Grounding the Symbols for Place Value: Evidence From Training and Long-Term Exposure to Base-10 Models

Abstract

Two experiments examined whether concrete models support place value learning. In Experiment 1 (N = 149), 7-year-olds were trained with either a) symbols alone or b) symbols and base-10 blocks. Children in both groups showed significant growth overall, but there were specific effects favoring one training type over another. Symbols-only training led to higher scores on a number line estimation task and was particularly effective among high-ability students, whereas blocks training led to better understanding of base-10 structure and was particularly effective among low-ability learners. In Experiment 2 (N = 68), Montessori students, for whom concrete models play a major role in mathematics instruction, also demonstrated better understanding of base-10 structure than did their matched peers enrolled in mainstream elementary schools.

ACKNOWLEDGMENTS

We are grateful to all of the children and school personnel who participated in this study, including those from the Holt Public Schools, Okemos Public Schools, Howell Public Schools, Fowlerville Public Schools, Leslie Public Schools, Montessori Children’s House of Lansing, Stepping Stones Montessori School of East Lansing, Montessori Radmoor School of Okemos, St. Thomas Elementary School, and community programs in the cities of Lansing, Mason, and Grand Ledge, MI.

FUNDING

This research was supported by a generous grant to the first and second authors from the Institute of Education Sciences (#R305A080287). The opinions and positions expressed in this article are the authors' and do not necessarily represent the opinions and positions of the Institute of Education Sciences or the U.S. Department of Education.

Notes

¹ We combined the data from two training conditions in this experiment. Children in one condition received more numeral interpretation items than children in the other condition. See the Appendix for details.

² The conditions were not represented equally in these ability groups because the median pretest score for the symbols-only group (median = 50%) was higher than that for the blocks group (median = 41%) and no training (median = 31%). This resulted in relatively more symbols-only children in the high-ability group versus the other conditions and relatively fewer symbols-only children in the low-ability group (symbols only, high ability, n = 30, low ability, n = 19; blocks, high ability, n = 26, low ability, n = 26; no training, high ability, n = 21, low ability, n = 27). Because such an imbalance might lead to spurious effects due to increased power in one ability group and decreased power in the other, we repeated the analyses after shifting the 5 symbols-only children who performed directly on the median (44%) from the high- to the low-ability group. This resulted in even ability group sizes (high ability, n = 25; low ability, n = 24) and yielded the same pattern of findings reported here.

³ Because scores on the school sale problem have a narrow range (0–2), we repeated the reported analyses using nonparametric ANOVAs (i.e., Kruskal-Wallis tests). One disadvantage to this approach is that it does not permit covariates, such as the pretest scores in Experiment 1 or the Peabody Picture Vocabulary Test-4 scores in Experiment 2, to be included. Still, the results of these analyses were the same as those reported here and in Experiment 2 using ANCOVAs.

Additional information

Funding

This research was supported by a generous grant to the first and second authors from the Institute of Education Sciences (#R305A080287). The opinions and positions expressed in this article are the authors' and do not necessarily represent the opinions and positions of the Institute of Education Sciences or the U.S. Department of Education.