Mathematical Modeling Although Even Complex

¶ … Mathematical Modeling

Although even complex mathematical modeling is certainly not new, the process has been facilitated enormously in recent years by the introduction of computer-based modeling applications. Despite these innovations, there are still some significant limitations to mathematical modeling that must be taken into account when using these techniques. To gain some additional insights in this area, this paper provides a review of the relevant literature to identify the benefits and limitations of mathematical modeling, a discussion concerning the use of mathematical modeling in the author's profession and the extent to which such modeling is used as value-added to other kinds of empirical research, and the extent to which it is used in place of other kinds of empirical research. A summary of the research and important findings are presented in the conclusion.

Review and Analysis

Serious interest in mathematical modeling emerged during the mid-20th century when computer science was in its infancy but the need for ways to simulate real-world situations became pronounced. According to Maxwell (2004), "The federal government and many private enterprises have used mathematical modeling since the late 1950s as aids in developing policies, conducting research and development, and engineering complex systems" (p. 67). Today, computer-driven mathematical modeling applications have a number of real-world applications, including gambling and sports simulations as well as modeling human interactions for couples therapy and other "people prediction" applications (Albert, 2002). In this regard, Oliver and Myers report that, "Game theory provides a rich history of considering the strategies derived from various payoff structures, rules about repeating the game, and how players communicate" (p. 34). Mathematical modeling has proven efficacy in other settings as well, including the entire range of economic analyses (Oliver & Myers, n.d.) and even enormously complex weather prediction applications (Kirlik, 2006). Moreover, mathematical modeling has been used to good effect in helping researchers better understand how physiological processes operate at the molecular level. For example, Peter (2008) reports that, "Mathematical models allow researchers to investigate how complex regulatory processes are connected and how disruptions of these processes may contribute to the development of disease" (p. 49).

Furthermore, mathematical modeling can facilitate the systematic analyses of various "what-if"-type scenarios (Oliver & Myers, n.d.), formulate new hypotheses to serve as the basis for regimens of therapeutic interventions and even to evaluate the appropriateness of specific molecules for therapeutic purposes (Peter, 2008). According to Peter, "Numerous mathematical methods have been developed to address different categories of biological processes, such as metabolic processes or signaling and regulatory pathways. Today, modeling approaches are essential for biologists, enabling them to analyze complex physiological processes, as well as for the pharmaceutical industry, as a means for supporting drug discovery and development programs" (2008, p. 50). In fact, some authorities suggest that the limits of mathematical modeling are fundamentally human-based rather than technologically restricted. In this regard, Maxwell (2004) points out that, "Mathematical modeling and computer simulation are limited only by the ingenuity of the person or team conducting the analysis. They have been used for such tasks as improving personnel scheduling for police services, allocating response areas for urban fire departments, and describing security systems to test their security posture" (p. 68)..

Notwithstanding these wide ranging applications, though, Kirlik (2006) emphasizes that, "Not all domains are amenable to mathematical modeling. This type of modeling requires a large set of data. For example, modeling algorithms perform best when provided with a data set that includes multiple performance examples across all possible combinations of cues, which is not always consistent with collecting data under representative conditions" (p. 161). Notwithstanding these constraints, researchers have found a number of ways to improve the mathematical modeling process and the techniques are commonly used to solve so-called word problems by using semantic cues in the construction of corresponding situation models (Martin & Bassock, 2005). The benefits of mathematical modeling are especially apparent when the textual descriptions of equations, for example, are unable to fully communicate the fundamental underlying concepts that are involved; in other words, why doing this mathematics problem in a certain way is the right way. In this regard, Oliver and Myers report that, "There is value in translating mathematics into words, so that those who do not readily grasp equations can appreciate the ideas they convey" (p. 33).

While there are increasingly sophisticated mathematical modeling applications available, a number of businesses use conventional spreadsheet programs such as Excel for a wide range of valuable mathematical modeling purposes (Ellington & Hardin, 2008). It should be pointed out, however, that Excel and more sophisticated mathematical modeling applications facilitate the analyses process, but they do not replace the need for a firm foundation in the fundamentals. In this regard, Ellington and Hardin (2008) add that, "A solid understanding of mathematical modeling requires a blend of mathematical skills and conceptual understanding. The technology available today may reduce, but does not eliminate the need for mathematical skills. Future analysts need a solid understanding of mathematical modeling when they enter the workforce" (p. 110).