The processing of Arabic numbers is under cognitive control

In this study, we evaluated whether the processing of two-digit Arabic numbers is subject to cognitive control. Participants performed a number comparison task with unit–decade compatible trials (36–47) and unit–decade incompatible number pairs (37–46). The unit–decade compatibility (incompatible minus compatible trials) was considered an index of conflict situation in numerical processing. To examine whether participants adapted to numerical conflict, we manipulated the percentage of incompatible trials relative to the percentage of compatible trials (20, 50, 80 % incompatible trials). The unit–decade compatibility effect was modulated by the amount of incompatible trials: the compatibility effect decreased progressively as the percentage of incompatible trials increased. This result indicated that participants adapted to numerical conflict. The findings are discussed in terms of cognitive control. Implications for models of two-digit number processing are also addressed.     

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.     

Proficiency in a second language influences the processing of number words

We evaluated whether proficiency in a second language (L2) influenced the processing of numerical information. In Experiment 1, two groups of German/English bilinguals, one less proficient in English (L2) and the other more proficient in L2, performed two-digit number comparison tasks while the unit–decade compatibility was evaluated. All participants presented compatibility effect with Arabic digits regardless of their L2 proficiency. However, when bilinguals with less proficiency in L2 performed verbal number comparison tasks they showed regular compatibility effect in German and reverse compatibility effect in English, whereas more proficient bilinguals did not show compatibility effects in either German or English. In Experiment 2, the same pattern of results was obtained with highly proficient bilinguals after controlling their working memory span. These results indicate that L2 proficiency influences the processing of two-digit number words.     

The processing of two-digit numbers in bilinguals

We explored possible between-language influences when bilinguals processed two-digit numbers. Spanish/English bilinguals and German/English bilinguals performed a number comparison task with Arabic digits and verbal numbers in their first language (L1) and second language (L2) while the unit-decade compatibility was manipulated. The two bilingual groups showed regular compatibility effect with Arabic digits. In L1, Spanish/English bilinguals showed reverse compatibility effect, while German/English bilinguals showed regular compatibility effect. However, both groups of bilinguals presented reverse compatibility effect in English (L2), which suggested that the bilinguals' L1 did not determine the processing of number words in their L2. The results indicated that bilinguals processed two-digit number words selectively in their L1 and L2 and that they did not transcode L2 numbers into Arabic format.     

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" ), 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

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.     

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.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

两位数的整体与局部加工

关于两位数的加工方式有整体加工说和局部加工说,实验证据主要来自数字数量控制/主动加工任务。本研究主要考察在数字数量自动加工任务中两位数的加工方式。实验一要求被试完成数量大小比较和物理大小比较两个任务,实验二只要求被试完成物理大小比较任务。结果是在数量比较任务和物理比较任务中都存在显著的个位十位一致性效应和数量物理一致性效应,这表明在两位数的数量主动和自动加工任务中均存在整体加工和局部加工两种方式。     

Flexible and unique representations of two-digit decimals

We examined the representation of two-digit decimals through studying distance and compatibility effects in magnitude comparison tasks in four experiments. Using number pairs with different leftmost digits, we found both the second digit distance effect and compatibility effect with two-digit integers but only the second digit distance effect with two-digit pure decimals. This suggests that both integers and pure decimals are processed in a compositional manner. In contrast, neither the second digit distance effect nor the compatibility effect was observed in two-digit mixed decimals, thereby showing no evidence for compositional processing of two-digit mixed decimals. However, when the relevance of the rightmost digit processing was increased by adding some decimals pairs with the same leftmost digits, both pure and mixed decimals produced the compatibility effect. Overall, results suggest that the processing of decimals is flexible and depends on the relevance of unique digit positions. This processing mode is different from integer analysis in that two-digit mixed decimals demonstrate parallel compositional processing only when the rightmost digit is relevant. Findings suggest that people probably do not represent decimals by simply ignoring the decimal point and converting them to natural numbers.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

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.     

In touch with numbers: Embodied and situated effects in number magnitude comparison

The idea of embodied numerosity denotes that seemingly abstract number concepts (e.g., magnitude) are rooted in bodily experiences and situated action. In the present study we evaluated whether there is an embodied representation of the place–value structure of the Arabic number system and if so whether this representation is influenced by situated aspects. In a two-digit number magnitude comparison task participants had to directly touch the larger of two numbers. Importantly, pointing responses were systematically biased toward the decade digit of the target number. Additionally, this leftward bias towards the tens digit was reduced in unit–decade incompatible number pairs. Thereby, we demonstrated an influence of place–value processing on manual pointing movement. Our results therefore corroborate the notion of an embodied representation of the place–value structure of Arabic numbers which is modulated by situated aspects.     

Decade breaks in the mental number line? Putting the tens and units back in different bins.

Most models of number recognition agree that among other number representations there is a central semantic magnitude representation which may be conceptualized as a logarithmically compressed mental number line. Whether or not this number line is decomposed into different representations for tens and units is, however, controversial. We investigated this issue in German participants in a magnitude comparison (selection) task in which the larger of two visually presented Arabic two-digit numbers had to be selected. Most importantly, we varied unit-decade-compatibility: a number pair was defined as compatible if the decade magnitude comparison and the unit magnitude comparison of the two numbers would lead to the same response (e.g. 52 and 67) and as incompatible if this was not the case (e.g. 47 and 62). While controlling for overall numerical distance, size and other variables, we consistently found compatibility effects. A control experiment showed that this compatibility effect was not due to perceptual presentation characteristics. We conclude that the idea of one single number line representation that does not additionally assume different magnitude representations for tens and units is not sufficient to account for the data. Finally, we discuss why decade effects were not found in other experimental settings.     

Is adding 48 + 25 and 45 + 28 the same? How addend compatibility influences the strategy execution in mental addition

A recent study revealed that adults frequently start to add two two-digit numbers from the larger one, suggesting that addend magnitudes are compared at an early stage of processing. However, several studies showed that symbolic number comparison involves compatibility effects: Such numerical comparison is easier when the larger number also contains the larger unit (48_25) than in the opposite, incompatible case (45_28). In this context, whether the compatibility between tens and units across operands affects the execution of arithmetic-solving strategies remains an open question. In this study, we used two kinds of verbal protocols to assess how addend compatibility influences the implementation of magnitude-based strategies. We observed that participants started their computations from the larger operand more frequently when solving compatible additions than they did when solving incompatible ones. The presence of a compatibility effect extends the view that multidigit number processing is componential rather than holistic, even in an arithmetic task that did not explicitly require a number magnitude comparison. Further, the findings corroborate the notion that number magnitude is used in mental calculation and influences the way calculation strategies are implemented.     

Speaking two languages with different number naming systems: What implications for magnitude judgments in bilinguals at different stages of language acquisition?

Differences between languages in terms of number naming systems may lead to performance differences in number processing. The current study focused on differences concerning the order of decades and units in two-digit number words (i.e., unit-decade order in German but decade-unit order in French) and how they affect number magnitude judgments. Participants performed basic numerical tasks, namely two-digit number magnitude judgments, and we used the compatibility effect (Nuerk et al. in Cognition 82(1):B25–B33, 2001) as a hallmark of language influence on numbers. In the first part we aimed to understand the influence of language on compatibility effects in adults coming from German or French monolingual and German–French bilingual groups (Experiment 1). The second part examined how this language influence develops at different stages of language acquisition in individuals with increasing bilingual proficiency (Experiment 2). Language systematically influenced magnitude judgments such that: (a) The spoken language(s) modulated magnitude judgments presented as Arabic digits, and (b) bilinguals’ progressive language mastery impacted magnitude judgments presented as number words. Taken together, the current results suggest that the order of decades and units in verbal numbers may qualitatively influence magnitude judgments in bilinguals and monolinguals, providing new insights into how number processing can be influenced by language(s).     

Exploring the developmental changes in automatic two-digit number processing

Even when two-digit numbers are irrelevant to the task at hand, adults process them. Do children process numbers automatically, and if so, what kind of information is activated? In a novel dot-number Stroop task, children (Grades 1-5) and adults were shown two different two-digit numbers made up of dots. Participants were asked to select the number that contained the larger dots. If numbers are processed automatically, reaction time for dot size judgment should be affected by numerical characteristics. The results suggest that, like adults, children process two-digit numbers automatically. Based on the current findings, we propose a developmental trend for automatic two-digit number processing that goes from decomposed sequential (activation of decade digit followed by that of unit digit) to decomposed parallel processing (simultaneous activation of decade and unit digits).     

The differential influence of decades and units on multidigit number comparison

What role do the magnitudes of the constituent digits play in three-digit number comparison (e.g., choosing the larger one of two numbers)? The present study addressed this question by examining compatibility effects between hundred and decade digits and between hundred and unit digits. For example, the number pair 372–845 is hundred–decade incompatible because the larger number contains the smaller decade digit, but hundred–unit compatible because the larger number contains the larger unit digit. We obtained significant effects of hundred–decade and hundred–unit compatibility on number comparison times. However, the effect of hundred–unit compatibility was largely restricted to the hundred–decade-compatible condition. These results suggest that place-value information, through decomposition, is automatically taken into account when multidigit numbers have to be compared. Implications of our findings for models of number processing are discussed.     

The role of emotional context in facilitating imitative actions

In two experiments, we explored whether emotional context influences imitative action tendencies. To this end, we examined how emotional pictures, presented as primes, affect imitative tendencies using a compatibility paradigm. In Experiment 1, when seen index finger movements (lifting or tapping) and pre-instructed finger movements (tapping) were the same (tapping–tapping, compatible trials), participants were faster than when they were different (lifting–tapping, incompatible trials). This compatibility effect was enhanced when the seen finger movement was preceded by negative primes compared with positive or neutral primes. In Experiment 2, using only negative and neutral primes, the influence of negative primes on the compatibility effect was replicated with participants performing two types of pre-instructed finger movements (tapping and lifting). This emotional modulation of the compatibility effect was independent of the participants' trait anxiety level. Moreover, the emotional modulation pertained primarily to the compatible conditions, suggesting facilitated imitation due to negatively valent primes rather than increased interference. We speculate that negative stimuli increase imitative tendencies as a natural response in potential flight-or-fight situations.     

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.