Unit-decade Compatibility Effect Research Articles

Working memory and two-digit number processing

The processing of two-digit numbers in comparison tasks involves the activation and manipulation of magnitude information to decide which number is larger. The present study explored the role of different working memory (WM) components and skills in the processing of two-digit numbers by examining the unit–decade compatibility effect with Arabic digits and number words. In the study, the unit–decade compatibility effect and different WM components were evaluated. The results indicated that the unit–decade compatibility effect was associated to specific WM skills depending on the number format (Arabic digits and number words). We discussed the implications of these results for the decomposed view of two-digit numbers.

Cognitive control in number processing: Evidence from the unit–decade compatibility effect

In this study we evaluated whether the reverse compatibility effect observed when participants compare two-digit Spanish number words might be modulated as a function of the percentage of filler trials (within-decade comparisons). The participants performed a comparison task with two-digit Spanish number words while the unit–decade compatibility in between decade comparison trials was manipulated. We biased the relevance of units by increasing the intra/inter decade comparison ratio (20%, 50%, and 70%). The compatibility effect shifted from reverse to regular depending on the unit relevance which indicates that the relative emphasis on the unit processing in two-digit number word can be modulated by the stimulus list. These results are discussed in terms of cognitive control in number processing.

Whorf reloaded: Language effects on nonverbal number processing in first grade—A trilingual study

The unit–decade compatibility effect is interpreted to reflect processes of place value integration in two-digit number magnitude comparisons. The current study aimed at elucidating the influence of language properties on the compatibility effect of Arabic two-digit numbers in Austrian, Italian, and Czech first graders. The number word systems of the three countries differ with respect to their correspondence between name and place value systems; the German language is characterized by its inversion of the order of tens and units in number words as compared with digital notations, whereas Italian number words are generally not inverted and there are both forms for Czech number words. Interestingly, the German-speaking children showed the most pronounced compatibility effect with respect to both accuracy and speed. We interpret our results as evidence for a detrimental influence of an intransparent number word system place value processing. The data corroborate a weak Whorfian hypothesis in children, with even nonverbal Arabic number processing seeming to be influenced by linguistic properties in children.

Is there cross-language modulation when bilinguals process number words?

ABSTRACTThis study explores the possibility of cross-language activation when bilinguals process number words in their first language (Italian) and their second language (German). Italian monolinguals (Experiment 1), German monolinguals (Experiment 2), and Italian/German bilinguals (Experiment 3) were required to decide the larger of two number words while the unit–decade compatibility effect was examined. For compatible trials the decade and unit comparisons lead to the same response (e.g., 24–67), whereas for incompatible trials the decade and unit comparisons lead to different responses (e.g., 27–64). The regular unit–decade compatibility effect was significant when bilinguals and monolinguals performed the comparison in German. However, this effect was not found when bilinguals and monolinguals performed the task in Italian. In addition, the decade distance played a major role when bilinguals processed in their first language, whereas the unit distance was more important when they worked in their second language. These results indicate that the processing of number words in one language is not modulated by the way bilinguals process number words in their alternative language.

Two-digit number processing: holistic, decomposed or hybrid? A computational modelling approach

Currently, there are three competing theoretical accounts concerning the nature of two-digit number magnitude representation: a holistic, a strictly decomposed, and a hybrid model. Observation of the unit-decade compatibility effect (Nuerk et al. in Cognition 82:B25-B33, 2001) challenged the view of two-digit number magnitude to be represented as one integrated entity. However, at the moment there is no study distinguishing between the decomposed and the hybrid model. The present study addressed this issue using a computational modelling approach. Three network models complying with the constraints of all three theoretical models were programmed and trained on two-digit number comparison. Models were compared as to how well they accounted for empirical effects in the most parsimonious way. Generally, this evaluation indicated that the empirical data were simulated best by the strictly decomposed model. Implications of these results for our understanding of the nature of human number magnitude representation are discussed.

Two-digit number comparison: Decade-unit and unit-decade produce the same compatibility effect with number words.

This study explored the processing of 2-digit number words by examining the unit-decade compatibility effect in Spanish. Participants were required to choose the larger of 2-digit number words presented in verbal notation. In compatible trials the decade and unit comparisons led to the same response (e.g., 53-68) while in incompatible trials the decade and unit comparisons led to different responses (e.g., 59-74). Participants were slower on compatible trials as compared to incompatible trials. In Experiments 2 and 3, we evaluated whether the reverse compatibility effect in Spanish was only due to a pure left-to-right encoding which favours the decade processing in this language (decade-unit order). When participants processed 2-digit number words presented in reverse form (in the unit-decade order), the same reverse compatibility effect was found. This pattern of results suggests that participants have learnt a language-dependent process for analysing written numbers which is used irrespective of the specific arrangement of units and decades in the comparison task.

Holistic and Componential Processing for Two-Digit Numbers

A great deal of research has been conducted to investigate how people process two-digit numbers. Researchers have found that the time to compare two two-digit numbers decreases logarithmically with the absolute difference of two numbers, which suggests a holistic processing for two-digit numbers. However, some evidence seems to support additional componential processing. A number pair can be defined as compatible if the decade-magnitude comparison and the unit-magnitude comparison of the two numbers lead to the same result (e.g., 23 vs. 78, 27 and 38), and as incompatible if this is not the case (e.g., 29 vs. 84, 28 but 94). Subjects took longer to respond the unit-decade-incompatible number pairs than to the unit-decade-compatible number pairs. The unit-decade compatibility effect suggests that the magnitude of the unit digits of numbers in addition to the whole magnitude of the numbers is activated. However, previous studies on two-digit number processing have typically used tasks involving intentional processing of numerical magnitude. To intentionally process two-digit numbers, subjects need to pay attention to decade and unit digits serially or simultaneously, which may lead to componential processing. This study addressed whether componential processing is also involved in unintentional processing of the magnitude of numbers. Stroop-like paradigm was applied. Sixteen undergraduate students participated in Experiment 1. They were asked to compare the physical size of two two-digit numbers, as well as the numerical magnitude of the two numbers (e.g., 23 vs. 78). Four hundred two-digit pairs consisting of the numbers between 21 and 98 were presented in Arabic notation, excluding numbers that contained 0 as unit (e.g., 30, 40) or the same digits for decade and unit (e.g., 22, 33). We manipulated the ratio of the physical size of two numbers (i.e., 1∶1.1 vs. 1∶1.2), the congruency of magnitude size and physical size, and the unit-decade-compatibility. To avoid confounding of the distance and problem size effect, we controlled overall distance and problem size between compatible and incompatible pairs. In addition, we also conducted Experiment 2 to avoid interference of intentional and unintentional processing. Twenty undergraduate students were asked to compare only the physical size of two two-digit numbers. Experimental design and procedure were same as Experiment 1, except the ratio of physical size of two numbers had three levels containing 1∶1.1, 1∶1.25 and 1∶1.5. The results from the magnitude comparison task in Experiment 1 replicated those found in previous studies involving the same tasks. Tasks involving physical size comparison in both Experiment 1 and Experiment 2 showed the effect of magnitude and physical size congruency. The effect supported holistic processing. The unit-decade-compatibility effect was evident for congruent number pairs, but there was no or a reverse effect for incongruent number pairs. In sum, two-digit number processing can be componential as well as holistic in both unintentional and intentional processing of numerical magnitude.

Decomposed but Parallel Processing of Two-Digit Numbers in 1st Graders

It has been suggested that decomposed processing of two-digit numbers develops from sequential (left-to-right) to parallel with age (Nuerk et al., 2004). However, task demands may have provoked sequential processing as a specific rather than a universal processing style. In the current study a standard unit-decade compatibility effect observed in two-digit number magnitude comparison indicated that first graders were already able to process the single digit magnitudes of tens and units separately and in parallel. Consequently, previous findings of sequential processing may be specific for stimulus characteristics in which such a processing style is useful. It is concluded that even first graders seem to be able to adapt their individual processing styles depending on stimulus properties. More generally, this suggests that the manner by which children process two-digit numbers is strategically adaptive rather than fixed at a particular developmental stage.

Neural representations of two-digit numbers: A parametric fMRI study

When participants are asked to decide which of a pair of two-digit Arabic numbers is larger, they compare units and decades, even when units are irrelevant. Typically, behavioral responses are slower when units are incongruent with the decade comparison (e.g. 81_26, because 8 > 2 but 1 < 6; unit–decade compatibility effect, [Nuerk, H.-C., Weger, U., Willmes, K. 2001. Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition 82, B25–B33.]). We defined parametric regressors to examine the effect of decade digit distance and unit distance-based compatibility processing on the fMRI signal to investigate the neural correlates of two-digit magnitude processing. Fourteen male right-handed volunteers (mean age = 27, range 21–38 years) took part in the study. In a rapid event-related design, participants had to decide which of two two-digit Arabic numbers was larger. Data were preprocessed and analyzed statistically in SPM2. Activation in the anterior portion of the right IPS was significantly modulated by compatibility-based unit distance processing. Furthermore, decade distance predicted an increase in the fMRI signal from cortex around the left IPS, right anterior and right posterior IPS. These results indicate that magnitude representations of unit–decade compatibility and decade distance are subserved by the intraparietal cortex and that the symbolic structure of the Arabic number system is an important determinant of multi-digit number magnitude processing. Implications of the present results in terms of general symbolic information processing are discussed.

Language effects in magnitude comparison: Small, but not irrelevant

It is assumed that number magnitude comparison is performed by assessing magnitude representation on a single analog mental number line. However, we have observed a unit–decade–compatibility effect in German which is inconsistent with this assumption ( Nuerk, Weger, & Willmes, 2001). Incompatible magnitude comparisons in which decade and unit comparisons lead to different responses (e.g., 37_52 for which 3 < 5, but 7 > 2) are slower and less accurately responded to than compatible trials in which decade and unit comparisons lead to the same response (e.g., 42_57, for which 4 < 5 and 2 < 7). As overall distance was held constant, a single holistic magnitude representation could not account for this compatibility effect. However, because of the inversion property of the corresponding German two-digit number words (“einundzwanzig” <one-and-twenty>), the language-generality of the effect is questionable. We have therefore examined the compatibility effect with native English speakers. We were able to replicate the compatibility effect using Arabic notation. Thus, the compatibility effect is not language-specific. However, in cross-linguistic analyses language-specific modulations were observed not only for number words but also for Arabic numbers. The constraints imposed on current models by the verbal mediation of Arabic number comparison are discussed.

On the Perceptual Generality of the Unit-Decade Compatibility Effect

On the Perceptual Generality of the Unit-Decade Compatibility Effect

On the perceptual generality of the unit-decade compatibility effect.

Number magnitude is assumed to be holistically represented along a single mental number line. Recently, we have observed a unit-decade-compatibility effect which is inconsistent with that assumption (Nuerk, Weger, & Willmes, 2001). In two-digit Arabic number comparison, we have demonstrated that compatible comparisons in which separate decade and unit comparisons lead to the same decision (32_47, 3 < 4 and 2 < 7) were faster than incompatible trials (37_52, 3 < 5, but 7 > 2). Because overall distance was matched, a holistic model could not account for the compatibility effect. However, one could argue that the compatibility effect was due to the specific vertical perceptual arrangement of the two-digit numbers in Nuerk et al.'s (2001) experiment where the decade digits and unit digits were presented column-wise above each other. To examine this objection, we studied the perceptual generality of the compatibility effect with diagonal presentation. We replicated the compatibility effect with diagonal presentation. It is concluded that the compatibility effect is not due to encoding characteristics imposed by the perceptual setting of the original experiment. In particular, the assumption of an overall analog magnitude representation for two-digit numbers is not consistent with these data.

On the Development of the Mental Number Line: More, Less, or Never Holistic With Increasing Age?

Magnitude is assumed to be represented along a holistic mental number line in adults. However, the authors recently observed a unit-decade compatibility effect for 2-digit numbers that is inconsistent with this "holisticness" assumption (H.-C. Nuerk, U. Weger, & K. Willmes, 2001). This study used the compatibility effect to examine whether the mental number line representation of magnitude changes toward greater or less holisticness in children from Grades 2-5. The results indicate that decades and units of 2-digit numbers are processed separately rather than holistically from Grade 2 on. However, this separate processing seems to develop from a more sequential (left-to-right) to a more parallel processing mode. Moreover, children may use different strategies depending on task demands. The results are interpreted in the framework of Siegler's overlapping waves model.

A Unit-Decade Compatibility Effect in German Number Words

A Unit-Decade Compatibility Effect in German Number Words