From Number Lines to Graphs in the Coordinate Plane: Investigating Problem Solving Across Mathematical Representations

Abstract

This article reports on students’ problem-solving approaches across three representations—number lines, coordinate planes, and function graphs—the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a hierarchical representational narrative (HRN), a discursive narrative around a set of representations that model conventional mathematics in structurally consistent ways. A paper-and-pencil assessment was administered to students in grades 5 and 8 along with videotaped interviews with a subset of students. Results revealed students’ application of particular meta-rules, which reflect their attempts to find and make use of recurring patterns in mathematics discourse. One such meta-rule, consistent with the HRN, was characterized by students’ coordination of geometric and numeric properties of an axis, whereas alternate meta-rules reflected coordinations inconsistent with conventional mathematics. Detailed analyses of problem-solving strategies are reported, and implications for theory, curriculum, and instruction are discussed.

Notes

Note that the narrative of coordinating linear and numeric units in this way is one endorsed narrative of the discipline, yet other narratives are also endorsed in which intervals take on a different mathematical meaning, such as logarithmic number lines (in which equal intervals represent exponential increases in the underlying quantity for a given base) or bar graphs (in which intervals may represent discrete rather than continuous quantity).

A common exception to this are graphs without any values at all that are used in instruction to highlight observable trends in the shape of a graph. I do not consider this type of graph in the present study.

The classroom is one important context of structured activity that supports the development of mathematical concepts, although mathematics may also be learned through out-of-school contexts, including (but not limited to) store exchanges (Brenner, 1998; Saxe & Esmonde, 2005; Taylor, 2009), street vending (Carraher, Carraher, & Schliemann, 1985; Saxe, 1988; Schliemann, Araujo, Cassundé, Macedo, & Nicéas, 1998; Sitabkhan, 2009), religious practices (Taylor, 2013), and sports (Nasir, 2000; Nasir & Hand, 2008).

All school names are pseudonyms.

One grade 5 interviewee that had accurately written point 5, 5 as a response did not provide a rationale during the interview and received a code of “Other.”