Meaning and Importance

When solving difficult problems, we frequently suggest to students that they try to solve a simpler problem, observe what happens in a few specific cases (that is, look for a pattern) and proceed from there. This pattern-based thinking, in which patterns are used to analyze and solve problems, is an extremely powerful tool for doing mathematics. Students who are comfortable looking for patterns and then analyzing those patterns to solve problems can also develop understanding of new concepts in the same way. Most of the major principles of algebra and geometry emerge as generalizations of patterns in number and shape. For example, one important idea in geometry is that: For a given perimeter, the figure with the largest possible area that can be constructed is a circle. This idea can easily be discovered by students in the middle grades by examining the pattern that comes from a series of constructions and measurements. Students can be given a length to use as the perimeter of all figures to be created, say 24 centimeters. Then they can construct and measure or compute the areas of a series of regular polygons: an equilateral triangle, a square, and a regular hexagon, octagon, and dodecagon (12 sides). The pattern that clearly emerges is that as the number of sides of the polygon increases (that is, as the polygon becomes more "circular"), the area increases.

All of the content standards have close interconnections, but this is one that is very closely tied to all of the others since an understanding of patterns can be either content or process. When the patterns themselves and their rules for generation are the objects of study, they represent the content being learned. However, when pattern-based thinking, or the search for patterns, is the approach taken to the discovery of some other mathematical principles, then patterning is a process, and the approach can easily be applied to content in numeration, geometry, operations, or the fundamentals of calculus. There is a very special relationship, though, between patterns and algebra. Algebra provides the language in which we communicate the patterns in mathematics. Early on in their mathematical careers, children must begin to make generalizations about patterns that they find and try to express those generalizations in mathematical terms. Examples of the different kinds of expressions they might use are given in the discussion below.

K-12 Development and Emphases

Young students should frequently play games which ask them to follow a sequence of rules (arithmetic or otherwise) or to discover a rule for a given pattern. Sequences which begin as counting patterns soon develop into rules involving arithmetic operations. Kindergartners, for example, will make the transition from 2, 4, 6, 8 ... as a counting by twos pattern to the rule "Add 2" or "+2." The calculator is a very useful tool for making this connection. Through the use of the calculator's constant function feature, any calculator can represent counting up or counting down by any constant amount. Students can be challenged to guess the number that will come up next in the calculator's display and then to explain to the class what the pattern, or rule, is.

At a slightly higher level, input-output activities which require recognition of relationships between one set of numbers (the "IN" values) and a second set (the "OUT" values) provide an early introduction to functions. One of these kinds of activities, the Function Machine games are a favorite among first through fifth graders. In these, one student has a rule in mind to transforms any number which is suggested by another student. The first number inserted into the imaginary Function Machine and another number comes out the other side. The rule might be plus 7, or, times 4 then minus 3, or even the number times itself. The class's job is to discover the rule by an examination of the input-output pairs.

In the middle grades, students begin to work with patterns that can be used to solve problems within the domain of mathematics as well as from the real world. There should also be a more obvious focus on relationships involving two variables. An exploration of the relationship between the number of teams in a round robin tournament and the number of games that must be played, or between a number of coins to be flipped and the number of possible resulting configurations, provides a real-world context for pattern-based thinking and informal work with functions. Graphing software is extremely valuable at this level to help students visualize the relationships they discover.

At the secondary level, students are able to bring more of the tools of algebra to the problem of analyzing and representing patterns and relationships. Thus we expect all students to be able to construct as well as to recognize symbolic representations such as y = f(x) = 4x+1. They should also develop an understanding of the many other representations and applications of functions as well as of a greater variety of functional relationships. Their work should extend to quadratic, polynomial, trigonometric, and exponential functions in addition to the primarily linear functions they worked with in earlier grades. They should be comfortable with the symbol f(x), both as the application of a rule of correspondence and as a "value" corresponding to x, in much the same way that elementary students have to view 3+2 both as a quantity and as a process.

The use of functions in modeling real-life observations also plays a central role in the high school mathematics experience. Line- and curve-fitting as approaches to the explanation of a set of experimental data go a long way toward making mathematics come alive for students. Technology must also play an important role in this process, as students are now able to graphically explore relationships more easily than ever before. Graphing calculators and computers should be made available to all students for use in this type of investigation.

In summary, an important task for every teacher of mathematics is to help students recognize, generalize, and use patterns that exist in number, shape, and the world around them. Students who have such skills are better problem solvers, have a better sense of the uses of mathematics, and are better prepared for work with algebraic functions than those who do not.