Children's number processing is context dependent

The aim of this paper was to test the hypothesis of a context dependence of number processing in children. Fifth-graders were given two numbers presented successively on screen through a self-presentation procedure after being asked either to add or subtract or compare them. We considered the self-presentation time of the first number as reflecting the complexity of the encoding for a given planned processing. In line with Dehaene's triple-code model, self-presentation times were longer for additions and subtractions than for comparisons with two-digit numbers. Alternative interpretations of these results in terms of more cognitive effort or more mental preparation in the case of addition and subtraction than comparison are discussed and ruled out.     

Better elementary number processing in higher skill arithmetic problem solvers: Evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Better elementary number processing in higher skill arithmetic problem solvers: evidence from the encoding step

This paper addresses the relationship between basic numerical processes and higher level numerical abilities in normal achieving adults. In the first experiment we inferred the elementary numerical abilities of university students from the time they needed to encode numerical information involved in complex additions and subtractions. We interpreted the shorter encoding times in good arithmetic problem solvers as revealing clearer or more accessible representations of numbers. The second experiment shows that these results cannot be due to the fact that lower skilled individuals experience more maths anxiety or put more cognitive efforts into calculations than do higher skilled individuals. Moreover, the third experiment involving non-numerical information supports the hypothesis that these interindividual differences are specific to number processing. The possible causal relationships between basic and higher level numerical abilities are discussed.     

Number line estimation and complex mental calculation: Is there a shared cognitive process driving the two tasks?

It is widely accepted that different number-related tasks, including solving simple addition and subtraction, may induce attentional shifts on the so-called mental number line, which represents larger numbers on the right and smaller numbers on the left. Recently, it has been shown that different number-related tasks also employ spatial attention shifts along with general cognitive processes. Here we investigated for the first time whether number line estimation and complex mental arithmetic recruit a common mechanism in healthy adults. Participants’ performance in two-digit mental additions and subtractions using visual stimuli was compared with their performance in a mental bisection task using auditory numerical intervals. Results showed significant correlations between participants’ performance in number line bisection and that in two-digit mental arithmetic operations, especially in additions, providing a first proof of a shared cognitive mechanism (or multiple shared cognitive mechanisms) between auditory number bisection and complex mental calculation.     

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.     

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.     

大小判断任务中正负号及其异同对SNARC效应的影响

SNARC效应(Spatial-Numerical Association of Response Codes)是指被试对数字做按键反应时,对于较小的数字,按左键的速度快于按右键;对于较大的数字,按右键的速度快于按左键。本研究以ERP作为测量手段,采用修正的大小判断任务,旨在探究数字正负号及其异同对SNARC效应的影响。行为结果发现,在反应时上,当目标数字与基线数字正负号相同且基线数字为+5时,一致条件显著快于不一致条件。ERP结果发现,当目标数字与基线数字正负号相同时,无论基线数字为+5还是–5,在反应选择阶段,不一致都比一致条件更负且均诱发了P3。当目标数字与基线数字正负号相异时,若基线数字为+5,一致比不一致条件在刺激呈现阶段诱发了波幅显著更小的N300;若基线数字为–5,一致比不一致条件在反应执行阶段诱发了更正的LPP。无论目标数字与基线数字正负号相同还是相异,在反应选择阶段,不一致都比一致条件更负且均诱发了P3,表明出现了SNARC效应。同时,SNARC效应的出现激活了额叶头皮位置,负数加工伴随左额叶的激活,而正数加工伴随右额叶的激活,溯源分析结果进一步表明SNARC效应定位于额叶与顶叶。这些结果说明负数按实际大小表征在心理数字线上,支持了负数空间表征的个体发展论假说;表明符号捷径机制会改变SNARC效应的发生时间;同时证明了负数与正数的空间表征具有不同的优势半球。     

Cognitive control in number magnitude processing: evidence from eye-tracking

The unit-decade compatibility effect describes longer response times and higher error rates for incompatible (e.g., 37_52) than compatible (e.g., 42_57) number comparisons. Recent research indicated that the effect depends on the percentage of same-decade filler items. In the present study, we further examined this relationship by recording participants’ eye-fixation behaviour. In four conditions, participants had to compare item sets with different filler item types (i.e., same-decade and same-unit filler items) and different numbers of same-decade filler items (i.e., 25, 50, and 75 %). We found a weaker unit-decade compatibility effect with most fixations on tens in the condition with same-unit filler items. Moreover, the compatibility effect increased with the percentage of same-decade filler items which was accompanied by less fixations on tens and more fixations on units. Thus, our study provides first eye-tracking evidence for the influence of cognitive control in number processing.     

The syllable-length effect in number processing is task-dependent

Two experiments were run in order to investigate the relationship between syllable length of number names and eye-fixation durations during silent reading of one- and two-digit numbers. In Experiment 1, subjects had to read a series of three numbers and recall them orally; in Experiment 2, subjects had to indicate manually whether the value of the middle number was between the values of the outer numbers. The effect of syllable length was controlled for possible confounding effects of number frequency and number magnitude. Findings indicated that fixation duration depended on syllable length of number names in the first task, but not in the second task. The results call into question the claim that phonological encoding is imperative in visual processing; phonological encoding was used only when the numbers had to be recalled, and not when they were coded for computational purposes.     

Gaze orientation interferes with mental numerical representation

Number comparison tasks are characterized by distance and size effects. The distance effect reveals that the higher the distance is between two numbers, the easier their magnitude comparison is. Accordingly, people are thought to represent numbers on a spatial dimension, the mental number line, on which any given number corresponds to a location on the line. The size effect, instead, states that at any given distance, comparing two small numbers is easier than comparing two large numbers, thus suggesting that larger numbers are more vaguely represented than smaller ones. In the present work we first tested whether the participants were adopting a spatial strategy to solve a very simple numbers comparison task, by assessing the presence of the distance and the magnitude effect. Secondarily, we focused on the influence of gaze position on their performance. The present results provide evidence that gaze direction interferes with number comparisons, worsening the vague representation of larger numbers and further supporting the hypothesis of the overlapping between physical and mental spaces.     

The syllable-length effect in number processing is task-dependent.

Two experiments were run in order to investigate the relationship between syllable length of number names and eye-fixation durations during silent reading of one- and two-digit numbers. In Experiment 1, subjects had to read a series of three numbers and recall them orally; in Experiment 2, subjects had to indicate manually whether the value of the middle number was between the values of the outer numbers. The effect of syllable length was controlled for possible confounding effects of number frequency and number magnitude. Findings indicated that fixation duration depended on syllable length of number names in the first task, but not in the second task. The results call into question the claim that phonological encoding is imperative in visual processing; phonological encoding was used only when the numbers had to be recalled, and not when they were coded for computational purposes.     

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.     

Use of indirect addition in adults' mental subtraction in the number domain up to 1,000

This study examined adults' use of indirect addition and direct subtraction strategies on multi-digit subtractions in the number domain up to 1,000. Seventy students who differed in their level of arithmetic ability solved multi-digit subtractions in one choice and two no-choice conditions. Against the background of recent findings in elementary subtraction, we manipulated the size of the subtrahend compared to the difference and only selected items with large distances between these two integers. Results revealed that adults frequently and efficiently apply indirect addition on multi-digit subtractions, yet adults with higher arithmetic ability performed more efficiently than those with lower arithmetic ability. In both groups, indirect addition was more efficient than direct subtraction both on subtractions with a subtrahend much larger than the difference (e.g., 713 - 695) and on subtractions with a subtrahend much smaller than the difference (e.g., 613 - 67). Unexpectedly, only adults with lower arithmetic ability fitted their strategy choices to their individual strategy performance skills. Results are interpreted in terms of mathematical and cognitive perspectives on strategy efficiency and adaptiveness.     

Measuring effects of voluntary attention: A comparison among predictive arrow,colour, and number cues

The Posner cueing paradigm is one of the most widely used paradigms in attention research. Importantly, when employing it, it is critical to understand which type of orienting a cue triggers. It has been suggested that large effects elicited by predictive arrow cues reflect an interaction of involuntary and voluntary orienting. This conclusion is based on comparisons of cueing effects of predictive arrows, nonpredictive arrows (involuntary orienting), and predictive numbers (voluntary orienting). Experiment 1 investigated whether this conclusion is restricted to comparisons with number cues and showed similar results to those of previous studies, but now for comparisons to predictive colour cues, indicating that the earlier conclusion can be generalized. Experiment 2 assessed whether the size of a cueing effect is related to the ease of deriving direction information from a cue, based on the rationale that effects for arrows may be larger, because it may be easier to process direction information given by symbols such as arrows than that given by other cues. Indeed, direction information is derived faster and more accurately from arrows than from colour and number cues in a direction judgement task, and cueing effects are larger for arrows than for the other cues. Importantly though, performance in the two tasks is not correlated. Hence, the large cueing effects of arrows are not a result of the ease of information processing, but of the types of orienting that the arrows elicit.     

Face recognition memory: The effects of exposure duration and encoding instruction

Two experiments test the effects of exposure duration and encoding instruction on the relative memory for five facial features. Participants viewed slides of Identi-kit faces and were later given a recognition test with same or changed versions of each face. Each changed test face involved a change in one facial feature: hair, eyes, chin, nose or mouth. In both experiments the upper-face features of hair and eyes were better recognized than the lower-face features of nose, mouth, and chin, as measured by false alarm rates. In Experiment 1, participants in the 20-second exposure duration condition remembered faces significantly better than participants in the 3-second exposure duration condition; however, memory for all five facial features improved at a similar rate with the increased duration. In Experiment 2, participants directed to use feature scanning encoding instructions remembered faces significantly better than participants following age judgement instructions; however, the size of the memory advantage for upper facial features was less with feature scanning instructions than with age judgement instructions. The results are discussed in terms of a quantitative difference in processing faces with longer exposure duration, versus a qualitative difference in processing faces with various encoding instructions. These results are related to conditions that affect the accuracy of eyewitness identification.     

It’s not just what you say but when you say it: Self-presentation and temporal construal

Three experiments demonstrated that the use and effectiveness of self-presentation strategies are affected by time. In Experiment 1, participants used more indirect self-presentation statements for the distant than near future, but used more direct self-presentation statements for the near than distant future. In Experiment 2, participants for whom indirect self-presentation strategies were made accessible rated a future interview as more temporally distant than those for whom direct self-presentation strategies were made accessible. In Experiment 3, participants rated their self-presentation attempts as more effective if they used direct strategies for the near future and indirect strategies for the distant future. Implications for studying the timing of self-presentation and its relation to temporal construal levels are discussed.     

Arithmetic split effects reflect strategy selection: an adult age comparative study in addition comparison and verification tasks.

We tested whether split effects in arithmetic (i.e., better performance on large-split problems, like 3 + 8 = 16, than on small-split problems, like 3 + 8 = 12) reflect decision processing or strategy selection. To achieve this end, we tested performance of younger and older adults, matched on arithmetic skills, on two arithmetic tasks: the addition/number comparison task (e.g., 4 + 8, 13; which item is the larger?) and in the inequality verification task (e.g., 4 + 8 < 13; Yes/No?). In both tasks, split between additions and proposed numbers were manipulated. We also manipulated the difficulty of the additions, which represents an index of arithmetic fact calculation (i.e., hard problems, like 6 + 8 < 15, are solved more slowly than easy problems, like 2 + 4 < 07, suggesting that calculation takes longer). Analyses of latencies revealed three main results: First, split effects were of smaller magnitude in older adults compared to younger adults, whatever the type of arithmetic task; second, split effects were of smaller magnitude on easy problems; and third, calculation processes were well maintained in older adults with high level of arithmetic skills. This set of results improves our understanding of cognitive aging and strategy selection in arithmetic.     

On the mental representation of negative numbers: Context-dependent SNARC effects with comparative judgments

In one condition, positive and negative number pairs were compared in separate blocks of trials. In another condition, the positive and the negative number pairs were intermixed. In the intermixed condition, comparisons involving negative numbers were faster with the left hand than with the right, and comparisons were faster with the right hand than with the left hand with the positive numbers; that is, a spatial numerical association of response codes (SNARC) effect was obtained, in which the mental number line was extended leftward with the negative numbers. On the other hand, in the blocked condition, a reverse SNARC effect was obtained with the negative numbers; that is, negative number pairs have the same underlying spatial representation as the positive numbers in this context. Nongraded semantic congruity effects, obtained in both the blocked and the intermixed conditions, are consistent with the idea that magnitude information is extracted prior to the generation of discrete semantic codes.     

On the perceptual generality of the unit-decade compatibility effect

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

Processing of order information for numbers and months

Despite a great deal of research on the processing of numerical magnitude (e.g., the quantity denoted by the number 5), few studies have investigated how this magnitude information relates to the ordinal properties of numbers (e.g., the fact that 5 is the fifth integer). In the present study, we investigated order-related processing of numbers, as well as months of the year, with a novel ordering task to see whether the processing of order information differs from the processing of magnitude information. In Experiments 1 and 2, participants were shown three numbers (Experiment 1) or three months (Experiment 2) and were required to indicate whether the stimuli were in the correct order. In Experiment 3, participants were again shown three numbers; however, now they were instructed to indicate whether the three numbers were ordered in a forward, backward, or mixed direction. Whereas number comparison tasks typically reveal distance effects (comparisons become easier with increased distance between two numbers), these three experiments reveal a different pattern of results. There were reverse distance effects when the stimuli crossed a boundary (i.e., when numbers crossed a decade or months crossed the year boundary) and no effect of distance when the stimuli did not cross a boundary (i.e., when numbers were within a decade and months were within the January—December calendar year). These data suggest that additional mechanisms are involved in the processing of order information: a scanning mechanism and a long-term memory checking mechanism.