Features of a pan balance that may support students’ developing understanding of mathematical equivalence

ABSTRACT

Elementary school students struggle in interpreting the equal sign as a symbol denoting equivalence. Although many have advocated using a pan-balance scale to help students develop this understanding, less is known about what features associated with this model support learning. To attempt to control and examine these features, the investigators developed a digital, pan-balance computer applet. This allowed for experimentally manipulating three relatively grounded instructional conditions (involving the core principle of making two sides the same; a balancing analogy; or both, along with a dynamic demonstration), compared to a relatively idealized control condition. Results indicated that the relatively more grounded conditions promoted a relational understanding of mathematical equivalence among 148 second- and third-grade students and further suggest that providing dynamic, grounded support may not be as optimal as less-enriched supports to promote students’ learning.

KEYWORDS:

• Mathematical equivalence
• equal sign
• balance analogy
• digital manipulative

Acknowledgments

The research reported in this article was supported by a grant from the Hardie Dissertation Fellowship program in the College of Education at the University of Illinois at Urbana-Champaign to the first author. We thank Lyn English and the reviewers whose insight helped shape and clarify this article. The authors also appreciate valuable feedback received from Sarah T. Lubienski, Robb Lindgren, Dan Hoffman and the STEM Development Research Group, with special thanks to Shuai Wang for his statistical expertise, on an earlier version of this work. Finally, we are grateful to the students who participated in this project; without them, none of this would have been possible.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1. We use the term dynamic to indicate movement of the pan balance; this is different than how the term is used elsewhere, e.g. in dynamic environments, such as in Geometer’s Sketchpad.

2. We note that we have identified four features that typically surface when using a pan-balance scale in instruction, which we thought were likely to have an effect on student learning, although we acknowledge that there might be other features that can have an effect on student outcomes. For example, we employ objects that look the same and weigh the same employing different weights may have a different (or similar results), as in Barlow and Harmon’s (2012) work.

3. We use the term manipulative broadly. Although some might argue that the term manipulative should be reserved for things that can be handled (either with hands, or virtually, with, e.g. a mouse), following Manches and O’Malley (2012), we use the term digital manipulative because we wanted to be transparent about the parallels we make between the actual pan-balance scale and its digital equivalent.

4. Much of our data were skewed and not distributed normally. In those cases when data violated assumptions for normality (Shapiro-Wilk, p < .001) and the Levene’s test for homogeneity of variances revealed unequal variances (p < .001), we conducted non-parametric tests, including Welch’s ANOVA, McNemar’s test, and Wilcoxon signed-rank test, which are appropriate (and robust) tests in such cases.

5. We also conducted analyses with data treated continuously and obtained comparable results. We report the categorical data because these illustrate the variation in the extent of student learning more clearly.

6. These questions are adapted from Matthews et al. (2012).

7. These questions are adapted from the questions by Rittle-Johnson et al. (2011). A trainer read a prompt and explained the task as described in the text.