Notation-dependent And Notation-independent Representation Of Common And Decimal Fractions

Numerical representation refers to the mental representation of a given number. Though the perspective of abstract representation has predominated for a long time, recent evidence shows the notation-dependent representation of number with the development of automatic processing paradigms and brain imaging techniques. There were two theories which assumed that the notation-independent and notation-dependent representations might coexist in the number processing. The computational model proposed that the summation code was notation-dependent, whereas the place code was notation-independent for both non-symbolic and symbolic number. The extension of dual code theory suggested that numerical information was represented internally by way of automatic and intentional codes. In the first stage, there was an automatic activation of the numerical quantity that was notation-dependent. Then the representation could be transferred to an on-line abstract representation on demands of tasks at the stage of intentional code. Although those assumptions were supported by extensive behavioral and neuropsychological studies, the internal mechanism of numerical representation focused on integer, little is known about that of fractions as the higher levels of numerals.This study is the extension of the numerical representation of integers to explore the issue of notation-dependent processing, by a comparison of numerical representations of common and decimal fractions. Therefore, event-related potentials (ERPs) were measured in the brains carrying out a magnitude matching task with the aim of investigating numerical processing of non-symbolic fractions, common and decimal fractions in study1. The study2includes two experiments, based on the two dimension of the "paradigm" and "awareness level of processing", using the numerical/physical comparison task and numerical/physical same-different judgment task to investigate the notation-dependent mechanism underlying the numerical representation, respectively under the modes of "automatic processing" and "intentional processing.Study1:A non-symbolic fraction was presented to participants, who were instructed to judge whether its magnitude matched the magnitude of a subsequent fraction presented in symbolic notation. Behavioral results showed significant main effects of notation and distance, with shorter RTs for decimal (compared to common) fractions and for far (compared to close) distances. No significant interaction of notation and distance was observed. Electrophysiological data reflected notation-dependent visual perceptual processing for both symbolic and non-symbolic fractions. Significant notation effects were observed on N1component with larger amplitudes or shorter latencies for common fractions. In addition, the effect of numerical distance as an index of numerical magnitude representation was observed on the amplitude of N1during120-170ms. This effect was also found on P2p and P3components, with larger amplitudes and longer latencies for numerically closer distances. Consistent with behavioral results, electrophysiological results did not reveal significant interaction of distance by notation which suggested that processing numerical magnitudes was notation-independent.Study2:a numerical/physical comparison paradigm was adopted in the experiment1. The two symbolic numbers were presented on the two sides at the same time. In the numerical comparison task, participants are instructed to judge the numerical magnitude while ignoring physical size (intentional processing of numerical magnitude). In the physical comparison task, participants are instructed to judge the physical size while ignoring numerical magnitude (automatic processing of numerical magnitude). The experimental results showed that:(1) common and decimal fractions exist notation-dependent numerical processing on the size congruency effect (SCE). Only the condition of decimal fractions was observed longer reaction time and lower accuracy rate in the incongruent condition (when the physically larger digit is numerically smaller) than in the congruent condition (when the physically larger digit is also numerically larger).(2) notation-dependent processing did not exist on the distance effect, which was detected only in the intentional processing task. Numerical/physical same-different paradigm was used in experiment2. In the numerical same-different task, participants were instructed to decide whether the two numbers successively presented were the same in terms of their values, irrespective of notations (intentional processing). In the physical same-different task, participants were instructed to decide whether the two numbers were physically identical (automatic processing). The experimental results showed that:(1) the notation-dependent processing for common and decimal fractions was observed on the distance effect, only absent in the condition of pure decimal fractions under the intentional processing and only existed in the condition of pure decimal fractions under the automatic processing.(2) The value interference for common fractions was observed in the condition without distance effect under the automatic processing. Because in the mixed conditions of common and decimal fractions, the numbers’numerical value impaired participants’"different" responses to different-notation pairs with the same numerical values compared with those with different numerical values.To sum up the results, we found that:(1) the notation-independent and notation-dependent representations coexist in the number processing.(2) Task paradigm and awareness level of processing would affect the notation-dependent representation of numbers.(3) The numerical representation includes two stages. Its semantic representation was vulnerable to be influenced by the task demand and manifest abstract representation. The automatic processing task was sensitive to the notation-dependent representation under the perceptual asemantic processing before the semantic stage.