Patterns and Change

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One of the ways in which mathematics is used extensively is to analyze change. Change often occurs in a predictable way, and if a mathematical pattern can be found to describe the change, we can use mathematics to predict accurately when certain kinds of events will occur. For example, one of the earliest uses of mathematical patterns to predict change was in the area of astronomy. Mathematicians and scientists discovered mathematical patterns that would allow them to predict eclipses of the sun and moon. Now we know exactly when and where these events will occur, and we can reliably plan for them. A mathematical study of the patterns of the motions of the sun, moon, and earth made this possible. (A middle school teacher might use the book A Connecticut Yankee in King Arthurs Court in order to integrate mathematics, science, and literature. In this book a modern-day man who knows a little astronomy finds himself transported in time to the days of King Arthur. He uses his knowledge of the timing of an eclipse to impress people and gain some influence among them.)

The formulation of a mathematical representation of the motions of the sun, moon, and earth would not have been possible if early mathematicians had not been looking for patterns in their data. The search for patterns, whether in data, or in graphical designs, or in music, or in daily events, is an essential component of inductive reasoning. Inductive reasoning is a crucial part of the scientific method. Students of mathematics should develop a habit of looking for patterns in their work with problems of all types. Most often patterns can be found, even though sometimes the patterns may be quite complex. In all of the work presented in this text, we recommend that pattern searching be used as a primary strategy for problem solving.