Inversion errors in Arabic number reading: is there a nonsemantic route?

Top-Down Number Reading: Language Affects the Visual Identification of Digit Strings

Reading numbers aloud involves visual processes that analyze the digit string and verbal processes that produce the number words. Cognitive models of number reading assume that information flows from the visual input to the verbal production processes-a feed-forward processing mode in which the verbal production depends on the visual input but not vice versa. Here, I show that information flows also in the opposite direction, from verbal production to the visual input processes. Participants read aloud briefly presented multi-digit strings in Hebrew, in which the order of words is congruent with the order of digits (21 = twenty-and-one), and in Arabic, in which the ones word precedes the tens word (one-and-twenty). The error-by-digit-position curve was affected by language: relative to Hebrew, in Arabic the error rate was slightly lower for the unit digit and slightly higher for the decade digit, indicating that in Arabic the unit digit was processed earlier and the decade digit later, in accord with the Arabic word order. This language-dependent processing order originated in the visual level and was not a verbal confound, because it persisted even when I controlled for the serial position of the decade/unit word in the verbal number by using numbers with 0 (two hundred three/two hundred thirty). I conclude that the visual analyzer's digit scanning order, decade-first or unit-first, is not fixed but affected by the language in which the number is produced-a top-down, verbal-to-visual information flow.

Syntactic chunking reveals a core syntactic representation of multi-digit numbers, which is generative and automatic

Representing the base-10 structure of numbers is a challenging cognitive ability, unique to humans, but it is yet unknown how precisely this is done. Here, we examined whether and how literate adults represent a number’s full syntactic structure. In 5 experiments, participants repeated number-word sequences and we systematically varied the order of words within each sequence. Repetition on grammatical sequences (e.g., two hundred ninety-seven) was better than on non-grammatical ones (hundred seven two ninety). We conclude that the participants represented the number’s full syntactic structure and used it to merge number words into chunks in short-term memory. Accuracy monotonously improved for sequences with increasingly longer grammatical segments, up to a limit of ~ 4 words per segment, irrespectively of the number of digits, and worsened thereafter. Namely, short chunks improved memorization, whereas oversized chunks disrupted memorization. This chunk size limit suggests that the chunks are not based on predefined structures, whose size limit is not expected to be so low, but are created ad hoc by a generative process, such as the hierarchical syntactic representation hypothesized in Michael McCloskey’s number-processing model. Chunking occurred even when it disrupted performance, as in the oversized chunks, and even when external cues for chunking were controlled for or were removed. We conclude that the above generative process operates automatically rather than voluntarily. To date, this is the most detailed account of the core representation of the syntactic structure of numbers—a critical aspect of numerical literacy and of the ability to read and write numbers.

Writing Units or Decades First in Two Digit Numbers Dictation Tasks: The Case of Arabic-An Inverted Language

This study investigated the effect of educational level and of the syntactic representation of numbers in Arabic on the task of transcoding two-digit numbers from dictation. The participants were primary, junior-high, and high school pupils and higher education students. All spoke Arabic as a mother tongue. They performed a transcoding task, namely writing two-digit numbers from dictation. Units first\decades first writing patterns were collected depending on the differential syntactic structures of the two-digit number dictated (decades first: whole tens; units first: teen numbers; identical units and decades, remaining two-digit numbers). The findings reveal that in general, Arabic speakers adopt a decades-first writing pattern for two-digit numbers, especially when it is consistent with the syntactic structure of two-digit numbers, as in whole-tens numbers. This decade-first writing pattern is more evident and consistent in junior-high school, high school, and higher education than in primary school due to the improvement in mathematical skills and second and third languages. However, this pattern is modulated by the syntactic complexity of the unit-decade structure. This complexity is more pronounced in two-digit numbers whose processing is more dependent on numerical syntax. Thus, whole-tens numbers, teen numbers, and identical-decade-unit numbers are less complex than the remaining two-digit numbers.

Syntactic priming reveals an explicit syntactic representation of multi-digit verbal numbers

When we say or understand verbal numbers, a major challenge to the cognitive system is the need to process the number's syntactic structure. Several studies showed that number syntax is handled by dedicated processes, however, it is still unclear how precisely these processes operate, whether the number's syntactic structure is represented explicitly, and if it is - what this representation looks like. Here, we used a novel experimental paradigm, syntactic priming of numbers, which can examine in detail the syntactic representation of multi-digit verbal numbers. In each trial, the participants - Arabic-Hebrew bilinguals and Hebrew monolinguals - heard a multi-digit number and responded orally with a random number. The syntactic structure of their responses was similar to that of the targets, showing that they represented the verbal number's syntax. This priming effect was genuinely syntactic, and could not be explained as lexical - repeating words from the target; as phonological - responding with words phonologically-similar to the target; or as a numerical distance effect - producing responses numerically close to the target. The syntactic priming effect was stronger for earlier words in the verbal number and weaker for later words, suggesting that the syntactic representation is capped by working-memory limits. We propose that syntactic priming could become a useful method to examine various aspects of the syntactic representation of numbers.

Dysnumeria in Sign Language: Impaired Construction of the Decimal Structure in Reading Multidigit Numbers in a Deaf ISL Signer

We report on the first in-depth analysis of a specific type of dysnumeria, number-reading deficit, in sign language. The participant, Nomi, is a 45-year-old signer of Israeli Sign Language (ISL). In reading multidigit numbers (reading-then-signing written numbers, the counterpart of reading aloud in spoken language), Nomi made mainly decimal, number-structure errors- reading the correct digits in an incorrect (smaller) decimal class, mainly in longer numbers of 5-6-digits. A unique property of ISL allowed us to rule out the numeric-visual analysis as the source of Nomi's dysnumeria: In ISL, when the multidigit number signifies the number of objects, it is signed with a decimal structure, which is marked morphologically (e.g., 84 → Eight-Tens Four); but a parallel system exists (e.g., for height, age, bus numbers), in which multidigit numbers are signed non-decimally, as a sequence of number-signs (e.g., 84 → Eight, Four). When Nomi read and signed the exact same numbers, but this time non-decimally, she performed significantly better. Additional tests supported the conclusion that her early numeric-visual abilities are intact: she showed flawless detection of differences in length, digit-order, or identity in same-different tasks. Her decimal errors did not result from a number-structure deficit in the phonological-sign output either (no decimal errors in repeating the same numbers, nor in signing multidigit numbers written as Hebrew words). Nomi had similar errors of conversion to the decimal structure in number comprehension (number-size comparison tasks), suggesting that her deficit is in a component shared by reading and comprehension. We also compared Nomi's number reading to her reading and signing of 406 Hebrew words. Nomi's word reading was in the high range of the normal performance of hearing controls and of deaf signers and significantly better than her multidigit number reading, demonstrating a dissociation between number reading, which was impaired, and word reading, which was spared. These results point to a specific type of dysnumeria in the number-frame generation for written multidigit numbers, whereby the conversion from written multidigit numbers to the abstract decimal structure is impaired, affecting both reading and comprehension. The results support abstract, non-verbal decimal structure generation that is shared by reading and comprehension, and also suggest the existence of a non-decimal number-reading route.

One language, two number-word systems and many problems: numerical cognition in the Czech language

Comparing numerical performance between different languages does not only mean comparing different number-word systems, but also implies a comparison of differences regarding culture or educational systems. The Czech language provides the remarkable opportunity to disentangle this confound as there exist two different number-word systems within the same language: for instance, "25" can be either coded in non-inverted order "dvadsetpät" [twenty-five] or in inverted order "pätadvadset" [five-and-twenty]. To investigate the influence of the number-word system on basic numerical processing within one culture, 7-year-old Czech-speaking children had to perform a transcoding task (i.e., writing Arabic numbers to dictation) in both number-word systems. The observed error pattern clearly indicated that the structure of the number-word system determined transcoding performance reliably: In the inverted number-word system about half of all errors were inversion-related. In contrast, hardly any inversion-related errors occurred in the non-inverted number-word system. We conclude that the development of numerical cognition does not only depend on cultural or educational differences, but is indeed related to the structure and transparency of a given number-word system.

Extending the Mental Number Line

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

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.

Deafness for the meanings of number words

We describe the performance of an aphasic individual who showed a selective impairment affecting his comprehension of auditorily presented number words and not other word categories. His difficulty in number word comprehension was restricted to the auditory modality, given that with visual stimuli (written words, Arabic numerals and pictures) his comprehension of number and non-number words was intact. While there have been previous reports of selective difficulty or sparing of number words at the semantic and post-semantic levels, this is the first reported case of a pre-semantic deficit that is specific to the category of number words. This constitutes evidence that lexical semantic distinctions are respected by modality-specific neural mechanisms responsible for providing access to the meanings of words.

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).

On the functional role of human parietal cortex in number processing: How gender mediates the impact of a ‘virtual lesion’ induced by rTMS

Areas around the horizontal part of the intraparietal sulcus (hIPS) have repeatedly been reported to participate in processing numerical magnitude. Using transcranial magnetic stimulation (TMS), we investigated the functional role of the hIPS by examining two effects from the domain of numerical cognition: in magnitude comparison tasks response latencies are inversely related to the numerical distance between two numbers. This distance effect indexes access to the mental number representation. In magnitude comparison tasks responses are faster when decade and unit comparison would lead to the same decision (e.g. 42_57, 4 < 5 and 2 < 7) than when they would not (e.g. 47_62, 4 < 6 but 7 > 2). This compatibility effect reflects unit-decade integration processes. Differential susceptibility of (fe)male participants to TMS was examined. We applied repetitive TMS (rTMS; 1Hz for 10 min) over the left hIPS in 12 participants (6 female). No stimulation and vertex stimulation served as control conditions. The effect of rTMS was mediated by gender: in male participants, the distance effect decreased after TMS over hIPS. For female participants distance and compatibility effect both increased. This modulation of the compatibility effect was limited in duration to no more than 4 min. The hIPS seems to be functionally involved both in number magnitude processing and in integrating unit-decade magnitude information of two-digit numbers. Relative hemispheric specialization of the hIPS with respect to two-digit magnitude comparison is discussed.

Number processing in posterior cortical atrophy--a neuropsycholgical case study

Posterior cortical atrophy (PCA) is an uncommon syndrome of dementia with early onset, characterised by disorders of higher visual function, variable symptoms of Balint's syndrome, visual agnosia, alexia, agraphia, finger agnosia, right-left disorientation and dyscalculia [Benson D. F., Davis R. J., & Snyder B. D. (1988). Posterior cortical atrophy. Archives of Neurology, 45, 789-793]. In a single case study specific numerical deficits were observed which may be predicted by parietal neurodegeneration (more pronounced on the right side; verified by SPECT). Besides impairments in all tasks involving visuo-spatial abilities (e.g., dot counting, analog number scale task), deficits appeared in tasks requiring access to an internal representation of numbers such as mental number bisection, approximation, estimation and semantic facts. In number comparison an increased distance effect was found. In simple arithmetic, a striking dissociation between operations was found-multiplication and addition facts being preserved at a superficial level, subtraction and division being severely impaired. The study confirms the close relation between spatial and numerical processing and highlights the modular organisation of the semantic system (number semantics impaired). Moreover, the study adds evidence about the clinical manifestation of the particular degenerative syndrome.

The role of the temporal lobe semantic system in number knowledge: evidence from late-stage semantic dementia

Previous reports have demonstrated that many aspects of number knowledge remain unimpaired in semantic dementia, despite severe comprehension problems in other domains. It is argued that this advantage for numbers arises because the disease spares the parietal lobe magnitude system thought to be critical for number processing. Models of numerical cognition that favour a separation between verbal and magnitude representations of number might, however, predict a restricted impairment of the verbal number code in this condition. We obtained support for this hypothesis in a patient with late-stage semantic dementia. She was impaired at a variety of tasks tapping the verbal number code; for example, reading and writing Arabic numerals, naming and word-picture matching with dot pictures, reading aloud number words, digit span and magnitude comparison/serial ordering tasks with number words. In contrast, she demonstrated good understanding of the magnitude and serial order of numbers when tested with Arabic numerals and non-symbolic representations. These findings suggest that although the magnitude meaning of numbers is isolated from the temporal lobe semantic system, the anterior infero-temporal lobe may play a critical role in binding English number words to their non-symbolic magnitude meaning.

Access to numerical information is dependent on the modality of stimulus presentation in mental addition: a combined ERP and behavioral study

We investigated whether the mental representation of numbers is abstract amodal or modality-dependent. Subjects verified simple additions. In an event-related potential (ERP) experiment, subjects added an Arabic digit (S2) to a preceding number (S1) offsetting 3 s before S2. S1 was either a visually shown Arabic digit, a written number word or an acoustically presented number word. In a behavioral experiment, we measured the speed of addition using a modified paradigm. In the ERPs to S2, the amplitude of the parietal N1, the fronto-central P2, and the late positivity between 320 and 460 ms were more positive and RTs were faster when S1 was a heard number word than when S1 was a written number word. ERP amplitudes and reaction times took intermediate positions between the other two conditions when S1 was an Arabic digit. Between the Arabic and heard number conditions, this so-called numeral modality effect (NME) was present at electrodes Pz, P4, P3 and Cz when number size was small, whereas it was significant over electrode C4 and P4 when number size was large. Our results suggest that numbers presented in different surface-formats have differential access to number representations. Conclusions for models of number processing are drawn and the possible role of parietal number representations is discussed. We replicated the N270 ERP component and elicited the ERP numerical distance effect in response to incongruent arithmetical results.

Code-dependent pathways for number transcoding: evidence from a case of selective impairment in written verbal numeral to arabic transcoding

This article presents a brain-damaged patient (RR) suffering from cognitive deficits following neurological insults, who showed a selective impairment in number transcoding. Except for written verbal numeral to arabic transcoding, his ability to transcode numerals, including writing arabic numerals to dictation, is largely preserved. Other number processing skills, including numeral recognition, numeral comprehension, and calculation, were unimpaired. Semantic and asemantic models of number processing cannot easily account for the patient's performance and it is suggested that the number transcoding system should include different code-dependent pathways for arabic transcoding from spoken verbal numerals and from written verbal numerals. Since the errors produced in the impaired transcoding rely upon the syntactical structure of numeral stimuli, it is also proposed that transcoding code-dependent pathways should reflect the structure of the verbal numeral system, especially the difference between sum and product relationships.