Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

Abstract

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.

Keywords:

• Numerical cognition
• Fractions
• Magnitude comparison
• Distance effect
• Congruity effect

Acknowledgments

This study was supported by a grant from Actions de Recherche Concertées 05/10–327 of the French-Speaking Community of Belgium. G.M. and M.P.N are supported by the Fund for Scientific Research of the French-Speaking Community of Belgium (FRS-FNRS).

Notes

¹ The magnitude of the common numerators in x/a_x/b fractions was smaller than the magnitude of the common denominators in a/x_b/x fractions. It was due to the use of irreducible fractions and therefore of fractions whose numerical value is smaller than 1 (i.e., with the denominator larger than the numerator). Control of simplification took priority over control of the size of the common components as the simplification could have introduced between- and within-participant variability.

² A given pair of fractions was not presented in the two left–right orders, and the pairs with a left response were not matched to the pairs with a right response. Therefore, we did not test for the SNARC effect.

³ The distance uncorrelated with RTs and error rates (i.e., the overall distance for a/x_b/x fractions and the distance between the denominators for x/a_x/b fractions) was identified as a suppressor variable by the method suggested by Tzelgov and Henik (1991). A suppressor variable is a variable that clears out residual variance from the other predictor in multiple regression. Therefore, when both distances were simultaneously entered in a regression, the coefficient of the valid distance was inflated, and the coefficient of the suppressor distance changed sign (i.e., became positive). However, we did not report multiple regressions as the suppressor distance never contributed to the improvement of the prediction (all p > .10 for R ² change) and as its coefficient was never significant (all ps > .10).

⁴ The same analyses were also run on the medians of RTs to the pairs associated with a correct response and preceded by a correct response to the prime in all four priming conditions to maintain a strict matching between conditions. This analysis led to the same significant effects in Experiment 1 and in Experiment 2. These results, however, were not reported because of the heavy loss of data for some participants. Moreover, this trimming could have induced an imbalance in the control of the lateralization of the motor response to the probe relative to the motor response to the prime, as a given trial could have the same response as the prime in a condition and a different response in another condition (this control was made within each priming condition).

⁵ The distance effect was tested on natural numbers to check that their magnitude was processed in all priming conditions. The distance effect was significant in the specific priming by x/a_x/b fractions, β = –.60, t(30) = –4.06, p < .01; in the unspecific priming by x/a_x/b fractions, β = –.49, t(30) = –3.12, p < .01; and in the unspecific priming by a/x_b/x fractions, β = –.51, t(30) = –3.28, p < .01; and it tended to be significant in specific priming by a/x_b/x fractions, β = –.34, t(30) = –1.98, p = .06. The correlations between RTs and distance did not differ significantly between the priming conditions with the t test suggested by Williams (1959, cited in Howell, 1997/1998; all ps > .10). These distance effects suggested that the magnitude was processed in all priming conditions.

⁶ See Footnote 4.

⁷ The distance effect was tested on natural numbers in order to exclude whether the facilitation effect was due to the episodic retrieval of the response activated during the processing of the previous fractions rather than due to the residual activation from the processing of the components during the fraction comparison. The distance effect on RTs was significant in all priming conditions: in the specific priming by a/x_b/x fractions, β = –.42, t(30) = –2.57, p = .02; in the unspecific priming by a/x_b/x fractions, β = –.43, t(30) = –2.64, p = .01; in the unspecific priming by x/a_x/b fractions, β = –.36, t(30) = –2.10, p = .04; and in the specific priming by x/a_x/b fractions, β = –.40, t(30) = –2.42, p = .02. The correlations between the RTs and the distance did not differ between the priming conditions (all ps > .10).