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.     

Effects of number of items on the baboon's discrimination of same from different visual displays

Three experiments explored the baboon's discrimination of visual displays that comprised 2 to 24 black-and-white computer icons; the displayed icons were either the same as (same) or different from one another (different). The baboons' discrimination of same from different displays was a positive function of the number of icons. When the number of icons was decreased to 2 or 4, the baboons responded indiscriminately to the same and different displays, exhibiting strong position preferences. These results are both similar to and different from those of pigeons that were trained and tested under comparable conditions. Accepted after revision: 23 May 2001 Electronic Publication     

Number sense and quantifier interpretation

We consider connections between number sense—the ability to judge number—and the interpretation of natural language quantifiers. In particular, we present empirical evidence concerning the neuroanatomical underpinnings of number sense and quantifier interpretation. We show, further, that impairment of number sense in patients can result in the impairment of the ability to interpret sentences containing quantifiers. This result demonstrates that number sense supports some aspects of the language faculty.

新蔡葛陵楚卜筮简中的繇辞解析

新蔡葛陵楚卜筮简中记录了战国时期楚人占筮时的卦画和繇辞,本文尝试利用卦象对繇辞进行解析,结果表明繇辞与卦象大体相符。这种解卦法,也见于《左传》和《国语》,可见战国楚地楚人的筮占是用《周易》,与春秋筮法有一脉相承的关系。竹简繇辞不见今本《周易》,当是民间的另一种传本。     

Basic numerical skills in children with mathematics learning disabilities: a comparison of symbolic vs non-symbolic number magnitude processing

Forty-five children with mathematics learning disabilities, with and without comorbid reading disabilities, were compared to 45 normally achieving peers in tasks assessing basic numerical skills. Children with mathematics disabilities were only impaired when comparing Arabic digits (i.e., symbolic number magnitude) but not when comparing collections (i.e., non-symbolic number magnitude). Moreover, they automatically processed number magnitude when comparing the physical size of Arabic digits in an Stroop paradigm adapted for processing speed differences. Finally, no evidence was found for differential patterns of performance between MD and MD/RD children in these tasks. These findings suggest that children with mathematics learning disabilities have difficulty in accessing number magnitude from symbols rather than in processing numerosity per se.     

The differentiation of response numerosities in the pigeon

Two experiments examined how pigeons differentiate response patterns along the dimension of number. In Experiment 1, 5 pigeons received food after pecking the left key at least N times and then switching to the right key (Mechner's Fixed Consecutive Number schedule). Parameter N varied across conditions from 4 to 32. Results showed that run length on the left key followed a normal distribution whose mean and standard deviation increased linearly with N; the coefficient of variation approached a constant value (the scalar property). In Experiment 2, 4 pigeons received food with probability p for pecking the left key exactly four times and then switching. If that did not happen, the pigeons still could receive food by returning to the left key and pecking it for a total of at least 16 times and then switching. Parameter p varied across conditions from 1.0 to .25. Results showed that when p= 1.0 or p=.5, pigeons learned two response numerosities within the same condition. When p=.25, each pigeon adapted to the schedule differently. Two of them emitted first runs well described by a mixture of two normal distributions, one with mean close to 4 and the other with mean close to 16 pecks. A mathematical model for the differentiation of response numerosity in Fixed Consecutive Number schedules is proposed.     

Levels of number knowledge during early childhood

Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children’s incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the ‘discontinuity’ account of number development in general and the ‘knower-levels’ account in particular.     

Operational momentum in large-number addition and subtraction by 9-month-olds

Recent studies on nonsymbolic arithmetic have illustrated that under conditions that prevent exact calculation, adults display a systematic tendency to overestimate the answers to addition problems and underestimate the answers to subtraction problems. It has been suggested that this operational momentum results from exposure to a culture-specific practice of representing numbers spatially; alternatively, the mind may represent numbers in spatial terms from early in development. In the current study, we asked whether operational momentum is present during infancy, prior to exposure to culture-specific representations of numbers. Infants (9-month-olds) were shown videos of events involving the addition or subtraction of objects with three different types of outcomes: numerically correct, too large, and too small. Infants looked significantly longer only at those incorrect outcomes that violated the momentum of the arithmetic operation (i.e., at too-large outcomes in subtraction events and too-small outcomes in addition events). The presence of operational momentum during infancy indicates developmental continuity in the underlying mechanisms used when operating over numerical representations.     

Place‐value understanding in number line estimation predicts future arithmetic performance

In multi‐digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place‐value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place‐value understanding in first grade. We hypothesized that estimating the position of two‐digit numbers requires place‐value understanding. Therefore, we fitted a linear function to children's estimates of two‐digit numbers and considered the resulting slope as a measure of children's place‐value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place‐value understanding. Additionally, the slope for two‐digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first‐graders' place‐value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place‐value understanding) may drive the association between number line estimation and arithmetic performance.