Two-digit Number Processing Research Articles

Multiple number-naming associations: How the inversion property affects adults' two-digit number processing.

Some number-naming systems are less transparent than others. For example, in Dutch, 49 is named "negenenveertig," which translates to "nine and forty," i.e., the unit is named first, followed by the decade. This is known as the "inversion property," where the morpho-syntactic representation of the number name is incongruent with its written Arabic form. Number word inversion can hamper children's developing mathematical skills. But little is known about its effects on adults' numeracy, the underlying mechanism, and how a person's bilingual background influences its effects. In the present study, Dutch-English bilingual adults performed an audiovisual matching task, where they heard a number word and simultaneously saw two-digit Arabic symbols and had to determine whether these matched in quantity. We experimentally manipulated the morpho-syntactic structure of the number words to alter their phonological (dis)similarities and numerical congruency with the target Arabic two-digit number. Results showed that morpho-syntactic (in)congruency differentially influenced quantity match and non-match decisions. Although participants were faster when hearing traditional non-transparent Dutch number names, they made more accurate decisions when hearing artificial, but morpho-syntactically transparent number words. This pattern was partly influenced by the participants' bilingual background, i.e., their L2 proficiency in English, which involves more transparent number names. Our findings suggest that, within inversion number-naming systems, multiple associations are formed between two-digit Arabic symbols and number names, which can influence adults' numerical cognition.

A model for two-digit number processing based on a joint Garner and system factorial technology analysis.

As a vital part of our daily lives, number processing has received much attention in current cognitive research. Most adults perceive with ease the numerical magnitude of a two-digit number and also the individual magnitudes of the component digits, yet the relation between the 2 is poorly understood. Application to two-digit numbers of Garner's speeded classification paradigm (Experiment 1) and of system factorial technology (Experiment 2) jointly reveals phases of independent and of coactive processing within a well-defined time-course. A new model, the interactive race coactive model, explains various facets of two-digit number processing. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

Linguistic Effects on the Processing of Two-Digit Numbers

The syntactic structure of numbers in Arabic mainly differs from that of Hebrew in terms of the order of units and decades. In Arabic (inverted), two-digit numbers are written and read from right to left, i.e. the first digit is the units and the second is the decades (24 = four and twenty), while in Hebrew (non-inverted), the reverse. Therefore, studying transcoding in the two languages may enable us to tease out the relative effects of linguistic experience (first versus second language) and counting system transparency i.e. the syntactic representation of numbers (inverted versus non-inverted number system). For this purpose, the paradigm of reading and writing two-digit numbers from dictation, in both languages was used. Sixty university bilingual students were given two tasks in both Arabic-L1 (First language) and Hebrew-L2 (Second language): One task involved writing two-digit numbers to dictation, and the other involved reading two-digit numbers aloud. Reading times and the error rates were calculated in both languages according to type of error—total errors, substitution errors (replacement of two-digit number units with decades, and vice versa; for example, 23 ? 32), change errors (change of one digit 23 ? 28), and omission errors (omission of one digit; for example 23 ? 2). The participants made some errors in reading and especially in writing two-digit numbers. Their commonest errors were substitution errors compared to change and omission errors. Such errors were commoner for numbers which require processing the numerical syntactic structure than for decade numbers, or numbers from 11 to 19, which require less attention to numerical syntax. Speed and accuracy are greater in Arabic-L1 than Hebrew-L2, although the Arabic counting system has the inversion feature, and is thus less transparent than Hebrew. It is concluded that familiarity with the verbal counting system of the first language has a greater influence on transcoding than the transparency of the counting system. Theoretical and practical implications are discussed.

Direct evidence for linguistic influences in two-digit number processing.

Language-specific differences in number words influence number processing even in nonverbal numerical tasks. For instance, the unit-decade compatibility effect in two-digit number magnitude comparison (compatible number pairs [42_57: 4 < 5 and 2 < 7] are responded to faster than incompatible pairs [47_62: 4 < 6 but 7 > 2]) was shown to be influenced by the inversion of number words (e.g., in German the number word for 42 is zweiundvierzig [literally: two-and-forty]). In two studies, we used articulatory suppression to investigate whether previously observed cross-linguistic differences in two-digit number processing are indeed driven by differences in number word formation. In a two-digit number comparison task, German- and English-speaking participants had to identify the larger of two numbers presented in Arabic digits. In Study 1, participants performed the same task twice, with and without articulatory suppression. In Study 2, the percentage of within-decade filler items (36_39) was manipulated additionally. As expected, in both studies between-groups differences in the compatibility effect disappeared under articulatory suppression irrespective of the percentage of fillers included. Furthermore, paralleling results of previous studies including 33% or less filler items, we found that the compatibility effect was larger in German compared with English speakers in the 20% filler condition. However, this pattern was reversed in the 50% filler condition in both studies. Thus, results provide first direct evidence for influences of verbal number word formation on symbolic number processing. Moreover, these new findings suggest that linguistic influences and those of cognitive control processes associated with characteristics of the stimulus set interact in symbolic number processing. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

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.

Two-Digit Number Processing during Childhood: A Mental Rotation Task?

The association between numerical and spatial thinking manifests itself throughout various cognitive observations, concerning arithmetic problem solving as well as simple number recognition tasks. The spatial component in numerical thinking is assumed to represent the semantic processing pathway, for example, the mental representation of numerical magnitude. There is an interconnection of semantic, linguistic, and visual/symbolic pathways, which is thought to be stronger in adults than in children. During childhood, the semantic pathway is supposed to play a major part in numerical processing. In this study, we conducted a functional magnetic resonance imaging paradigm combining a mental rotation task with a number processing task. The behavioral data showed the highest error rate for two-digit numbers (not for three-digit numbers). In regard to two-digit numbers, we also found the strongest neuronal activation overlap for number processing and mental rotation in the left intraparietal sulcus. This pattern suggests that the mental inversion of digits in native German speakers might be a spatial function related to mental rotation. In terms of overcoming opposed informational input for symbolic and linguistic representation and therefore impeded informational extraction the spatial processing pathway might play a major role. To conclude, spatial training could be an effective educational approach, supporting the interconnection of numerical processing pathways.

Conflict resolution in two-digit number processing: evidence of an inhibitory mechanism

We investigated the mechanism involved in conflict resolution when individuals processed two-digit numbers. Participants performed a comparison task in blocks of two trials. In the first trial, between-decade two-digit numbers were used in a compatible condition where the decade and the unit of one number were larger than those of the other number (i.e., 21-73) and an incompatible condition where the decade of one number was larger but the unit was smaller than those of the other number (i.e., 61-53). In the second trial, within-decade two-digit numbers were presented in a related condition where the numbers contained the units presented previously (i.e., 41-43) and an unrelated condition with units that did not appear before (i.e., 48-49). In the first trial, participants responded more slowly in incompatible trials relative to compatible trials. In the second trial, participants were slower in the related condition relative to unrelated trials only after incompatible trials. These results suggest that participants experienced conflict in the incompatible condition of first trial and that they inhibited irrelevant units to resolve conflict.

Fluency in two reading systems and the processing of two-digit numbers

Research question: We evaluated whether the processing of two-digit numbers was determined by the direction of reading in the two languages of bilingual individuals. Methodology: A group of Arabic (L1)/Spanish (L2) proficient bilinguals performed a comparison task with Arabic digits, Arabic number words and Spanish number words while the unit–decade compatibility effect was evaluated. Data: Participants showed a regular compatibility effect with Arabic digits. However, no compatibility effect was observed when they performed the task with verbal numbers, regardless of whether numbers were presented in Arabic or in Spanish. Conclusion: The similar pattern of results across the bilinguals’ two languages suggests that the direction of the reading (right-to-left in Arabic and left-to-right in Spanish) did not influence the processing of verbal numbers. Originality: This is the first study in which the unit–decade compatibility effect has been explored with Arabic/Spanish bilinguals. Significance: The main point to highlight from this study is that bilinguals are not influenced by the reading direction of their languages when they process two-digit verbal numbers.

Automatic two-digit number processing by people of different age groups and sex

Currently, there is a growing body of papers on the topic about two-digit (2D) number processing. Separate studies revealed the importance of subjects’ age (e. g., Nuerk, Kaufmann et al., 2004) and sex (Pletzer et al., 2013) in a 2D number representation. The aim of this study was to investigate an automatic 2D number processing in people of different age and sex. 2nd graders (average age 8 years and 5 months), 4th graders (average age 10 years and 5 months), and adults (average age 27 years and 2 months) performed two experimental tasks in which they had to judge the numeric or monetary value of presented stimuli (e. g., which is more, 82 ct or 14 Lt). Results were obtained by totalizing the reaction time and error rate. In statistical analysis, the Wilcoxon criteria were used to distinguish the difference in each participants’ group between blocks of different stimuli. In addition, the results were counted for each participant individually. As in previous studies (e. g., Chan et al., 2011), 2nd graders, 4th graders and adults processed 2D numbers automatically. In this study, 2nd graders processed two-digit numbers in a holistic fashion, while 4th graders and adults used a decomposed parallel processing. The ‘holistic’ 2D number processing in children can be explained as a ‘transition’ stage between decomposed sequential and decomposed parallel. The ‘holistic’ result may derive from approximately equal frequency in using sequential and parallel ways of processing.The data did not support the hypothesis that during adulthood, males process two-digit numbers in a different fashion from females. In this study, both sexes processed 2D numbers in the same way (decomposed parallel). In contrast, Pletzer et al. (2013) found the holistic 2D number processing to prevail in males. In this paper, potential influences on this difference are discussed. First, it could be due to different languages, or it could be due to experiment arrangement. The probability of the same participant using different strategies in response to different (simple or more complex) stimuli is suggested.

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.

On the development of Arabic three-digit number processing in primary school children

The development of two-digit number processing in children, and in particular the influence of place-value understanding, has recently received increasing research interest. However, place-value influences leading to decomposed processing have not yet been investigated for multi-digit numbers beyond the two-digit number range in children. Therefore, we evaluated the separate influences of hundreds, tens, and units on three-digit number processing by means of the hundred distance effect, the decade-hundred compatibility effect, and the unit-hundred compatibility effect in a longitudinal design from Grade 2 to Grade 4. In a number magnitude comparison task, a strong hundred distance effect indicated that the magnitudes of the hundreds digits were predominantly processed. We also observed indexes of decomposed parallel processing of hundreds and units digits but not of hundreds and tens digits. Regarding the developmental trajectories, the hundred distance effect and the unit-hundred compatibility effect showed a reliable trend to increase with grade level. However, both the significance and the increase with grade level of decomposed parallel processing were observed to be less consistent than expected. The latter is discussed in terms of different processing strategies as well as specificities differentiating between two- and three-digit numbers. Taken together, these are the first data showing decomposed processing of three-digit numbers in children. Yet, it must be noted that the results also indicate that findings from two-digit number processing cannot simply be generalized to the three-digit number range.

Across-Notation Automatic Processing of Two-Digit Numbers

The existence of across-notation automatic numerical processing of two-digit (2D) numbers was explored using size comparisons tasks. Participants were Arabic speakers, who use two sets of numerical symbols—Arabic and Indian. They were presented with pairs of 2D numbers in the same or in mixed notations. Responses for a numerical comparison task were affected by decade difference and unit-decade compatibility and global distance in both conditions, extending previous findings with Arabic digits (Nuerk, Weger, & Willmes, 2001). Responses for a physical comparison task were affected by congruency with the numerical size, as indicated by the size congruency effect (SiCE). The SiCE was affected by unit-decade compatibility but not by global distance, thus suggesting that the units and decades digits of the 2D numbers, but not the whole number value were automatically translated into a common representation of magnitude. The presence of similar results for same- and mixed-notation pairs supports the idea of an abstract representation of magnitude.

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.

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 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.

Sequential Processing of Two-Digit Numbers

Two experiments using a number matching task (NMT) explored whether two-digit numbers are processed holistically or in a compositional fashion. In the NMT participants are required to decide whether one of the two numbers initially provided (cues) is presented some milliseconds later or not (probe). Probes which have some arithmetic relationship to the cues (e.g., cues: 2 3, probe: 6) are rejected more slowy than probes unrelated to their cues (e.g., cues: 2 3, probe: 7) – interference effect –, and this is considered as evidence of the automatic activation of that arithmetic relationship. Participants were presented with two-digit cues and probes which had an arithmetic progression relationship only detectable once the numbers were decomposed (Experiment 1: cues: 56 7, probe: 89; Experiment 2: cues: 45 67, probe: 89). Results showed longer response times in these conditions compared to unrelated conditions. Data support componential processing even when the numbers to be matched are presented serially.

The Processing of Two-Digit Numbers Depends on Task Instructions

Recently, a lot of research has focused on resolving whether two-digit numbers are processed holistically or compositionally. This has led to inconsistent results. In the present study we investigated effects of task instructions. Subjects performed magnitude or parity judgments on targets preceded by masked primes containing parts of the target at a task-congruent (3#_37) or task-incongruent (#3_37) position. Priming effects were influenced by the instructions: In the magnitude task, the priming effects were primarily mediated by the congruency of the decade digit, whereas in the parity task they were elicited by the congruency of the unit digit, which is in line with a flexible compositional processing style. These and previous findings show that two-digit numbers can be processed in a very flexible way, depending on the task context.

Attentional Strategies in Place-Value Integration

To process a multi-digit number its constituting digits need to be integrated into the place-value structure of the Arabic number system. For two-digit numbers, processes of unit-decade integration are reflected by the compatibility effect in magnitude comparison. Recent research in adults indicated that the size of the compatibility effect increases when stimuli prevent to focus on the decade digits as achieved by the inclusion of within-decade items (43_47). In the present study within- and between-decade items (47_62) were used to assess the compatibility effect in children. We observed reliable compatibility effects that increased with grade level and that were larger than in a previous study without within-decade stimuli. Furthermore, evaluation of the developmental trajectories showed that two-digit number processing develops to more automatic parallel processing of the constituent digits of tens and units. From these results we conclude that (i) even for children attentional processes can strongly influence multi-digit number processing and (ii) with increasing age and experience more parallel and automated understanding of two-digit numbers develops which seems to remain relatively stable once achieved.

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.

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.