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Metaphors in Knowledge and Metaphors of Knowledge: Notes on the

Constructivist View of Learning

CARLO TARSITANI University of Rome "'La Sapienza"

ABSTRACT: As the scientific change problem is at the center of epistemological research, the conceptual change problem is widely discussed in didactical research. In this paper common features of both problems will be discussed. Recent ideas on the role of analogies and metaphors in the growth of scientific knowledge are analyzed in order to show their relevance for scientific education problems. These ideas are also confronted with the image of scientific learning put forward by pedagogical and epistemological eonstrtmtivism. In this perstxx~tive, the analysis of Maxwelrs views about metaphors and analogies seems to give a deeper insight as to the problem of scientific change and scientific abstraction which still require clarification in educational and epistemological reflection.

KEYWORDS: Abstraction, accommodation, analogy, assimilation, conceptual change, construetivism, growth of knowledge, history, learning, metaphor.

Introduction: The Scientific Change Problem

One of the main factors of the present crisis of the standard image of science, that is the empiricist-inductivist image of science stated essentially by the neopositivistic school (and also widespread in a simplified version among scientists, science teachers, and science textbooks) is its inability to cope with scientific change. Here scientific change means, in a very general fashion, the transformation of the criteria of explanation and description of phenomena, that is the emerging of the new in

science, as is usually stated in terms of a new way of seeing old things or, more radically, seeing new things in the place of the old ones. In particular, this idea embodies the birth of new concepts or the meaning variance of the old ones, that is conceptual changes in general.

The classical distinction between the context of discovery and the context of justification confined scientific change (seen simply as discovery) to an irrational, psychological zone, inaccessible to logical analysis - while postulating that procedures based on apparently neutral empirical data were available for its

Interchange, Vol. 27/1, 23-40, 1996. ©Kluwer Academic Publishers. Pdnted in the Netherlands.

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subsequent logical justification. Indeed, in the traditional view of teaching, a justification of the same kind is considered as a sufficient reason for learning, where learning is conceived as the result of adding new information to a previously empty mental space.

Now, the dogma "the process of change in science is irrational" seems to persist even in postmodem, or relativist representations of science in progress (Laudan, 1990).

In the first representations of science to diverge from the justificationist model and to gain a wide audience - that is, on the one hand, Popper's falsificationism and, on the other hand, Kuhn's historical relativism - the dynamic or historical character of science became central. The Popperian logic of research, however, excluded a logic of discovery, discovery being considered a free creative act of the imagination. Kuhnian scientific revolution was metaphorically described as a collective gestaltic change, or a kind of religious conversion, not easily analyzable in rational terms. Moreover, the meaning variance problem seemed to undermine any objectivistie- rationalistic view of the growth of knowledge.

So both Pepper's and Kulm's images of science share with the standard image of science an inherent incapability to deal with the process of conceptual change. Thus, if the history of science is indeed a history of processes of conceptual change, the essential tension between history and philosophy of science cannot be relaxed. Similarly, in recent developments in educational theory, the learning process is conceived as a change from conceptions that preexist in the learner's mind, so therefore the idea of the irrationality, or illogicality, of conceptual changes becomes a serious obstacle to a clear statement of educational strategies. For instance, Nussbaum (1989) sees a strong correlation between the new view of scientific change triggered by the works of Popper, Kulm, Lakatos, and Toulmin and the spreading of the constructixdst fashion both in epistemology and in science education, and claims that this is the main reason why the concepttml change problem has become central in educational research.

To find a systematic attempt to give a rational foundation to the description of change, we have to go back to classical German philosophy, that is to the idealistic, then historical materialistic view of the Hegel-Marx-Engels philosophy. In this, dialectical logic was a general scheme to encompass the essentially historical character of human, social, and intellectual evolution. The general metaphysical character of dialectics was denounced by Popper as being too generic, all the more suspect because it was able to cope with everything that might happen in history: indeed, the dialectical model leads to a kind of historical justificationism, inasmuch as everything in history must follow a rational post hoc plan. I think, however, that some of the present attempts to give a representation of scientific change resemble some aspects of the dialectical view.

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Currently the most common form of representation of historical change in scientific schemes uses the evolutionary metaphor. Popper too described his view" of science as evolutionary, scientific change being analogous to Darwinian casual mutations, that should be thereafter subjected to severe selection by the experimental environment (Popper, 1976).

The evolutionary metaphor brings us to recent trends both in the theory of knowledge and in the theory of leaming, namely the contructivistic theory of knowledge, with its assumption of a parallelism in the process of growth of knowledge both for a learning individual and for scientific communities. In this theory, acquisition of knowledge is seen as a progressive adaptation of mental constructs, cxmceptions, or representations in continuous interaction with the environment)

However it is worth noting that, while Popper emphasizes the struggle for life and the natural selection of altemative hypotheses, constructivists insist on the conservationist effort of mental constructs to preserve their individuality in the face of extemal and internal perturbations. This idea, in effect, provided the basis for the constructivist movement, in Piaget's genetic epistemology (Piaget, 1970), where a rich and nontrivial image of the growth of knowledge is founded on a highly developed evolutionary metaphor. Indeed, from Piaget's point of view, epistemology itself tends to become a part of a complex evolutionary process that includes the psycho-biological behavior of human beings. Adopting most of Piaget's theses, pedagogical and epistemological constructivism attributes to the knowing-operating (i.e., experiencing) subject to the spontaneous tendency to preserve viable representations of the world in a changing environment (Glasersfeld, 1989).

I think that Piaget's view (and, consequently, the constructivist view) can be further developed: in particular, the problem of conceptual change deserves better attention, since without a full theory of conceptual change Piaget's genetic epistemology may appear to retain a justificationist view of the growth of knowledge. In fact Piaget saw the growth of knowledge as a process towards necessary conceptual structures that are thus, in a certain sense, absolutised. At the same time, and for analogous reasons, because it lacks an explicit analysis of the dynamic of mental constructs, the constructivist vision tends to assume the flavor of a conservative version of classical conventionalism. 2

In the following I will discuss the cognitive role of metaphor and try to resolve, at least partially, the above shortcomings. In speaking of metaphor, I will be concerned with the main result of analogical thinking, in the sense that the discovery of an analogy becomes scientifically productive when the analogy implies a transfer of conceptual schemes bringing a metaphorical description to the field of phenomena seen in an analogical way.

Pointers will be taken fi'om the recent debate on analogy and metaphor in epistemology and the history of physics. I will deal in particular with Maxwelrs ideas on the function of analogies and metaphors in the development of knowledge. I think that through this analysis it is possible to enrich both the pedagogical and

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epistemologic~ aspects of the constnlctivist view. The main issue is this: how can the essential function of metaphors in scientific change be incorporated into the evolutionary metaphor used by the constructivist movement to depict both the individual's and the scientific community's growth of knowledge? And, eventually, what kind of suggestions earl be made, on the basis of this, regarding pedagogical constructivism? 3

The Role o f Metaphor From the Epistemological Point o f View

Various recent works have stressed the importance of the role of metaphors in the growth of scientific knowledge. I will discuss what I consider to be the most promising line of thought: that stemming from Max Black's interactive conception of metaphor (Black, 1962), and developed in particular by Mary Hesse's and Stephen Boyd's approaches to the role of metaphors, analogies, and models. Both have seen in the use of metaphors and in the related use of analogies and models a way to represent the intrinsic dynamics of scientific knowledge, which is so difficult to describe through rational representations.

Let us state briefly, using Hesse's words, this interaction view of metaphors. The traditional views of the function of metaphors stressed their substitutive role. Starting from two systems, the primary and the secondary (each supposed to be describable in literal language), a metaphorical operation consists of a description of the primary system in words normally used in connection with the secondary system, Obviously, in this substitutive use of metaphors, some kind of similarity or analogy between the two systems is presumed. Then, on the basis of this analogy, one can infer that some further properties of the secondary system may find their correlate in the primary one (this is the classical "argument by analogy"). Starting from Black's analysis, Hesse claims that there are at least two respects in which such a conception of a metaphor may be insufficient. First, since "the metaphor works by transferring the associated ideas and implications of the secondary to the primary system" (Hesse, 1966, pp. 162-163), it may produce a change also in the description of the secondary system. As Hesse says,

in accordance with the doctrine that even literal expressions are understood partly in terms of the set of associated ideas carried by the system they describe, it follows that the associated ideas of the primary are changed to some extent by the use of metaphor and that, therefore, even its original literal description is shifted in meaning. The same applies to the secondary system, for its association come to be affected by assimilation to the primary [italics added]. (1966, p. 163 )

In this sense metaphor is not only a comparison between two systems, but it also creates a semantic resonance between them, that is an interaction between their conceptual representations.


Second, it may happen that the metaphor itself creates the similarity: it gives a new way of looking at a primary system, making it possible to see new essential features of it.

The main argument used by Hesse is based on a critique of the hypothetical- deductive conception of scientific explanation (the covering law deductive model), in the sense that every genuine explanation, constructing a model of the system to be explained, actuaUy gives a metaphorical transcription of the explanandum domain by changing its meaning.

Thus metaphors are very important factors in conceptual change. As Hesse states,

the interaction view sees language as dynamic: an expression initially metaphoric may become literal (a "dead" metaphor), and what is at one time literal may become metaphoric ....What is important is not to try to draw a line between the metaphoric and the literal, but rather to trace out the various mechanisms of meaning-shift and their interaction. (1966, p. 166)

Leaving aside tlesse's defense of the function of models (against the purely formal view of theories), as nontrivial ways to give theories a real predictive power and to enlarge their phenomenic field of application, let us continue to develop the function of metaphors in scientific change.

It is here that Boyd's contribution becomes relevant. Boyd criticizes the view that the function of metaphors lies in a pretheoretical stage of the development of a discipline, or, for mature sciences, in heuristics, pedagogy, or informal exegesis. He instead argues that there exists an important class of metaphors that play a constitutive role. More precisely, there are "cases in which there are metaphors which scientists use in expressing theoretical claims for which no adequate literal paraphrase is known" (Boyd, 1979, p. 360). One of the examples quoted by Boyd is the representation of the brain's behavior in the terminology of computer science.

Indeed, a metaphorical description may function as a sort of catachresis, that is, it may be used to introduce theoretical terminology where none previously existed. The open-endness of metaphors of this kind, given that they are unable to precisely specify the relevant aspects of similarity or analogy, is for Boyd a very important factor in theory change.

Indeed, the utility of theory-constitutive metaphors seems to lie largely in the fact that they provide a way to introduce terminology for features of the world whose existence seems probable, but many of whose fundamental properties have yet to be discovered. Theory-constitutive metaphors, in other words, represent one strategy for the accommodation of language to as yet undiscovered causal features of the world [italics added]. (1979, p. 364)

The use of constitutive metaphors of this kind, including those specialized metaphors that we call models, demonstrates the tension that exists between the "it is so" and the "it is not so" that is typical of analogical thinking. 4 Moreover, terms like "assimilation" (Hesse) and "accommodation" (Boyd) lead us back to Piaget's theory of cognitive processes.

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In Piaget's complex representation of the growth of knowledge, as in that of his constructivist followers, the evolutionary metaphor appears to be constitutive in the Boydian sense. Speaking of the cognitive process, Piaget uses concepts such as equilibrium, structural stability, and depicts psychogenesis as a process aimed at constructing formal structures of intelligence which are increasingly equilibrated and adapted to the external environment. As Ceruti (one of Piaget's disciples) states, the concept of"amelioration" of the equilibrium forms is central to the representation of cognitive development, as it accounts for the continuous tension between conservation and novelty, between invariance and change (Ceruti, 1989, p. 185). From this point of view, for Piaget, the evolution of cognitive structures is ruled by the same dialectics of conservation and change which is typical of biological evolution.

Indeed, the metaphor adopted by Piaget, creates a semantic resonance between biological evolution and psychogenetic evolution by describing the adaptation process in terms of assimilation and accommodation of structures or schemes.

Le fair essentiel dont il convient de partir est qu'aucune connaissance meme perceptive, ne constitue une simple topic du r6el, parce qu'eUe comporte tojours un processus d'assimilation ~ des structures ant6rieures ... qui peuvent demeurer inchang6es ou sont plus ou moins modifi6es par eette integration meme, mais sans etre d6truites et en s'accomodant simplement ~ la nouvelle situation. (Piaget, 1967, p. 20)

It is impossible not to recognize in concepts such as assimilation and accommodation a semantic link with what, in the above arguments by Hesse and Boyd (which use the same terms), is seen as metaphorical process. Furthermore, Piaget's subsequent specification of the interplay between assimilation and accommodation has many features resembling what, in the studies on metaphor, is called the interaction and the constitutive view of metaphors.

For Piaget the very attribution of a meaning to new external information is due to an assimilation process. This process is essentially active, in the sense that conceptual schemes are the abstract correlate of action schemes. So, any assimilation scheme tends to simply incorporate external elements, but in doing so it is obliged to accommodate itself to those elements (or to the other co-existing internal schemes) - without losing its identity - and it is obliged to preserve a general condition of equilibrium. Because of the continuous processes of assimilation and accommodation the equih'brium condition is not static but dynamic: the amelioration of this condition is just what may be called the progressive adaptation of the intellectual organism to the environment, which cannot be seen as a one way process, but as process of coevotution consisting both of adaptation to and adaptation of the environment.

The use of the concept of adaptation is the distinguishing feature of the constructivist movement. A comment is necessary here: in using the term adaptation, constructivists emphasize the role of the subject, whereas the traditional views of evolutionary epistemology, such as Popper's, emphasize the role of the external environment. The central point is the substitution of the idea of external cause with


the idea ofintemal and external constraints (Ceruti, 1989, p. 222). Adaptation is not a simple effect of the action of the environment: rather it is an active response of a system that constructs viable schemes of action and representation to preserve its identity and to survive within those constraints. Radical constructivism deprives the evolutionary metaphor of the objectivistic view of adaptation, in the sense that there is no way to establish independent criteria that allow adaptation to be judged in terms of better and better correspondence between internal structures and external environment. The only criterion that is retained is that of a success, seen merely as survival along viable paths within the constraints imposed by experience. However, although the evolutionary metaphor has become the new fashion in the epistemological representation of knowledge, it has lost the important details that allowed Piaget to construct a richer conceptual scheme for the description of the scientific change process.

In conclusion, I think that the epistemological analysis of the role of metaphors is an attempt to give a linguistic version of conceptual change in science that is parallel to the psychogenetic version stated in the evolutionary metaphor of the constructivistic movement. I think also that the concepts of assimilation and accommodation posit a link between the two versions that may allow at least a reciprocal partial translation. Proceeding with the analysis of the role of analogies and metaphors, I will examine whether this common area may be further enlarged and made still more manageable.

An Exemplary Historical Case: Maxwell on Analogy and Metaphor

To complete my argument I have now to go back to the view of the function of analogies and metaphors put forward more than a century ago by Maxwell, who felt the need to develop such ideas even in a mature science like physics. ~ Indeed, for Maxwell, the analogical method is one of the most important factors in the growth of knowledge. And his theorization is far from being a trivial one. Rather it affords us a picture which is in some sense richer and more convincing (being related to actual theoretical developments in physics) than the somewhat abstract ones that we have considered so far.

Maxwelt's work had a central role in the radical transformations of both conceptual and methodological standards of theoretical thinking in physics in the second half of the 19th century. His initial concern was to overcome the constraints imposed on the conceptualization of experience by the French mathematical physics tradition, and to construct new conceptual and formal instruments to fully develop the English tradition of the dynamical theor ies . 6 This general aim found its methodological expression in the use of analogies and models. 7

Maxwell (who may considered one of the first theoretical physicists in the modern sense) uses analogies and models in two situations: In'st, to fmd a new way of giving a conceptual representation to an already developed field of research (for

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instance Laplace's and Poisson's electrostatics, or Amp6re's electrodynarnies); second, to construct a theoretical representation for a new field of inquiry (for instance the kinetic theory of gases). In the first case, the discovery of an analogy is the discovery of an equivalence between the formal schemes that describe the laws of two physical systems (the primary and the secondary) which appear prima facie different regarding their actual physical nature; in the second case the theory of the primary system is not yet completely formulated, just some experimental properties of it being known: in this case the analogy becomes constitutive in the Boydian sense, and the search for an analogical system is devoted mainly to the construction of a theoretical model for the primary system.

Let me state, briefly, some differences between the two situations, even if later I will show that they have converging outcomes. Maxwell begins by using analogies of the first kind: in his first paper on electromagnetic phenomena, he says that a physical analogy is simply "that partial similarity between the laws of one science and those of another which makes each of them illustrate the other," and that its function is simply "to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena" (Maxwell, 1856/1890a, pp. 156, 159). So, the formal similarity between the primary system and the secondary one (the fact that the two systems share the same geometry) means that the two systems exhibit an analogy and that the formal and conceptual schemes adopted in the first one may be used to illustrate the features of the sexond one. As Maxwell will say later,

now a truly scientific illustration is a method to enable the mind to grasp some conception or law in one branch of science, by placing before it a conception or a law in a different branch of science, and directing the mind to lay hold of that mathematical form which is common to the corresponding ideas in the two sciences, leaving out of account for the present the difference between the physical nature of the real phenomena. (1870/1890c, p. 219)

This procedure may have, at a first level, only a didactic purpose: it may, for instance, "assist the imagination" of a learner unable to retain in his mind "the unembodied symbols of the pure mathematician" (Maxwell, 1870/1890c, p. 220). And the metaphorical function of the new conceptual scheme is merely substitutive. But obviously this is not the most interesting function. Let us proceed with the more promising consequences of the discovery of an analogy.

On the one hand, the formal scheme of the secondary system transfers its set of associated ideas and implications to the primary one, which now can be seen in a different light. The hydrodynamic analogy of the first Maxwell paper on electromagnetism (Maxwell, 1856/1890a) in which an electromagnetic system is assimilated to a perfect fluid system, gives a new version of electromagnetic phenomena in which they are the outcome of something happening in the space, later called field, that surrounds electric and magnetic bodies. That this kind of transfer had a great heuristic value, in the Boydian sense, is confn-med by the subsequent work and discoveries by Maxwell himself.


On the other hand, when the primary system does not yet have a theoretical representation, the transfer of schemes extracted from another field of inquiry directly assumes the form of the construction of a model. Here, Maxwell again speaks in terms of analogical method: for instance, introducing his first paper on the dynamical theory of gases, Maxwell says:

In order to lay the foundation of such investigations on strict mechanical principles, I shall demonstrate the laws of motion of an indefinite number of small, hard, and perfectly elastic spheres acting on one another only during impact. If the properties of such a system of bodies are found to correspond to those of gases, an important physical analogy will be established, which may lead to more accurate knowledge of the properties of matter. If experiments on gases are inconsistent with the hypothesis of these propositions, then our theory, though consistent with itself, is proved to be incapable of explaining the phenomena of gases [italics added]. (1856/1890b, pp. 377-378)

Leaving apart a consideration of the meaning given by Maxwell to terms like theory and explanation (I will come later to the methodological consequences of his view), one could think that, so far, we have just some indirect hints to cope with the problem of conceptual change. Indeed, Maxwelrs views may be seen as conservatory insofar as they privilege particular forms of mechanical conceptualization. But, taking a second look at the above observations, one realizes that, in Maxwell's work, a simple assimilating procedure tends to become an accommodating one. In other words, in the transfer to the primary system, the associate ideas and implications of the hydrodynamic secondary system assume an abstract and artificial character. Here one can see a continuity between the discovery of an analogy and the construction of a model. As Maxwell had already said in 1856, referring to his hydrodynamic analogy,

the substance here treated of must not be assumed to possess any of the properties of ordinary fluids except these of freedom of motion and resistance to compression. It is not even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a collection of imaginary properties which may be employed for establishing certain theorems in pure mathematics in a way more intelligible to many minds and more applicable to physical problems than that in which algebraic symbols alone are used. The use of the word "Fluid" will not led us into error, if we remember that it denotes a purely imaginary substance, (1856/1890a, p. 160)

Therefore, the transferred conceptual scheme gains a relative autonomy from the field of inquiry (the secondary system) for which it was originally introduced. Thus, from the methodological point of view, the theoretical physicist can work on schemes or representations that are in a sense detached from the physical system for which they were originally elaborated. His main ~ is to construct consistent representations of phenomena, without claiming to reveal their essential features (since one may find different representations of the same phenomena and the same representation may be used for different phenomena). Therefore such schemes are like tools that make physical reality observable from various points of view and describable in their own

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language. Each scheme may reveal different aspects of the same physical reality, just as an optical lens can focus on different parts of the same object. As soon as the conceptual scheme becomes a theoretical representation, it looses the link with the particular concrete reality from which it was originally extracted and assumes an idealized meaning: the aim of theoretical physics is to adapt our representations to reality and not the reality to our representations.

Now, conceptual change comes about when one reflects on the metaphorical value of the description by analogy. When applied to the primary system, the collection of imaginary properties extracted from the secondary one assumes the character of an interactive metaphor, and the analogical transfer produces a change of level in the theoretical representation. A favorite example given by Maxwell is the new classification of physical quantifies afforded by hydrodynamic analogies: the distinction between forces and fluxes, intensities and quantifies, becomes of general application: it operates on a meta-theoretical level, extending the conceptual hierarchies of physical representations. This classification was to be applied by Maxwell not only in electromagnetism, but also in thermodynamics. To give another example, let us return to the development of the kinetic model of a gas, where one has to remember the long-standing dittieulty of attributing elasticity to hard spheres. The conception of atoms as elastic hard spheres was considered seN-contradictory by Newton, and remained a widespread obstacle to the acceptance of the atomic theory even after Maxwell (for instance Kelvin based his lasting objections to the kinetic theory of gases on this ditticulty), s

So abstraction and generalization of conceptual schemes are the most important result of the use of analogies. The presumed natural connection between a particular theoretical structure with a particular phenomenic field is broken, and through this generalizing procedure, concepts, precisely because they are used metaphorically, shift in their meaning.

Maxwell is very clear, about this process also:

The figure of speech or of thought by which we transfer the language and ideas of a familiar science to one with which we are less acquainted may be called Scientific Metaphor. Thus the words Velocity, Momentum, Force, etc. have acquired certain precise meanings in Elementary Dynamics. They are also employed in the Dynamics of a Connected System 9 in a sense which, though perfectly analogous to the elementary sense, is wider and more general. The generalized forms of elementary ideas may be called metaphorical in the sense in which every abstract term is metaphorical. The characteristic of a truly scientific system of metaphors is that each term in its metaphorical use retains al the formal relations to the other terms of the system which it had in its original use. The method is then truly scientific - that is, not only a legitimate product of science, but capable of generating science in its turn [italics added]. (1870/1890c, p. 227)


In the same way as for concepts like velocity, momentum, and so on, the metaphorical use of concepts like flux or hard elastic sphere make their user free from intuitive constraints (Maxwell never uses models that are "intuitive" in the ordinary sense). Metaphor and abstraction are deeply connected: "every abstract term is metaphorical" (Maxwell, 1870/1890c, p. 219). Detached from the occasional concrete reference, not only do conceptual schemes become more manageable and controllable, but they also reveal deeper underlying structures and implications. In this sense, Maxwell anticipates the interactive view:

The illustration is not only convenient for teaching science in a pleasant and easy manner, but the recognition of the formal analogy between the two systems of ideas leads to a knowledge of both, more profound than could be obtained by studying each system separately. (1870/1890c, p. 219)

The conclusion of MaxweU's discourse on the function of metaphors and analogies is also rich in methodological insight. This emerges from those instances "in which two different explanations have been given of the same thing" (Maxwell, 1870/1890c, p. 227). The instances given by Maxwell are the conflicting representations (undulatory and corpuscular) of the nature of light and the conflicting representation of the electromagnetic phenomena in the action at a distance and action through a medium. Now, for Maxwell, whose main theoretical interest was an authentic growth of knowledge (by which he meant a change to existing schemes), the question was not to be solved by a competition between the two schemes, but with the attainment of"a scientific altitude from which the true relation between hypotheses so different can be seen" (p. 228). Indeed, in one of the above instances, that regarding the nature of light, Hamilton's discovery of the fact that "to every brachistochrone problem there corresponds a problem of free motion, involving different velocities and times, but resulting in the same geometrical path" (p. 228) allows us to understand the true relation between the contrasting representations. Maxwell considered a similar development to be of paramount importance for electromagnetism too. Here again Maxwell sees in the discovery of deeper and more abstract analogies the way to gain more complete and powerful knowledge.

The other face of this complex methodological problem is the long-standing question of the relations between mathematics and models. It is well known that, since Duhem's critique of English imaginative physicists, models and mathematical description have been seen as two conflicting modes of description, the first intuitive and pictorial, the second abstract and formal. In the neopositivistic tradition this tension was solved giving to mathematics the role of framing the logical structure of a theory- the only level of the theory directly related to empirical facts - and giving to models the role of furnishing an intuitive interpretation of that structure, related to the subjective need for a mental picture.

I think that Maxwell's work shows that the above view is too schematic. On the one hand, the use of the formal structures of the mathematics of continuous media is not neutral to the conceptual representation of the physical reality of electromagnetic interactions. On the other hand, the growth of mathematical schemes (of differential

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and vectorial calculus) is deeply connected to the growth of physical models: that is to say, in the continuous path towards increasingly abstract models, we also have a process of growth of mathematical conceptions.

Finally, let us compare Maxwell's view of the process of abstraction with Piaget's idea of reflecting abstraction. First, from the constructivist perspective, abstraction is not simply an extraction of common features from a class of objects; rather it is a kind of emancipation of conceptual schemes from the initial field of application. Its correlate in the metaphorical proce~ is the progressive fading out of the link between the representation (or model) and the secondary system, caused not only by its success in describing the primary one, but also by its further generalization to other fields. This process is essentially interactive, because it is based on a radical change of meaning of the terms related to the secondary system. To make an analogy, it resembles the process by which an instrument invented for a particular investigation, reveals new possibilities of application. For instance radio-antennae are no longer used only to receive radio-station signals, but are also used to detect the behavior and the structure of galaxies, which in turn become radio-sources.

During the process of reflecting abstraction the subject becomes aware of the representative nature of what previously appeared to be inherent and implicit in a particular cognitive operation. It is that change of level, that Watzlawick (1977) sees as the primary factor of an actual intellectual or psychological change. As Ceruti (1989) underlines, reflecting abstraction, for Piaget, assumes "as its object the cogrn'tive activities of the subject, no longer through observable results on an empirical object external to them, but through direct attention to the purely formal and operational aspects of such activities" (p. 166). And,

inside this experimental (interdisciplinary and evolutionary) context there emerges the possibility (and the utility) of substituting the traditional conception of the relations between subject and object (knowledge and reality) based on the idea of representation, with a conception which understands such relations in terms of the idea of construction. (Ceruti, 1989, p. 170)

For Maxwell, it is precisely the metaphorical use of scientific terms that is the basis for the jump of level from assimilation and accommodation, on the one side, to reflecting abstraction, on the other side. What Piaget sees as a transition, appears, in MaxweU's view, as the essential feature of a process in which cognitive structures (schemes), freeing themselves from the reference of the original environmental constraints, create for themselves new possible constraints. Therefore, the process is both endogenous and exogenous (remember that in the 1850s and 1860s there was no cogent external reason to adopt a field theory of electromagnetic phenomena).

METAPHORS IN KNOWLEDGE

Conclusions

Let's summarize Maxwell's argument in the following scheme:

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I same mathematical laws~ a~~ I Representation I \ I and/or model: I I physical.anal°gYl

I Secondary System L= asaimilat" I - Primary System , ,,(known fie!d of phenomena)l T M I (new field of phenomena)l

' ~ semantic r e s o n a n c e : / metaphorical use /

of the primary concepts 1

I I k~ abs~action

physical a n alo~y1~"

i; I accomodation: I conceptual change

We can see how Maxwell's methodology clarifies the dynamics of three levels of the conceptual change process, that is, assimilation, abstraction, accommodation.

As yon Glaserfeld (1980) says, "what an organism learns is retained for the very reason that it leads to satisfactory results... [thus] adaptation is again viability... [and]

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if something has been found to work, it is likely to work again" (p. 69). Therefore we have in learning a spontaneous tendency to conservation through assimilation.

In this sense pedagogical constructivism gives due importance to the analysis of the so called mental representations of a learner. Scientific learning has to be grafted onto preexisting schemes of common sense knowledge and presupposes an increasing tension between conservation and change. The teacher's role is restricted to providing external stimuli to a substantially autonomous process, or, at best, to promoting a negotiation between different possible representations.

I think that, according the Maxwell's view, once one realizes the deep entailments of the metaphorical thinking, one can how an internal conceptual dynamics could work, just the kind of dynamics that eonstructivists would aim to trigger.

At the same time, the learner can recognize some of the essential features of scientific knowledge itself, features that are sometimes very different from those of common sense knowledge. Indeed, at the same moment in which the learner begins to consider his or her mental representations as a scientific description of an external reality, he or she not only sets in motion a deep change of contexts, but also gives a new epistemological status to his or her ideas, concepts, and words. For instance, he or she begins to confront problems of coherence, of meaning, of logical structure, and passes from "things" to "ways of describing things" with a reflective abstraction. On the other hand, scientific learning presupposes the constitution of a specific system of attitudes and values. The learner has to be pushed to test and extend his or her schemes, and he or she must be prepared to correct and change them. Thus, a prerequisite for learning should be the assumption of an interested, knowledge- oriented, critical (we could say scientific) attitude.

Does this mean that pedagogical eonstructivism leaves much of the science teaching problem unsolved? I don't think so: the central idea of learning as a process in which one deals with the change of mental representations is of great importance. However, as we have seen, the description of such a change is grounded on a constitutive biological metaphor which, without any specific application to effective conceptual change processes, appears somewhat abstract. For instance, there is no ground to choose a "Darwirfian" scheme for adaptation instead of a '°Lamarkian" one.

My purpose in this paper has been to show that the epistemotogical analysis of the function of analogies and metaphor can make the constructivist evolutionary metaphor more transparent. For instance, such analysis earl bring the research on learning p r ~ from a vague reference to mental behaviors, to a more controllable reference to linguistic behaviors. In this approach, the interplay between assimilation, aecommod~on, and reflective abstraction takes on concrete features, and one can see it at work in particular cases.

Moreover, in descnbing Maxwelrs image of science, I have presented a strategic view of the function of analogies and metaphors that not only gives several pointers


for a theory of conceptual change, but may also help to clear up some of the central concerns of constructivist epistemology. Anticipating some issues of postmodem epistemology, Maxwell did not see the growth of scientific knowledge as caused by a merely passive response by scientists to new empirical data (or new sources of information), rather he gave to scientists an active role in constructing conceptual schemes that recast both theory and experience. Maxwell thus gets rid of any essentialistic view of natural things and/or empirical findings and shi~s attention to the coevolution of our representations, ideas, and experience in the continuous effort to extend our horizons of reality.

The proce~ of growth of knowledge described by Maxwell goes beyond a strictly functional adaptive process: rather he stresses the internal dynamic factors that make scientific knowledge something more than a merely reactive process. The search for tmification, coherence and simplicity establishes a more extended correlation between a plurality of levels of representation, giving to scientific research a wider cultural meaning. I am worried that most of the present constructivist didactic proposals, in using ideas such as viability, tend to lose sight of the importance of these cultural aspects, depriving science of one of its most attractive features. For instance, constmctivist epistemology seems to give an exceedingly narrow representation to the historical character of the scientific enterprise. Indeed, it seems to forget that any scientific concept has a long historical process beyond itself. Without richer ar-ticulation, the concept of viability seems too restrictive: if the Darwinian struggle for life is seen, in the Popperian sense, as purely a competition between alternative schemes, one cannot explain the process of synthesis which occurs at a more abstract level and which is an essential aspect of the growth of knowledge. I think, therefore, that in the ~ t i v i s t perspective it persists as a sort of post hoc justification of the successive stages of knowledge: the success in surviving is both the reason for constructing representations and the reason why they resist.

Therefore, the above reinterpretation of the constuctivist view of conceptual change may show the way to overcome two of its main epistemological shortcomings: a) the neo-justificationist attitude, and b) the overall subjectivistic attitude.

NOTES

1. As far as I know, a systematic treatment of the constructivist point of view (in epistemology and/or in science education) is still missing. I may quote the work of Arbib and Hesse (1986), but I don't think that it is fully representative. Most works treat particular topics, as for instance, the paper by Anderson (1992), which tried to correlate cognitive and neurobiologieal processes in a constmetivist perspective, and the papers by Ernest (1992, 1993), which analyze the relevance and the limitations of the conslmetivist hypotheses as regards the nature of Mathematics. It may be interesting to point out that in the anthological volume The Philosophy of Science, edited by R. Boyd (1991a), the term "constmetivism" is vaguely defined as "the view that the subject matter of scientific researeh is wholly or partly constructed by the background theoretical assumptions of the scientific community and thus is not, as realists claim, largely

38 CARLO TARSITANI

independent of our thoughts and theoretical commitments" (p. 775); in this perspective eonstruetivism is seem by Boyd as a "nee-Kantian philosophy" (Boyd, 199 l b). 2. N. Marini (1994) defends the Piaget experimental discoveries in the process of learning of the mechanical concepts against the constructivity alternative conceptions movement, giving at the same time a rich bibliography on that issue. 3. As far as I know, the authors who stress the importance of metaphors (Munby & Russell, 1990; Riehtie, 1994) confine their use into the teaching environment. 4. The shift from theories to models in educational research (see Hestenes, 1992) may be understood in the light of the new epistemologieal mood: theories can be considered as true or false, models cannot. 5. While this paper was in preparation, I realized that the relevance of Maxwell's epistemological thought as regards educational problems has also been stressed by Nancy Neressian (in press), even if this author does not underline the semantic role of metaphors in the conceptual change process. 6. For a broad account of the context and purpose of Maxwellian works (and related bibliography) see Wise (1982). 7. My reeonslnmtion of Maxwell's methodological thinking will be somewhat different from, even if not contradicting, the one given by Mary Hesse (1973). Hesse's concern was to find in Mam~ll's analogical method an alternative to the hypothetical deductive image of theories, and to the implicit dualistic image of science based on the logical interplay between experimental facts and hypothetical theories. The analogical method could in this way be used to confront the subjectivistie flavor of the above dualism, and to overcome the embarrassing results of the meaning variance thesis. My analysis is, however, more oriented towards the dynamic role of analogy in scientific change (a problem that may also be of interest for theories of learning), and has no other epistemological concerns. 8. This problem has been historicaUy treated by Scott (1970) and by Smith and Wise (1989). 9. Here, Maxwell is speaking of the analytical formulation of mechanics in Lagrangean or Hamiltonian form.

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Author's address:

University of Rome "La Sapienza" Faculty of Science, Department of Physics Piazzale Aldo Moro, 2 1 - 00185 Rome Italy