Patterns and Change

If the study of relationships is important to a deep understanding of mathematics, a study of patterns may be even more so. Some mathematicians believe that mathematics is essentially a study of patterns. When a person looks for a repetition of an event or a repetition of a sequence of events, he is involved in pattern searching. When a person who is searching for a pattern succeeds in finding a repeated unit he is, once again, involved in a process of composing units. This is one of the ways in which various areas of mathematics overlap and interconnect.

A number of areas of mathematics come together under a study of patterns. Often a visual pattern is represented using letters. A pattern of shapes such as: square, circle, circle, square, circle, circle, might be represented as the "A-B-B pattern." The letter "A" represents the square, and the letter "B" is used to represent each circle. "A-B-B" is a more general way to describe the type of pattern shown, and the same representation can be used for other occurrences of the same pattern. For example, 3-8-8-3-8-8 could also be represented as an "A-B-B" pattern. Students who are exposed to this sort of thinking at an early age come to expect patterns to be represented through some form of symbolic abstraction. The data from the table below might make it easy to find a pattern. A student might describe this pattern with words ("As the weight goes up by one, the price goes up by $0.15.") or by a formula such as cost = weight X 0.15. A study of patterns easily leads into a study of relationships as well as a study of representation. The various areas of mathematical processes are deeply interwoven and interconnected.

Table

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.