The concept of function is one of the most important ideas in the learning of mathematics (Dubinsky & Harel, 1992). Yet it is considered by many researchers to be the least understood by high-school and college students (Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Sfard, 1992). Reforming early mathematics curricula in algebra, therefore, is justified. To this end, the National Council of Teachers of Mathematics (2000) called for a longitudinal view of algebra, from elementary to advanced mathematics education. As a strand in the Rutgers-Kenilworth longitudinal study in 1993, Robert B. Davis introduced early algebra ideas to eleven-year-old students during the sixth grade. Research prior to Davis’ intervention with the students showed how they built their understanding of linear, quadratic, and exponential functions (Spang, 2009; Giordano, 2008; Mayansky, 2007). Building on Davis’ approach to early algebra and the learning of function, Emily Dann designed a study to determine whether these students, now seventh graders, could extend their understanding to the concept of inverse function.

The present study analyzes videotaped work of seventh-grade students who were engaged in a series of activities that Dann had devised. The Guess-My-Rule activities, as they were called, were conducted over three consecutive days. Using the model that Powell, Francisco, and Maher (2003) described for analyzing videotaped data, this study examines in detail the students’ work as they collaborated in small groups to develop rules for function and inverse; the study also investigates the obstacles students had experienced.

This research demonstrates that seventh-graders understood the idea of function by writing rules, symbolically, to describe the relationships of quantities. Understanding function as action, they progressed to the process concept when creating their own function tables and corresponding rules. Using inverse operations, students wrote inverse rules; however, due to difficulties with integer and fraction arithmetic, they needed to adjust their initial attempts in order to be successful.

This study maintains that having facility with function and inverse function concepts will permit students to learn the subject matter, to communicate ideas and solutions, and to interconnect mathematical ideas. In the process of exploring these related concepts, students will be encouraged to think independently and to devise original strategies in their work with function and inverse.

The results demonstrate to researchers and educators how students build the concepts of function and of inverse function through group work in a specific environment. Seventh-grade students can engage in activities, similar to those described above, that are essential to the study of algebra.