The Effects of Physical Manipulatives on Children's Numerical Strategies

ABSTRACT

This article focuses on how the representational properties of manipulatives affect the strategies children employ in problem solving. Two studies examined the effect of physical materials on 4–7-year-old children's problem solving strategies in a numerical (i.e., additive composition) task. The first study showed how children not only identified more solutions using physical materials compared with no materials, but that using manipulatives fostered conceptually more developed strategies: relating consecutive solutions to each other systematically in exploring the space of permutations. The second study demonstrated the unique benefits of physical manipulation by comparing children's solutions and strategies using materials they could or could not spatially manipulate (i.e., physical versus pictorial). As with the first study, children using the physical materials had more solutions and showed more conceptually developed strategies compared with the children in the pictorial condition. There was no advantage in providing children with a record of all their solutions.

Acknowledgments

We thank all the children who have taken part in these studies. We also give our thanks to reviewers for their insightful comments and support that has significantly developed this article and the thinking of the authors.

Funding

This research was supported by grants from the UK Economic and Social Research Council (ES/K001531/1; PTA-026-27-2642).

Notes

1. Although, as Uttal et al. (1997) argue, children's prior experiences may lead them not to interpret physical materials as numerical representations, citing the example of using blocks to recreate notation in an addition problem.

2. Strategy 2 is arguably more developed than Strategy 1 but is considered comparable, mainly for the practical difficulty of identifying when children are estimating as small quantities can be enumerated visually (i.e., subitized).

3. The blocks therefore provided a record of the last solution created. It was decided not to recollect blocks as this was considered more ecologically valid, and this was considered an important representational affordance of manipulatives not to design out of the study.

4. This was not counterbalanced due to sample size. We recognize this limitation, although counterbalanced conditions should have mitigated possible effects.

5. Providing this amount likely offloaded the demands of using physical materials. However, the focus of the study was on if and how materials influenced children's strategies, where offloading demands might arguably encourage a less efficient strategy. It is interesting to reflect upon how various manipulatives offer this representational benefit (e.g., bead string or abacus when partitioning 10).

6. In all studies carried out on this problem, scores were categorized to compare performance on problems of different total amount accordingly: no solutions, single solution, more than one but less than half total solutions, and more than half solutions. It was found that analyses using these coded scores revealed differences in the same direction and magnitude. Therefore, the analyses reported henceforth just examined the absolute number of correct solutions.

7. Identifying 3 + 4 following 4 + 3 (or visa versa) falls under commutative and compensation; yet it is difficult to know which strategy children are employing. Solutions of this pattern were coded as compensation for several reasons: (a) this was suggested by the patterns of solutions (e.g., preceded by and/or followed by a compensation solution); (b) observations in the physical condition (e.g., moving one block not swapping groups of blocks as associated with other commutative solutions; and (c) we had no reason to believe that any coding errors would not be as likely to occur in either condition.

8. Considering the median scores of zero in the no materials condition, another way to approach analysis would have been to categorize scores according to whether children identified at least one solution or not, and then carry out paired sampled tests on the binomial distributions. However, Wilcoxon tests will be reported in this article as significance levels for differences between conditions were unchanged, and this acknowledges the interval data for the majority of children in one of the within subjects conditions.

9. As described, the total possible number of compensation and other solutions is greater than the total number of commutative solutions.

10. We use the term record to describe the representational feature enabling children to refer back to previous solutions during action. However, we recognize that manipulatives do provide a record of the last solution before subsequent action, and later discuss this study design limitation.

11. Unlike Study 1, this meant children began with an odd number to partition; however, Study 1 found that children partitioned equally for odd and even numbers. Moreover, children in Study 2 received the same problem order in all four conditions.

12. National Library of Virtual Manipulatives is available at http://nlvm.usu.edu.