Comma N' cents in pricing: The effects of auditory representation encoding on price magnitude perceptions

Numbers and prices can be processed and encoded in three different forms: 1) visual [based on their written form in Arabic numerals (e.g., 72)], 2) verbal [based on spoken word-sounds (e.g., “seventy” and “two”), and 3) analog (based on judgments of relative “size” or amount (e.g., more than 70 but less than 80)]. In this paper, we demonstrate that including commas (e.g., $1599 vs. $1599) and cents (e.g., $1599.85 vs. $1599) in a price's Arabic written form (i.e., how it is perceived visually) can change how the price is encoded and represented verbally in a consumer's memory. In turn, the verbal encoding of a written price can influence assessments of the numerical magnitude of the price. These effects occur because consumers non-consciously perceive that there is a positive relationship between syllabic length and numerical magnitude. Three experiments are presented demonstrating this important effect.     

Naming two-digit arabic numerals: evidence from masked priming studies

The authors investigated how 2-digit Arabic numerals are named by looking at the effects of masked primes on the naming latencies. Target numerals were named faster when prime and target shared a digit at the same position (e.g., the target 28 primed by 18 and 21). In contrast, naming latencies were slower when prime and target shared 1 or 2 digits at non-corresponding places (e.g., the target 28 primed by 82, 86, or 72). Subsequent experiments showed that these priming effects were situated at the level of the verbal production of the Arabic numerals. The data point to a non-semantically mediated route from visual input to verbal output in the naming of 2-digit Arabic numerals.     

Transcoding abilities in typical and atypical mathematics achievers: The role of working memory and procedural and lexical competencies

Transcoding between numerical systems is one of the most basic abilities acquired by children during their early school years. One important topic that requires further exploration is how mathematics proficiency can affect number transcoding. The aim of the current study was to investigate transcoding abilities (i.e., reading Arabic numerals and writing dictation) in Brazilian children with and without mathematics difficulties, focusing on different school grades. We observed that children with learning difficulties in mathematics demonstrated lower achievement in number transcoding in both early and middle elementary school. In early elementary school, difficulties were observed in both the basic numerical lexicon and the management of numerical syntax. In middle elementary school, difficulties appeared mainly in the transcoding of more complex numbers. An error analysis revealed that the children with mathematics difficulties struggled mainly with the acquisition of transcoding rules. Although we confirmed the previous evidence on the impact of working memory capacity on number transcoding, we found that it did not fully account for the observed group differences. The results are discussed in the context of a maturational lag in number transcoding ability in children with mathematics difficulties.     

Verbal structure of numerals and digits handwriting: New evidence from kinematics

Two experiments used a digitizing tablet to analyse the temporal, spatial, and kinematic characteristics of handwritten production of arabic numbers. They addressed a specific issue of the numerical domain: Does the lexical and syntactic structure of verbal numerals influence the production of arabic numerals (Experiments 1 and 2), even after enforced semantic processing in a comparison task (Experiment 2)? Subjects had to write multi-digit arabic numerals (e.g., 1200) presented in two different verbal structures: a multiplicative one (e.g., teen-hundred, douze cents (twelve hundred)) or an additive one (e.g., thousand-unit-hundred, mille deux cents (one thousand two hundred)). Results show differences in the inter-digit jumps that reflect the influence of the structure of verbal numerals, even after the semantic task. This finding is discussed with regard to different models of number transcoding (McCloskey, Caramazza, & Basili, 1985; Power & Dal Martello, 1990, 1997).     

On the language specificity of basic number processing: transcoding in a language with inversion and its relation to working memory capacity

Transcoding Arabic numbers from and into verbal number words is one of the most basic number processing tasks commonly used to index the verbal representation of numbers. The inversion property, which is an important feature of some number word systems (e.g., German einundzwanzig [one and twenty]), might represent a major difficulty in transcoding and a challenge to current transcoding models. The mastery of inversion, and of transcoding in general, might be related to nonnumerical factors such as working memory resources given that different elements and their sequence need to be memorized and manipulated. In this study, transcoding skills and different working memory components in Austrian (German-speaking) 7-year-olds were assessed. We observed that inversion poses a major problem in transcoding for German-speaking children. In addition, different components of working memory skills were differentially correlated with particular transcoding error types. We discuss how current transcoding models could account for these results and how they might need to be adapted to accommodate inversion properties and their relation to different working memory components.     

Mapping Among Number Words,Numerals, and Nonsymbolic Quantities in Preschoolers

In mathematically literate societies, numerical information is represented in 3 distinct codes: a verbal code (i.e., number words); a digital, symbolic code (e.g., Arabic numerals); and an analogical code (i.e., quantities; Dehaene, 1992). To communicate effectively using these numerical codes, our understanding of number must involve an understanding of each representation as well as how they map to other representations. In the current study, we looked at 3- and 4-year-old children’s understanding of Arabic numerals in relation to both quantities and number words. The results suggest that the mapping between quantities and numerals is more difficult than the mapping between numerals and number words and between number words and quantities. Thus, we compared 2 competing models designed to investigate how children represent the meanings of Arabic numbers—whether numerals are mapped directly to the quantities they represent or instead if numerals are mapped to quantities indirectly via a direct mapping to number words. We found support for the latter suggesting that children may first map numerals to number words (another symbolic representation) and only through this mapping are numerals subsequently tied to the quantities they represent. In addition, unlike both mappings involving quantity, the mapping between the 2 symbolic representations of number (numerals and number words) was not set-size-dependent, therefore providing further evidence that children may map symbols to other symbols in the absence of a quantity referent. Together, the results provide new insight into the important processes involved in how children acquire an understanding of symbolic representations of number.     

Two routes for the processing of verbal numbers: evidence from the SNARC effect

The functional locus of the semantic system is an important issue in number processing. In the present article, the necessity of addressing a central semantic magnitude system in the processing of printed verbal number words is evaluated by looking at the presence of a spatial-numerical association of response codes or SNARC effect. This effect consists of an association of number magnitude and response-preference (preferred responses to small numbers with the left hand and to large numbers with the right hand) and reflects semantic access. Two experiments were run. In Experiment 1, participants performed a parity judgment task which requires access to number semantics. A SNARC effect was observed. In Experiment 2 a phoneme monitoring task was used, which can, in principle, be performed through direct asemantic transcoding. No SNARC effect occurred. Apparently, written number words access the semantic system only if this is necessary for correct task completion. Hence, a semantic and an asemantic route can be postulated for the processing of word numerals. These observations contrast with the processing of Arabic numerals for which semantic effects are omnipresent. Implications of this explicit demonstration of a dissimilarity between the processing of digits and of number words are discussed. Received: 25 October 2000 / Accepted: 21 May 2001     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

Syntactical knowledge in a case of agrammatism: Evidence from transcoding Roman and Arabic numerals

The ability of an aphasic subject with agrammatism in both comprehension and production to transcribe quantities from Roman numerals to Arabic and the reverse was investigated. Systematic errors in the transcoding processes were observed that could not be accounted for by the peculiarities of the two ideographic coding systems or by difficulties with direct transcoding rules. The results are discussed in the framework of the current debate on preserved/impaired hierarchical syntactical knowledge of agrammatic subjects. The findings paralleled the results of previous studies on the transcoding skills of agrammatics from/to alphabetic numerals to/from digital forms. In the case of this particular patient, it is therefore tentatively concluded in favor of preserved syntactical knowledge.     

Are Arabic numerals processed as pictures in a Stroop interference task?

In a picture-word interference task, picture naming is interfered by an incongruent word, but word naming is hardly hindered by the presence of an incongruent picture. In this study, we investigated whether Arabic digits are processed more like pictures or like words. We report two experiments in which Arabic digits and verbal numerals were confronted in a Stroop task. Arabic digit naming is interfered by the presence of an incongruent verbal numeral, while naming the verbal numeral is not influenced by the presence of an incongruent Arabic digit. In a second experiment, we excluded the hypothesis that the results are due to ignoring the Arabic digits: interferences from an incongruent distracter were similar for both notations in a semantic classification task. It seems that an asemantic conversion for Arabic digits is too slow to influence naming times, and that Arabic digit naming, like picture naming, is semantically mediated.     

Cross-notation number priming investigated at different stimulus onset asynchronies in parity and naming tasks

In this paper, a parity judgment task and a number naming task were used to investigate cross-notational number priming. Primes and targets could be verbal (e.g., seven) or Arabic numbers (e.g., 7), and were always presented in a different notation within the same trial (either a verbal prime and an Arabic target or an Arabic prime and a verbal target). Previous experiments showed that response latencies increase when the distance between prime and target increases (for example, in a naming task, seven is pronounced faster after 6 than after 5). This semantic distance priming effect was the same for Arabic and verbal targets and was the same for within-notation trials as for cross-notation trials. In the present experiments, we wanted to investigate whether the cross-notational priming effect also occurs at SOAs shorter than the ones used in previous experiments. Therefore, we used SOAs of 43, 57, 86, and 115 ms. Semantic distance effects were indeed present at these shorter SOAs: Processing times in the semantic parity judgment task and in the non-semantic naming task increased when the distance between prime and target increased. The results are discussed and integrated within an interactive dual-route model of number processing that postulates that the impact of the semantic and the non-semantic route depends on the task and the notation of the stimuli.     

Automatic stimulus-response associations may be semantically mediated

Three experiments on numerical odd/even judgment are presented. In the first experiment, we show that tachistoscopically presented Arabic primes influence the reaction latencies (RTs) to Arabic targets in two ways: First, RTs to targets are longer when the prime and the target have a different parity status than when they share the same parity status, and second, on compatible trials, RTs are longer when the absolute distance between the prime and the target is larger. Experiments 2 and 3 extend the first finding by showing that the response compatibility effect is also obtained (1) when the primes are not part of the target set and the participants never reacted to them and (2) when the primes are presented in a different modality (verbal numerals) than are the targets (Arabic numerals). On the basis of these results, we conclude that, in masked priming, response codes are automatically activated by stimulus characteristics of the prime and that the activation of response codes is semantically mediated when the primes are meaningful.     

Flexible mental processes in numerical size judgments: The case of hebrew letters that are used to convey numbers

In addition to its primary linguistic function, the Hebrew alphabet is sometimes used as a means of number notation (i.e., the system of gematria). Hebrew letters, Arabic numerals, Hebrew number names, and Hebrew letter names were used in a numerical size comparison task, in which two visually presented symbols were compared for numerical value while irrelevant variations in their physical size had to be ignored. A size congruity effect, indicated by faster responses when differences in physical and numerical size were consistent, was larger for Arabic numerals than for number names. The effect for Hebrew letters was similar to that for Arabic numerals and was stronger than that observed for letter names. These results suggest flexible processing of Hebrew letters, so that they function as ideographic symbols in an arithmetic context. A distance effect, indicated by an inverse relationship between reaction time and numerical distance, was found for all notations but was particularly strong for Hebrew letters.     

Semantic processing in the production of numerals across notations

In the present work, we conducted a series of experiments to explore the processing stages required to name numerals presented in different notations. To this end, we used the semantic blocking paradigm previously used in psycholinguist studies. We found a facilitative effect of the semantic blocked context relative to the mixed context for Arabic digits and number words. However, the blocked context produced an interference effect for physical numerosity and Roman numbers. Our results provided further evidence to models suggesting that Arabic digits may be named through an asemantic route similar to that of number words, whereas a semantic route is mandatory to name physical numerosity and Roman numerals.     

Low working memory capacity impedes both efficiency and learning of number transcoding in children

This study aimed to evaluate the impact of individual differences in working memory capacity on number transcoding. A recently proposed model, ADAPT (a developmental asemantic procedural transcoding model), accounts for the development of number transcoding from verbal form to Arabic form by two mechanisms: the learning of new production rules that enlarge the range of numbers a child can transcode and the increase of the mental lexicon. The working memory capacity of 7-year-olds was evaluated along with their ability to transcode one- to four-digit numbers. As ADAPT predicts, the rate of transcoding errors increased when more production rules were required and when children had low working memory capacity, with these two factors interacting. Moreover, qualitative analysis of the errors produced by high- and low-span children showed that the latter have a developmental delay in the acquisition of the production rules.     

The effects of familiarity in a perceptual matching task

Latencies of same-different judgments to pairs of two-digit numerals were recorded for stimuli presented in familiar or unfamiliar (inverted) orientation. Familiar stimuli were responded to more quickly than unfamiliar. For both stimulus types, latencies were correlated with the syllable length of the verbal representation of the numerals, allowing the interpretation that the effect of stimulus orientation is on encoding processes. In two other experiments, it was found that familiarity had no effect on different judgments when the stimuli were relatively simple (e.g., a single digit), but did affect different judgments with more complex stimuli. These results were related to the hypothesis that the complexity of verbal material determines whether different judgments are instigated by visual or by verbal representations of the stimuli.     

Grey parrot number acquisition: The inference of cardinal value from ordinal position on the numeral list

A Grey parrot (Psittacus erithacus) had previously been taught to use English count words ("one" through "sih" [six]) to label sets of one to six individual items (Pepperberg, 1994). He had also been taught to use the same count words to label the Arabic numerals 1 through 6. Without training, he inferred the relationship between the Arabic numerals and the sets of objects (Pepperberg, 2006b). In the present study, he was then trained to label vocally the Arabic numerals 7 and 8 ("sih-none", "eight", respectively) and to order these Arabic numerals with respect to the numeral 6. He subsequently inferred the ordinality of 7 and 8 with respect to the smaller numerals and he inferred use of the appropriate label for the cardinal values of seven and eight items. These data suggest that he constructed the cardinal meanings of "seven" ("sih-none") and "eight" from his knowledge of the cardinal meanings of one through six, together with the place of "seven" ("sih-none") and "eight" in the ordered count list.     

Pure misallocation of "0" in number transcoding: a new symptom of right cerebral dysfunction

To account for the mechanism of number transcoding, many authors have proposed various models, for example, semantic-abstract model, lexical-semantic model, triple-code model, and so on. However, almost all of them are based on the symptoms of patients with left cerebral damage. Previously, I reported two Japanese patients with right posterior cerebral infarction showing pure misallocation of "0" (omission: "40,265"-->"4,265," addition: "107"-->"1,007," transposition: "4,072"-->"4,702") both in writing and oral reading of Arabic numerals. To examine whether the pure misallocation of "0" is commonly observed in patients with right cerebral damage, I investigated writing and oral reading of Arabic numerals in 18 patients with right cerebral damage and 16 healthy controls. All patients with right cerebral damage showed pure misallocation of "0" both in writing and reading. The pure misallocation of "0" due to right cerebral damage cannot be explained by current models. It may be more useful to explain the phenomenon by regarding an Arabic numeral as graph on a two-dimensional plane composed of two axes (place-holding values and digits).