Notes on Modeling: Session I
York College (CUNY)
Jamaica, NY 11451
1. Many people are so used to thinking that mathematical questions have a single answer that when they first address modeling questions it makes them nervous or uncomfortable. However, even in traditional domains of mathematical questions things are not as simple as some may think.
For example, a genre of "school problems" is to give the next term in a sequence, the first few terms of which are revealed. Thus, given the sequence which starts, 1, 2, 4, 8, 16, what is the next term of the sequence? While the designer of this question may be expecting the response 32, can you see how to find a function f(n) such that f(1) = 1, f(2) = 2, ..., f(5) =16 but f(6) is not 32? We can use:
What is the value of f(6)?
When one teaches mathematical skills one can organize the presentation in a structured logical way. Mathematical modeling is intrinsically more unstructured. Historically, there have been two approaches to teaching mathematical modeling. One approach starts with a body of mathematical knowledge, game theory, difference equations, Markov processes, and then gives examples of how one can use this tool to get insight into various problems which arise outside of mathematics. In the other approach, one starts with an applications situation and constructs the model using ideas that are part of one's tool kit already. Sometimes one must develop new mathematical tools to take this approach, so pure mathematics and applied mathematics go hand in hand here, as they always have.
2. In problems such as finding out what is the surface area of the earth, often using data which is very precise garners no real value. Rough estimates typically will do because perhaps all one is looking for is the first significant figure for an answer. One wants to get a rough idea of what is going on.
3. Write down the for "formula" for the area A of a Euclidean circle of radius r, and the formula for V the volume of a Euclidean sphere of radius r. Now compute dA/dr and dV/dr. What do you notice?
4. Can you make up a "story" where the goal is get a student to estimate the circumference of a Euclidean circle (rather than the surface area of a Euclidean sphere)?