## Mathematical Models

There are several situations in which mathematical models can be used very effectively in introductory education.

- Mathematical models can help students understand and explore the meaning of equations or functional relationships.
- Mathematical modeling software such as Excel, Stella II , or on-line JAVA /Macromedia type programs make it relatively easy to create a learning environment in which introductory students can be interactively engaged in guided inquiry, heads-on and hands-on activities.
- After developing a conceptual model of a physical system it is natural to develop a mathematical model that will allow one to estimate the quantitative behavior of the system.
- Quantitative results from mathematical models can easily be compared with observational data to identify a model's strengths and weaknesses.
- Mathematical models are an important component of the final "complete model" of a system which is actually a collection of conceptual, physical, mathematical, visualization, and possibly statistical sub-models.

## Statistical Models

A solid statistical background is very important in the sciences. But the extent to which statistical ideas are appropriate in an introductory course depends on specific course objectives and the degree or institutional structure. Here we list several examples showing why and when statistical models are useful.

Statistical models or basic statistics can be used:

- To characterize numerical data to help one to concisely describe the measurements and to help in the development of conceptual models of a system or process;
- To help estimate uncertainties in observational data and uncertainties in calculation based on observational data;
- To characterize numerical output from mathematical models to help understand the model behavior and to assess the model's ability to simulate important features of the natural system(model validation). Feeding this information back into the model development process will enhance model performance;
- To estimate probabilistic future behavior of a system based on past statistical information, a
**statistical prediction model**. This is often a method use in climate prediction. A statement like 'Southern California will be wet this winter because of a strong El Nino' is based on a statistical prediction model. - To extrapolation or interpolation of data based on a linear fit (or some other mathematical fit) are also good examples of statistical prediction models.
- To estimate input parameters for more complex mathematical models.
- To obtain frequency spectra of observations and model output.