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

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.

Extending the Mental Number Line

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

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.

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.

Digit Repetition Effect in Two-Digit Number Comparison

Previous data from two-digit number naming show that when target and prime numbers share one digit at different positions (e.g., target 28 and prime 86 or 72) an inhibition effect may be observed (Ratinckx, Brysbaert, & Fias, 2005). Such an effect has been ascribed to the mechanism of morpho-phonological transcoding from Arabic to verbal format. We evaluate the alternative hypothesis of an inhibition effect arising during the Arabic form processing when two different syntactic values have to be assigned to the same single digit. In Experiments 1 and 3, a digit repetition effect was observed in number comparison tasks, even when phonological transcoding is blocked by an articulatory suppression task. Conversely, Experiment 2 showed that no digit repetition effect can be found with number comparison of verbal written numbers. Results are compatible with an Arabic-syntactic processing stage, where units and decades are decomposed and where each single digit is activated and recognized.

Holistic Representation of Negative Numbers is Formed When Needed for the Task

Past research suggested that negative numbers are represented in terms of their components—the polarity marker and the number (e.g., Fischer & Rottmann, 2005; Ganor-Stern & Tzelgov, 2008). The present study shows that a holistic representation is formed when needed for the task requirement. Specifically, performing the numerical comparison task on positive and negative numbers presented sequentially required participants to hold both the polarity and the number magnitude in memory. Such a condition resulted in a holistic representation of negative numbers, as indicated by the distance and semantic congruity effects. This holistic representation was added to the initial components representation, thus producing a hybrid holistic-components representation.

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.

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.

[Rounding off positions used in number comparisons]

In everyday situations, people round off multidigit numbers to what they consider an appropriate place. This study examined how many places people rounded to in a situation where they were required to choose between higher payments with long delays and lower payments with short delays. The payments were 4-digit numbers of yen and the ordinal sequences of the digits in the second, third, and fourth places were manipulated. In the compatible conditions, the ordinal direction in the second place of the low-order numbers was the same as that of highest-order numbers, for example 23xx and 48xx yen. In the incompatible conditions, the ordinal direction of the low-order digits was reversed, for example 28xx and 43xx yen. The results showed that rounding to a higher payment was more prevalent in the compatible conditions than in the incompatible conditions for the second place, but there was little effect in the third and fourth places. These results suggest that people compare the first two digits and make little reference to the third or later ones in rounding numbers.

Language effects in addition: How you say it counts

This study aimed to investigate the role of language in calculation. Two populations were compared, one with a base-10 language, and another (Basque) in which number words are constructed by combining multiples of 20 and units or teens (e.g., “35” is said “twenty and fifteen”). Experiment 1 asked participants to verbally solve additions presented as Arabic digits. Basque participants solved the additions that consisted of a multiple of 20 and a teen (e.g., 20 + 15) faster than controls with identical answers (e.g., 25 + 10). No differences were found in the base-10 language group. Experiment 2 replicated this result even if participants had to type the answer on a numerical keypad, instead of saying it. Hence, the structure of number words in each of the languages influenced the way additions were solved, even if language was not necessary for conducting the task. Finally, in Experiment 3, both language groups performed a numerical comparison task in which no effects of the structure of number words were obtained. Results of the three experiments are discussed in light of current models of numerical cognition.

Numerical representation in the parietal lobes: abstract or not abstract?

The study of neuronal specialisation in different cognitive and perceptual domains is important for our understanding of the human brain, its typical and atypical development, and the evolutionary precursors of cognition. Central to this understanding is the issue of numerical representation, and the question of whether numbers are represented in an abstract fashion. Here we discuss and challenge the claim that numerical representation is abstract. We discuss the principles of cortical organisation with special reference to number and also discuss methodological and theoretical limitations that apply to numerical cognition and also to the field of cognitive neuroscience in general. We argue that numerical representation is primarily non-abstract and is supported by different neuronal populations residing in the parietal cortex.

Short Article: 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.

The Representation of Negative Numbers: Exploring the Effects of Mode of Processing and Notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive 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.

Rational numbers: componential versus holistic representation of fractions in a magnitude comparison task

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.

Not all basic number representations are analog: Place coding as a precursor of the natural number system

Rips et al.'s arguments for rejecting basic number representations as a precursor of the natural number system are exclusively based on analog number coding. We argue that these arguments do not apply to place coding, a type of basic number representation that is not considered by Rips et al.

Automaticity for numerical magnitude of two-digit Arabic numbers in children

This paper examines the automatic processing of the numerical magnitude of two-digit Arabic numbers using a Stroop-like task in school-aged children. Second, third, and fourth graders performed physical size judgments on pairs of two-digit numbers varying on both physical and numerical dimensions. To investigate the importance of synchrony between the speed of processing of the numerical magnitude and the physical dimensions on the size congruity effect (SCE), we used masked priming: numerical magnitude was subliminally primed in half of the trials, while neutral priming was used in the other half. The results indicate a SCE in physical judgments, providing the evidence of automatic access to the magnitude of two-digit numbers in children. This effect was modulated by the priming type, as a SCE only appeared when the numerical magnitude was primed. This suggests that young children needed a relative synchronization of numerical and physical dimensions to access the magnitude of two-digit numbers automatically.

Using parametric regressors to disentangle properties of multi-feature processes

FMRI data observed under a given experimental condition may be decomposed into two parts: the average effect and the deviation of single replications from this average effect. The average effect is represented by the mean activation over a specific condition. The deviation from this average effect may be decomposed into two components as well: systematic variation due to known empirical factors and pure measurement error. In most fMRI designs deviations from mean activation may be treated as measurement error. Nevertheless, often deviation from the average also may contain systematic variation that can be distinguished from simple measurement error. In these cases, the average fMRI signal may provide only a coarse picture of real brain activation. The larger the variation within-condition, the coarser the average effect and the more relevant is the impact of deviations from it. Systematic deviation from the mean activation may be examined by defining a set of parametric regressors. Here, the applicability of parametric methods to refine the evaluation of fMRI studies is discussed with special emphasis on (i) examination of the impact of continuous predictors on the fMRI signal, (ii) control for variation within each experimental condition and (iii) isolation of specific contributions by different features of a single complex stimulus, especially in the case of a sampled stimulus. The usefulness and applicability of this method are discussed and an example with real data is presented.

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 laterality effect: Myth or truth?

Tzelgov and colleagues [Tzelgov, J., Meyer, J., and Henik, A. (1992). Automatic and intentional processing of numerical information. Journal of Experimental Psychology: Learning, Memory and Cognition, 18, 166-179.], offered the existence of the laterality effect as a post-hoc explanation for their results. According to this effect, numbers are classified automatically as small/large versus a standard point under autonomous processing of numerical information. However, the genuinity of the laterality effect was never examined, or was confounded with the numerical distance effect. In the current study, I controlled the numerical distance effect and observed that the laterality effect does exist, and affects the processing of automatic numerical information. The current results suggest that the laterality effect should be taken into account when using paradigms that require automatic numerical processing such as Stroop-like or priming tasks.

Negative Numbers Are Generated in the Mind

Abstract. The goal of the present study was to disentangle two possible representations of negative numbers - the holistic representation, where absolute magnitude is integrated with polarity; and the components representation, where absolute magnitude is stored separately from polarity. Participants' performance was examined in two tasks involving numbers from -100 to 100. In the numerical comparison task, participants had to decide which number of a pair was numerically larger/smaller. In the number line task, participants were presented with a spatial number line on which they had to place a number. The results of both tasks support the components representation of negative numbers. The findings suggest that processing of negative numbers does not involve retrieval of their meaning from memory, but rather the integration of the polarity sign with the digits' magnitudes.

Neighborhood consistency in mental arithmetic: Behavioral and ERP evidence

BACKGROUND

Recent cognitive and computational models (e.g. the Interacting Neighbors Model) state that in simple multiplication decade and unit digits of the candidate answers (including the correct result) are represented separately. Thus, these models challenge holistic views of number representation as well as traditional accounts of the classical problem size effect in simple arithmetic (i.e. the finding that large problems are answered slower and less accurate than small problems). Empirical data supporting this view are still scarce.

METHODS

Data of 24 participants who performed a multiplication verification task with Arabic digits (e.g. 8 x 4 = 36 - true or false?) are reported. Behavioral (i.e. RT and errors) and EEG (i.e. ERP) measures were recorded in parallel.

RESULTS

We provide evidence for neighborhood-consistency effects in the verification of simple multiplication problems (e.g. 8 x 4). Behaviorally, we find that decade-consistent lures, which share their decade digit with the correct result (e.g. 36), are harder to reject than matched inconsistent lures, which differ in both digits from the correct result (e.g. 28). This neighborhood consistency effect in product verification is similar to recent observations in the production of multiplication results. With respect to event-related potentials we find significant differences for consistent compared to inconsistent lures in the N400 (increased negativity) and Late Positive Component (reduced positivity). In this respect consistency effects in our paradigm resemble lexico-semantic effects earlier found in simple arithmetic and in orthographic input processing.

CONCLUSION

Our data suggest that neighborhood consistency effects in simple multiplication stem at least partly from central (lexico-semantic') stages of processing. These results are compatible with current models on the representation of simple multiplication facts - in particular with the Interacting Neighbors Model - and with the notion of decomposed representations of two-digit numbers in general.

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.

The nonintentional processing of Arabic numbers in children

This paper aims at determining when magnitude is automatically activated from Arabic digits in children and whether this automatic activation depends on the number size of the stimuli considered. Second, third, and fourth graders performed a physical Stroop task in which pairs of single- and two-digit Arabic numerals in a ratio 1:2 or 2:3 were used. In addition, the speed of perceptual and semantic information was balanced: Instead of presenting the two members of each pair directly with a physical size difference, they first appeared in the same intermediate physical size, the perceptual difference occurring only after some time. The results indicated a significant effect of congruity at all grades, with faster reaction times and lower error rates for congruent trials than for incongruent trials, for both one- and two-digit numerals. Furthermore, this congruity effect was affected by the ratio of the numbers in the pairs. Only pairs of ratio 1:2 were affected by this congruity effect. These findings strongly support the fact that children as young as 7 years old can automatically access the magnitude information of one- as well as two-digit Arabic numerals.

Can synaesthesia research inform cognitive science?

The renaissance of synaesthesia research has produced many insights regarding the aetiology and mechanisms that might underlie this intriguing phenomenon, which abnormally binds features between and within modalities. Synaesthesia is interesting in its own right, but whether it contributes to our knowledge of neurocognitive systems that underlie non-synaesthete experience is an open question. In this review, we show that results from the field of synaesthesia can constrain cognitive theories in numerical cognition, automaticity, crossmodal interaction and awareness. Therefore, research of synaesthesia provides a unique window into other domains of cognitive neuroscience. We conclude that the study of synaesthesia could advance our understanding of the normal and abnormal human brain and cognition.

Bilateral Processing of Redundant Information: the Influence of Stimulus Notation and Processing Speed in Number Comparison

Improved performance is generally observed when identical copies of a stimulus are presented bilaterally to both visual half fields (VHFs), relative to unilateral single presentations. We investigated the influence of stimulus notation and processing speed on this bilateral redundancy gain (B-RG) in number comparison. Two experiments were performed with consistent (i.e., two identical copies of Arabic digits or word numerals) and inconsistent (i.e., an Arabic digit in one VHF with a word numeral of the same value in the other VHF) displays (see also Marks and Hellige, 2003 for a similar design in number naming). In Experiment 1, the processing speeds of the Arabic digits were faster than the ones of the word numerals while in Experiment 2, the processing speeds of the two stimulus notations were kept equal. A significant difference between consistent and inconsistent conditions was restricted to Experiment 1 in which the processing speeds of the two stimulus notations diverged. These results are discussed in the framework of a simple race model in which two independent hemispheric processes compete for the control of response, and the faster process wins.

Interference effects in a numerical Stroop paradigm in 9- to 12-year-old children with ADHD-C

Attention deficit hyperactivity disorder (ADHD) is characterized by deficient self-regulation, poor attentional control, and poor response inhibition. To date, however, the extent to which these deficits affect basic interference control remains a matter of controversy. Secondly, ADHD has been reported to be associated with arithmetic deficits. It remains unclear whether such deficits are a secondary consequence of the above-mentioned characteristics of ADHD or whether basic numerical magnitude representations are also affected. In the present study we attempted to investigate these issues using a basic numerical interference paradigm. Nine- to twelve-year-old children with ADHD-C (attention-deficit hyperactivity-disorder combined type) and control children without ADHD (each n = 16) were presented with two digits of possibly different physical sizes (e.g., 3 7). This numerical Stroop task requires subjects to make a magnitude classification concerning either the physical or the numerical stimulus dimension. The irrelevant dimension can be congruent (same response), incongruent (different response), or neutral (no response association). Children with ADHD-C performed worse than control children in most analyses. The most important finding was a significant interaction of congruity effects with group in the numerical comparison task. Children with ADHD-C tended to show larger congruity and interference effects than controls, and these were not attributable to a speed-accuracy trade-off. The results might reflect differential processing speeds, or a different degree of automatic activation of physical and numerical magnitudes in children with and without ADHD-C. Alternative explanations, such as insufficient inhibition of selective (domain-specific) attention are also discussed.

The Involvement of the Inferior Parietal Cortex in the Numerical Stroop Effect and the Distance Effect in a Two-digit Number Comparison Task

The neural mechanism of number representation and processing is currently under extensive investigation. In this functional magnetic resonance imaging study, we designed a number comparison task to examine how people represent and compare two-digit numbers in the brain, and whether they process the decade and unit digits in parallel. We manipulated the decade-unit-digit congruency and numerical distance between the pairs of numbers. We observed both Stroop-like interference and the distance effect in the participants' performance. People responded more slowly to incongruent pairs of numbers and pairs of a smaller distance. The inferior parietal cortex showed common and distinct patterns of activation for both attentional selection and number comparison processes, and its activity was modulated by the Stroop-like interference effect and the distance effect. Taken together, these results support both parallel and holistic comparison of two-digit numbers in the brain.

Size congruity effects with two-digit numbers: Expanding the number line?

The size congruity effect is the interesting result that comparisons of the sizes of the physical formats in which numerals appear are affected by the numerical magnitudes of the respective numerals. We demonstrated that separating the physical and the numerical attributes in space leaves the effect unchanged. We then applied the spatially separated version to two-digit numerals and showed the effect to be comparable to that obtained with single numerals. We showed further that the effect is sensitive to the relative salience of the numeric and physical dimensions, to the extent that when the latter is the more salient dimension, a reverse effect obtains by which physical size interferes with number comparison. The results can be explained by a relative speed of processing account, but they are also compatible with an attention account that does not appeal to the notion of automaticity.

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.

What do Semi-Illiterate Adults Know About 2-Digit Arabic Numbers?

Brazilian semi-illiterates (n=19) with 1 to 4 years of academic training compared 2-digit Arabic numbers. Like competent readers, semi-illiterates responded faster and more accurately when the larger number also had the larger unit digit (unit-decade compatibility). Even semi-illiterates with minimal academic training were sensitive to the unit-decade compatibility on number comparison. Probably semi-illiterates derive their ability to automatically activate unit-decade segmentation directly from the oral/verbal number representation. Interestingly, processing speed had an effect on two-digit number comparison. Faster participants showed a smaller or even inverted unit-decade compatibility. Processing speed reflects increased automatization of number segmentation and better selection of the decade-digits for comparison, what eliminates the semantic interference of units. In line with previous research, our data suggest that unit-decade segmentation plays a pervasive role in Arabic number comparison.

What Makes Multiplication Facts Difficult

Two current models of arithmetic fact retrieval, the network interference theory (NIT; Campbell, 1995 ) and the interacting neighbors (IN) model ( Verguts & Fias, 2005a ), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are “consistent,” compared with “inconsistent” errors). In a reanalysis of error data originally reported by Campbell (1997) , we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001) . However, the NIT and IN‐model differ in their explanations of the problem‐size effect, a hallmark finding robustly observed in arithmetic fact retrieval. Only the IN‐model predicts that a correct answer's neighborhood consistency can fully account for the problem‐size effect, which was confirmed in our analysis.

On the functional role of human parietal cortex in number processing: How gender mediates the impact of a ‘virtual lesion’ induced by rTMS

Areas around the horizontal part of the intraparietal sulcus (hIPS) have repeatedly been reported to participate in processing numerical magnitude. Using transcranial magnetic stimulation (TMS), we investigated the functional role of the hIPS by examining two effects from the domain of numerical cognition: in magnitude comparison tasks response latencies are inversely related to the numerical distance between two numbers. This distance effect indexes access to the mental number representation. In magnitude comparison tasks responses are faster when decade and unit comparison would lead to the same decision (e.g. 42_57, 4 < 5 and 2 < 7) than when they would not (e.g. 47_62, 4 < 6 but 7 > 2). This compatibility effect reflects unit-decade integration processes. Differential susceptibility of (fe)male participants to TMS was examined. We applied repetitive TMS (rTMS; 1Hz for 10 min) over the left hIPS in 12 participants (6 female). No stimulation and vertex stimulation served as control conditions. The effect of rTMS was mediated by gender: in male participants, the distance effect decreased after TMS over hIPS. For female participants distance and compatibility effect both increased. This modulation of the compatibility effect was limited in duration to no more than 4 min. The hIPS seems to be functionally involved both in number magnitude processing and in integrating unit-decade magnitude information of two-digit numbers. Relative hemispheric specialization of the hIPS with respect to two-digit magnitude comparison is discussed.

A special role for numbers in working memory? An fMRI study

Although numbers carry averbal semantics (i.e., magnitude), they are often utilized in verbal Working Memory (vWM) tasks. However, vWM is thought to rely on a purely phonological code. Here, we tested the influence of (a) averbal semantics and (b) different tasks on performance in a vWM context by examining stimulus and task-specific variation of activity in the horizontal parts of the intraparietal sulcus (hIPS). The hIPS has previously been shown to subserve magnitude processing modulated by (a) specific stimuli and (b) specific tasks. Two variants of an n-back paradigm (comparison and identity match tasks) utilizing letters and numbers as stimulus material at different levels of vWM load were administered in 16 participants. Behavioral and functional imaging data reveal stimulus-specific modulation of activity in the hIPS suggesting a semantic influence of numbers. In the identity match task, numbers induced additional hIPS activation compared to letters while letters never induced additional hIPS activation when compared to numbers. Letters as compared to numbers only induced additional hIPS activation in the comparison task. These results question the assumption of a purely phonological code in vWM because hIPS activation subserving magnitude processing is modulated by stimulus semantics and task demands.

Making sense of number sense: implications for children with mathematical disabilities

Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of "number sense." I expand on their preliminary effort by examining in detail the diverse set of components purported to be encompassed by this construct. My analysis reveals some major differences between the ways in which number sense is defined in the mathematical cognition literature and its definition in the literature in mathematics education. I also present recent empirical evidence and theoretical perspectives bearing on the importance of measuring the speed of making magnitude comparisons. Finally, I discuss how differing conceptions of number sense inform the issue of whether and to what extent it may be teachable.

Brain mappings of the arithmetic processing in children and adults

Despite the increasing number of experimental mapping showing that human arithmetic cognition is supported by widely spread neural circuits; the theoretical reasoning about these data remains mostly metaphorical and guided by a connectionist approach. Although neurons at distinct areas in the brain are assumed to take charge of different duties in the solution of the experimental task, the results are always discussed by hypothesizing some association between the different areas without questioning any difference of behavior at the level of the neurons at each of these areas. Here, the brain is assumed as Distributed Intelligent Processing System (DIPS) formed by collections of loosely interacting specialized agents (neurons), each agent specializing, for example, in data collection (sensors), problem solving (associative neurons), data communication (interneuronal systems) and in acting upon the surrounding environment (motorneurons). A new technique for EEG brain mapping is proposed and used to study arithmetic cognition in elementary school aged children and adults. Factor analysis showed three distinct patterns of neuronal recruitment for arithmetic calculations in all experimental groups which varied according to the type of calculation, age and sex.