There was a significant main effect of task, such that the standardised performance score in the physical size task was significantly higher than in the mathematical value task, F (1,18) = 184.124, p < .001 (Figure 2e). The lower score for incongruent compared with congruent and neutral trials (-.395 vs. .217 vs. .105;
respectively) led to a significant main effect of congruity, F (2,36) = 7.480, p < .01.
Moreover, the main effect of distance was also significant, such that the standardised performance score for close distance (-.542) was lower than that for longer distance (.494), F (1,18) = 62.268, p < .001. There was a significant three-way interaction effect between task, congruity, and distance, F (2,36) = 4.280, p < .05. The distance effect was only absent for the congruent condition in the mathematical value task. Furthermore, the congruity effect was significant for close distance rather than far distance in the physical size task. Notably, in the mathematical value task, the score difference between the far distance & neutral condition and far distance & incongruent condition was marginally significant, p = .084.
Discussion
In the present experiment we examined the role of mathematical values of C/Ms using the number-size comparison task. In terms of accuracy, first, we observed that participants responded more accurately when comparing the physical size of the C/M phrases than the mathematical values of C/M phrases. Moreover, as predicted, we found that participants answered more accurately when making a judgment between stimuli with farther distance (i.e. either mathematical value or physical size)
regardless of the tasks. Furthermore, incongruent trials led to the lowest accuracy rate than neutral and congruent trials in both tasks. This reflected that the mathematical values of C/Ms and physical size interfered with the other. Last but not least, distance and congruity showed an interaction effect. Participants were more easily affected by the information from the irrelevant dimension (size/value) when comparing the stimuli with closer value/size distance. To be more specific, when participants make a judgment between a pair with small numerical/physical distance, accuracy of
congruent pairs was facilitated while that of incongruent trials was hindered. On the other hand, congruity did not have such an impact on stimuli with far value/size distance.
Not surprisingly, participants spent longer time comparing the mathematical values of C/Ms than physical size. In addition, similar to the accuracy rate, distance effect of RT was observed. Although the distance effect was more profound in the physical size task, mathematical value task also revealed the same trend that the closer the distance, the longer the RT. However, congruity did not have a significant
influence on RT. Notably, the mean RT in the mathematical value task was relatively long (1344.57 ms). As literature suggested, the congruity effect occurs due to
automatic processing of irrelevant information (26, 27). It was possible that the long RT in the mathematical value task provide more time to inhibit the response to
irrelevant information. Though not significant, physical size task exhibited the pattern of facilitation for congruent trials (shorter RT than neutral trials) and interference for incongruent trials (longer RT than neutral trials) when comparing stimuli with close physical size. This suggests that the mathematical value of C/Ms may have some interference effect on RT of physical size. Nonetheless, it can be seen that the mean RT for far distance in the physical size task was extremely short and almost the same among the three conditions of congruity. This may indicate that comparing physical sizes with far distance was too easy to respond to before the irrelevant information (i.e., mathematical value of C/Ms) could interfere.
Since accuracy and RT did not display the same pattern, we also looked into the standardised performance score, which is an index of overall performance. The results were very similar to the findings of accuracy. Firstly, participants performed better in the physical size task compared to the mathematical task. Second, except for the congruent trials in the mathematical value task, distance effect emerged in all other conditions. This was probably because that the consistent physical size facilitated the performance for comparing C/Ms with close distance in the mathematical value task, making it as comparatively easy as comparing C/Ms with far distance and resulting in non-significant difference between them. However, marginal interference effect (p
= .084) was observed in comparing C/Ms with far distance between the neutral and incongruent conditions. As found in RT results, congruity effect was not as apparent in the mathematical value task as in the physical size task. One possible reason may lie on the limitation that there was a fundamental difference between semantic processing and perceptual processing. Moreover, although previous studies showed that numerical value interfered with physical size (13, 23-24), their stimuli were digits instead of Chinese C/M phrases, which indeed had higher processing requirements and even longer RT. Since congruity effect was manifested when participants could not ignore the irrelevant information, longer RT in the mathematical value task may instead be beneficial to participants for having more time to inhibit the influence by the irrelevant information. This may explain why interference effect from the physical
size was alleviated in the mathematical value task. On the other hand, the congruity effect was remarkable when comparing physical sizes with close distance. This demonstrated that the mathematical value of C/Ms impeded the process of comparing similar physical sizes.
Taken together, if C/Ms denote mathematical values and form a multiplicative relation with Num, we should be able to observe typical features of magnitude
processing, i.e., distance effect (12, 16-20) and congruity effect (13, 23-24). Certainly, participants performed better at comparing the two distant stimuli than the proximate ones. This was consistent with the view that mental representation of adjacent
numbers overlap to some degree. It is thus more difficult to distinguish them than remote numbers (15). Moreover, participants' performance was affected by the irrelevant information from the other dimension, suggesting that the mathematical values of C/Ms and physical size interfere with each other mutually. This was in line with Walsh (22), who suggested that there is a common coding system of numerical and physical dimensions. Moreover, accuracy rates showed interaction between congruity and distance, such that the congruity effect was more pronounced when participants compared stimuli with closer distance. To be more specific, when the task on hand was at a higher difficulty level, influence from the irrelevant dimension was crucial. Performance of congruent trials may be facilitated whereas that of
incongruent trials could be worsened.
To summarise, these findings supported Her's (6) theory, which indicated that C and M converge as the multiplicand –– with Num as the multiplier –– and diverge with different mathematical values, i.e., C = 1, M ≠ 1. If C/Ms did not denote mathematical values, characteristics of number processing would not have been observed in the current study. Moreover, if the relation between Num and C/M was not multiplication, we should not have found the distance effect because the distance between the stimuli was manipulated as Her's (6) theory provided, which is [Num X N] = [[Num × X] N].
In conclusion, this study contributed to the literature by offering empirical evidence of quantity processing of C/Ms. While Cui et al. (8) reported that the neural correlates of processing numeral classifiers was similar to that of tool nouns instead of that of numbers and dot arrays using the semantic distance comparison task, Her et al.
(7, 9) followed the same paradigm but showed different behavioural and
neuroimaging results, which suggest that C/Ms represent quantity. Moreover, our results also showed that the mathematical value of C/M and physical size interfere with each other, suggesting that these two dimensions may share common cognitive mechanisms (22). This was consistent with the neural evidence that processing numeral classifiers, numbers, dot arrays, and number words elicited conjunct
activations in the IPS (9). Last but not least, results from the current experiment not only demonstrated with a different task that C/Ms denote mathematical values (C = 1, M ≠ 1) but also verified that the relation between Num and C/M is multiplication.
Because previous studies only used 1 as Num in C/M phrases (7-9), the relation between Num and C/M remained unknown. However, our experiment found the distance effect and congruity effect with a range of combinations of Num and C/Ms as stimuli, verifying that Num and C/M form a multiplicative relation (6). In sum, the linguistic system of C/M interacts with magnitude cognition.
Acknowledgments
We thank Brian Butterworth and Carlo Semanza for helpful comments on experimental design.
Funding
This work was supported by the Ministry of Science and Technology (MOST 103-2410-H-004-136-MY3).
Author contributions
O.-S.H. and N.-S.Y. conceived the study. Y.-C.C., O.-S.H., and N.-S.Y. designed the study. Y.-C.C. developed stimuli, collected and analyzed the data. Y.-C. C., N.-S.Y., and O.-S.H., interpreted the data. Y.-C.C. and O.-S.H. wrote the paper.