Abstract
The present study concerns preschoolers’ understanding of the middle concept as it applies to numerical sequences. Previous research using implicit psychophysical assessment suggests that the numerical midpoint is embedded within numerical representations by 4 years of age. Here, we examined 3- to 5-year-olds’ ability to identify the midpoint value in triplets of non-symbolic numbers when explicitly probed to do so. We found that whereas 4- and 5-year-olds were capable of explicit access to numerical midpoint values and showed ratio-dependent performance, a signature of the approximate number system (ANS), 3-year-olds performed at chance. Children’s difficulty in identifying numerical midpoint values was not due to comparing multiple arrays, nor was it entirely due to a spatial association with the word “middle” used in the task. We speculate that explicit access to numerical midpoint values may be jointly supported by endogenous control of attentional mechanisms and the development of a mental number line.
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Notes
Sera and Smith (1987) found that even 4-year-olds have difficulty with the word “medium.” They argued that this difficulty stems from the low word frequency of “medium” in the English language. We thus opted to use “middle” instead of “medium” in our instructions, given its greater relative frequency in English (Davies and Gardner 2010).
Two 3-year-olds and two 4-year-olds failed the lowest number (i.e., 2). They were coded as “1” in this analysis. The same coding scheme was used in the subsequent experiments.
We acknowledge that not all non-numerical cues were controlled in Experiment 2. Indeed, as is the case in other studies, it is difficult to control for all such cues on a given task. Nevertheless, if children had relied on non-numerical properties on our task, then there should have been a significant difference between accuracy in Experiment 1 and 2 (because one of the non-numerical properties, cumulative area, was incongruent with number in Experiment 2). This was not the case, however. Moreover, if children relied on cumulative area (CA) information for identifying numerical midpoints, then their performance should have been at chance on CA-controlled trials. Comparisons to chance, however, revealed that 4-year-olds performed significantly above chance on these trials in this experiment: CA-controlled trials: M = 47.78 %, SD = 26.27 %, t(19) = 2.46, p < .05, d = 0.55; and in Experiment 4:
condition, CA-controlled trials: M = 45 %, SD = 23.77 %, t(19) = 2.2, p < .05, d = 0.49 
condition CA-controlled trials: M = 50 %, SD = 29.83 %, t(19) = 2.5, p < .05, d = 0.56.This child was in the
condition. Including her in the analyses, however, did not affect the pattern of results.Readers may ask whether participants performed better on trials in which numerical values were oriented from left-to-right (i.e., trials with the smaller number on the left, middle number in the middle, and the largest number on the right, S–M–L) compared to the reverse arrangement (i.e., trials with the larger number on the left, middle number in the middle, and the smallest number on the right, L–M–S) and, in particular, whether 3-year-olds’ failure on the NMT was due to poor performance on trials with the reverse arrangement (L–M–S). An analysis of 3-year-olds performance on these trials revealed that they performed comparably to when the trials were oriented as S–M–L (Experiment 1: M S–M–L = 47.5 %, SD S–M–L = 44.35 %, M L–M–S = 52.5 %, SDL–M–S = 37.08 %, t(19) = −0.47, p > .6). Note that trials with the middle numerical value on the left or right were not included in these analyses. In the interest of full disclosure, we should also note that there were no differences across S–M–L and L–M–S trials for the 4- or 5-year-olds in Experiment 1 (ps > .8). However, there was a significant difference in Experiment 2, but in the opposite direction than might be expected; that is, 4-year-olds performed worse on S–M–L than L–M–S trials, M S–M–L = 42.5 %, SD S–M–L = 43.75 %, M L–M–S = 67.5 %, SD L–M–S = 34.51 %, t(19) = −2.48, p < .05, d = 0.55. This effect is difficult to interpret, though, because of the limited number of trials (S–M–L trials: 2; L–M–S trials: 4) and a confound with ratio. L–M–S trials consisted of easier ratios (3 trials with the 4:1 ratio and 1 trial with the 2:1 ratio) than S–M–L trials (1 trial with the 3:1 ratio and 1 trial with the 2:1 ratio). These experiments were not designed to test for differences between S–M–L and L–M–S arrangements in numerical midpoint identification and thus were not fully crossed with ratio.
condition, CA-controlled trials: M = 45 %, SD = 23.77 %, t(19) = 2.2, p < .05, d = 0.49 
condition CA-controlled trials: M = 50 %, SD = 29.83 %, t(19) = 2.5, p < .05, d = 0.56.References
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Acknowledgments
We would like to thank Edmund Fernandez, Vladislav Ayzenberg, and members of the Spatial Cognition Laboratory for their help with data collection. This research was supported by a Scholars Award from the John Merck Fund to Stella F. Lourenco.
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Cheung, CN., Lourenco, S.F. Representations of numerical sequences and the concept of middle in preschoolers. Cogn Process 16, 255–268 (2015). https://doi.org/10.1007/s10339-015-0654-4
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DOI: https://doi.org/10.1007/s10339-015-0654-4