Repeating Pattern or Number Pattern: The Distinction Is Blurred
Liljedahl, Peter, Focus on Learning Problems in Mathematics
"Life itself is a creator of patterns." (Piaget, 1950, p. 167)
Many argue that patterns are the cornerstone of mathematics. They are the foundation that the whole of the subject is built on. From the earliest tally systems to the development of differential calculus to modern mathematics, patterns, were and are the genesis, the motivation, and the foundation of mathematical knowledge. As such, mathematics is often referred to as the science of patterns (Borwein & Jorgenson, 2001; Resnick, 1997). Steen (1988) articulates this relationship between patterns and mathematics thus:
"Mathematical theories explain the relations among patterns; functions and maps, operators and morphisms bind on type of patterns to another to yield lasting mathematical structures. Applications of mathematics use these patterns to "explain" and predict natural phenomena that fit the patterns. Patterns suggest other patterns, often yielding patterns of patterns." (p. 612)
Even the very description of what it means to do mathematics can be defined in the context of patterns--"mathematicians observe patterns; they conjecture, test, discuss, verbalize, and generalize these patterns" (National Council of Teachers of Mathematics--NCTM, 1991).
However, the role of patterns in mathematics is an ironic one. While much of mathematics has its roots in patterns, there is no place for patterns in the formal representation of mathematics. The contemporary view is that mathematics is axiomatic in nature. As such, convention dictates that mathematics is presented in a linear and deductive argument, in the form of theorems and proofs. Patterns, on the other hand, are not axiomatic, nor are they necessarily linear. By nature, a pattern is inductive, and thus has no place in mathematical proof. "Much of what is "pattern" in the knowledge of mathematics is instead encoded in a linear textual format born out of the logical formalist practice that now dominates mathematics." (Borwein & Jorgenson, 2001, p. 897).
In the teaching and learning of mathematics this irony is extended. While a mathematical proof, with its unfaltering deductive logic, contains within its structure the truth about a mathematical concept, it is often inappropriate for conveying mathematical concepts in that it may, in fact, be conveying the wrong message to our students--that full rigor is the core of mathematics (Hanna, 1989). Furthermore, it is often the use of a pattern that unlocks that truth and both presents it to the student and convinces them of it (Harel & Sowder, 1998; Mason, 2002; Rowland, 2002; Tahta, 1980). Consider, for example, the properties of negative exponents. Although they can be shown to be true using deductive reasoning these concepts initially defy students' intuition. It is the use of patterns which most often facilitates the conceptual change necessary to create the understanding.
[2.sup.4] = 16 [2.sup.3] = 8 [2.sup.2] = 4 While the exponent decreases by one as you move down [2.sup.1] = 2 the left column, the value of the exponential [2.sup.0] = 1 expression is divided by two as you move down the [2.sup.-1] = 1/2 right column. [2.sup.-2] = 1/4
Unfortunately, the way in which patterns are used in the teaching of mathematical concepts can create a whole new set of misunderstandings--not of the mathematical content but of the patterns themselves. If through the pedagogical use of patterns in the teaching and learning of mathematics due care is not taken to preserve distinction between the types of patterns used then there is a risk that students' understanding of patterns can become blurred. This article examines how the lack of explicit attention to the distinction between repeating patterns and number patterns leads to difficulties for students engaged in problem solving activities that involve investigation of patterns and offers a pedagogical solution to the prevention of this blurring. …