into play during problemsolving and how they might affect the involvedconstruction and coordination mechanisms.

The rest of this article is organized as follows.We begin with a brief description of the applica-tion domain and then put forward a conceptu-alization of qualitative and quantitative problemrepresentations as complementary. Sub-sequently, the cognitive simulation model Sepiaand its functioning are delineated. The model-based prediction of how misconceptions affectphysics problem solving as well as an empiricalinvestigation conducted to test this predictionare presented next. A discussion of how a learn-ing environment could benefit from the informa-tion encoded and mechanisms employed inSepia concludes this article.

QUALITATIVE ANDQUANTITATIVE PROBLEM

REPRESENTATIONS INCLASSICAL MECHANICS

Sepia's application domain is made up ofstandard textbook problems (e.g., Halliday &Resnick, 1981) which involve one-dimensionalmotion with constant acceleration. The focus ison problems which ask for a precise quantitativesolution. Table 1 shows three problems out ofSepia's application domain. These problems ad-dress various concepts in dynamics (e.g., theconcepts of mass, gravitational force, normalforce and friction force) and kinematics (e.g., theconcepts of displacement, time, velocity andacceleration).

We conceptualize qualitative and quantitativeproblem representations as complementaryrepresentations based on qualitative and quan-titative physics knowledge, respectively. Thisconceptualization does not aim at replacingquantitative reasoning by qualitative reasoning.Rather, we investigate how qualitative andquantitative problem representations have to be

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Constructing Quantitative Problem Representations 97

Table 1. Three problems out of Sepia's application domain

Problem 1: What is the minimum stopping distance for a car travelling along a flat horizontal road with thevelocity v = 30 mis, if the coefficient of friction between tires and road is f = 0.6?

Problem 2: A coin is tossed straight up into the air with the velocity v = 1 m/s. How far up does the coin gountil its velocity is reduced to v = 3 mlsl

Problem 3: A block of massm = 10 kg is projected up an inclined plane with the velocity v = 5 mis. Theplane is inclined at an angle a = 30°. How far up the plane does the block go, if the coefficient of friction be-tween block and plane is f = 0.3?

coordinated in order to achieve successful andefficient problem solving performance.

In Sepia, qualitative physics knowledge en-codes information about three different aspectsof physics concepts. The first aspect is theconditions under which physics concepts canlegitimately be applied. The second aspect is theattributes possessed by physics concepts. Thethird aspect is the values concept attributesmight have. Qualitative physics knowledgemight additionally include information aboutthe abstractions involved in physics concepts orabout their time-dependency. For example,qualitative knowledge about the kinetic frictionforce specifies that there is such a force on anobject, whenever there is a normal force on theobject and the object is moving on a surfacewhich is not frictionless.

Quantitative physics knowledge encodes in-formation about interaction laws and motionlaws. It relies on mathematical formalisms todescribe definitions of and functional relation-ships between physics concepts by means ofalgebraic and vector-algebraic equations. Forexample, quantitative knowledge about the ki-netic friction force specifies that the magnitudeof the kinetic friction force Ff on an objectequals the magnitude of the normal force Fn onthe same object multiplied by the coefficient offriction f: Ff = Fn x f. However, as de Kleer(1975) points out, qualitative and quantitativephysics knowledge cannot be separated by aclear-cut boundary. Rather, they are as extremeends of a continuum, between which consider-able body of physics knowledge might lie (cf.Ploetzner, Spada, Stumpf & Opwis, 1990; VanJoolingen, 1994; White & Frederiksen, 1990).

The qualitative and quantitative physicsknowledge encoded in Sepia provides z. generaldomain model. If the domain model is related toa specific problem, qualitative and quantitativeproblem representations are successively con-structed and coordinated by copying over andinstantiating parts of the general domain modeluntil they yield the problem's solution. Accord-ing to this view, problem solving refers to theconstruction, coordination and consecutivemodification of problem representations on thebasis of a general domain model.

If qualitative and quantitative problem repre-sentations are conceptualized in the way justdescribed, then the former is on no account amere subset of the latter. Furthermore, qualita-tive problem representations are not part of thealgebraic closure of quantitative problem repre-sentations. Instead, qualitative and quantitativeproblem representations are complementary. Aqualitative problem representation includes in-formation about the concepts to be consideredand the distinctions to be drawn. While a quan-titative problem representation allows one toresolve remaining ambiguities frequently in-volved in qualitative problem representations,the construction of an appropriate quantitativeproblem representation needs to be ba:;ed on a(partial) qualitative problem representation.

THE SIMULATION MODEL SEPIA

Sepia's conceptualization is based on twokinds of analyses. Firstly, a computationallysufficient task analysis has been conducted todetermine the qualitative and quantitativeknowledge structures as well as the inference

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mechanisms that are sufficient to solve theproblems in question. Secondly, a reanalysis ofthink-aloud protocols has been carried out tofurther constrain the qualitative and quantita-tive knowledge structures and inference mecha-nisms which were finally implemented into themodel (for details see Ploetzner, 1995). Theprotocols were taken from subjects who partici-pated in an empirical study by Chi, Bassok,Lewis, Reimann and Glaser (1989).1 In thisstudy, nine subjects solved 19 problems whilethinking aloud. The problems were taken fromthe same application domain that makes upSepia's application domain.

Sepia has been implemented in Prolog as aknowledge-based system. It comprises four ma-jor components:

1. An interpreter,2. A knowledge base for qualitative and

quantitative physics knowledge,3. A knowledge base for qualitative and

quantitative vector-algebraic knowledge,4. A knowledge base for geometric and al-

gebraic knowledge.

The interpreter employs domain-independentas well as domain-specific control knowledge tofacilitate the appropriate use of qualitative andquantitative physics knowledge during problemsolving. For instance, the domain-specific con-trol knowledge includes a procedure made upof six steps to construct so-called free-bodydiagrams: (1) identify the object whose motionhas to be analyzed, (2) determine all the forceson the object, (3) draw an arrow for each forceon the object, (4) choose a suitable coordinatesystem as a reference frame, (5) resolve theforces for their components along each coordi-nate axis and (6) apply Newton's second law tothe resultant force along each axis. Though thisprocedure aims at the construction of more andmore complete problem representations, itleaves nevertheless unspecified which physicsknowledge actually has to be applied to achieveeach step when solving a certain problem.2

In Sepia, qualitative and quantitative physicsknowledge is formalized by means of an

equation-based representation language. Asimilar representation language has been em-ployed earlier by VanLehn, Jones and Chi (1992)in implementing the simulation program Cas-cade, a cognitive model of the self-explanationeffect as discovered by Chi et al. (1989). Twobasic kinds of entities are distinguished: physi-cal objects and physics concepts. The relevantphysical objects such as planes and blocks areusually explicitly mentioned in a problem state-ment. The functions and relations defined onphysical objects designate various attributes ofand relations between them.

Further functions and relations are definedon physics concepts such as gravitational force,normal force, kinetic friction force, velocity andacceleration. Three important functions definedon vectorial physics concepts are the functions'magnitude' (possible values are real numbers),'inclination' (possible values are real numbersbetween 0 and 180) and 'sense' (possible valuesare 'up' and 'down'). These functions map avector to its magnitude and direction, respec-tively. Functions and relations on physical ob-jects as well as on physics concepts may beformulated as being time-dependent by mak-ing use of an extensional temporal operator'value (T, F/R)' which denotes the value of afunction F or relation R at time T which mightbe a point in time or an interval of time (cf.Davis, 1990).

Each expression which encodes a piece ofqualitative or quantitative physics knowledge ismade up of a left- (i.e., the condition) and aright-hand side (i.e., the conclusion). An expres-sion's left-hand side is a conjunction of a possi-bly empty set of atomic sentences [S1 . . . Sk].An expression's right-hand side consist of ex-actly one atomic sentence S which is always anequation. The left-hand side and the right-handside of an expression are connected by theimplication operator: SI A. . .A Sk => S.

In the case of qualitative physics knowledge,arithmetic operators must not occur in theexpression's left- or right-hand side. Qualitativeknowledge about physics concepts is formalizedby means of two kinds of expressions. The firstkind of expression states the conditions under

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Constructing Quantitative Problem Representations 99

which a physics concept can legitimately beapplied. The second kind of expression con-strains the values of the various attributes aphysics concept has.

Table 2 shows how qualitative knowledgeabout the kinetic friction force is formalized inSepia. Whenever there is an object which is aninstance of a body, and there is a normal forceon the object due to a plane, and the object ismoving on the plane which is not frictionless,then there is a kinetic friction force on theobject due to the plane (cf. expression Ff2 inTable 2). If there is a kinetic friction force on anobject due to a plane, then the inclination of thekinetic friction force equals the inclination ofthe object's velocity (cf. expression Ff2 in Table2) and the sense of the kinetic friction force isopposite to the sense of the object's velocity (cf.expression Ff3 in Table 2).

In the case of quantitative physics knowledge,an expression's left-hand side might be empty,which is denoted by the symbol ' φ ' . An expres-sion's right-hand side consists of exactly oneatomic sentence which is always an algebraicequation. Table 2 shows how quantitativeknowledge about the kinetic friction force isformalized in Sepia (cf. expression Eq in Table

2): the magnitude of the kinetic friction forceon an object due to a plane equals the magni-tude of the normal force on the object times thecoefficient of friction between object and plane.In its current implementation, the model com-prises 44 expressions which encode qualitativephysics knowledge and 13 expressions whichencode quantitative physics knowledge. Ofcourse, additional vector-algebraic, geometricand algebraic knowledge is required to make themodel executable.

THE CONSTRUCTION ANDCOORDINATION OF CORRECT

QUALITATIVE ANDQUANTITATIVE PROBLEM

REPRESENTATIONS

Sepia is not tailored to the specifics of anindividual subject's problem solving behavior.Instead, it reconstructs characteristic differ-ences in the problem solving behavior of thosesubjects who construct and coordinate correctqualitative and quantitative problem repre-sentations and those who do not (for the recon-struction of problem solving behavior resultingfrom incomplete qualitative problem repre-

Table 2. Qualitative and quantitative knowledge about the kinetic friction force as forma izedin Sepia

Qualitative Knowledge:

Ffl: value(7", instancelObject, body)) = true Avaluejr, instancejforcelObject,, Object2/h), normal_force)) = true A

value(7", instance(Object2, plane)) = true A

value(F, moves_on(Object,, Object2)) = true A

0 (value(7", frictionless(Object2) = true) =>

value(7", instancelforcelObject,, Object2, ff, frictionforce)) = true

Ff2: value(7", instancetforcelObject,, Object2, ff), frictionforce)) = true A

value(7", instance(Object2, plane)) = true =>

value(7", inclination(force(Object], Object2, ff)) — value(7", inclinationlvelocityfObject!)))

Ff3: value(7", instance(force(Object1# Object2, ff), friction_force)) = true A

valueir, instance(Object2, plane)) = true =>

value(7", senselforcelObject,, Object2 ff)) = opposite(value(7", sensefvelocitylObject,)))

Quantitative Knowledge:

Eq: 0 = >

value(7", magnitude(force(Objectt, Object2, ff))) =

value(7", magnitudetforcelObject,, Object2, fn))) * value(7", frictionlObject,, Object2))

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sentations see Ploetzner, 1995). Sepia is appliedto problem descriptions encoded in the samerepresentation language which has been em-ployed to formalize qualitative and quantitativephysics knowledge. For instance, Problem 1 inTable 1 is encoded in the following way:

given (value (1, instance (car_l, car))= true)given (value (1, instance (road_l, plane))"

true)given (value (1, moves_on (car_l, road_l))=

true)given (value (1, inclination (plane_l))= 0)given (value (1, magnitude (veloc-

ity(block*)))= 30::m/s)given (value (1, sense (velocity (block*)))= up)given (value (1, friction (car_l, road_l))= 0.6)sought (value (1 —> 2, magnitude(displace-

ment(car_l))))

Table 3 summarizes-in a more familiar nota-tion—how Problem 1 is solved by Sepia on thebasis of sufficient and correct qualitative andquantitative physics knowledge. Initially, quali-tative reasoning ascertains that there are threedifferent forces on the car. Subsequently, afree-body diagram is constructed.3 This diagramenables the resultant forces on the car alongeach coordinate axis to be specified in qualita-tive terms. Afterwards, these specifications areexpressed in algebraic terms by applying vector-algebraic knowledge and quantitative physicsknowledge about Newton's second law. Theresulting algebraic expressions extend the quan-titative information available to the model. Ad-ditional qualitative reasoning further restrictsthe quantitative knowledge to be applied. Fi-nally, the problem's solution is derived by suc-cessively applying the relevant dynamics andkinematics laws.

Two kinds of coordination between qualita-tive and quantitative problem representationstake place within the model. By drawing onvector-algebraic knowledge, the first kind ofcoordination is that information included in aqualitative problem representation expressed inalgebraic terms to construct additional quanti-tative information not available to the model

beforehand for example. It may algebraicallyspecify the resultant force on an object whosemotion has to be analyzed. In many cases, suchadditionally constructed quantitative informa-tion is a necessary prerequisite if subsequentquantitative problem solving is to be successful.

The second kind of coordination is that theinformation included in a qualitative problemrepresentation provides constraints for the useof already available quantitative knowledge. Forinstance, if qualitative reasoning leads to theconclusion that an object moves with constantacceleration, then only those kinematics lawsare considered which also refer to constantacceleration. This kind of coordination makesquantitative problem solving more efficient, be-cause how quantitative physics knowledge isapplied depends more on the way the problemis represented qualitatively and less on what theunknown variable is.

Thus, if Sepia is applied to a problem on thebasis of sufficient and correct qualitative andquantitative physics knowledge, its problemsolving behavior corresponds to the problemsolving behavior of those subjects who guidethe construction of their quantitative problemrepresentation by means of an initially con-structed (partial) qualitative problem repre-sentation (cf. Chi, Glaser, & Rees, 1982; Larkin,1983).

Preconceptions as IncorrectQualitative Problem Representations

One of the major findings in research onscience education in formal sciences such asphysics is that students before instruction de-velop preconceptions about the phenomenathat physicists explain (for a bibliography seePfundt & Duit, 1994). Preconceptions aremainly made up of qualitative common senseknowledge. Because very often they differ fromthe concepts taught, they are frequently namedmisconceptions. Figure la shows a problemthat is representative of one of the type whichhas frequently been cited by investigators ofmisconceptions in physics (e.g., Clement, 1982;McCloskey, 1983).

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Constructing Quantitative Problem Representations 101

Table 3. How qualitative and quantitative problem representations are coordinated in Sspia

Qualitative reasoning: Quantitative reasoning:

the object whose motion has to be analyzed is the car the forceson the car are Fg, Fn and Ff

+

direction of Ff is opposite to direction of v0 magnitude of Fxequals negative magnitude of Ff

direction of F~ is opposite to direction of Fnmagnitude ofFy equals difference between magnitude of Fn andmagnitude ofFg

motion with constant acceleration and with initial velocity

direction of a is opposite to direction of v0 magnitude of finalvelocity v equals zero

Fx = -Ff=m * ax

V^-V"1*0,v2 = vl + 2 * a * Ar

v = 0 m/s

magnitude of ay equals zero a~ 0 m/s2

ay = a

= -Ff/m

= -(Fn * f)lm

= -(m * g * cos a f)/m

= -g * cos a * f

Ar = vf/(-2 * g * cos a * f)

Figure lb presents a frequently observed in-correct solution to the problem shown in Figurela. In addition to the gravitational force, iterroneously involves an upward-pointing anddecreasing force which does not exist in Newto-nian physics. Since this misconception resem-

bles in many ways the concept of impetus as itwas discussed in the middle ages by Philoponusand others (for a discussion see Franklin, 1976;Szabo, 1976), it is usually named impetus con-cept, one of the most prominent misconceptionsin classical mechanics. Typically, students apply

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102 INTERACTIVE LEARNING ENVIRONMENTS Vol. 5

A coin is thrown straight up in the air and comes straight back down again.Using the dots on its path, draw the forces acting on the coin at that pointin its path.

C«...

• • D b)

- • E IA A

Figure 1. A Physics Problem to Pinpoint Misconceptions (a) and aFrequently Observed Incorrect Solution (b).

the impetus concept to moving objects charac-terized by the absence of a proper force in thedirection of motion. They often assume that animpetus is imparted during the act of setting anobject in motion, that it serves to keep an objectin motion, and that it dissipates or builds upover time in order to account for changes in anobject's motion.

Further examples of well-documented mis-conceptions (e.g., Halloun & Hestenes, 1985) inphysics are the assumptions that the gravita-tional force on a falling object increases overtime, that heavier objects fall faster and that aconstant force imparts a constant speed to anobject. After many different misconceptionshave been documented, it has been recognizedthat they are not easily removed by traditionalinstructional techniques. In particular, instruc-tional techniques which are based on the ideaof confronting students with the logical inade-quacies of their beliefs while at the same timeproviding them with accurate principles appearto be only moderately successful (e.g., Joshua &Duprin, 1987; White, 1993).

In recent years, two theoretical approachesemerged aiming at answering the question ofhow correct and incorrect preconceptions arise.Di Sessa (1988, 1993) characterizes preconcep-tions as being based on a large number of

fragmentary pieces of knowledge called pheno-menological primitives. These fragments aresimple abstractions from countless everydayexperiences and are taken as primitive in thesense that they need no further explanation.Phenomenological primitives are only weaklyorganized and serve as the elementary buildingblocks for common sense predictions and expla-nations. In di Sessa's (1988, 1993) framework,misconceptions are considered as preliminaryconstructs of proper physics concepts. For in-stance, the physics concept of momentumshares several properties with the impetus mis-conception. Consequently, misconceptionsshould not be replaced during instruction, butbe adapted and extended to construct correctphysics knowledge (see also Smith, di Sessa &Roschelle, 1993).

In a different framework (e.g., Carey, 1991;Chi, 1992; Chi, Slotta & de Leeuw, 1994) it isassumed that misconceptions are rooted in mis-categorizations of physics concepts. In particu-lar, it is supposed that students frequently (mis-)assign physics concepts to the familiar categoryof matter whereas physicists assign them to theabstract category of constraint-based interac-tions. For instance, very often students materi-alize the concept of force when they believe thatit is a property of a moving object and that it is

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Constructing Quantitative Problem Representations 103

used up due to the object's motion. Accordingto this view, accurate learning of physics con-cepts is so difficult, because it requires the (re-)assignment of concepts to the unfamiliar andabstract category of constraint-based interac-tions.

Research on students' preconceptions hascertainly shed light on how preconceptionsmight arise and on potential sources for theirpersistence during instruction. However, it hasnot much to offer when their role in solvingproblems which ask a precise quantitative solu-tion is at stake. On the basis of Sepia it cannotonly be hypothesized where misconceptionscome into play during the construction andcoordination of qualitative and quantitativeproblem representations, but also which correctqualitative physics knowledge should be appliedinstead in order to guide the construction ofquantitative problem representations success-fully and efficiently.

HOW MISCONCEPTIONS AFFECTTHE CONSTRUCTION AND

COORDINATION OFQUALITATIVE AND

QUANTITATIVE PROBLEMREPRESENTATIONS

To predict how misconceptions such as theimpetus concept affect the construction andcoordination of qualitative and quantitative

problem representations Sepia's qualitativeknowledge has been extended with incorrectqualitative knowledge which encodes the mis-conceptions under scrutiny. Table 4 shows howan impetus concept is formalized in S;pia. Thefirst expression (cf. expression II in Table 4)states that whenever an object has velocity andno force acts on the object in the direction ofits velocity, then the object has an impetus. Theremaining two expressions (cf. expression 12 and13 in Table 4) specify that the direction of anobject's impetus equals the direction of its ve-locity. Note that this formalization of impetusdoes not include an agent or a recipient of animpetus whereas the formalization of forcesdoes (cf. Table 2). Rather, impetus is conceptu-alized as a property of an object.

After the extension of Sepia's qualitativephysics knowledge, the model has been appliedto the problems presented in Table 1. Sincethese problems are characterized by the ab-sence of a force in the direction of motion, theyare of special interest with respect to the ques-tion of how misconceptions affect the construc-tion and coordination of qualitative andquantitative problem representation:.. In themodel, the application of misconceptions affectsboth kinds of coordination as described above.

With respect to the first kind of coordination,the application of misconceptions such as animpetus concept results in qualitative problemrepresentations which cannot be utilized to

Table 4. The formalization of an impetus concept in Sepia

/,: value(7", instance(Object, body)) = true Avalue(7", instance(velocity(Object), velocity)) = true A- i (value(7"instance(Force, force)) = true Apatient(Force) = Object Avalued, inclination(Force)) = /1 Avaluefr, inclination(velocity((Object))) = l2 AEqualjnclinesl/j,, /2) Avalued, sense(Force)) = S Avalued, sense(velocity(Object))) = S) =>value(f, instance(impetus(Object), impetus)) = true

/2: value(rinstance(impetus(Object), impetus)) = true =>

value(7", inclination(impetus(Object))) = value(f, inclination(velocity(Object)))

/3: value(7", instance(impetus(Object), impetus)) = true =>value(7", sense(impetus(Object))) = value(7", sense(velocity(Object)))

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derive additionally required quantitative infor-mation. This is mainly due to the fact thatmisconceptions usually have no quantitativecounterparts. If the information included in aqualitative problem representation encom-passes a misconception and this information isto be expressed algebraically, an impasse isreached. For instance, if an impetus concept isapplied to Problem 1, the resultant force on thecar along the x-axis is constructed in the follow-ing way: Fx = I - Ff= m* ax. Thereafter, Sepiaattempts to apply dynamics laws to furtherspecify the magnitudes of the involved kineticfriction force as well as of the car's impetus.However, this attempt fails, since the modelcannot compute the magnitude of an object'simpetus. In the model, such an impasse can onlybe overcome by dropping the misconception andby applying correct qualitative knowledge in-stead. Otherwise, additionally required quanti-tative information cannot be derived andproblem solving fails.

With respect to the second kind of coordina-tion, the application of misconceptions resultsin qualitative problem representations whichcan only partially be exploited to constrain thesubsequent construction of quantitative prob-lem representations. In solving simple problemswhich do not require the first kind of coordina-tion this (merely) leads to less efficient problemsolving behavior.

In order to test the predictions derived on thebasis of Sepia, we conducted an empirical studywith 28 subjects (6 females and 22 males) whowere students of a technically oriented highschool. The subjects' grade corresponded tograde 13. At the time the investigation tookplace, the physics of motion in one dimensionhad been taught to the subjects about 9 monthsbefore. In the course of the study, the subjectshad to work on the problems presented in Table1. During problem solving, the subjects hadavailable two tables. The first table showed allthe quantitative physics relevant to the posedproblems. The second table presented a shortsummary of how to resolve vectors for theircomponents. The subjects were urged to useonly the provided materials, to apply no energy

principles, to make notes of all their attempts tosolve a problem as detailed as possible and towork on the problems as in a classroom exami-nation.

The observed number of correct, incorrectand no solutions with respect to Problem 1 was14, 7 and 9, respectively. With respect to Prob-lem 2 it was 4, 13 and 14, and with respect toProblem 3 it was 10, 8 and 5. The application ofan impetus concept has been diagnosed in 7cases with respect to Problem 1, in 5 cases withrespect to Problem 2 and in 16 cases withrespect to Problem 3. Generally, the applicationof an impetus concept has been diagnosed,whenever a subject drew or verbally referred toa force in the direction of motion. One subjectapplied an impetus concept to all three prob-lems, five subjects applied an impetus conceptto Problem 1 and 3, four subjects to Problem 2and 3, one subject only to Problem 1 and sixsubjects only to Problem 3.

Further misconceptions which have been di-agnosed (cf. Halloun & Hestenes, 1985) are thebeliefs that an object subjected to a constantforce moves with constant velocity (three appli-cations) and that acceleration and velocity arethe same concepts (four applications). Unlikewhat is commonly believed, these results clearlydemonstrate that students frequently make useof their misconceptions such as an impetusconcept even in solving problems which ask fora precise quantitative solution.

However, had the application of an impetusconcept any consequences with respect to thecorrectness of the problem solutions? Table 5reveals that the number of correct problemsolutions decreased significantly if an impetusconcept had been applied (x2 = 13.26; df = 2; p< .01).4 The use of misconceptions obviouslyaffected the subjects' problem solving attemptsin that it frequently led to incorrect or noproblem solutions at all.

As predicted by the model, in most of thecases, subjects who applied an impetus conceptencountered severe difficulties in their problemsolving attempts exactly when the results oftheir (incorrect) qualitative problem analyseshad to be coordinated with the use of their

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Constructing Quantitative Problem Representations 105

Table 5. How the use of an impetus concept affects the correctness of problem solutions

Correct Solution Incorrect Solution No Solution

impetus appliedno impetus applied

32730

121931

131023

285684

quantitative physics knowledge. In 25 cases(89%) in which an impetus concept had beenapplied, subjects were not able to express theresults of their (incorrect) qualitative reasoningalgebraically and reached an impasse. 12 sub-jects resolved this impasse by means of illegalalgebraic transformations, for example, leadingto incorrect problem solutions.

Thirteen subjects, however, stopped makingany further effort to solve the respective prob-lem. Only in three cases subjects who appliedan impetus concept came up with correct prob-lem solutions. After one subject failed to alge-braically express an arrow drawn in thedirection of motion, he simply dropped theimpetus concept represented by that arrow andsuccessfully resumed his problem solving at-tempts. Two subjects determined the correctproblem solutions by treating the assumed im-petus concepts algebraically as if they wereresultant forces.

DISCUSSION AND POTENTIALBENEFIT FOR LEARNING

ENVIRONMENTS

In this article, we set forth the instructionallyand psychologically motivated cognitive simula-tion model Sepia in which qualitative and quan-titative problem representations areconceptualized as complementary repre-sentations. The focus has been on how theconstruction of quantitative problem repre-sentations can be rooted in information whichresults from constructing qualitative problemrepresentations. The knowledge that underliesqualitative problem representations includesknowledge about the conditions under whichphysics concepts can legitimately be applied, theattributes possessed by physics concepts and

the values these attributes might have. Theknowledge that forms the basis of quantitativeproblem representations is made up of knowl-edge about the functional relationships be-tween physics concepts.

A learning environment could benefit from asimulation model such as Sepia in various re-spects. Reif and Allen (1992) as well as Vander-Stoep and Seifert (1994) demonstrated that theteaching of the conditions under which physicsconcepts have to be applied is of great impor-tance to learning and problem solving perform-ance in physics. If embedded in a learningenvironment, Sepia can serve to explicate theconditions under which physics concepts haveto be applied as well as to demonstrate how theinformation they provide can subsequently betaken advantage of.

Though many students are able to construct(partial) qualitative problem representations aswell as (partial) quantitative problem repre-sentations, only a few of them are able to flexiblycoordinate both representations to solve theproblems in question successfully (e.g., Chi,Glaser & Rees, 1982; Larkin, 1983). The inspec-tion of how qualitative and quantitative problemrepresentations are constructed and coordi-nated in Sepia can support the design of instruc-tions which explicitly teach the involvedmechanisms. It can draw the students' attentionto instructionally relevant transitions in theproblem solving process which demand the co-ordination of qualitative and quantitative prob-lem representations. In addition, the model canbe utilized to decide which problems are suit-able to exercise the coordination of qualitativeand quantitative problem representations aswell as to generate worked-out examples more

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106 INTERACTIVE LEARNING ENVIRONMENTS Vol. 5

completely than those commonly presented inphysics textbooks.

Physics instruction frequently focuses onquantitative problem solving. One might as-sume that misconceptions are not applicableunder these conditions. On the basis of thepredictions derived by means of Sepia as well asthe empirical findings, however, it is not safe tomake this assumption. Students apply theirmisconceptions even to problems which ask fora precise quantitative solution. In most of thecases, these misconceptions are extremely harm-ful to the subjects' problem solving attempts. Ifthe intention is to prevent students from apply-ing their misconceptions, the posed problemshave to be designed very carefully in a way thatthe qualitative as well as the quantitative prob-lem representations do not meet the applicationconditions of the students' misconceptions. Se-pia offers a tool to support the design of prob-lems which fulfill these requirements.

According to Sepia, the students most af-fected by their misconceptions are the studentswho master textbook physics halfway through.On the one hand, students who have just startedto learn physics very often approach textbookproblems by constructing only a quantitativeproblem representation (e.g., Chi, Glaser &Rees, 1982; Larkin, 1983). In this context, theirmisconceptions are in fact not applicable. On thebasis of this approach, they are able to solve onlyvery simple problems which do not require theconstruction and coordination of a qualitativeand quantitative problem representation.

On the other hand, advanced students fre-quently possess sufficient and tightly relatedqualitative and quantitative physics knowledge.They are able to flexibly construct and coordi-nate multiple problem representations and thusto solve even demanding textbook problems.However, those students who have just learnedthat many textbook problems which ask for aquantitative problem solution nevertheless re-quire the initial construction of a qualitativeproblem representation are bad off. Since theyvery often still hold strong beliefs in their mis-conceptions, they might apply them and, as aconsequence, might even fail to solve simple

problems they were able to solve earlier. Thus,on the basis of Sepia one would predict that thedevelopment of problem solving performance inphysics over time follows some U-shaped func-tion.

Since Sepia cannot only be taken advantageof to hypothesize where students' misconcep-tions might come into play during solving text-book problems, but also to determine thecorrect qualitative physics knowledge to be ap-plied instead, we suspect that instructions de-signed on the basis of Sepia are especiallyhelpful for students who would otherwise fall inbetween the two sides of the U.

Acknowledgments: The research described in thisarticle was supported by the German National Re-search Foundation (Deutsche Forschungsgemein-schaft) under contract Sp 251/6-1. We thankJohannes Bellert and Michael Bosniak for their assis-tance in conducting the empirical investigations.Three anonymous reviewers provided many helpfulcomments on an earlier version of this article. Finally,it is largely due to the many competences of MichaelStumpf that this kind of research is possible at theDepartment of Psychology at the University ofFreiburg.

NOTES

1. We thank Michelene Chi and her colleagues for pro-viding us with their data.

2. The functioning of Sepia's interpreter is described indetail in Ploetzner (1995).

3. Sepia constructs such a free-body diagram only interms of symbolic descriptors. Graphical elements are notinvolved.

4. This is not to say that the subjects would have beensuccessful without applying an impetus concept. Many sub-jects made more than just this error.

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