Models in science come in different forms. A physical model that you probably are familiar with is an anatomically detailed model of the human body. Mathematical models are less commonly found in science classes, but they form the core of modem cosmology. Mathematical models are extremely powerful because they usually enable predictions to be made about a system. The predictions then provide a road map for further experimentation. Consequently, it is important for you to develop an appreciation for this type of model as you learn more about cosmology. Two sections of the activity develop mathematical models of direct relevance to cosmology and astronomy. The math skills required in the activity increase with each section, but nothing terribly advanced is required. A very common approach to the mathematical modeling of a physical system is to collect a set of experimental data and then figure out a way to graph the data so that one gets a straight line. Once a straight line is obtained, it is possible to generalize the information contained in the straight line in terms of the powerful algebraic equation:
y = mx + b
You probably are familiar with this equation. In it y represents a value on the y-axis, x represents a value on the x-axis, m represents the slope of the straight line, and b represents the value of the intercept of the line on the y-axis. In all sections of this activity, your goal will be to analyze and then graph a set of data so that you obtain a straight line. Then you will derive the equation that describes the line, and use the equation to make predictions about the system. So relax and have fun with math!
Mathematical modeling is the process of creating a mathematical representation of some phenomenon in order to gain a better understanding of that phenomenon. It is a process that attempts to match observation with symbolic statement. During the process of building a mathematical model, the model will decide what factors are relevant to the problem and what factors can be de-emphasized. Once a model has been developed and used to answer questions, it should be critically examined and often modified to obtain a more accurate reflection of the observed reality of that phenomenon. In this way, mathematical modeling is an evolving process; as new insight is gained, the process begins again as additional factors are considered. “Generally the success of a model depends on how easily it can be used and how accurate are its predictions.” (Edwards & Hamson, 1994, p. 3)
Mathematics – The Language of Modelling
Like other languages, the essence of mathematics is the way it enables us to express, communicate, and reason about ideas and, especially, ideas about our world. The word “black” in English is important because it describes the color below. Without seeing this color one misses a great deal about the word “black.”
We are interested in using mathematics to talk about meaningful problems. For this reason, laboratory equipment like the Texas Instrument CBL that allows us to collect and record quantitative information about the real world, and sources like the United States Census are especially important to us.
Working with real problems requires the full power of mathematics — the ability to work with symbols, with graphics, and with numerical calculations. For this reason computer algebra systems like MathCad, Maple, Mathematica, and the CAS built into the TI-92 are an integral part of our tool kit. They give us powerful environments for doing mathematics. And together with a browser, like Netscape, some cables, and equipment like the TI-CBL they give us the ability to use the full power of mathematics with real data from the real world.
Building a Mathematical Model
Building a mathematical model for your project can be challenging, yet interesting, task. A thorough understanding of the underlying scientific concepts is necessary and a mentor with expertise in your project topic is invaluable. It is also best to work as part of a team to provide more brainstorming power. In industry and engineering, it is common practice for a team of people to work together in building a model, with the individual team members bringing different areas of expertise to the project.
Although problems may require very different methods of solution, the following steps outline a general approach to the mathematical modeling process:
1. Identify the problem, define the terms in your problem, and draw diagrams where appropriate.
2. Begin with a simple model, stating the assumptions that you make as you focus on particular aspects of the phenomenon.
3. Identify important variables and constants and determine how they relate to each other.
4. Develop the equation(s) that express the relationships between the variables and constants.