Date of Award
Doctor of Philosophy (PhD)
Educational Theory, Policy, and Practice
The rational number domain is known for the great difficulties it presents for students and teachers. As students construct knowledge of fractions, they search for connections to existing knowledge. Procedures from the whole number domain are tried, but many do not work on the new numbers! Recent research has focused on the unit as a way to link whole number and rational number understanding (Behr, Harel, Post & Lesh, 1992). Studies have suggested that students intuitively form units (Lamon, 1994), and that this intuitive knowledge can serve as a foundation for rational number understanding (Lamon, 1994; Mack, 1990). Golding (1994) found that the unit concept can link whole number and rational number domains for addition and subtraction. This study examined the role of the unit as a link between whole and rational number domains for multiplication and division. Further it explored whether students' learning could carry over from the group setting to individual performance, and whether their new understandings could be applied to standard school tasks. This study describes the evolving cognitive processes of four seventh-grade students of varying mathematical ability selected from a seventh-grade class of a rural K-12 school. A fifteen lesson teaching experiment was designed to build on students' existing knowledge of unit and extend this to the rational number domain. Data were collected through videotapes, audiotapes, researcher journal, students' written work, and individual student interviews. Several conclusions were drawn: (a) students developed a flexible concept of unit; (b) modeling provided continuity between conceptual domains; (c) equipartitioning remained a persistent difficulty; (d) sustained focus on the measuring unit is difficult; (e) faulty selection and use of measurement units handicap development of models; (f) unitizing skills endure and are extendable; and (g) models can inform procedural methods and/or provide alternative solution methods. The study points to the need for more school practice in partitioning and measurement activities and more extensive use of modeling to facilitate development of unit concepts. Future research should investigate strategies for enabling students to overcome constraints of primitive models of division.
Alexander, Nancy Sutton, "The Role of the Unit as a Cognitive Bridge Between Additive and Multiplicative Structures." (1997). LSU Historical Dissertations and Theses. 6456.