Notes on Modeling: Session I; Part II
York College (CUNY)
Jamaica, NY 11451
When one creates a model it is often necessary to use data to construct the model. For example to determine how much corn one might be able to grow on the Earth one would need to have "data" related to treating the Earth as a perfect sphere and what portion of the Earth's surface is covered by water. Furthermore, one would also need to know how much corn one might be able to grow on an acre of land. What numbers should one use and where might one find numbers to use?
For the diameter of the Earth, since the earth is not a perfect sphere it is not totally clear what number to use. If one examines a plane that cuts the Earth through its presumed center, the shape one gets will not be a perfect circle since the earth has mountains and valleys on its surface. However, the difficulty is more complex because even if one could agree that the best choice of a diameter would be to use the Earth's diameter at the equator (assuming for the moment this to be a perfect circle) where could one find this information? One could look on the internet, one could look in a "trustworthy" encyclopedia, or one one might try a geology text book or world almanac. As an experiment you might want to try going to a variety of such different sources and see if you get the same number. What should be done if you do not? Of course, for this particular situation the stakes are not high in getting an answer which might not be "true" but there are situations where life and death depend on the accuracy with which the numbers one works with are used. For example, the manufacturer of a new drug must determine a non-toxic amount of a drug to be taken so that the drug is efficacious, since nearly all drugs become toxic when too much of the drug is given.
Constructing mathematical models is allied to problem solving in general but it requires a somewhat different set of skills from problem solving of the kind that comes up in technique driven approaches to the teaching mathematics.