Generative Analogies as Mental Models
Dedre Gentner
Subject explaining electric current: If you in- to predict the set of inferences that should follow crease resistance in the circuit, the current slows from use of a given analogy; (2) contrast two analogi- down. Now that s like a high--cars on a highway cal models for the domain and show that they lead to where you--if you notice as you close down a lane, different indirect inferences; (3) to show that people s you have cars moving along. Okay, as you go down inferences concerning simple circuits vary according to into the thing, the cars move slower through that which of these models they use; and finally (4) to narrow point. discuss the general issue of analogical models and structure-mapping. When people are reasoning about an unfamiliar do- main, they often appear to use analogies, as in the A structure-mapping theory of analogy. The claim above example from a protocol. Analogies are also used here is that analogies select certain aspects of exist- in teaching, as in the following excerpt: ing knowledge, and that this selected knowledge can be The idea that electricity flows as water does is a structurally characterized. First, let s consider what Picture the wires as pipes carrying good analogy. an analogy is not. An analogy such as water (electrons). Your wall plug is a high-pres- (1) An electric circuit with a battery and resistor sure source which you can tap simply by inserting much like a plumbing system with a reservoir a plug. The plug has two prongs--one to take the and a constricted section of pipe. flow to the lamp, radio, or air conditioner, the clearly does not convey that all of one s knowledge second to conduct the flow back to the wall. A about the plumbing system should be attributes! to the valve (switch) is used to start or stop flow. circuit. The inheritance of characteristics is only If implicit or explicit analogies are an important partial. This might suggest that an analogy is a weak determinant of the way people think about complex sys- similarity statement, conveying that some but not tems, then it becomes crucial to know exactly how such all of the characteristics of the base system apply to analogies work. This paper considers the psychological the target system. But this weak characterization fails role of these analogies in structuring the target to capture the distinction between literal similarity domain. and analogical relatedness. Contrast statement (1) with The first question that must be posed is whether a literal similarity statement like such seeming analogical models do in fact strongly (2) A hose is like a pipe. affect the person s conceptualization of the target The literal domain (the Generative Analogy hypothesis), or whether similarity statement (2) conveys that the pipe and they are merely convenient ways of talking about the the hose share object attributes--e.g. cylindrical domain (the Mere Terminology hypothesis). The mere use shape--as well as sharing similar relationships with of terms borrowed from a given domain--as, for example, other objects--e.g. CONVEY (hose, water)/CONVEY (pipe, when electricity is discussed in terms of moving vehi- water). Statement (1) also conveys considerable overlap cles or flowing water--is not in itself proof that the in functional relations: e.g. I MPEDE (resistor, current) speaker is conceiving of electricity as deeply analo- /IMPEDE (constriction, water). However, it does not gous to traffic or to water flow. convey overlap of objects and their attributes. The To demonstrate that an analogy has generative con- resistor as a separ to object need not have any quali- ceptual power, we must show that nontrivial inferences ties in common with A constriction. The analogy, in specific to the base occur in the target. These infer- short, conveys overlap in the system of relations ences must be such that they cannot be attributed to among objects, but no particular overlap in the charac- shallow lexical associations; e.g. it is not enough to teristics of the objects themselves. The literal find that the person who speaks of electricity as similarity statement conveys overlap both in relations "flowing" also uses terms such as "capacity" or among the objects and in the attributes of the indivi- "pressure". Such usage is certainly suggestive of a dual objects. Generative Analogy, but it could also occur under the The analogical models used in science can be Mere Terminology hypothesis. characterized as structure-mappings between complex The goal here is to show that, at least some of systems. In these analogies, the objects of the known the time, the Generative Analogy hypothesis holds: domain, the base domain, are mapped onto the objects of that deep, indirect inferences in the target follow the domain of inquiry, the target domain; the predicates from use of a given base domain as an analogical model. of the base domain--particularly the relations that hold To do this, we must first decide what inferences should among the nodes--are then applied in the target domain. follow from use of a given analogy, and then observe Structure-mapping analogy asserts that identical opera- whether the analogies people adopt appear to affect the tions and relationships hold among nonidentical things. set of inferences they readily make. The relational structure is preserved, but not the The plan of this paper is (1) to propose a objects. structure-mapping theory of analogy that will allow us Given a particular propositional representation of knowledge we can proceed with an explicit charac- terization of analogical mapping. A structure-mapping analogy between a target system T and a base system B This research was supported by the Office of Naval Re- search and was carried out at Bolt Beranek and Newman and at U.C.S.D. Donald Gentner collaborated on all aspects of this work, but particularly in developing the two analogical models of electricity. I thank Allan Collins, Ken Forbus, Ed Smith and Al Stevens for many insightful comments on this research. Please address correspondence to Dedre Gentner, Bolt Beranek Newman, 50 Moulton Street, Cambridge, MA 02138.
9 7 T o model of imple elec ic ci c i . One common analog ed o each imple elec ici i ba ed on pl mbing em . Fig e 1 ho he c e-mapping con e ed b hi analog . The objec -node of he h d a lic ba e domain (e.g. he e e oi and con ic- ion) a e mapped on o he objec -node ( he ba e and e i o ) of he ci c i . Gi en hi co e pondence of node , he analog con e ha he ela ion hip ha hold be een he objec and objec -a ib e of he h d a lic em al o hold be een he node of he elec ic em: fo e ample, ha c en inc ea e i h ol age j a a e of a e flo inc ea e i h Combina ional p oblem . One a o ob e e deep p e e; and ha c en dec ea e i h e i ance indi ec infe ence in he a ge domain i o a k j a he a e of a e flo dec ea e i h deg ee abo diffe en combina ion of componen . Fo e ample, of con ic ion. e can a k ho he c en in a ci c i i h o e i - A econd kind of analog fo elec ic ci c i i o in e ie o in pa allel compa e i h ha in a ba ed on objec mo ing h o gh ch e . C en i imple one- e i o ci c i . The an e a e no ob- een a a mo ing c o d of mall objec : ol age i he io . The fo ci c i gene a ed b e ie and pa a- fo a d p e e o p hine of he objec . Like he llel combina ion of ba e ie and e i o a e non- pl mbing model, he mo ing-objec model p o ide ela- an pa en . The p o ide an e cellen a o ob e e ion ha map ef ll in o he elec ical em: If e infe ence , a oppo ed o hallo e bal a ocia- e imagine a o ce of p hine co e ponding o he ion . To ded ce he c en in he e fo ci c i , ba e , and ga e in he ch e co e ponding o he he pe on m mo e be ond he fi - age nai e model e i o , hen he mo e p hine , he highe he a e of ci c i ( ho n in Fig e 1). Thi fi le el of of agg ega e mo ion; he na o e he ga e , he lo e i n igh i ha ba e ie make fo mo e c en , e i - he a e of agg ega e mo ion. o make fo le c en . The e le hold fo ba e ie and e i o in e ie , b no fo pa allel combina ion , a ho n in Fig e 2. Pa allel ba e ie Al ho gh he e o analogie con e man of he gi e he ame c en a a ingle em, no mo e; ame ela ion , in ome e pec he diffe in he hile pa allel e i o allo mo e c en han in a ap ne of he ela ional ma ch i h he a ge domain, imple ci c i , no le . pa ic la l if e con ide ligh l mo e comple ci c i ( ee Gen ne and Gen ne , in p e .)
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Figure 2. CROWD MODEL Current and voltage in simple serial and parallel Again I have all these people coming along configuration circuits. here. I have this big area here where people are milling around. I think it is crucial
SIMPLE CIRCUIT CURRENT VOLTAGE that I separate them though before they get x At x At x to the gates . . . I can model the two gate system by just putting the two gates right into the arena just like that. So this is one possible model of the two gate system. There are two gates instead of one which seems to imply that the resistance would be y I V half as great if there were only one gate R for all those people. SERIAL' BATTERIES This protocol suggests that models do affect infer- x ences. The following study tests this possibility more on a larger scale. In this study, fairly naive high I school and college students were first shown a simple ~ T circuit with a battery and a resistor, and then asked T I to give qualitative solutions for the four combination y circuits shown in Figure 2. They were asked to circle PARALLEL BATTERIES whether the current (and voltage) in each of the combin- x ation circuits would be greater than, equal to, or less than that of the simple battery-resistor circuit. After they gave their answers for all four combination cir- cuits, they were asked to describe the way they thought about electricity. Then, for each of the four circuit problems, they were asked to circle whether they had SERIAL RESISTORS thought about flowing fluid, moving objects, or some x other way of conceiving of electricity. They were also asked questions about water, to be sure that they under- V stood the base domain. Figure 3 shows a schematic diagram of parallel and y serial resistors In the target and in the two base systems. PARALLEL RESISTORS Figure 3
SERIAL RESISTORS PARALLEL RESISTORS 21 V V
Do analogies make a difference. If subjects are really using their analogical models to understand electronics they should draw on their knowledge of com- W W W W F ~ Q N binational relations among the corresponding components 2 H 3- N in their respective base domains. Even though the com- plex circuit problems are couched purely in terms of electronics, we should see differences in subjects predictions depending on which model they use. For example, here are two sections of a protocol of a subject trying to predict the current in a parallel-resistor circuit. In the first section, she uses a hydraulics model with reservoirs for batteries, which was her initial model, and derives the wrong t7 F answer of less current. In the second section, she L7 U Z U u W uses a crowT model suggested to her by the experimenter C) m to derive the correct answer of more current: 2_- C), f o
HYDRAULICS MODEL WITH RESERVOIR
We started off as one pipe, but then we split into two. Now does that make any difference? I guess it seems to me that this does make a J J difference. So what we have here is one pipe, U one sort of line coming off and then we let it I- u go like that for a while. We let it split off. J U J U We have a different current in the split-off W W section, and then we bring it back together. TH That s a whole different thing. That just functions as one big pipe of some obscure description. So you should not get as much current.
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Figure 4 Subject who used the fluid model should do well on Proportion correct on different kinds of circuits for the battery questions. This is because serial and subjects using different models of electricity parallel reservoirs combine in the same manner as ser- ial and parallel batteries, and the combinational dis- CURRENT tinctions are spatially quite distinct in the water domain. Two equal reservoirs in series (one above the WATER Moue. other) give more pressure and hence a greater rate of 1e water flow than a single reservoir. However, since water pressure depends only on height, not on volume, BATTERIES two reservoirs in parallel (side by side) yield only a .8 rate of flow equal to that of a single reservoir. Thus, if the flowing fluid analogy is generative, then sub- jects with this model (assuming they know the way .6 RESISTORS pressure works in the base) should be able to differ- entiate serial and parallel configurations of batteries. For resistors, however, the fluid flow model with its constrictions should not, in general, lead to a strong differentiation between serial and parallel resistors. As the protocol above shows, the different be- .2 tween serial and parallel constrictions is fairly opaque in the fluid flow domain; therefore subjects with this model cannot import the correct combination- 0 al distinction into the electricity domain. Thus, the SERIAL PARALLEL prediction is that subjects with the fluid flow model should do better with batteries than with resistors. MOVING OBJECTS MODEL For subjects with the moving-objects model, the 1.0 - pattern should be quite different. In this model, configurations of batteries should be relatively diffi- cult to differentiate, since analogies for batteries .8 are hard to find. In contrast, resistors should be RESISTORS better understood. This is because in the moving- objects model, resistors are often seen as gates. Con- . 6 ceiving of resistors as gates should lead to better differentiation between the parallel and serial config- urations. If all the objects must pass through two . 4 gates one after the other (serial) then the rate of 0 flow should be lower than for just one gate. On the a BATTERIES other hand, if the flow splits and moves through two . 2 parallel gates, then the rate of flow should be twice the rate for a single gate. Thus subjects using this model should correctly respond that parallel resistors 0 give more current than a single resistor; and serial SERIAL PARALLEL resistors, less. Overall, if these models are truly generative lations from the base domain are mapped into the target analogies, we should find that the fluid-flow people domain, where they genuinely affect the person s con- do better with batteries than resistors, and the ceptual view of the domain. moving-object people do better with resistors than with The structure-mapping interpretation of the pro- batteries. cess avoids two extreme positions that often arise in Results. Figure 4 shows the results for subjects discussions of analogy as explanation: the "vague who used either the fluid-flow analogy or the moving- metaphoricizing" position, which holds that analogy is objects analogy consistently, on all four problems. inherently illogical and unhelpful, and the "appropri- For the fluid-flow subjects, only those who correctly ate abstractions" position, which emphasizes the fact answered the latter questions about the behavior of that analogies convey correct knowledge about the reservoirs were included. This was to insure that target domain. The structure-mapping view is neutral subjects possessed the requisite knowledge in the base with respect to whether analogy per se is helpful or domain. There were nine fluid-flow subjects and seven harmful. According to the structure-mapping view, the moving-object subjects. inferences conveyed by a given analogy are not neces- The patterns of combinational inference are dif- sarily either correct or incorrect; they are predicta- ferent depending on which model the subject had. As ble from the predicate structure in the two domains. predicted, people who used the fluid-flow model per- Relational predicates, particularly those that parti- formed better on batteries than on resistors. The cipate in higher-order systems of relations, are most reverse is true for the moving-object people. In a likely to be mapped; but these may be either correct Model type x Component type x Topology analysis of (as in the case of the moving-object model applied to variance, the interaction between model type and cir- parallel resistors) or incorrect or indeterminant (as cuit component is significant; F(/,l3) =,4s3; p < . 05 when the moving-object model is applied to batteries). The more we know about the structure of analogy, the Conclusions. The results of the study indicate better we can design good educational analogies and that, for our subjects, the analogies used for elec- predict the problem that will occur in use of any given tricity were truly generative. Use of different ana- analogy. logies led to systematic differences in the patterns References of correct and incorrect inferences in the target Gentner, D. and Gentner, D.R. Flowing waters or teem- domain. Moreover, these combinatorial differences are i ng crowds: Mental models of electric circuits. not easily attributable to shallow verbal associations I n D, Gentner A, Stevens (Eds.) Mental models. and communicative patterns. These analogies seem to Hillsdale, New Jersey: Lawrence Erlbaum Associates be truly generative for our subjects; structural re- i n press.
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