Resumen

We describe 24 third (8–9 years old) and 24 fifth (10–11 years old) graders’ generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students work with two or more covarying quantities rather than isolated computations, and focusing on representations can help to identify the type of representations useful to them. The goals of this study are to (1) describe the functional relationships evidenced in students’ responses and (2) describe the representations that the students use. In addressing these research objectives, we describe student responses drawn from a Classroom Teaching Experiment in each grade. We analyzed students’ written responses to different questions designed to generalize the relationships in a problem that involves the function y = 2x + 6. Our findings illustrate that 11 third graders and 19 fifth graders provide evidence of functional relationships in their responses. Three third graders and all fifth graders generalized the relationship. We conclude that these differences may be due to the students’ previous classroom mathematical experiences, since students in higher grades would be more likely to focus on the relationships between variables, whereas third graders would focus on the details of arithmetic computations. In addition, we find that natural language is the main vehicle used to generalize in both grades. Unlike third graders, fifth graders perceive general rules from the numerical calculation and express these generalizations even when not explicitly requested to do so.