This research extended the validation of a framework for assessing and describing children's thinking in multidigit number situations and used this framework to generate and evaluate different versions of an instructional program. The key constructs of the framework—counting, partitioning, grouping, and number relationships—appeared to be highly stable within each of the five levels and across the full range of thinking exhibited by 12 case studies. Results suggest that the levels may generate a hierarchy of thinking. Teachers were effective in implementing two versions of the framework-driven instructional program. Any differences could be attributed to the quality of problem-solving experiences, the level of student interactions, and, in essence, to differences in teachers' familiarity with the program.
Graham A. Jones, Vi iting Professor, 4520 Mathematics Department. Illinois State University, Nonnal, lL 61790-4520; e-mail: firstname.lastname@example.org
Carol A. Thornton, Distinguished University Professor. 4520 Mathematics Department, lllinois State University, Nonnal. lL 61790-4520; e-mail: email@example.com
Ian J. Putt, Senior Lecturer, School of Education, James Cook University, Townsville, Queensland, Australia 4814; e-mail: lan.Putt@jcu.edu.au
Kevin M. Hill, Graduate Research Assistant. 4520 Mathematics Department, IUinois State University, Normal, IL 61790-4520
A. Timothy Mogill, Instructor, 4520 Mathematics Department, Jllinois State University, Normal, lL 61790-4520; e-mail: mogill @math.ilstu.edu
Beverly S. Rjch, Assistant Professor, 4520 Mathematics Department. Illinois State University, Normal, lL 6 1790-4520; e-mail: firstname.lastname@example.org
Laura R. Van Zoest, Assistant Professor, Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, Ml49008; e-mail: email@example.com