Acquiring Conceptual Expertise from Modeling: Th e Case of Elementary Physics

In many domains, the real world is modeled with systems of equations. Such a model uses variables to represent domain properties and equations to represent applications of domain principles. Given a set of true domain relationships expressed as equations, one can deduce new equations from them using only the rules of mathematics, and the new equations will also be true domain relationships. Th e latter step, wherein mathematical implications are derived from the initial model, can oft en be done mechanically, for example, by mathematical symbol manipulation programs, spreadsheets, calculators, etc. Given a real world situation that is amenable to such analysis, experts and novices understand them quite diff erently. Whereas novices must go through the whole modeling process by writing equations on paper and solving them, experts can generate many conclusions about the same situations without having to commit anything to paper. For the expert, many domain relationships are just “obvious” or can be easily inferred “by inspection.” Th ere are limits to the experts’ abilities. Although experts usually cannot mentally infer quantitative relationship, such as the exact numerical value for an energy or a velocity, they can infer qualitative relationships, such as whether a quantity is zero, increasing or greater than some other quantity. Th us, it is oft en said that expertise in such domains is characterized by a conceptual or qualitative understanding of real world situations (VanLehn, 1996 ). It is sometimes said that they have developed domain-specifi c intuitions (Simon & Simon, 1978 ). Th is ability of the experts is called conceptual expertise , and it is the focus of this chapter. Interest in conceptual expertise has increased in recent years with the discovery that in some surprisingly simple situations, novices have intuitions that confl ict with the experts’ intuitions (McCloskey, Caramazza, & Green, 1980 ). For instance, suppose a bowling ball and a golf ball are dropped from 2 meters above the Earth’s surface. An expert will know immediately that the two balls strike the earth at exactly the same time, whereas novices usually say that the balls land at slightly diff erent times. Novice intuitions about such situations 16