The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults

Abstract

This study mainly investigated the specificity of the processing of fraction magnitudes. Adults performed a magnitude-estimation task on fractions, the ratios of collections of dots, and the ratios of surface areas. Their performance on fractions was directly compared with that on nonsymbolic ratios. At odds with the hypothesis that the symbolic notation impedes the processing of the ratio magnitudes, the estimates were less variable and more accurate for fractions than for nonsymbolic ratios. This indicates that the symbolic notation activated a more precise mental representation than did the nonsymbolic ratios. This study also showed, for both fractions and the ratios of dot collections, that the larger the components the less precise the mental representation of the magnitude of the ratio. This effect suggests that the mental representation of the magnitude of the ratio was activated from the mental representation of the magnitude of the components and the processing of their numerical relation (indirect access). Finally, because most previous studies of fractions have used a numerical comparison task, we tested whether the mental representation of magnitude activated in the fraction-estimation task could also underlie performance in the fraction-comparison task. The subjective distance between the fractions to be compared was computed from the mean and the variability of the estimates. This distance was the best predictor of the time taken to compare the fractions, suggesting that the same approximate mental representation of the magnitude was activated in both tasks.

Keywords:

• Fractions
• Nonsymbolic ratios
• Continuous versus discrete magnitudes
• Estimation
• Mental magnitude representation

Acknowledgments

Gaëlle Meert and Marie-Pascale Noël are supported by the Fund for Scientific Research of the French-Speaking Community of Belgium (FRS–FNRS). We would like to thank Pierre Mahau (Institut de Recherche en Sciences Psychologiques (IPSY), Université Catholique de Louvain) for having made the potentiometer, Sandrine Mejias (IPSY, Université Catholique de Louvain) for having had the idea of the response modality, and Jonathan Jaeger (Support en Méthodologie et Calcul Statistique (SMCS), Université Catholique de Louvain) for having done the Bayesian analyses.

Notes

¹LMMs were used because, unlike ordinary least-squares regressions, they allow the dependence between observations due to repeated measures to be taken into account (Van den Noortgate & Onghena, 2006). They can include a random intercept and a random slope for participants, which allow the intercept and the slope to vary between participants. In the present study, these random factors were entered into the model only if they significantly contributed to the variance, which was tested by the likelihood ratio test (Norusis, 2008).

² Given the limited number of fractions employed in the magnitude-estimation task, we had to use one pair with common denominators, one pair with multiple denominators, and one pair with common numerators and with multiple denominators in the magnitude-comparison task. Nevertheless, these pairs were a minority of those used (3 out of 30 pairs).

³Subitizing refers to the fast, accurate, and seemingly effortless enumeration of small sets of items (Kaufman, Lord, Reese, & Volkmann, 1949; see also Mandler & Shebo, 1982; Trick & Pylyshyn, 1994). Subitizing is involved in the enumeration of sets from one to four or five items, with an increase of about 40–100 ms per item in the RT. Subitizing is commonly distinguished from counting, which allows a larger set of items to be enumerated, but which is slow, effortful, and error prone. For counting, RTs increase by about 200–400 ms per item.

⁴An additional ANOVA was run to test the effect of the format and the size of the components on SDs without the rectangle condition. Since this condition was found not to be sensitive at all to the size of the components, it was affecting the interaction and might mask other interesting results for the conditions that were sensitive to the size of the components (i.e., fractions, homogeneous dots, and heterogeneous dots). This additional analysis did not reveal any new interesting results: The main effects and the interaction remained significant (all ps < .01). The interaction was explained by the fact that the effect of the format, which was significant for all sizes of the components (all ps < .05), depended on the size of the components (see the main text for a comparison of the three formats for each size of the components).

⁵Given that the same pairs of fractions were presented four times and that the participants got a feedback on their response, we tested whether the participants improved their performance throughout the task. We compared the mean RT and the percentage of correct responses in the first part of the task (i.e., the first two presentations of the 30 pairs) with those in the second part of the task (i.e., the last two presentations of the 30 pairs). We ran an ANOVA for repeated measures with the part as a within-participant variable. Four participants were excluded from this analysis because their responses to the last two presentations of the pairs were not recorded due to a technical problem. The participants responded more slowly in the first part (M = 1,832, SD = 509) than in the second part (M = 1,552, SD = 410), F(1, 28) = 26.77, p < .01; and they made more errors in the first part (M = 19.5%, SD = 12.0%) than in the second part (M = 13.2%, SD = 8.8%), F(1, 28) = 18.07, p < .01. These results showed that the participants improved their performance during the task. We tested the effect of each distance separately for the two parts. The results that were reported for the whole task were replicated for each part: Only the distance between the fractions predicted the RT in the first part, β = –.51, t(28) = –3.11, p < .01, and in the second part, β = –.74, t(28) = –5.96, p < .01.