Multiple representations and rate of change: the nature of diverse Students' Initial Understanding.

Abstract

Access to quality mathematics content and instruction has been equated as a civil right (Moses & Cobb, 2001), as it is a gatekeeper to higher education and lucrative careers. Unfortunately, access to empowering mathematics is not a reality for many urban youth. Data show an achievement gap between Black, Hispanic and American Indian students and their peers which, in turn, result in unequal access to education and economic opportunities (NAEP, 2007; U.S. Department of Education, 1997). This evaluative case study focused on eight racially diverse high school students from an urban charter school who were studying Algebra 1 in classroom contexts that were student-centered and discourse-based. The study took place after a sequence of six lessons that introduced representations of linear functions through geometric patterns. Pairs of students participated in a series of interview tasks, largely modeled from problems in their curriculum Algebra Connections (Dietiker, Kysh, Sallee, Hoey, 2005), that focused on their understanding of representations of functions (tables, graphs, equations and context), the embodiment of rate of change in the representations and translating among the representations. The study sought to understand the nature of the students’ initial understandings of representations and translations including their natural language when describing observations, the perspectives from which they approached the tasks (explicit or recursive) and how their thinking compares to literature in the field. The students were able to translate linear functions directly between any two representations and many were able to extend their thinking to novel problems involving non-linear functions.