Processing multi-digit numbers: a translingual eye-tracking study

Age-related effects in magnitude and place-value processing

While general cognitive skills decline during aging, numerical skills seem to be mainly preserved. Such skills are essential for an independent life up to old age, e.g., when dealing with money or time. Operating with numbers usually requires number magnitude and place-value processing. The question is whether these processes are negatively affected by aging due to the general cognitive decline or positively affected due to lifelong experience with numbers. Therefore, we investigated age-related changes in the distance and compatibility effects in single-digit, two-digit, and four-digit number comparison. On the one hand, older adults took longer for number processing and showed a smaller distance effect, indicating altered number magnitude representations. On the other hand, older adults were better in place-value processing as indicated by a smaller compatibility effect than in younger adults. We conclude that aging differentially affects basic numerical skills.

Children's comparison of different-length numbers: Managing different attributes in multidigit number processing

In everyday life the comparison of numbers usually occurs between numbers with different numbers of digits. However, experimental research here is scarce. Recent research has shown that adults respond faster to congruent pairs (the initial digit in the number with more digits is larger, e.g., 2384 vs. 107) than to incongruent pairs (the initial digit is larger in the number with fewer digits, e.g., 2675 vs. 398). This has been interpreted as support for the processing of multiple attributes in parallel and against serial accounts. The current research asked whether there is a change in the relevance of these attributes as school grades increase. School-age children from the second to sixth grades (N = 206) were presented with pairs of numbers that had either the same number of digits (3 vs. 3 or 4 vs. 4) or a different number of digits (3 vs. 4). In this latter condition, the stimuli, matched by distance, could be either length/digit congruent (e.g., 2384 vs. 107) or length/digit incongruent (e.g., 2675 vs. 398). Linear mixed models showed a length/digit congruity effect from second graders. Interestingly, in the response time measure, congruity interacted with school grade and the side in which the larger number of the pair was presented. Whereas these results support a model that considers number comparison as a process that weighs different attributes in parallel, it is also argued that developmental changes are associated with differences in the level of automatization of the componential skills involved in the comparison.

Place-value and physical size converge in automatic processing of multi-digit numbers

Previous research has shown that multi-digit number processing is modulated by both place-value and physical size of the digits. By pitting place-value against physical size, the present study examined whether one of the attributes had a greater impact on the automatic processing of multi-digit numbers. In three experiments, participants were presented with two-digit number pairs that appeared in frames. They were instructed to select the larger frame while ignoring the numbers within the frames. Importantly, we manipulated the physical size of the digits (i.e., both decade/unit digits were physically larger) within the frames, the unit-decade compatibility (i.e., the relationship between the numerical values of both decade and unit digits was consistent or inconsistent), and the congruity between the numerical values of the decade digits and the frames' physical size (i.e., decade-value-frame-size congruity). In Experiment 1, where all pairs were unit-decade compatible, a decade-value-frame-size congruity effect emerged for pairs with physically larger decade, but not unit, digits. However, when adding unit-decade incompatible pairs (Experiments 2-3), in unit-decade compatible pairs, there was a decade-value-frame-size congruity effect regardless of the digits' physical size. In contrast, in unit-decade incompatible pairs, there was no decade-value-frame-size congruity effect, even when the physically larger digit (i.e., unit) contradicted the place-value information, presumably due to the cancellation of the opposing influences of the digits' physical sizes their place-values. Overall, these findings suggest that place-value and physical size are intertwined in the Hindu-Arabic numerical system and are processed as one.

Reading numbers is harder than reading words: An eye-tracking study

We recorded the eye movements of adults reading aloud short (four digit) and long (eight to 11 digit) Arabic numerals compared to matched-in-length words and pseudowords. We presented each item in isolation, at the center of the screen. Participants read each item aloud at their pace, and then pressed the spacebar to display the next item. Reading accuracy was 99 %. Results showed that adults make 2.5 times more fixations when reading short numerals compared to short words, and up to 7 times more fixations when reading long numerals with respect to long words. Similarly, adults make 3 times more saccades when reading short numerals compared to short words, and up to 9 times more saccades when reading long numerals with respect to long words. Fixation duration and saccade amplitude stay almost the same when reading short numerals with respect to short words. However, fixation duration increases by ∼50 ms when reading long numerals (∼300 ms) with respect to long words (∼250 ms), and saccade amplitude decreases up to 0.83 characters when reading long numerals with respect to long words. The pattern of findings for long numerals-more and shorter saccades as well as more and longer fixations-shows the extent to which reading long Arabic numerals is a cognitively costly task. Within the phonographic writing system, this pattern of eye movements stands for the use of the sublexical print-to-sound correspondence rules. The data highlight that reading large numerals is an unautomatized activity and that Arabic numerals must be converted into their oral form by a step-by-step process even by expert readers.

Length is not all that matters: testing the role of number identity and the ratio of fillers in comparisons of multi-digits with different digit length

Research in multi-digit number comparison usually considers stimuli with the same number of digits (e.g., 3452 vs. 7831). Surprisingly, there is almost no research on the comparison of numbers that differ in length (e.g., 995 vs. 1000), which demands a focus on the number of digits in each multi-digit, despite the fact that the role of number length has been explicitly acknowledged in componential models of multi-digit processing. Our study explores whether the comparison of pairs of natural numbers that differ in length is affected by the identity of the leftmost digit of each multi-digit, and asks what is the effect of having variable proportions of trials with pairs of numbers of the same-length in the task. Across three studies participants compared numbers in blocks with different proportions of same-length multi-digit pairs (Experiment 1 and 2: 25% vs. 50% vs. 75%; Experiment 3: 0% vs. 50%). Stimuli in the different-length condition were length-digit congruent (the number with more digits starting with a larger digit: 2384 vs. 107) or length-digit incongruent (the number with more digits starting with a smaller number: 2675 vs. 398). Response times were shorter in length-digit congruent pairs than in the incongruent pairs. Unexpectedly, this effect was only slightly modulated by the proportion of same-/different-length multi-digit pairs in the experimental set. Despite its perceptual saliency, length is not the only information considered when comparing different-length numbers. The leftmost-digit is also taken into account, with variable relevance here, depending on the characteristics of the stimuli set.

Serial and syntactic processing in the visual analysis of multi-digit numbers

The visual analysis of letter strings and digit strings is done by two separate cognitive processes. Recent studies have hypothesized that these processes are not only separate but also qualitatively different, in that they may encode information specific to numbers or to words. To examine this hypothesis and to shed further light on the visual analysis of numbers, we asked adults to read aloud multi-digit strings presented to them for brief durations. Their performance was better in digits on the number's left side than in digits farther to the right, with better performance in the two outer digits than their neighbors. This indicates the digits were processed serially, from left to right. Visual similarity of digits increased the likelihood of errors, and when a digit migrated to an incorrect position, it was most often to an adjacent location. Interestingly, the positions of 0 and 1 were encoded better than the positions of 2-9, and 2-9 were identified better when they were next to 0 or 1. To accommodate these findings, we propose a detailed model for the visual analysis of digit strings. The model assumes imperfect digit detectors in which a digit's visual information leaks to adjacent locations, and a compensation mechanism that inhibits this leakage. Crucially, the compensating inhibition is stronger for 0 and 1 than for the digits 2-9, presumably because of the importance of 0 and 1 in the number system. This sensitivity to 0 and 1 makes the visual analyzer specifically adapted to numbers, not words, and may be one of the brain's reasons to implement the visual analysis of numbers and words in two separate cognitive processes.

Individual differences influence two-digit number processing, but not their analog magnitude processing: a large-scale online study

Symbolic magnitude comparison is one of the most well-studied cognitive processes in research on numerical cognition. However, while the cognitive mechanisms of symbolic magnitude processing have been intensively studied, previous studies have paid less attention to individual differences influencing symbolic magnitude comparison. Employing a two-digit number comparison task in an online setting, we replicated previous effects, including the distance effect, the unit-decade compatibility effect, and the effect of cognitive control on the adaptation to filler items, in a large-scale study in 452 adults. Additionally, we observed that the most influential individual differences were participants' first language, time spent playing computer games and gender, followed by reported alcohol consumption, age and mathematical ability. Participants who used a first language with a left-to-right reading/writing direction were faster than those who read and wrote in the right-to-left direction. Reported playing time for computer games was correlated with faster reaction times. Female participants showed slower reaction times and a larger unit-decade compatibility effect than male participants. Participants who reported never consuming alcohol showed overall slower response times than others. Older participants were slower, but more accurate. Finally, higher grades in mathematics were associated with faster reaction times. We conclude that typical experiments on numerical cognition that employ a keyboard as an input device can also be run in an online setting. Moreover, while individual differences have no influence on domain-specific magnitude processing-apart from age, which increases the decade distance effect-they generally influence performance on a two-digit number comparison task.