A given situation is built from many attributes, relationships, and measurements. A model is a collection of related mathematical objects that encode this information.

A model might include definitions, functions, equations, diagrams, measurements, etc. The model may hold restrictions on the objects that reflect the constraints of the situation. Or, it may not. The model may be lacking information available in the real situation.

The model is the model.

With the model in hand, we shelve the situation and enter the model’s world. By analyzing the model, we mean to expand its collection of objects to include related objects, which are logically derived from the initial objects. The collection should always hold objects which are logically consistent with each other.

• These related objects might be new functions we derived from the existing functions.
• These new objects might be import function values at important domain values.
• These new objects might be equations describing further relationships we discover.

As the analysis continues, the model expands and conclusions may be drawn. These become part of the model. Eventually, the model is compared back to the real situation.

• Does the expanded model still accurately describe the situation?
• Does the model need revision?
• What information about the real situation does the new model suggest that we were not thinking before?

Often, there is a question about the real situation and we are using the model to respond to the question. In this case, the initial model has encoded the question into a question about the model. When we compare back to the real situation, we must decode the model information back into situational information.

In any case, we must separate the model from the situation. We allow the model analysis to go wherever the mathematics logically takes it.

Learning Outcomes

In this section, students will

• encoding and decoding information.