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Open Access Publications from the University of California

Supporting Generative Thinking about Number Lines, the Cartesian Plane, and Graphs of Linear Functions

  • Author(s): Earnest, Darrell Steven
  • Advisor(s): Saxe, Geoffrey
  • et al.

This dissertation explores fifth and eighth grade students' interpretations of three kinds of mathematical representations: number lines, the Cartesian plane, and graphs of linear functions. Two studies were conducted.

In Study 1, I administered the paper-and-pencil Linear Representations Assessment (LRA) to examine students' understanding of the three representations. The LRA had an experimental component that compared performance on routine problems to non-routine problems (problems not amenable to routine solution procedures). I administered the assessment to Grade 5 students (n=126) who had no formal instruction involving function graphs, and I compared their performances with those of Grade 8 students (n=131) enrolled in Algebra 1. A repeated measures ANOVA revealed students in each grade performed better on routine problems compared to non-routine problems, suggesting that routine problems may falsely indicate greater competence. Paired samples t-tests indicated no differences in performance between Grades 5 and 8 students on number line items, though Grade 8 students outperformed fifth graders on Cartesian plane and function graph items. Videotaped interviews with a subset of Grades 5 and 8 students revealed that students in each grade approached tasks across representations in similar ways, suggesting persisting misconceptions. Interviews also revealed patterns unique to each grade.

In Study 2, I examined the efficacy of a tutorial intervention. The intervention introduced written definitions to support principled understandings of the number line, the Cartesian plane, and function graphs. A repeated measures ANOVA that compared pre/posttest scores of Grade 5 students (n=20) to a matched control group (n=20) revealed significant gains from pre- to posttest in the experimental group, with no detectable gains in a control. At posttest, Grade 5 tutorial students performed significantly better on non-routine LRA problems than Grade 8 students who did not receive the tutorial. Video analysis revealed a correlation between tutorial students' appropriate uptake of definitions and gains from pretest to posttest.

Analyses across the two studies indicate that instruction that supports students' coordination of linear and numerical units can support students' learning with understanding. Potential applications include the development of curricula to support students' learning with understanding related to these representations and teacher professional development interventions.

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