Sequential effects in number comparison

The modular framework of number processing (e.g., S. Dehaene & R. Akhavein, 1995) was applied to study sequential trial-to-trial effects in a number comparison task. In Experiment 1, numbers were always presented as digits. Responses were faster when the same number was repeated, but this effect was additive with the numerical distance effect. In Experiment 2, numbers were presented either as digits or as words. The authors found significant effects of repeating (a) the same physical stimulus, (b) the same number but in a different notation, and (c) the same notation but a different number. Again, all 3 effects were additive with the numerical distance effect. The authors' results provide strong evidence against accounts according to which, on stimulus repetition trials, the comparison stage is bypassed (as proposed by S. Dehaene, 1996), and the results clearly favor an early, precomparison locus of repetition effects.     

Notional number agreement in English

To investigate the contested role of notional number in English subject-verb agreement, we used a sentence completion task to examine agreement with minimally different subject noun-phrases, such as the gang on the motorcycles and the gang near the motorcycles. These contrasting phrases biased different notional construals of collective nouns, such as gang, which are normally ambiguous between plural (distributed) and singular (collected) construals. With subjects biased toward spatial distribution, such as gang on motorcycles, more plural verbs occurred in speakers’ sentence completions than in sentence completions with a bias toward spatial collection such as gang near motorcycles. This offers strong evidence regarding both the existence and the magnitude of notional effects on subject-verb number agreement in English.     

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.     

Semantic priming in number naming

The issue of semantic and non-semantic conversion routes for numerals is still debated in numerical cognition. We report two number-naming experiments in which the target numerals were preceded by another numeral (prime). The primes and targets could be presented either in arabic (digit) notation or in verbal (alphabetical) notation. The results reveal a semantically related distance effect: Latencies are fastest when the prime has the same value as the target and increase when the distance between prime and target increases. We argue that the present results are congruent with the idea that the numerals make access to an ordered semantic number line common to all notations, as the results are the same for within-notation priming (arabic-arabic or verbal- verbal) and between-notations priming (arabic-verbal or verbal-arabic). The present results also point to a rapid involvement of semantics in the naming of numerals, also when the numerals are words. As such, they are in line with recent claims of rapid semantic mediation in word naming.     

Ordinal knowledge: number names and number concepts in Chinese and English.

Previous research has demonstrated cross-language variation in early counting associated with linguistic differences in number-naming systems. Ordinal number names are typically learned later than cardinal names, but languages also differ in the regularity with which they form these names. Elementary school children in China and the U.S. showed differences in the acquisition and use of ordinal numbers corresponding to linguistic differences in ordinal names in their native languages. On tasks assessing children's conceptual knowledge of ordinal relations, a more complicated picture emerged. These results suggest that (a) children induce their language's set of ordinal number names by generalization based on rules sanctioned by early examples, and (b) the relation between ordinal names and ordinal concepts is a complex one, with language only one source of difficulty in understanding ordinal relations. Implications for studies of the relation between linguistic structure and cognitive development are discussed, in particular the possibility that effects of linguistic differences may vary for different levels of development and for different aspects of cognition.     

The approximate number system is not predictive for symbolic number processing in kindergarteners

The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.     

Facilitation of attention to number and conservation of number

This study examined the relation between number conservation and attention to number, density, and length or area in kindergarteners (age 5 years, 8 months) and third graders (age 8 years, 11 months). Attention was assessed by the dimensional preference technique, using triads. Both the attention and conservation tasks included stimulus factors known to facilitate or hinder conservation: small or large numbers of objects, nonlinear or linear arrays, and trials ordered from easiest to hardest, hardest to easiest, or randomly. The number of objects and type of array affected both tasks similarly and there was an order effect in the attention task. There was an increase in attention to number from kindergarten nonconservers to kindergarten conservers to third-graders. Possible developmental interactions between attention and conservation were suggested.     

Place information in multidigit number comparison

How do we use place information in a number comparison task involving multidigit numbers? In four experiments, subjects identified stimulus numbers as larger or smaller than the number 5,000 in a choice reaction time task. As the difference between the number of places in the stimulus number and the number of places in the standard decreased, response time and errors significantly increased. When the number of places was held constant, the type of numeric information extracted depended on the value of the standard (5,000 or 5,555). In some cases, irrelevant place information affected choice time.     

Synesthesia and number cognition in children

Grapheme-color synesthesia, when achromatic digits evoke an experience of a specific color (photisms), has been shown to be consistent, involuntary, and linked with number concept in adults, yet there have been no comparable investigations with children. We present a systematic study of grapheme-color synesthesia in children aged between 7 and 15 years. Here we show that such children (but not children with phoneme-color synesthesia) experience involuntary difficulties in numerical tasks when digits are presented in colors incongruent with their photisms. Synesthesia in children may thus have important consequences for certain aspects of numerical cognition.     

Activation of number facts in bilinguals

Two experiments examined the effect of the presentation format of numbers—digits versus word format in the first and in the second languages of bilinguals—on mental arithmetic. Speed of number-fact retrieval and the presence of interference produced by numbers that were either numerically close to or associatively related to the correct answers of stored arithmetic problems (e.g., 2+5 and 7×8) were compared across formats. The verification of true problems was increasingly slower and less accurate from the digit condition to the second-language condition. Interference was produced by both types of incorrect answers in the digit and first-language conditions, whereas in the second-language condition, it was constrained to answers that were numerically close to correct answers. Together, the results suggest that the retrieval of arithmetic facts and the automatic spreading of activation within the network of numerical facts are not only language-sensitive, but format-sensitive in general.     

Size invariance in visual number discrimination

Summary This study deals with the observer's ability to discriminate the numerosity of two random dot-patterns irrespective of their relative size. One of these two patterns was a reference one that was always composed of 32 dots randomly distributed within aK ×K invisible square window (K = 1.92°). The second one was the test pattern with one of the five magnifications (K = 0.64°, 1.28°, 1.92°, 2.56°, 3.20°) and the relative number of dots varied on 11 levels (N = –15, –12, –9, –6, –3, 0, 3, 6, 9, 12, or 15 dots). The observer's task was to indicate which of the two patterns contained more dots. The results show that the stimulus size, as an irrelevant stimulus attribute, can be ignored in the judgements about relative numerosity. This means that the perceived numerosity is size invariant, at least for a 1.6-times magnification and a 3-times reduction of the test pattern. The size invariance observed constrains the range of potential models, since the perceived numerosity can be identified only by means of a feature of the stimulus that will remain invariant after any change in the absolute stimulus size.     

Hierarchical number estimation

We investigated number estimation using dot patterns grouped by proximity into larger clusters. Participants estimated the number of dots and clusters in separate trials. Estimation was most accurate when the numbers of elements on both scales were the same. When the number of elements on the unattended scale was higher, overestimation occurred. Conversely, when the number of elements on the unattended scale was lower, underestimation occurred. In Experiment 2, response cues were blocked to reduce any tendency toward attending the irrelevant level. The results were essentially unchanged, indicating response confusion alone cannot account for the effect. The data support the existence of an opposite scale effect in which the number of elements at the unattended level influence the processing of number.The preliminary results of the two experiments were presented at the Annual Meeting of The Association for Research in Vision and Ophthalmology, April 29–May 4, 2001, in Fort Lauderdale, Florida, USA     

Possible number systems

Number systems—such as the natural numbers, integers, rationals, reals, or complex numbers—play a foundational role in mathematics, but these systems can present difficulties for students. In the studies reported here, we probed the boundaries of people’s concept of a number system by asking them whether “number lines” of varying shapes qualify as possible number systems. In Experiment 1, participants rated each of a set of number lines as a possible number system, where the number lines differed in their structures (a single straight line, a step-shaped line, a double line, or two branching structures) and in their boundedness (unbounded, bounded below, bounded above, bounded above and below, or circular). Participants also rated each of a group of mathematical properties (e.g., associativity) for its importance to number systems. Relational properties, such as associativity, predicted whether participants believed that particular forms were number systems, as did the forms’ ability to support arithmetic operations, such as addition. In Experiment 2, we asked participants to produce properties that were important for number systems. Relational, operation, and use-based properties from this set again predicted ratings of whether the number lines were possible number systems. In Experiment 3, we found similar results when the number lines indicated the positions of the individual numbers. The results suggest that people believe that number systems should be well-behaved with respect to basic arithmetic operations, and that they reject systems for which these operations produce ambiguous answers. People care much less about whether the systems have particular numbers (e.g., 0) or sets of numbers (e.g., the positives).     

Cross-lingual numerical distance priming with second-language number words in native- to third-language number word translation

An important question in research about bilingualism is whether translation from a second language (L2) to the native language (L1) is semantically mediated or whether it occurs through word associations at the lexical level. Recent research has shown that both L1 and L2 number word translation imply semantic access, suggesting strong L2 lexicosemantic mappings for number words (Duyck & Brysbaert, 2004). In the present study, this assumption was further explored by means of a cross-lingual number distance priming paradigm with Dutch–English–French trilinguals. We found that number word translation from L1 to the third language (L3), and vice versa, was facilitated by L2 number masked word primes that represented the same magnitude as the target (translation equivalents), relative to primes that are numerically less close. This confirms the existence of strong L2 lexicosemantic mappings for number words and generalizes previous semantic effects in L1–L2 translation to translation processes between L1 and L3.     

How Spencer made number: First uses of the number words

This article describes the development of number concepts between infancy and early childhood. It is based on a diary study that tracked number word use in a child from 12 to 38 months of age. Number words appeared early in the child’s vocabulary, but accurate reference to specific numerosities evolved gradually over the entire 27-month period. Reliable performance on standardized number and counting tasks lagged behind accurate number use in informal settings by many months. Although there was evidence that this child labeled mentally represented sets, these instances comprised a very small proportion of the total sample and were usually references to anticipated sets rather than remembered sets.     

Large number discrimination in 6-month-old infants

Two experiments with the Stroop paradigm were conducted to investigate the role of phonological codes in access to the meaning of Chinese characters. Subjects named the ink color of viewed characters or color patches. Key items were color characters, their homophones with the same tone, homophones with different tones, and semantic associates. Apart from finding the usual Stroop interference effect, homophones produced significant interference in the incongruent condition, provided that they had the same tone as the color characters. The interference effect from homophones, however, was significantly smaller than that from color characters. Semantic associates generated an interference effect in the incongruent condition, an effect of the same magnitude as the effect from the same-tone homophones. Finally, in the congruent conditions, all the key items yielded facilitations compared to neutral controls, though the facilitation from color characters was larger than the facilitations from other types of characters. These findings suggest that phonological codes in Chinese are activated obligatorily and provide early sources of constraint in access to meaning.     

Temporal integration in duration and number discrimination

Temporal integration in duration and number discrimination by rats was investigated with the use of a psychophysical choice procedure. A response on one lever ("short" response) following a 1-s white-noise signal was followed by food reinforcement, and a response on the other lever ("long" response) following a 2-s white-noise signal was also followed by food reinforcement. Either response following a signal of one of five intermediate durations was unreinforced. This led to a psychophysical function in which the probability of a long response was related to signal duration in an ogival manner. On 2 test days, a white-noise signal with 5, 6, 7, 8, or 10 segments of either 0.5-s on and 0.5-s off or 1-s on and 1-s off was presented, and a choice response following these signals was unreinforced. The probability of a long response was the same function of a segmented signal and a continuous signal if each segment was considered equivalent to 200 ms. A quantitative fit of a scalar estimation theory suggested that the latencies to initiate temporal integration and to terminate the process are both about 200 ms, and that the same internal accumulation process can be used for counting and timing.