# Tasks that promote functional reasoning in early elementary school

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Nancy Tilley Payne (Creator)
Institution
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Sarah Berenson

Abstract: Algebra is often described as the gateway to higher mathematics (Carpenter, Franke, & Levi, 2003; Kaput, 2008; Kaput & Blanton, 2001; Mason, 2008). Unfortunately, many students do not navigate this gateway successfully. Kaput (2008) and Mason (2008) suggested that this is due in part to the abrupt switch from arithmetic to algebra that occurs in late middle school to early high school. Many called for a change in the way mathematics is taught from kindergarten through high school (Blanton, 2008; Kaput, 2008; NCTM, 2000), which included a change to the introduction of early algebra. Carraher, Schliemann and Schwartz (2008) caution that early algebra is different from algebra early. They suggested that this was a switch in the way the basic tenants of algebraic thinking is understood, including generalizing arithmetic and functional thinking (Brizuela & Lara-Roth, 2002; Warren, 2005b). Given this focus on functional reasoning in early elementary school, I conducted a study of students in the third through fifth grades. My purpose was to identify the characteristics of functional reasoning tasks that promoted the development of functional reasoning. I also wanted to discover the ways in which students used these characteristics as they worked on a task. Another important aspect of this study was to find any other influences on the types and frequencies of generalizations the students constructed. My study lasted for 10 weeks during an afterschool enrichment program called On Track Learn Math. The program occurred on two days during the week and students worked on a different task per week across two five-week sessions. This study analyzed 10 of those tasks. The data demonstrated that the tasks contained specific characteristics that promoted the development of functional reasoning. The two most noteworthy considerations were that grade level and the complexity of the function rule affected the frequency with which students constructed explicit rules. The results of this study have implications on student learning, teacher practice, and curricular changes.