Tracing students' growing understanding of rational numbers

This research, a component of a year long National Science Foundation funded study, traces and documents how rational number ideas are built by students as they move from placing fractions on a line segment (finite concept) to placing fractions on an infinite number line (infinite concept). The evidence is supported by representations used by students to express their ideas, explanations given by students and student justifications about their reasoning. The study was guided by the following research questions: 1. What evidence, if any, is there of the students' understanding of the idea of fraction as number? 2. How do students extend their understanding of fraction ideas to rational numbers? 3. What representations do students use to express their fraction ideas and extend these ideas to rational numbers? The subjects consisted of a heterogeneous class of twenty-five, fourth grade (nine and early ten year old) students. Digitized videos, transcripts, student work, observation notes, and student overhead transparencies comprised the data from extended classroom sessions, videotaped with three cameras. The study gives evidence that the students built understanding of fraction ideas such as equivalence and extended these ideas to negative fractions and improper fractions. It also showed that students successfully ordered fractions on line segments, then number lines, after working out distinctions between operator and number ideas. Student ideas revealed in these sessions showed that they were comfortable and successful with basic fraction operations. Lively classroom discussions and arguments worked out obstacles in the placement of fractions on a number line. Engagement in discussions about fraction ideas and negative fractions extended to rational numbers to include improper fractions as students identified equivalent number names for fractions. In the active student-centered environment the students worked together on tasks and shared their personal representations of rational number ideas and density of the rationals. This study provides detailed evidence that students can build understanding of fraction as number and successfully make connections to extend their understanding of number, generating and interest and understanding of fraction ideas that generally are not made accessible to students of this age.Ed.D.Includes abstractIncludes bibliographical referencesby Suzanna E. Schmeel