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A useful model fits many real situations to a good approximation.  Some useful models used by physicists to think about the physical include: motion with constant acceleration, the harmonic oscillator, the two level quantum system, Feynman diagrams and the Schwartzschield metric (which applies the law of General Relatively to find the warping of space-time due to a central sphere of matter).  A list of models of mechanics is available.

Why we use Models in this Course

The key pedagogical reason for using models in this course is to provide a framework within which students can organize the many facts and procedures they learn in introductory physics into a small number of useful models, relate these models to the few overall theories that underlie the material, and think about the real world by recognizing situations or simplifications where these models apply.   This leads to an understanding of the world, and the ability to simplify and model physical situations that are new.  From an expert/novice perspective models organize the many formulae and graphs on the novice's formulae sheet into a much smaller number of "chunks" of related things that are of reflective of nature's organization. 
Models are idealizations of physical reality involve a particular structure or pattern.  Models can be mathematical, logical, pictoral, or a physical object (but not usually in physics).  Models only approximate reality; they represent an idealization of reality (e.g. with no friction, ignore bending of rigid bodies, etc.), but generally they are applicable to many situations (else the model is not useful).  Models generally involve a cluster of several concepts and theories (e.g. harmonic motion involves kinematics, F=ma, and a linear restoring force).  Models almost always may be expressed in several representations, and the cross connection of these representations provides a richness for the models.  For example, motion with constant acceleration may be represented with standard equations, strobe pictures of the object, graphs or tables of kinematic variables vs. time, or a concise verbal description.  The two level system may be represented using the equations following from time-dependent perturbation theory, the Bloch vector, and the density matrix.   
Physicists and educational psychologists agree that understanding a model implies fluency with, and ability to transfer between, all of its commonly used representations.  A physicist familiar with any model can recognize/describe/understand/quantitatively predict situations that fit within the model's assumptions with little effort (as an exercise), but is typically confronted with a "problem" if even a small discrepancy exists between the situation at hand and the relevant model. 
Being able to understand and use a model involves:

1. understanding the various representations and their interrelationships
2. developing an ability to recognize (even novel) physical situations where the model applies
3. being able to map the reality onto the model (i.e. to ignore the unimportant)
4. being able to carry through the solution in any of the model's representation(s)
5. at a high level, being able to generalize the model

Understanding/Learning a Model

Understanding a model requires each student to reconstruct and interrelate its components in their own minds, and to understand the relationships among its representations.  This is usually achieved by a laboratory in which a group of several students follows a more or less guided discovery procedure (but never a completely cookbook prescription) followed by discussion conducted skillfully by a trained person.  DEP feels that interactive lecture demonstrations together with problems involving transfer of representations should be able to perform much of this function.   
However it is done, successful learning involves the student understanding the pieces of the model and being able to use it in context-rich problems (i.e. extract the relevant variables from a real world story as well as the minimalist presentation typically found in textbook problems).  Hestenes would contend that a key to modeling is that students become skillful at constructing models for new situations, so he and many true modelers would disprove of our approach.  Making a model and learning to apply it typically takes two weeks, limiting the number of models to ~6 for the course.  Our hope is that we can teach ~ one model/week by starting with only 4 general models and indicating how several other models are subcases of the general models.

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