Do First Graders Make Efficient Use of External Number Representations? The Case of the Twenty-Frame

Abstract

External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one counting. We compared first-grade children's performance on two types of computerized enumeration tasks, in which between one and 20 dots were presented in random arrangements or on a twenty-frame. The number of dots was a strong predictor of response times and accuracy rates in the enumeration task with random arrangements but not in the twenty-frame task. Performance on the twenty-frame task was correlated with performance on a number and arithmetic test, even when other cognitive variables were statistically controlled. We discuss these findings in the light of theories on utilizing external representations to support numerical learning.

Notes

There are also grids with five (five-frame), 10 (ten-frame), or 100 (hundred-square) items (e.g., Beishuizen, 1993; McGuire, Kinzie, & Berch, 2012). Note that other authors have used the term arithmetic rack for the physical material that is similar in structure to the twenty-frame (e.g., Gravemeijer, 1994; Verschaffel & De Corte, 1996).

We defined the subitizing range a posteriori according to our empirical data, which suggested a subitizing range of 1–4 dots.

Note that high efficiency on the twenty-frame task indicates low response times, so that the expected relation to high performance on the number and arithmetic test is expressed by a negative correlation coefficient.

The time limits for this and all other tasks of this study were adjusted according to pilot studies. For arguments supporting the suitability of these time limits, see the section of this article titled “Do children make use of the twenty-frame structure?”

Wald statistics of the linear estimation of equation model that accounts for repeated measurements within subjects.

As accuracy rates were generally high, the main focus here is on the analysis using response times (rather than accuracy rates) as the dependent variable. For this reason, these analyses are reported first.

For the reasons why accuracy rates were not included in this analysis, see the section of this article titled “The present study.”

Note that as for the twenty-frame task (see footnote 3), the expected correlation to performance on the number and arithmetic test is negative for the response times on computerized tasks.

Additional information

Funding

This work was financially supported by research grant 01 JG 0922 from the German Federal Ministry of Education and Research (BMBF).