Automaticity and two-digit numbers.

Automatic processing of 2-digit numbers was demonstrated using the size congruency effect (SiCE). The SiCE indicates the processing of the irrelevant (numerical) dimension when 2 digits differing both numerically and physically are compared on the relevant (physical) dimension. The SiCE was affected by the compatibility between unit and decade digits but was unaffected by the global magnitude of the numbers. Together these results suggest automatic processing of the magnitudes of the components of the 2-digit numbers but not of whole numbers. Finally, the SiCE was affected more by the magnitude of the decade digits compared with the unit digits, indicating that the syntactic roles of the digits were represented. The implications of these results for understanding the numerical representations are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Are Arabic and verbal numbers processed in different ways?

Four experiments were conducted in order to examine effects of notation--Arabic and verbal numbers--on relevant and irrelevant numerical processing. In Experiment 1, notation interacted with the numerical distance effect, and irrelevant physical size affected numerical processing (i.e., size congruity effect) for both notations but to a lesser degree for verbal numbers. In contrast, size congruity had no effect when verbal numbers were the irrelevant dimension. In Experiments 2 and 3, different parameters that could possibly affect the results, such as discriminability and variability (Experiment 2) and the block design (Experiment 3), were controlled. The results replicated the effects obtained in Experiment 1. In Experiment 4, in which physical size was made more difficult to process, size congruity for irrelevant verbal numbers was observed. The present results imply that notation affects numerical processing and that Arabic and verbal numbers are represented separately, and thus it is suggested that current models of numerical processing should have separate comparison mechanisms for verbal and Arabic numbers. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Attention, automaticity, and levels of representation in number processing.

Participants performed same–different judgments for pairs of numerals in 2 conditions: numerical matching (responding "same" to pairs such as 2–TWO), or physical matching (responding "different" to pairs such as 2–TWO). In most cases, a distance effect was obtained, with the different responses being slower when the 2 numbers were numerically close together (e.g., 1–2) than when they were further apart (e.g., 1–8). This indicates that numbers were automatically converted mentally into quantities, even when the participants had been told to attend exclusively to their physical characteristics. As postulated by several models of number processing, (e.g., Dehaene, see PA, Vol 80:4300; McCloskey, see PA, Vol 80:6390) Arabic and verbal numerals thus appear to converge toward a common semantic representation of quantities. However, the present results suggest that an asemantic transcoding route might allow for a direct mapping of Arabic and verbal numbers, perhaps by means of a common phonological representation. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Primed numbers: Exploring the modularity of numerical representations with masked and unmasked semantic priming.

In Experiment 1, participants classified a 1st number (prime) as smaller or larger than 5 and then performed the same task again on a 2nd number (target). In Experiment 2, participants classified a target number as smaller or larger than 5, while unknown to them a masked number was displayed for 66 ms prior to the target. Primes and targets appeared in Arabic notation, in verbal notation, or as random dot patterns. Two forms of priming were analyzed: quantity priming (a decrease in response times with the numerical distance between prime and target) and response priming (faster responses when the prime and target were on the same side of 5 than when they were not). Response priming transferred across notations, whereas quantity priming generally did not. Under conditions of speeded processing, the internal representation of numerical quantities seems to dissociate into multiple notation-specific subsystems. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Count me in! On the automaticity of numerosity processing.

Extraction of numerosity (i.e., enumeration) is an essential component of mathematical abilities. The current study asked how automatic is the processing of numerosity and whether automatic activation is task dependent. Participants were presented with displays containing a variable number of digits and were asked to pay attention to the number of digits or to their numerical value, in separate blocks. Effects of task were tested with a comparative judgment task and a parity judgment task. In the comparative judgment task, participants had to indicate whether the numerosity or the numerical value of the digits was smaller or larger than 5. In the parity judgment task, participants had to indicate whether this value was odd or even. Irrelevant numerical values modulated performance regardless of task. In contrast, irrelevant numerosities modulated performance only in the comparative judgment task. These results suggest that numerical value is activated automatically, whereas activation of numerosity is modulated by task. We suggest that the differences observed are related to different cognitive and neural mechanisms recruited by these tasks. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A generalized fraction: An entity smaller than one on the mental number line.

The representation of fractions in long-term memory (LTM) was investigated by examining the automatic processing of such numbers in a physical comparison task, and their intentional processing in a numerical comparison task. The size congruity effect (SiCE) served as a marker of automatic processing and consequently as an indicator of the access to the primitives of numerical representation in LTM. Mixed pairs composed of a natural number and a fraction showed both a SiCE and a distance effect. The SiCE for mixed pairs was stable across relative sizes of natural numbers compared to the fraction digits (Experiment 4). However, comparing pairs of fractions revealed a strong influence of fractional components: An inverse SiCE was found for pairs of unit fractions (Experiment 1), while no SiCE was found for pairs of non-unit fractions (Experiments 2–3). This leads to the conclusions that: (1) there are no unique representations of distinct fraction values in LTM, and (2) there is a representation of a “generalized fraction” as an “entity smaller than one” that emerges from the notational structure common to all fractions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Sequential analysis of the numerical Stroop effect reveals response suppression.

Automatic processing of irrelevant stimulus dimensions has been demonstrated in a variety of tasks. Previous studies have shown that conflict between relevant and irrelevant dimensions can be reduced when a feature of the irrelevant dimension is repeated. The specific level at which the automatic process is suppressed (e.g., perceptual repetition, response repetition), however, is less understood. In the current experiment we used the numerical Stroop paradigm, in which the processing of irrelevant numerical values of 2 digits interferes with the processing of their physical size, to pinpoint the precise level of the suppression. Using a sequential analysis, we dissociated perceptual repetition from response repetition of the relevant and irrelevant dimension. Our analyses of reaction times, error rates, and diffusion modeling revealed that the congruity effect is significantly reduced or even absent when the response sequence of the irrelevant dimension, rather than the numerical value or the physical size, is repeated. These results suggest that automatic activation of the irrelevant dimension is suppressed at the response level. The current results shed light on the level of interaction between numerical magnitude and physical size as well as the effect of variability of responses and stimuli on automatic processing. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Automatic Activation of Internal Magnitudes: A Study of Developmental Dyscalculia.

The association between Arabic numerals and the representations of magnitude in adult developmental dyscalculia was examined. University students compared physical size, vertical positioning (height), or grayness (different shades of gray) of 2 Arabic numerals. The numerical values could produce a Stroop-like numerical congruity effect (NCE; 3-5 vs. 3-5). The dyscalculia group did not show NCE in the grayness task, and their physical comparisons produced a significantly smaller NCE compared with that produced by the control group. Whereas previous research suggested that Arabic numerals activate representations of magnitude automatically, the results of this study indicate that this is not the case because (a) people with developmental dyscalculia require attention to associate internal representations of magnitude with Arabic numerals, and (b) activation of internal magnitudes depends on context (task). (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

An Encoding-Complex Approach to Numerical Cognition in Chinese-English Bilinguals.

We present a model of the cognitive architecture of basic numerical skills in adult Chinese-English bilinguals. The model is based on data reported by Campbell, Kanz, and Xue (1999) and combines Dehaene and Cohen's triple-code theory with Campbell and Clark's encoding-complex approach to modeling number processing. Participants were required to name, add or multiply Arabic or Mandarin numerals and to respond in English or Chinese. They also performed magnitude comparisons on pairs of Arabic or Mandarin numerals. The proposed model of their performance on this set of tasks assumes 1) that number processing is modular with respect to representational code (e.g., visual, visuo-spatial, verbal) rather than with respect to numerical function, 2) task-specific communication between representational codes is interactive rather than additive, and 3) memory for arithmetic facts is at least partially language- based and our Chinese-English bilinguals possessed both Chinese and English-language number-fact representations. We provide new analyses of the arithmetic data and a review of research on the role of language in simple arithmetic to substantiate our claims about linguistic codes for number-fact memory. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Interhemispheric stroop-like interference in number comparison: Evidence for strong interhemispheric integration of semantic number information.

Three experiments investigated interhemispheric interactions in number comparison using the interhemispheric Stroop-like paradigm (E. Ratinckx, M. Brysbaert, & B. Reynvoet, 2001). In all experiments, a target was presented in 1 visual field simultaneously with a distractor in the other visual field. In Experiment 1, both target and distractor were of the same modality (Arabic digits), whereas in Experiment 2, target and distractor were of different modalities (Arabic digits and word numerals). In Experiment 3, the interhemispheric Stroop-like task of Experiment 1 was combined with intrahemispheric conditions to evaluate the strength of the interhemispheric interactions. Overall, the results point to strong interhemispheric integration during semantic access and response preparation with very weak lateralization of the semantic number system. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

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. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Getting a grip on numbers: Numerical magnitude priming in object grasping.

To investigate the functional connection between numerical cognition and action planning, the authors required participants to perform different grasping responses depending on the parity status of Arabic digits. The results show that precision grip actions were initiated faster in response to small numbers, whereas power grips were initiated faster in response to large numbers. Moreover, analyses of the grasping kinematics reveal an enlarged maximum grip aperture in the presence of large numbers. Reaction time effects remained present when controlling for the number of fingers used while grasping but disappeared when participants pointed to the object. The data indicate a priming of size-related motor features by numerals and support the idea that representations of numbers and actions share common cognitive codes within a generalized magnitude system. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Stroop and Garner effects in comparative judgment of numerals: The role of attention.

In 7 experiments, participants selected the larger member of pairs of digits that differed in numerical magnitude as well as in physical size. Selective attention to the relevant dimension (number or size) was gauged by Garner and Stroop interference, both of which varied considerably as a function of the number and relative discriminability of values along the constituent dimensions. When the to-be-ignored dimension was more discriminable, sizable Garner and Stroop effects affected performance on the relevant dimension. When it was less discriminable, Garner and Stroop effects were considerably smaller regardless of whether the relevant dimension was numerical or physical size. The sensitivity of Stroop interference to manipulations of discriminability is accounted for by the allocation of attention to the constituent dimensions. The demonstrated malleability of the Stroop effect is incompatible with claims of strong automaticity in numerical processing. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Serial and parallel processes in working memory after practice.

Six young adults practiced for 36 sessions on a working-memory updating task in which 2 digits and 2 spatial positions were continuously updated. Participants either did 1 updating operation at a time, or attempted 1 numerical and 1 spatial operation at the same time. In contrast to previous research using the same paradigm with a single digit and a single dot, dual-task costs were not eliminated with practice. Costs of switching between digits and between spatial positions were found throughout practice, supporting the existence of a focus of attention in working memory that can hold 1 digit and 1 spatial position simultaneously, but is not expanded to hold 2 elements of the same kind. The results can be understood by assuming that observed limits on parallel processing, as well as on the capacity of the focus of attention, arise not from structural constraints but rather reflect the optimal configuration of the cognitive system for avoiding information cross-talk in a given task. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Symbolic distance between numerosity and identity modulates Stroop interference

Thirty undergraduates participated in an experiment investigating the effect of the arithmetic difference between stimulus identity and stimulus numerosity in a numerical version of the Stroop task. It was found that digits symbolically close to the enumeration response reliably produced larger interference than digits that were farther from the enumeration response. This semantic distance effect was found with different numerosities (1-9) and different enumeration processes (counting and subitizing), and it increased as a function of numerosity in the subitizing range. These findings suggest that digit identity autonomously activates a magnitude representation organized as a compressed number line.     

Interactive Effects of Numerical Surface Form and Operand Parity in Cognitive Arithmetic.

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Role of an abstract order schema in conceptual judgment.

Ss performed same-different judgments for which order information was logically irrelevant. In Exps 1 and 2, the stimuli for the judgment task were derived from an ordered set of concepts from long-term memory (US presidents ordered on the dimension of historical time); in Exps 3 and 4 the stimuli were derived from 2 ordered sets of concepts. In the stimulus set for each experiment, there were several associate phrases for each concept and the task was to judge whether the phrases were paired with the same concept (Exps 1, 2, and 3) or the same ordering (Exp 4). The time to respond "different" decreased with the ordinal distance between the concepts even when the concepts belonged to different orderings. It is concluded that same-different judgments are based in part on amodal order information (i.e., not tied to any particular dimension). Two models of the implicated order schema are proposed and tested. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Separating consciously controlled and automatic influences in memory for new associations.

The process dissociation paradigm was applied to investigate the contributions of automatic and consciously controlled processes to the repetition priming effect for new associations, under elaborative encoding (Experiments 1 and 2) and copy instructions (Experiment 3). Semantically unrelated context–target word pairs were presented during study, and context words and stems were presented during test. Target word stems were paired with the same context words as at study (intact), paired with different context words from study (recombined), or were the stems of unstudied words (control). Participants had to complete stems with the first word that came to mind (indirect), with studied words (inclusion), or with new, unstudied words (exclusion). Results indicated that consciously controlled processes mediated the associative repetition effect under elaborative encoding, whereas automatic processes were implicated under copy instructions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Developing representations of quantity.

Two studies were conducted to investigate how children use written notations to represent information about quantity. 73 children (aged 3–5 yrs) who could count and recognize numerals were asked to produce or select a written notation to represent a specified quantity and to then interpret this representation later. Their productions or selections were classified as being numeral, analogue or global representations. The results show a developmental increase in the use of numerals accompanied by a systematic decline in the use of the alternatives. Furthermore, only the numerals could be reliably interpreted later for quantity. The results point to weaknesses in children's understanding of number concepts in terms of how these relate to counting procedures and written notations. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

LEMMA: a language for easy medical models analysis

The quality of health care systems and processes is becoming a prominent problem and more and more efforts are devoted to define methodologies and tools to measure and assure quality of care. New methods are required to optimize health care processes to guarantee high quality standards within (limited) available resources. Resource optimizations able to preserve the quality of treatments require good models of medical processes. This paper presents LEMMA, a new notation to model medical processes. LEMMA provides physicians with intuitive graphical elements to design their models. At the same time a high level timed Petri net corresponding to the designed model is built automatically. In this way, LEMMA models are ascribed formal semantics and can be executed and analyzed automatically. The dual language approach followed in this paper allows physicians to gain all the benefits of formal methods without being proficient in them. Medical users manage simple graphical elements, while Petri nets ensure formality and validation capabilities. In this way LEMMA mixes formal and informal notations, overcoming the problems of both the approaches. The definition of the notation has been supported by the development of an environment to design LEMMA models. The environment, besides letting us experiment with the notation, has been employed to define and analyze real case studies.