The role of fingers in the development of numerical cognition is one of the most controversial in the didactics of elementary mathematics (e.g., Fischer, 2015). Therefore, each time a new demonstration—generally a demonstration of its importance and of its positive consequence (due to a well-known publication bias)—is reported in the literature, this demonstration should be examined scrupulously (e.g., Gracia-Bafalluy and Noël 2008; discussed in Fischer 2010). The recent paper by Newman (2016) presents a demonstration that finger gnosia impacts addition performance in an older group of children (N = 42; M age = 10.2 ± 1.0) but not in a younger group (N = 34; M age = 6.7 ± 1.1). What are the data?
In the full dataset, the correlation between finger gnosia and addition performance was significant before controlling for age (r = 0.36), but only trending (r = 0.19, p = 0.099) after controlling for age. This suggests a possible differential effect of age, but not demonstrated it. The author therefore notes that the two factors—finger gnosia and addition performance—were correlated in the older group (r = 0.32, p = 0.04) but not in the younger group (r = 0.17, p = 0.35). Furthermore, the author seems to reinforce this result by exploring interaction with a regression analysis qualified as atypical (p. 142): Finger gnosia significantly predicts addition performance for the older group [F(1,40) = 4.41, p < 0.05; accounted for 10 % of the variance], but not for the younger group [F < 1; accounted for 3 % of the variance]. Unfortunately, this regression analysis is exactly the same as the preceding correlation analysis if one knows the mathematical relationship between r(x), t(x), F(1, x), and explained variance: if the correlations r = 0.32 and r = 0.17 are the rounding of r = 0.315 and r = 0.165, respectively, r(40) leads to t(40) = 2.10, F(1,40) = [t(40)]2 = 4.41, and [r(40)]2 = 10 % in the older group, and r(32) leads to t(32) = 0.95, F(1,32) = [t(32)]2 = 0.90, and [r(32)]2 = 3 % in the younger group.
Given the comments included in the summary of the data just above, the main “demonstration” of the differential effect of age in the two groups lies in the significant correlation in the older group (p = 0.04) contrasted with the nonsignificant correlation (p = 0.35) in the younger group. Given this latter result, a common error is to think that because one correlation is significant and the other not, the difference between the two correlations is significant. However, to “demonstrate” the differential effect of age, one must test the significance of the difference between the two correlations. This was not done in the paper by Newman. And if it is done the difference is far from reaching significance, with a test for equality of two correlation coefficients from two independent populations (see Marascuilo and Serlin 1988; or Howell 1998): ΙzΙ = 0.665, p > 0.50, with a bilateral test.
This error is appealing in the case of a substantial difference between two correlation coefficients. However, even a substantial difference does not ensure its statistical significance. This is especially true when the samples are small (which is often the case in psychology). For example, most people would agree that a low correlation of 0.18 in one group of ten participants differs considerably from a high correlation of 0.84 in another group of ten participants also. But this difference is not significant at the two-tailed usual level 0.05: ΙzΙ = 1.944, p = 0.052. With Newman’s sample sizes, and r = 0.17 in the younger group, the correlation in the older group should be greater than 0.56 to be significant.
The conclusion that “Fingers sense impacted the performance in the older group but not in the younger group” (Newman, p. 144) is correct, strictly speaking. However, the relevance of the following considerations that try to explain the origin and the mechanism of this delayed effect of finger sense depends on the scientific demonstration that there is a difference between the correlation in the younger and the older children. In the absence of this demonstration, the suggested explanations have a non-negligible probability to explain a non-existing phenomenon.
In this short comment, my general message was that “the difference between significant and not significant is not itself necessarily significant” (Nieuwenhuis et al. 2011, p. 1107). The unawareness of this fact leads to a very common statistical error, as that was demonstrated for the neuroscience literature by Nieuwenhuis et al. This error plays an important role in the conclusion of Newman’s paper. However, other results in the paper are not concerned by this criticism. Notably the results of an initial stepwise multiple regression analysis support an interesting model in which age and general cognitive ability (including phonological processing) predict timed addition performance. In the present discussion of the finger role, it is noteworthy that finger gnosia was not a significant predictor of the addition performance in this model tested with the whole sample (younger and older groups) of children.
References
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Handling editor: Marta Olivetti Belardinelli (Sapienza University of Rome).
Reviewers: Catherine Thevenot (University of Lausanne), Marco Fabbri (University of Naples 2).
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Fischer, JP. Does finger sense really have a delayed effect on children’s addition performance?. Cogn Process 18, 105–106 (2017). https://doi.org/10.1007/s10339-016-0782-5
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DOI: https://doi.org/10.1007/s10339-016-0782-5