Magnitude representation in sequential comparison of two-digit numbers is not holistic either

There is accumulating evidence suggesting that two-digit number magnitude is represented in a decomposed fashion into tens and units rather than holistically as one integrated entity. However, recently, it has been claimed that this property does not hold for the case when two to-be-compared numbers are presented sequentially. In the present study, we pursued this issue in two experiments by evaluating perceptual as well as strategic aspects arising for sequential stimulus presentation in a magnitude comparison task. We observed reliable unit-decade compatibility effects indicating decomposed processing of tens and units in a magnitude comparison task with sequential presentation of the to-be-compared numbers. In particular, we found that both confounding low-level perceptual features and stimulus set characteristics determining cue validity of the units influenced the compatibility effect. Taken together, our results clearly indicate that decomposed representations of tens and units seem to be a general characteristic of multi-digit number magnitude processing, rather than an exception occurring under very specific conditions only. Implications of these results for the understanding of number magnitude representations are discussed.     

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.     

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.     

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.     

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。     

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.     

Comparing two-digit numbers: the importance of being presented together

The effect of presentation mode on magnitude comparisons of two-digit (2D) numbers was examined using the stimuli set developed by Nuerk, Weger, and Willmes (2001). In Experiment 1, only number pairs from difference decades were presented either simultaneously or sequentially. In the former case there was evidence for the parallel processing of both the units and decades digits and for a components representation, consistent with previous findings. In contrast, in the latter case there was evidence for the processing of mainly the decades digits. In Experiment 2, within-decade number pairs were added to make both digits task relevant. The results from the simultaneous condition were again consistent with a components representation, while results from the sequential presentation were in line with a holistic representation, in line with Zhang and Wang's (2005) research. Results therefore suggest that the processing of 2D numbers depends on the way they are presented.     

How do we convert a number into a finger trajectory?

How do we understand two-digit numbers such as 42? Models of multi-digit number comprehension differ widely. Some postulate that the decades and units digits are processed separately and possibly serially. Others hypothesize a holistic process which maps the entire 2-digit string onto a magnitude, represented as a position on a number line. In educated adults, the number line is thought to be linear, but the “number sense” hypothesis proposes that a logarithmic scale underlies our intuitions of number size, and that this compressive representation may still be dormant in the adult brain. We investigated these issues by asking adults to point to the location of two-digit numbers on a number line while their finger location was continuously monitored. Finger trajectories revealed a linear scale, yet with a transient logarithmic effect suggesting the activation of a compressive and holistic quantity representation. Units and decades digits were processed in parallel, without any difference in left-to-right vs. right-to-left readers. The late part of the trajectory was influenced by spatial reference points placed at the left end, middle, and right end of the line. Altogether, finger trajectory analysis provides a precise cognitive decomposition of the sequence of stages used in converting a number to a quantity and then a position.     

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.     

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.     

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.     

Multi-digit number processing beyond the two-digit number range: a combination of sequential and parallel processes

Investigations of multi-digit number processing typically focus on two-digit numbers. Here, we aim to investigate the generality of results from two-digit numbers for four- and six-digit numbers. Previous studies on two-digit numbers mostly suggested a parallel processing of tens and units. In contrast, the few studies examining the processing of larger numbers suggest sequential processing of the individual constituting digits. In this study, we combined the methodological approaches of studies implying either parallel or sequential processing. Participants completed a number magnitude comparison task on two-, four-, and six-digit numbers including unit-decade compatible and incompatible differing digit pairs (e.g., 32_47, 3<4 and 2<7 vs. 37_52, 3<5 but 7>2, respectively) at all possible digit positions. Response latencies and fixation behavior indicated that sequential and parallel decomposition is not exclusive in multi-digit number processing. Instead, our results clearly suggested that sequential and parallel processing strategies seem to be combined when processing multi-digit numbers beyond the two-digit number range. To account for the results, we propose a chunking hypothesis claiming that multi-digit numbers are separated into chunks of shorter digit strings. While the different chunks are processed sequentially digits within these chunks are processed in parallel.     

The effect of external representations on numeric tasks

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: The comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect is discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations, and the behaviour in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical cognition should consider external representations as an essential component.     

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.     

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.     

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).     

To carry or not to carry — Is this the question? Disentangling the carry effect in multi-digit addition

Recent research has suggested addition performance to be determined by both the need for a carry operation and problem size. Nevertheless, it has remained debatable, how these two factors are interrelated. In the current study, this question was pursued by orthogonally manipulating carry and problem size in two-digit addition verification. As the two factors interacted reliably, our results indicate that the carry effect is moderated by number magnitude processing rather than representing a purely procedural, asemantic sequence of processing steps. Moreover, it was found that the carry effect may not be a purely categorical effect but may be driven by continuous characteristics of the sum of the unit digits as well. Since the correct result of a carry problem can only be derived by integrating and updating the magnitudes of tens and units within the place-value structure of the Arabic number system, the present study provides evidence for the idea that decomposed processing of tens and units also transfers to mental arithmetic.     

Numerical comparison of two-digit numbers: How differences at encoding can involve differences in processing

The study of two-digit numbers processing has recently gathered a growing interest. Here, we examine whether differences at encoding of two-digit oral verbal numerals induce differences in the type of processing involved. Twenty-four participants were submitted to a comparison task to 55. Differences at encoding were introduced by the use of dichotic listening and synchronous (synchronous condition) or asynchronous presentation (tens-first and units-first conditions) of the two-digit numerals' components. Our results showed that differences at the encoding stage of two-digit numerals involve: (1) different comparison processes (tens-first and units-first conditions: parallel comparison; synchronous condition: parallel and holistic comparison); and (2) differences in the weight of the tens- and units-effects. Therefore, attentional mechanisms determining at the encoding stage how much attention is paid to the two-digit numerals' components might account for the different types of processing found with two-digit numbers.     

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.     

Two-digit comparison: decomposed, holistic, or hybrid?

We investigate whether two-digit numbers are decomposed for purposes of numerical comparison (e.g., choosing the larger one). Earlier theorists concluded that numbers are processed holistically (Brysbaert, 1995; Dehaene, Dupoux, & Mehler, 1990), or that holistic and decomposed processes operate in parallel (Nuerk, Weger, & Willmes, 2001). In the present experiment, we presented pairs of two-digit numbers with a decade distance of either zero (e.g., 54-57) or one (54-61). If a holistic process contributes to two-digit comparison, there should be an overall distance effect for number pairs with a decade distance of one. On the other hand, if numbers are decomposed and a holistic comparison does not contribute, this overall distance effect should be absent for these number pairs. Evidence is found that, at least in the present task settings, numbers are not compared holistically. The results are interpreted in terms of a recently proposed theory of numerical cognition (Verguts & Fias, 2004; Verguts, Fias, & Stevens, in press).