Science & Education 12: 773–777, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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What Are Models and Why Do We Need Them? RICHARD E. GRANDY Philosophy Department, Rice University MS 14, 6100 Main Street, Houston, TX 77005, USA; E-mail:
[email protected]
Abstract. Educators and philosophers tout the virtues of the ‘new’ view of theories. Unfortunately, there is no agreed-on name for the new view, but a recently a favored term has been ‘model based’. I address what a model might be in the context of science education. I am concerned about when and why we need to make the transition from implicit mental models to explicit external models. I explore two theories/models in physics. One draws on misunderstandings of rotating objects. The second concerns density and flotation. I offer two morals: 1. ‘Models’ in physics are often mathematical, and more attention needs to be devoted to integrating mathematics and science 2. Models are required in physics when we need an equation in which two variables occur and which is not linearly additive.
The official demise of the ‘Received View’ of scientific theories occurred at a conference at the University of Illinois in Champaign-Urbana in March 1969 (Suppe 1974, 2000, p. S102). Since then, science educators (Duschl 1990) and philosophers of science concerned with science education have touted the virtues of the new view of theories as an important tool in improving science education. Surely having an erroneous view of the nature of scientific theories was an impediment to science education, and having a correct one will be a boon. There is no generally agreed on name for the new view, unfortunately. It has been called ‘semantic’, ‘structuralist’, ‘model-theoretic’, ‘set theoretic’, ‘non-statement’, and more recently a favored term has been ‘model based’. This may just be an issue of nomenclature, but it may also reflect some important uncertainty about details. The best statement of the philosophers’ new views is probably Suppe’s. His ‘nuanced sloganistic analysis’ of the new view is that ‘Scientific theories are causally-possible collections of state-transition models of data for which there is a representation theorem’ (Suppe 2000, p. S111). Developmental psychologists have also shown an interest in the new view and used it in arguing for cognitive continuity between infant and adult science. The prospect of bringing together philosophy of science and cognitive developmental psychology in the service of science education is exciting and promising. However, to fulfill the promises, the various parties must pull in the same direction.
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Gopnik is one of the most explicit of the cognitive psychologists, and her account is that ‘A person’s theory is a system that assigns representations to inputs . . . .’ These representations are distinctive in having ‘. . . abstract, coherent, causal, ontologically committed, counterfactual supporting entities and laws . . . .’ Moreover, ‘The representations are operated on by rules that lead to new representations; for example, the theory generates predictions’ (Gopnik 1997, 43). The most sophisticated attempt to bring together psychology with the new view is Giere (1988, 1999). His account is that a theory is a family of models and that a model is an abstract entity having all and only the properties ascribed to it by an accompanying representation. I do not want to worry here whether Suppe’s models are the same kind of creature as Giere’s, nor whether Gopnik’s representations have anything to do with Suppe’s representation theorems. Instead, I want to consider what a model might be in the context of science education and to think about how the implications of the newer and older approaches differ. I would also like to investigate why and when we need to make the transition from implicit mental models to explicit external models. According to the Received View, a theory was an axiomatized set of sentences in the formalized symbolic language of first order logic. (Any damage done by the Received View to science education must, however, have been indirect since I know of no textbooks that present theories in this form.) Since the objections to the Received View seem to center on its axiomatic and linguistic form, consider an alternative, the Received View∗ : A theory is the class of models of an axiomatized set of sentences in the formalized symbolic language of first order logic.
On this alternative, a theory is a non-linguistic abstract collection of models. Presumably, there is still something wrong with this view, or else the whole fuss about overthrowing the Received View was just hype. And presumably it must be either that these are the wrong kind of models or else they are the wrong kind of collection of models. Or both. Rather than worry about this problem in the abstract, the remainder of my paper explores two specific instances of theories/models in basic physics. The first draws on the work of Profitt and Gilden (1989) on the understanding and misunderstanding of rotating objects. The simplest problems in mechanics have only one relevant parameter, the location of the center of mass of the object. However, when rotating objects are considered, the distribution of mass sometimes becomes relevant. Given two objects sliding down a frictionless plane, their distribution of mass is irrelevant. In contrast, given two cylinders of the same size, shape and mass, on an inclined plane, one still needs to know the distributions of mass in order to predict which cylinder will roll to the bottom first. In their studies, a variety of subjects were able to provide correct answers much more frequently to the one-parameter problems than to the two-parameter problems. Most telling was that when the two-parameter problems were presented to
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high-school physics teachers and the teachers were forced to answer rapidly, their answers were no better than those of undergraduates without time constraints. Summarizing: • Given problems about rolling cylinders in which mass distribution is a relevant parameter, over 80% of U Virginia undergraduates gave incorrect responses • Given the same problems, members of the UVa Bicycle club did no better • Given the same problems and required to answer immediately without calculations, UVa physics faculty did no better • Given the same problems and required to answer immediately without calculations, high school physics teachers did no better To quote from Profitt and Gilden’s conclusion, . . . physicists have a dual awareness of the characteristics of mechanical systems. This awareness is quite interesting to observe and it is easily elicited . . . . Prevent competent physicists from making explicit calculations about such events as rolling wheels, and they exhibit the basic confusions that are found in naive observers . . . . However, most physicists could work the problem out in a few minutes. (Profitt & Gilden 1989, 391)
What is required for the experts to solve the problems is the opportunity to make explicit calculations. This suggests that the most pragmatic understanding of ‘having a model’ is ‘having the ability to deploy relevant equations to describe the problem’. My second example is drawn from a piece of middle-school curriculum devised by Duschl and collaborators in SEPIA (Science Education Portfolio Instructional Assessment). Students were given a square piece of aluminum foil and were asked to produce a ‘vessel’, that is, to fold the aluminum foil so that it would float and carry a weight. The goal for them was for them to investigate how to choose the shape that would carry the maximum weight. This was done with 6th-grade students, and a large part of the instructional objective was for them to learn that the shape with the greatest volume would carry the greatest weight. Auxiliary experiments and repeated trials were provided. Students worked in groups and many groups eventually converged to near optimal solutions (Duschl & Gitomer 1997, Goldman & Duschl 2000). Even after coming to the conclusion that the maximum volume will displace the maximum amount of water and thus will carry the greatest weight, the problem is not trivial. It is non-trivial because of the constraints of using the single piece of foil, which means that any increase in the height of the sides of the vessel decreases the size of the base. We let S be the length of a side of the piece of aluminum foil, H the height of the side of a vessel obtained by folding the corners inward, and L the length of the base. (For reasons of time, we will just assume that a symmetric shape will be optimal.) Given this set of labels for the problem, students could measure the length S of the given piece of aluminum foil and then construct by calculation a chart that
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would give the volume of the vessel for each choice of H . Of course this would require the equation for volume, which in this case turns into V = H · L2 . Is this equation a theory? A model? I don’t know, but I want to underline that what the equation provides is the ability to calculate the result of constructing various shapes without having to actually produce them. Students could arrive at approximately the same results by constructing vessels with various S and H and measuring how much water they hold. And perhaps the equation could be discovered that way. But having the equation is a major conceptual and practical step because it gives the beginning of an exact articulation of the trade off between increasing H and decreasing L. The more exact articulation comes from more mathematics, from the fact that S = L + 2H, so L = S − 2H, and with a little algebra we find that for this problem V = H (S − 2H )2 = 4H 3 − 4H 2 S + H S 2 . Students with this equation now have a more articulated model of the problem giving an exact description of the tradeoff between increasing H and decreasing L. For students with slightly more sophistication or technology, a graph of the equation could be drawn or computergraphed This much could be accomplished at the level where students have Algebra 101, but of course, with calculus the next step can be taken of differentiating the equation with respect to H , which gives dV /dH = 12H 2 − 8H S + S = (6H − S)(2H − S). This gives extreme values when H = S/2 and H = S/6. The first is the minimum in which the L = 0, and the second is the desired solution. Unpacking the solution just a little, we notice that L = 2S/3, i.e., the maximum is achieved by putting one third of the square into the height of the vessel and two thirds into the base. One advantage of this equation/model is that it shows that the ratio of height to length is independent of the size of the square of foil. Calculating specific numbers might eventually suggest that this was the case, if you represented the data in the right way, but would hardly give a proof. Certainly, you would not arrive at the conclusion that the optimum ratio is exactly 1.666666666666666666 . . . . However, the model only works for the particular problem that we chose, that of using a square piece of foil to begin with. If we consider the more general case of a rectangular piece of foil, then the solution is somewhat more complicated to
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arrive at and certainly not as elegant. With this in mind, if one were designing a spiraling curriculum that would begin this problem in 6th grade and rework it two or three more times later, then it would probably be optimal to start with the square piece of foil knowing the outcome that is wanted at the end. So this problem also illustrates the value of the teacher having in mind an explicit model of the solution at the very beginning. Another point of this little problem is equally important, and quite different. Our solution has effectively assumed that the vessel, once folded into the desired shape is rigid. We have ignored the effects of water pressure on the sides of the vessel. In the real non-mathematical world, we ignore water pressure at our peril, and in fact, the sides of vessels collapse under the pressure of water relatively quickly. Contrary to our model the effect, and thus the optimum height, depends on the size of the piece of foil we start with. For larger pieces of foil, the optimum solution involves double-folding the sides of the vessel to resist water pressure. Of course this point about the limitations of models and their assumptions is a very important lesson, and the observation can be used in more advanced classes to begin pursuit of a more complex mathematical model that takes water pressure into account. In conclusion, I offer two morals: 1. My analysis suggests that ‘models’ in physics are very often mathematically represented, and if this is correct, more careful attention needs to be devoted to integrating mathematical and scientific curricula. 2. Models are required in physics when the proper representation of reality requires an equation in which two variables occur and which is not linearly additive. References Duschl, R.: 1990, Restructuring Science Education: The Importance of Theories and Their Development, Teachers College Press, New York. Duschl, R.A, & Gitomer, D.H.: 1997, ‘Strategies and Challenges to Changing the Focus of Assessment and Instruction in Science Classrooms’, Educational Assessment 4(1), 37–73. Giere, R.: l988, Explaining Science: A Cognitive Approach, University of Chicago Press, Chicago. Giere, R.: l999, Science without Laws, University of Chicago Press, Chicago. Goldman, S. & Duschl, R.A.: 2000, ‘Formative and Summative Assessments of Inquiry Science Deliverable #2: Report on the Implementation of the SEPIA Vessels Unit’, www.cilt.org/images/Goldman-Duschi-report-2.doc. Gopnik, A. & Meltzoff, A.N.: 1997, Words, Thoughts and Theories, MIT Press, Cambridge. Profitt, D.R. & Gilden, D.L.: 1989, ‘Understanding Natural Dynamics’, Journal of Experimental Psychology: Human Perception and Performance 15, 384–393. Suppe, F.: 1974, The Structure of Scientific Theories, University of Illinois Press, Urbana. Suppe. F.: 2000, ‘Understanding Scientific Theories: An Assessment of Developments, 1969-1998’, Philosophy of Science 67, 5102–5115.
