Early Algebra: Expressing Covariation and Correspondence

In a classroom teaching experiment, we explored fifth graders’ ability to identify relationships in and between two patterns—what we may refer to as functional relationships. Conventional functions curricula are dominated by defining relationships between two patterns via a rule, describing how to find y or f(x) given a particular value for x (e.g., y = 4x + 1). This type of relationship is a correspondence description of the relationship between two patterns (Confrey and Smith 1995). On the basis of research in which students identified patterns of values in individual columns of a table and coordinated corresponding values in two columns, Confrey and Smith (1995) suggested a second type of functional relationship: The covariation between quantities of two sequences of data (how a quantity in one pattern changes at the same time as a quantity in the other pattern changes), should also be emphasized in instruction to support development of functional thinking. The aim of our study is to illustrate how these ideas are likely to evolve for students “when they are provided appropriate curriculum tasks, instruction, and opportunities for discourse” (Confrey 2012, p. 4).