The Role of Generic Models in Conceptual Change

We hypothesize generic models to be central in conceptual change in science. This hypothesis has its origins in two theoretical sources. The first source, constructive modeling, derives from a philosophical theory that synthesizes analyses of historical conceptual changes in science with investigations of reasoning and representation in cognitive psychology. The theory of constructive modeling posits generic mental models as productive in conceptual change. The second source, adaptive modeling, derives from a computational theory of creative design. Both theories posit situation independent domain abstractions, i.e. generic models. Using a constructive modeling interpretation of the reasoning exhibited in protocols collected by John Clement (1989) of a problem solving session involving conceptual change, we employ the resources of the theory of adaptive modeling to develop a new computational model, ToRQUE. Here we describe a piece of our analysis of the protocol to illustrate how our synthesis of the two theories is being used to develop a system for articulating and testing ToRQUE. The results of our research show how generic modeling plays a central role in conceptual change. They also demonstrate how such an interdisciplinary synthesis can provide significant insights into scientific reasoning. 1. Conceptual Change in Science In many instances, solving novel or difficult problems leads to conceptual change. Such conceptual change can range from minor changes in existing concepts to the radical kind of change one associates with “scientific revolutions”. A significant issue in modeling conceptual change is how existing knowledge can be used in creating genuinely novel understandings. We hypothesize that generic models play a key role creating these new understandings. These models encompass domain properties, relations, principles, and mechanisms. To explore this hypothesis we analyze the role of generic models in a problem solving protocol collected by John Clement (1989). Our analysis makes use of the “cognitive-historical” theory of constructive modeling (Section 3) to provide a conceptual interpretation of the problem-solving session (Section 4). We then join this analysis with the computational theory of adaptive modeling (Section 5) that we believe provides the resources necessary to model the protocol as so analyzed. Together, the conceptual interpretation and the computational theory enable the development of a new computational model we call ToRQUE (Theory Revision through Questions, Understanding, and Evaluation) and a system which instantiates this model. (Section 7). 2. The Clement Protocol The problem posed in the Clement protocol is as follows: “... a weight is hung from a spring. The original spring is replaced with a spring made of the same kind of wire; with the same number of coils; but with coils that are twice as wide in diameter. Will the spring stretch form its natural length more, less, or the same amount under the same weight? (Assume the mass of the spring is negligible compared to the mass of the weight.) Why do you think so?” (Figure 1 a & b) In the study, subjects were asked to assess their confidence in their answer and in their understanding. We focus on one subject, S2, who changed his concept of a spring by incorporating the physical principle of torque into his understanding of how springs function. Unable to solve the problem directly, S2 began by reasoning that a spring when it is unwound is like a flexible rod (Figure 1c). He then reasoned that a spring of twice the diameter can be unwound into a longer rod, which will bend farther given equal force (Figure 1d). From this he concluded (correctly) that a spring of twice the diameter will stretch farther given equal force. S2, however, unlike most of the participants in the study, was not confident of this answer. He noticed that a significant difference between the stretched spring and the bent rod is that the bent rod has a varying slope, while the spring has a constant slope, i.e., the space between the coils is uniform both before and after the spring is stretched. At this point S2 constructed the models that are the primary focus of our modeling effort (Figure 1e-i). These models were constructed based on salient differences between the spring and the flexible rod, and are designed to resolve what S2 regarded as an anomaly: the nonuniform slope of the bending rod (see Darden 1991 on anomaly resolution). He eventually constructed a model of a hexagonal coil (Figure 1g) that led to the understanding that a spring maintains its constant slope through the twist of the coil wire during stretching. The notion of torque was not present in S2's original model of spring, so we contend that S2's concept of a spring is changed in the problem solving process. Although we are modeling the whole protocol, given space limitations we will focus on just this final piece of reasoning and how we interpret it as employing “generic models”. 3. Constructive Modeling Nersessian (1992, 1995, in press) has argued that general modes of reasoning such as visual reasoning, thought experiment, analogy, and generic abstraction play significant roles in scientific conceptual change. These various modes often are employed together in an iterative reasoning process we call “constructive modeling.” Constructive modeling is a semantic process in which the models produced are proposed as interpretations of the target satisfying specific constraints. Figure 2 provides a schematic representation of such a process. Constructing a model starts with properties and relations of a target system that serve as constraints to be satisfied by the initial model. A source domain satisfying some initial target constraints is selected. From this domain an initial analog model is retrieved or is constructed in the case where no direct analogy exists. This initial model and each constructed model serves as a source of additional constraints that interact with those provided by the target system to create an enhanced understanding of the target, in particular by making explicit further target constraints. The constraints can be supplied in different informational formats, including equations, texts, kinesthetic, diagrams, pictures, maps, and physical models. The model construction process involves different forms of abstraction (limiting case, idealization, generalization, generic abstraction), constraint satisfaction, adaptation, simulation, and evaluation. Additional source domains may be called upon throughout the iterations. This cycle is repeated until a satisfactory representation of the target problem is achieved. This representation is a model of the same type as the target problem with respect to the salient target constraints. We interpret S2's reasoning to be a case of constructive modeling . Clearly, to engage in constructive modeling the reasoner needs to know the generative principles and constraints for physical models in one or more domains. This is why analogy plays such a significant role in the constructive modeling process. On our account, the function of analogies is to provide constraints and generative principles for building models. This view is in contrast to the direct transfer view of most computational models (See for example Falkenhainer et al., 1989; Holyoak & Thagard 1989) Thus we view relations between domains in terms of the constraints they share. These constraints and principles may be represented in the different informational formats and knowledge structures that act either as explicit or tacit assumptions employed in constructing and adapting models during problem solving. Since these constraints are domain-specific they need to be understood at a sufficient level of abstraction in order for retrieval, transfer and integration to be possible. We call this level of abstraction “generic”. What we mean can easily be conveyed by looking at a simple example taken from Polya (1954). Polya considered two cases, abstracting from an equilateral triangle to a triangle-in-general and from it to a polygon-in-general (Figure 3). Loss of specificity is the central aspect of this kind of abstraction process. We call this process “generic abstraction”. The generic triangle created in this abstraction process is understood to represent those features that all kinds of triangles have in common. Although the figure entertained by the mind is specific, some of its salient features, the lengths of the sides and the degrees of (a) (b)