Supporting Generative Thinking About the Integer Number Line in Elementary Mathematics

Abstract

This report provides evidence of the influence of a tutorial “communication game” on fifth graders’ generative understanding of the integer number line. Students matched for classroom and pretest score were randomly assigned to a tutorial (n = 19) and control group (n = 19). The tutorial group students played a 13-problem game in which student and tutor each were required to mark the same position on a number line but could not see one another's activities. To resolve discrepant solutions, tutor and student constructed agreements about number line principles and conventions to guide subsequent placements. Pre-/posttest contrasts showed that (a) tutorial students gained more than controls and (b) agreement use predicted gain. Analyses of micro-constructions during play revealed properties of student learning trajectories.

ACKNOWLEDGMENTS

The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant R305B070299 to Geoffrey B. Saxe, University of California, Berkeley and through the Institute of Education Sciences pre-doctoral training grant R305B090026, also to the University of California, Berkeley. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. Thanks are due to Maryl Gearhart and Meghan Shaughnessy for consultation on the project from its inception, to Hyman Bass for comments on an earlier draft of the article, and to Deborah Ball who seeded some of the ideas in this work though the Elementary Math Laboratory at the University of Michigan. We are appreciative of the participation of teachers and students at multiple local elementary schools in Berkeley and Oakland, CA.

Notes

For problems that required use of Cuisenaire Rods (Problem Sets II–VI), we altered the colors (and hence the lengths) of the rods across problem sets. For Problem II the rods were of only one color, and the particular color choice emerged out of a cooperative agreement by the student and tutor. Problem Sets III and V required a red-purple pairing, and Problem Sets IV and VI required a light green-dark green pairing. For each rod pair, there was a 2:1 ratio in lengths between rod color (i.e., 2 reds = 1 purple; 2 light greens = 1 dark green). In Problem Sets VII and VIII, no rods were used.

After the student's spontaneous evaluation of each line, if agreements were not used at all or used inappropriately, the tutor engaged the student with an evaluation of the line in relation to the previously established agreements. The procedure ensured that the evaluation was achieved using agreements appropriately by the conclusion of the session.

One student did not respond on problem 1.

In some cases, students failed to construct the target length in rods, instead using a truncated length to correspond rods with each of the two interval sizes on the line; students then inscribed numerals that treated each tick mark as a count irrespective of interval size.