When students hear the word “model”, the first thing they usually think of is a physical model, such as an anatomical model of a heart, or a three dimensional model of the solar system. However, when we, as scientists and educators, talk about models, we are talking about conceptual models, about how ideas fit together to provide an explanation for a given phenomenon. Mental models are often constructed through analogy, particularly for visualization purposes (Collins & Gentner, 1987). Mathematical models, as discussed above, are of primary importance to a physicist, but scientific models can take many forms. A circuit diagram is a visual representation of a system. In kinematics, we often use pictorial representations such as vectors or graphical representations to show how an object moves over time. What all of these models have in common is that they are representations of scientific concepts. (See Harrison & Treagust (2000) for a typology to distinguish various modeling tools.)
The process of having students create their own conceptual models is what is referred to as “model-based inquiry” in the science education literature. This pedagogical framework is based on a constructivist learning theory, and in particular a model of conceptual change. Conceptual change is broadly defined as a change in the way one conceptualizes or understands a given phenomenon. Students learn by constructing their own understanding of the world, a process which requires them to evaluate and sometimes reject their own ideas. Several theories of conceptual change are represented in the literature, and each has a slightly different definition of what they mean by conceptual change.
The definition I will focus on here was first presented by Posner, Strike, Hewson, and Gertzog (1982). Their model applies work from the philosophy of science (Kuhn, 1970) to a theory of learning. The argument is that students learn through a process of “conceptual change” that is similar to a paradigm shift that occurs in scientific research. According to Kuhn, a paradigm shift occurs when new ideas or phenomena cannot be explained by the previous understanding, and thus new theories or conceptual structures must be invoked. In a similar way, dissatisfaction with existing explanations is critical for conceptual change to occur. This usually occurs when students are confronted with new data or phenomena that cannot be explained (this is sometimes referred to as cognitive conflict). They must recognize that there is some problem with their previous explanation, and consciously decide to employ a new one. The theory of conceptual change therefore describes learning as an active process in which students construct an understanding of the world based on their experiences.
This is effectively what we ask students to do when they engage in model-based inquiry (Stewart et al, 2005; Windschilt et al, 2008). They begin to construct a conceptual model based on their current understanding of the world. Then we present them with new data, or a new piece of information that they need to integrate into their model. One possibility is that their model accommodates the new data, which might provide further evidence for the validity of the model. Another possibility is that the model cannot explain the new data. This requires students to re-evaluate their model. This cycle of building conceptual models in the classroom mirrors the scientific inquiry process, as described above.
Posner and colleagues (1982) articulate two key ideas which explain how this learning process takes place: status and conceptual ecology. The conceptual ecology is the collection of current concepts or ideas held by a student. It is made up of different types of knowledge, and influences whether or not he or she will accept a new idea. Posner et al (1982) define several features of a conceptual ecology: anomalies, or failures of given ideas; analogies and metaphors; epistemological commitments, such as ways of explaining new phenomena; metaphysical beliefs and concepts; and other knowledge, such as knowledge in other fields or competing ideas. Learning then takes place as these new explanations are integrated into a student’s conceptual ecology.
Status is a way to measure the extent to which a student accepts a new idea (Hewson & Lemberger, 2000; Hewson & Hewson, 1992). In order for a new idea to be incorporated into a student’s conceptual ecology, it must meet three criteria. The idea must be intelligible, plausible, and fruitful (Posner et al, 1982). These are referred to in the literature as “status terms,” as they can be used to measure the status of a new idea or explanation; that is, the extent to which a student believes the idea to be true. The first step towards conceptual change is for the student to see an idea as intelligible; the student must understand this new explanation enough to further explore the possibilities. He or she also must see this as a plausible explanation for the phenomenon. The new explanation therefore must be consistent with the students’ prior experiences and existing knowledge. Finally, the student must see this new concept as being fruitful; it can be further applied to new situations and extend learning. Conceptual change learning occurs when the status of a given idea has been elevated (i.e. it goes from being merely intelligible to fruitful), and becomes a part of the student’s conceptual ecology.
What would it look like to apply this idea of conceptual change to the process of building conceptual or mathematical models? Ideally, we would like students to use mathematical models in physics to make predictions about the behavior of a system; we want them to be able to apply models to novel situations in way that demonstrates their fruitfulness. To do this, students must be convinced that a mathematical model is in fact a plausible representation of reality. But before we get to this, the students must find an equation or mathematical model intelligible, which requires they have a level of proficiency and comfort with mathematics. Often this is where we see our students get stuck; they don’t see the equations as conceptual models because the mathematics is not intelligible.
Although they may not be aware of it, students harbor unique mental models that they use to explain the world around them. The difficult task for a teacher is to access these mental models. The key to uncovering students’ models is language -- “Language does not refer directly to the world, but rather to mental models and components thereof! Words serve to activate, elaborate or modify mental models, as in comprehension of a narrative” (Hestenes, 2006). Thus understanding the language used to communicate the model is an important piece of understanding the student’s mental model. Throughout this curriculum, we aim to provide opportunities for students to express their ideas both mathematically and verbally so that the teacher has multiple ways to assess their understanding of a given topic.
Collins, A. & Gentner, D. (1987). How people construct mental models. Accessed on June 23, 2010 at http://groups.psych.northwestern.edu/gentner/papers/CollinsGentner87.pdf.
Harrison, A.G. & Treagust, D.F. (2000). A typology of schools science models. International journal of science education, 22(9), 1011-1026.
Hewson, P., & Hewson, M. (1992). The status of student conceptions. In R. Duit, F. Goldberg, & H. Niedderer (Eds.), Research in physics learning: Theoretical issues and empirical studies. proceedings of an international workshop held at the University of Bremen, March 4-8, 1991.
Hewson, P.W. & Lemberger, J. (2000). Status as the hallmark of conceptual learning. In R. Millar & J. Leach & J. Osborne (Eds.), Improving science education: The contribution of research (pp110-125). Buckingham: Open University Press.
Hestenes, D. (2006). Notes for a Modeling Theory of Science, Cognition and Instruction. Proceedings of the 2006 GIREP conference: Modelling in Physics and Physics Education.
Kuhn, Thomas. (1970). Structure of Scientific Revolutions, 2nd edition. University of Chicago Press: Chicago.
Stewart, J. Cartier, J. & Passmore, C. (2005). Developing Understanding through Model-Based Inquiry. In National Research Council. How Students Learn: Science in the Classroom. Washington, DC: National Academy Press.
Posner, G.J., Strike, K.A., Hewson, P.W., & Gertzog, W.A. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66(2), 211-227.
Windschitl, M., Thompson, J. & Braaten, M. (2008). Beyond the scientific method: Model‐based inquiry as a new paradigm of preference for school science investigations. Science Education, 92, 941–967.