What Are Mathematical Models, and Why Are they important in Environmental Science?

Some of the most powerful technologies invented by humans are mathematical models, which we use to supplement our mental models. Mathematical models are equations that help us perceive and predict various things. For example, equations that describe the movement of air, moisture, and heat in the atmosphere (weather models) are used to predict the weather for the next few days. Sometimes, television weather forecasters refer to such models when they tell us what to expect from tomorrow's weather; usually, they translate the results of these mathematical models into weather maps, a type of graphic model.

The big difference between mental models and mathematical models is the way we get information from them. To get a perception or prediction from a mental model, we need only think; we get estimates or predictions from such models without even being conscious of them. In contrast, to get a prediction from a mathematical model, we must do some calculating perhaps in our head, perhaps with pencil and paper, perhaps With a computer.

Mathematical models vary in size and sophistication. Weather models are fairly complicated; to calculate predictions from them takes the use of powerful computers. Other mathematical models are simpler, such as the rule of 70 equation (70/percentage growth rate = doubling time in years). Using this model (de- rived from the mathematical equation for exponential growth), we can make a simple calculation to estimate how many days remain until we run out of money or how long it might take for the world's population to double.

Like mental models, mathematical models are imperfect approximations of reality. They tend to make predictions ranging from fairly accurate to very accurate, depending on the model and the data used to formulate it.

The process for developing mathematical models is essentially trial and error, Similar to the processes depicted.

Making a mathematical model usually requires going through three familiar steps many times:

(1) Make a guess write down some equations;

(2) compute the predictions implied by the equations;

(3) compare the predictions with observations, mental models' predictions, and existing experimental data and scientific hypotheses, laws, and theories.

Mathematical models are important because they can improve perceptions and predictions, especially concerning matters for which our mental models are weak. Research has shown that people's mental models tend to be especially unreliable

(1) when there are many interacting variables;

(2) when we attempt to extrapolate from too few experiences to a general case;

(3) when consequences follow actions only after long delays;

(4) when the consequences of actions lead to other consequences;

(5) when responses are especially variable in response from one time to the next; and

(6) when controlled experiments (Connections, p. 55) are impossible, too slow, or too expensive to conduct. Under such conditions, a good mathematics cal model can do better than most mental models.

Most of our effects on the environment have the following characteristics:

(1) We have only a few experiences upon which to base the generalizations;

(2) the responses occur only after long delays (such as possible ozone depletion);

(3) the responses cause other, delayed responses (such as the development of skin cancers 10-20 years after exposure to increased ultraviolet radiation); and

(4) there is a great deal of variability in what happens because the environment is full of diversity. These are exactly the kinds of situations in which our mental models tend to be unreliable and mathematical. If they are sufficiently good approximations of reality can help us make better predictions. Better predictions can lead to better decisions.