Internal number magnitude representation is not holistic,either

Over the last years, evidence has accumulated that the magnitude of two-digit numbers is not only represented as one holistic entity, but also decomposed for tens and units. Recently, Zhang and Wang (2005) suggested such separate processing may be due to the presence of external representations of numbers, whereas holistic processing became more likely when one of the to-be-compared numbers was already internalised. The latter conclusion essentially rested on unit-based null effects. However, Nuerk and Willmes (2005) argued that unfavourable stimulus selection may provoke such null effects and misleading conclusions. Therefore, we tested the conclusion of Zhang and Wang for internal standards with a modified stimulus set. We observed reliable unit-based effects in all conditions contradicting the holistic model. Thus, decomposed representations of tens and units can also be demonstrated for internal standards. We conclude that decomposed magnitude processing of multidigit numbers does not rely on external representations. Rather, even when two-digit numbers are internalised, the magnitudes of tens and units seem to be (also) represented separately.     

Number games, magnitude representation, and basic number skills in preschoolers

The effect of 3 intervention board games (linear number, linear color, and nonlinear number) on young children's (mean age = 3.8 years) counting abilities, number naming, magnitude comprehension, accuracy in number-to-position estimation tasks, and best-fit numerical magnitude representations was examined. Pre- and posttest performance was compared following four 25-min intervention sessions. The linear number board game significantly improved children's performance in all posttest measures and facilitated a shift from a logarithmic to a linear representation of numerical magnitude, emphasizing the importance of spatial cues in estimation. Exposure to the number card games involving nonsymbolic magnitude judgments and association of symbolic and nonsymbolic quantities, but without any linear spatial cues, improved some aspects of children's basic number skills but not numerical estimation precision.     

Common spatial organization of number and emotional expression: a mental magnitude line

Converging behavioral and neural evidence suggests that numerical representations are mentally organized in left-to-right orientation. Here we show that this format of spatial organization extends to emotional expression. In Experiment 1, right-side responses became increasingly faster as number (represented by Arabic numerals) or happiness (depicted in facial stimuli) increased, for judgments completely unrelated to magnitude. Additional experiments suggest that magnitude (i.e., more/less relations), not valence (i.e., positive/negative), underlies left-to-right orientation of emotional expression (Experiment 2), and that this orientation accommodates to the context-relevant emotion (e.g., happier faces are more rightward when judged on happiness, but more leftward when judged on angriness; Experiment 3). These findings show that people automatically extract magnitude from a variety of stimuli, representing such information in common left-to-right format, perhaps reflecting a mental magnitude line. We suggest that number is but one dimension in a hyper-general representational system uniting disparate dimensions of magnitude and likely subserved by common neural mechanisms in posterior parietal cortex.     

非符号数量表征和符号分数表征的关系

个体学习符号分数的一个关键是能对其数值形成准确表征。现有研究假设符号分数表征的认知基础是人类自婴幼儿期就具有的非符号数量表征(如表征两个集合各自的数量, 或两个数量的比例)。其证据包括表征非符号数量(尤其是非符号数量比例关系)和表征符号分数在行为和大脑神经活动层面上都表现出相关性。然而要说明非符号数量表征是符号分数表征的认知基础, 还需更多研究表明两者在数量概念上的独特相关和因果联系, 并阐明符号分数表征形成的认知机制。     

Extracting parity and magnitude from Arabic numerals: developmental changes in number processing and mental representation.

Children from Grades 2, 3, 4, 6, and 8 (7.8, 9.2, 9.8, 11.7, and 13. 6 years old, respectively) made speeded, bimanual parity (odd/even) judgments of the Arabic numerals 0-9. Analysis of response times indicated that from fourth grade on, parity information is retrieved directly from memory rather than being extracted by means of a mental calculation strategy. As early as Grade 3, children exhibited the SNARC (Spatial-Numerical Association of Response Codes) effect, where small numbers were responded to faster with the left hand than with the right hand, while the converse held true for large numbers. This finding, previously demonstrated only in adults, confirms that (a) children represent magnitude information in the form of a left-to-right oriented mental number line, and (b) this information is accessed obligatorily even when irrelevant. Finally, although the SNARC effect remained strong at Grade 4, it was attenuated at Grades 6 and 8 by a linguistic effect based on associations between the unmarked adjectives "even" and "right" and between the marked adjectives "odd" and "left." Copyright 1999 Academic Press.     

Sensori-motor synchronisation variability decreases as the number of metrical levels in the stimulus signal increases

Timing performance becomes less precise for longer intervals, which makes it difficult to achieve simultaneity in synchronisation with a rhythm. The metrical structure of music, characterised by hierarchical levels of binary or ternary subdivisions of time, may function to increase precision by providing additional timing information when the subdivisions are explicit. This hypothesis was tested by comparing synchronisation performance across different numbers of metrical levels conveyed by loudness of sounds, such that the slowest level was loudest and the fastest was softest. Fifteen participants moved their hand with one of 9 inter-beat intervals (IBIs) ranging from 524 to 3125 ms in 4 metrical level (ML) conditions ranging from 1 (one movement for each sound) to 4 (one movement for every 8th sound). The lowest relative variability (SD/IBI < 1.5%) was obtained for the 3 longest IBIs (1600–3125 ms) and MLs 3–4, significantly less than the smallest value (4–5% at 524–1024 ms) for any ML 1 condition in which all sounds are identical. Asynchronies were also more negative with higher ML. In conclusion, metrical subdivision provides information that facilitates temporal performance, which suggests an underlying neural multi-level mechanism capable of integrating information across levels.     

Effect of reference number on magnitude estimation

The psychophysical function obtained by the method of magnitude estimation was influenced by the reference number (modulus) assigned to a “standard” line and the position of this standard in the range of comparison stimuli. Data from two experiments with judgments of apparent length of lines show how both variables systematically affect the slope of the power function. AllowingO to choose his own reference numbers, even though these numbers varied among as, tended to produce less variability in slope than ifE imposed fixed reference numbers forO to use.     

Spontaneous number representation in mosquitofish

While there is convincing evidence that preverbal human infants and non-human primates can spontaneously represent number, considerable debate surrounds the possibility that such capacity is also present in other animals. Fish show a remarkable ability to discriminate between different numbers of social companions. Previous work has demonstrated that in fish the same set of signature limits that characterize non-verbal numerical systems in primates is present but yet to provide any demonstration that fish can really represent number rather than basing their discrimination on continuous attributes that co-vary with number. In the present work, using the method of ‘item by item’ presentation, we provide the first evidence that fish are capable of selecting the larger group of social companions relying exclusively on numerical information. In our tests subjects could choose between one large and one small group of companions when permitted to see only one fish at a time. Fish were successful when both small (3 vs. 2) and large numbers (8 vs. 4) were involved and their performance was not affected by the density of the fish or by the overall space occupied by the group.     

Polarity correspondence in comparative number magnitude judgments

When asked which of two digits is greater, participants respond more quickly if physical size corresponds to number magnitude, such as in 3 7, than when the two attributes contradict each other, such as in 3 7. This size congruence effect in comparative number judgments is a well-documented phenomenon. We extended existing findings by showing that this effect does not depend on physical size of the number alone but can be observed with number symmetry. In addition, we observed that symmetric numbers are judged as being smaller than asymmetric numbers, which renders an interpretation of the number symmetry congruence effect in terms of physical size implausible. We refer to the polarity correspondence principle (Proctor & Cho, 2006) to explain the present findings.     

A scale for the psychological magnitude of number

A scale of the “psychological magnitude” of number was constructed from similarity ratings of the 45 number pairs that can be obtained from a set of 10 integers. A nonmetric analysis of these similarity ratings showed that “psychological number” was a power function of number.     

The development of automaticity in accessing number magnitude

This study traces developmental changes in automatic and intentional processing of Arabic numerals using a numerical-Stroop paradigm. In Study 1, university students compared the numerical or physical size of Arabic numerals varying along both dimensions. In Study 2, first graders (mean age = 6 years 6 months), third graders (mean age = 8 years 4 months), and fifth graders (mean age = 10 years 3 months) were tested to examine developmental changes in numerical and physical comparisons. In the numerical comparison task, a size congruity effect was found at all ages (i.e., relative to a neutral control, congruent physical sizes facilitated, and incongruent sizes interfered with, the numerical comparison). The pattern of facilitation and interference, however, was modulated by age. In the physical comparison task, the incongruity between physical and numerical size affected only older children and adults. These findings strongly suggest that the automatization in number processing is achieved gradually as numerical skills progress.     

Forming a stable spatial representation

The visual system sometimes fails, partially or completely, to encode and/or retrieve spatial relations among parts of an object. For example, targets can easily be confused with their mirror images, especially when they must be retained in memory. In the current experiments we ask whether our representations of spatial relations can be amended by information from different cognitive domains. Specifically, we ask whether failure to form a stable representation of spatial relations among parts can be overcome by the use of linguistic information. Four year-olds saw squares split by color and matched them after delay. In Experiment 1, children saw the target and were told either “Look, this is a blicket” (Label Condition) or “Look!” (NoLabel Condition). Then, three choices appeared: the target (e.g. vertical split with red left, green right), its mirror image, and another square that had a different internal split (e.g. horizontal). Overall, children performed better than chance. However, their errors were almost exclusively mirror image confusions, suggesting that children failed to bind color and location (e.g. red left, green right). There was no difference between the NoLabel and Label conditions, suggesting the whole-object novel label did not help children form a stable representation of the spatial relation among the parts. Experiment 2 tested whether color–location binding can be improved by providing language that might bind these features. Children were shown a target and were told, e.g. “The red is on the left.” Performance was reliably better than in Experiment 1, suggesting language did help children bind color and location. Experiments 3 and 4 explored whether the same performance improvement could be accomplished by increasing non-linguistic attention to the target (i.e. flashing the red part, Experiment 3) or by using neutral relational language (e.g. “The red is touching the green”). Neither experiment showed enhanced performance, suggesting that language can augment visual–spatial representations only if it conveys very specific information (e.g. direction). Generally, the results suggest that specific linguistic information can help form a stable representation of spatial relationship and that this effect is not attributable to general attentional effects.     

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction–decimal mixed pairs and decimal–fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.     

Usefulness and limitation of spatial representation

What discussed herein is not an “open problem” in the sense of mathematics. It is a problem that psychologists should keep in mind when presenting a formal model. A model will be useful for phenomena on which the model has been formulated. However, the model may contain a number of remaining properties that not necessarily represent related psychological phenomena adequately. The situation is analogous to that the particle model of light does not represent diffraction whereas the wave model of light is not adequate for the Compton effect. When presenting a model, mathematical psychologists should be especially keen about this point. The problem is discussed with a concrete example.     

Rapid communication The mental number line dominates alternative,explicit coding of number magnitude

Numerical judgments are facilitated for left-space responses to a smaller number and right-space responses to a larger number (the spatial–numerical association of response codes, SNARC, effect). Despite support for a mental number line (i.e., spatial) explanation of the SNARC effect, this account has been challenged by an intermediate-coding theory that makes use of a polarity-correspondence principle. The latter is a general explanatory framework whereby stimulus and response dimensions are represented in a categorical, binary manner, with complementary categories coded as having either positive or negative polarity. When stimulus and response polarity match, responding is facilitated. In the present experiment we pitted explicitly presented close–far coding against an implicit mental number line (i.e., left–right coding). Subjects categorized numbers (1, 4, 6, and 9) as greater or less than a standard (5) using keys defined only as close to and far from a starting key. We found that, despite instructing subjects to use a close–far coding scheme, they exhibited a typical SNARC effect, with small-number responses facilitated on the left and large-number responses on the right. These results are discussed in the context of results supporting the polarity explanation and with respect to representational pluralism.     

数字空间表征的在线建构：来自干扰情境中数字SNARC效应的证据

尽管已有研究发现数字以空间方式表征在人类记忆系统, 但是人脑如何完成数字的空间表征尚存争议。本研究两个实验在不同比例的数字字母(实验1)和不同比例的数字汉字(实验2)混合情境中考察了数字空间表征特点及其机制, 对上述争议进行了深入研究。结果发现：(1)当数字字母比例为“1 : 1”时, 数字加工中不出现SNARC效应。当数字字母比例为“1 : 6”和“6 : 1”时, 数字加工中均出现SNARC效应。即数字字母比例与数字SNARC效应之间呈倒“U”型关系。(2)数字汉字混合情境中数字汉字比例与数字SNARC效应之间同样呈倒“U”型关系。结果说明：(1)干扰刺激与数字混合呈现会影响数字SNARC效应。(2)干扰刺激加工对数字SNARC效应的影响受到数字与干扰刺激比例的调节, 且具有跨干扰材料的稳定性。研究结果意味着数字的空间表征是人类通过统计学习在线建构的, 支持了工作记忆理论。