Arabic Notation Research Articles

Stimulus frequency alone can account for the size effect in number comparison

In a number comparison task, the size effect (i.e, smaller values are easier to compare than larger values) is usually attributed to a psychophysics-based representation. However, alternative models assume that the size effect is a frequency effect: Smaller numbers are easier to process because they are observed more frequently. Previous studies have demonstrated that the frequency of the digits fundamentally influences the comparison size effect: In new number symbols, the frequency entirely determines the size effect. In contrast, in Arabic notation, the size effect aggregates the frequency in the actual session and the previous regular size effect. Here, we investigate whether the previously acquired regular size effect can depend on the frequency of the stimuli as well or on a psychophysics-based representation that is not yet active in new symbols. Participants in the study compared numbers that were denoted with new symbols, with the frequency of the symbols being changed throughout the session. We found that the frequency of the stimuli in both halves of the session was aggregated in the size effect. In addition, no psychophysics-based size effect was found throughout the session. These results confirm that the size effect can be created and shaped purely by the frequency of the symbols, while a psychophysics-based representation is not necessary to account for these size effect-related phenomena.

Non-symbolic and symbolic notations in simple arithmetic differentially involve intraparietal sulcus and angular gyrus activity

Addition problems can be solved by mentally manipulating quantities for which the bilateral intraparietal sulcus (IPS) is likely recruited, or by retrieving the answer directly from fact memory in which the left angular gyrus (AG) and perisylvian areas may play a role. Mental addition is usually studied with problems presented in the Arabic notation (4+2), and less so with number words (four+two) or dots (:: +·.). In the present study, we investigated how the notation of numbers influences processing during simple mental arithmetic. Twenty-five highly educated participants performed simple arithmetic while their brain activity was recorded with functional magnetic resonance imaging. To reveal the effect of number notation, arithmetic problems were presented in a non-symbolic (Dots) or symbolic (Arabic; Words) notation. Furthermore, we asked whether IPS processing during mental arithmetic is magnitude specific or of a more general, visuospatial nature. To this end, we included perception and manipulation of non-magnitude formats (Colors; unfamiliar Japanese Characters). Increased IPS activity was observed, suggesting magnitude calculations during addition of non-symbolic numbers. In contrast, there was greater activity in the AG and perisylvian areas for symbolic compared to non-symbolic addition, suggesting increased verbal fact retrieval. Furthermore, IPS activity was not specific to processing of numerical magnitude but also present for non-magnitude stimuli that required mental visuospatial processing (Color-mixing; Character-memory measured by a delayed match-to-sample task). Together, our data suggest that simple non-symbolic sums are calculated using visual imagery, whereas answers for simple symbolic sums are retrieved from verbal memory.

Difficulties of 6th Grade Elementary School Students in Solving the Four Basic Fundamental Operations: Addition, Subtraction, Multiplication and Division of Natural Numbers

The results of this study point out the contributions of the theory of Conceptual Fields to identify the nature of the mistakes made by students of the 6th year of elementary school in solving mathematical questions with ten natural numbers. We ask: What are the concepts and theorems put into action to carry out the operations of addition, subtraction, multiplication and division with natural numbers? The concepts and theorems in action related to the understanding of the decimal numbering system structure are responsible for the errors? The mistakes are due to the understanding of the organization of representation registers (word and Arabic notation)? As a result, we find that students have the addition of errors and subtraction errors by misunderstanding of the structure of the Numbering Decimal Positional System, errors due to lack of meaning assignment to the operations of addition, subtraction, multiplication and division, and also the numbers of Arabic script which induce errors in the handling of algorithms.

Language affects symbolic arithmetic in children: The case of number word inversion

Specific language influences have been observed in basic numerical tasks such as magnitude comparison, transcoding, and the number line estimation task. However, so far language influences in more complex calculations have not been reported in children. In this translingual study, 7- to 9-year-old German- and Italian-speaking children were tested on a symbolic addition task. Whereas the order of tens and units in Italian number words follows the order of the Arabic notation, the order is inverted in German number words. For both language groups, addition problems were more difficult when a carry operation was needed, that is, when a manipulation within the place-value structure of the Arabic number system was particularly important. Most important, this carry effect was more pronounced in response latencies for children speaking German, a language with inverted verbal mapping of the place-value structure. In addition, independent of language group, the size of the carry effect was significantly related to verbal working memory. The current study indicates that symbolic arithmetic and the carry effect in particular are modulated by language-specific characteristics. Our results underline the fact that the structure of the language of instruction is an important factor in children’s mathematical education and needs to be taken into account even for seemingly nonverbal symbolic Arabic tasks.

Holistic and Componential Processing for Two-Digit Numbers

A great deal of research has been conducted to investigate how people process two-digit numbers. Researchers have found that the time to compare two two-digit numbers decreases logarithmically with the absolute difference of two numbers, which suggests a holistic processing for two-digit numbers. However, some evidence seems to support additional componential processing. A number pair can be defined as compatible if the decade-magnitude comparison and the unit-magnitude comparison of the two numbers lead to the same result (e.g., 23 vs. 78, 27 and 38), and as incompatible if this is not the case (e.g., 29 vs. 84, 28 but 94). Subjects took longer to respond the unit-decade-incompatible number pairs than to the unit-decade-compatible number pairs. The unit-decade compatibility effect suggests that the magnitude of the unit digits of numbers in addition to the whole magnitude of the numbers is activated. However, previous studies on two-digit number processing have typically used tasks involving intentional processing of numerical magnitude. To intentionally process two-digit numbers, subjects need to pay attention to decade and unit digits serially or simultaneously, which may lead to componential processing. This study addressed whether componential processing is also involved in unintentional processing of the magnitude of numbers. Stroop-like paradigm was applied. Sixteen undergraduate students participated in Experiment 1. They were asked to compare the physical size of two two-digit numbers, as well as the numerical magnitude of the two numbers (e.g., 23 vs. 78). Four hundred two-digit pairs consisting of the numbers between 21 and 98 were presented in Arabic notation, excluding numbers that contained 0 as unit (e.g., 30, 40) or the same digits for decade and unit (e.g., 22, 33). We manipulated the ratio of the physical size of two numbers (i.e., 1∶1.1 vs. 1∶1.2), the congruency of magnitude size and physical size, and the unit-decade-compatibility. To avoid confounding of the distance and problem size effect, we controlled overall distance and problem size between compatible and incompatible pairs. In addition, we also conducted Experiment 2 to avoid interference of intentional and unintentional processing. Twenty undergraduate students were asked to compare only the physical size of two two-digit numbers. Experimental design and procedure were same as Experiment 1, except the ratio of physical size of two numbers had three levels containing 1∶1.1, 1∶1.25 and 1∶1.5. The results from the magnitude comparison task in Experiment 1 replicated those found in previous studies involving the same tasks. Tasks involving physical size comparison in both Experiment 1 and Experiment 2 showed the effect of magnitude and physical size congruency. The effect supported holistic processing. The unit-decade-compatibility effect was evident for congruent number pairs, but there was no or a reverse effect for incongruent number pairs. In sum, two-digit number processing can be componential as well as holistic in both unintentional and intentional processing of numerical magnitude.

Effects of Regular Switching between Languages during Random Number Generation

Random number generation is a task that engages working memory and executive processes within the domain of number representation. In the present study we address the role of language in number processing by switching languages during random number generation (numbers 1-9), using German (L1) and English (L2), and alternating L1/L2. Results indicate large correspondence between performance in L1 and L2. In contrast to nonswitching performance, randomization with alternating languages showed a significant increase of omitted responses, whereas the random sequences were less stereotyped, showing significantly less repetition avoidance and cycling behavior. During an intentional switch between languages, errors in language sequence appeared in 23% of responses on the average, independently of the quality of randomization but associated with a clear persistence of L2. These results indicate that random number generation is more closely linked to auditory-phonological representation of numerals than to visual arabic notation.

Language effects in magnitude comparison: Small, but not irrelevant

It is assumed that number magnitude comparison is performed by assessing magnitude representation on a single analog mental number line. However, we have observed a unit–decade–compatibility effect in German which is inconsistent with this assumption ( Nuerk, Weger, & Willmes, 2001). Incompatible magnitude comparisons in which decade and unit comparisons lead to different responses (e.g., 37_52 for which 3 < 5, but 7 > 2) are slower and less accurately responded to than compatible trials in which decade and unit comparisons lead to the same response (e.g., 42_57, for which 4 < 5 and 2 < 7). As overall distance was held constant, a single holistic magnitude representation could not account for this compatibility effect. However, because of the inversion property of the corresponding German two-digit number words (“einundzwanzig” <one-and-twenty>), the language-generality of the effect is questionable. We have therefore examined the compatibility effect with native English speakers. We were able to replicate the compatibility effect using Arabic notation. Thus, the compatibility effect is not language-specific. However, in cross-linguistic analyses language-specific modulations were observed not only for number words but also for Arabic numbers. The constraints imposed on current models by the verbal mediation of Arabic number comparison are discussed.

Modulation of Parietal Activation by Semantic Distance in a Number Comparison Task

The time to compare two numbers shows additive effects of number notation and of semantic distance, suggesting that the comparison task can be decomposed into distinct stages of identification and semantic processing. Using event-related fMRI and high-density ERPs, we isolated cerebral areas where activation was influenced by input notation (verbal or Arabic notation). The bilateral extrastriate cortices and a left precentral region were more activated during verbal than during Arabic stimulation, while the right fusiform gyrus and a set of bilateral inferoparietal and frontal regions were more activated during Arabic than during verbal stimulation. We also identified areas that were influenced solely by the semantic content of the stimuli (numerical distance between numbers to be compared) independent of the input notation. Activation tightly correlated with numerical distance was observed mainly in a group of parietal areas distributed bilaterally along the intraparietal sulci and in the precuneus, as well as in the left middle temporal gyrus and posterior cingulate. Our results support the assumption of a central semantic representation of numerical quantity that relies on a common parietal network shared among notations.

Number Processing in Chinese-English Bilinguals

To investigate effects of surface notation on basic numerical skills we examined number naming, magnitude selection, and simple arithmetic performed by adult Chinese-English bilinguals born and educated in China. Stimuli were presented using either arabic (7 + 8) or “mandarin” (㈦ + ㈧)symbols and participants were cued to respond either in English or Chinese. The naming task demonstrated that the mandarin characters were as easy to identify as the arabic digits, but for both arithmetic and magnitude selection there were faster RTs and fewer errors overall with arabic notation. Arabic notation also produced smaller problem-size effects in arithmetic and a smaller split effect in magnitude selection relative to mandarin notation. These results suggest that retrieval processes in the arithmetic and selection tasks were more efficient with arabic than mandarin stimuli. Arithmetic RTs were substantially slower with English than Chinese responses given either arabic or mandarin stimuli, but the English-language ...

Some Methods in Subtraction

In teaching subtraction three factors of the subject present themselves to us in different ways, and the inter-relation of these factors by a series of permutations produces a variety of methods sometimes widely different, and sometimes very similar, but still enough unlike to prove confusing to children. First, there is the nature of the subject itself, second, the terminology to be employed, and third, the matter of “carrying” in the Arabic notation. The terminology depends in some cases directly upon the first factor and in others upon the third.

La Disme of Simon Stevin—The First Book on Decimals

The history of the decimal fraction has two distinct sections: the evolution of the idea of decimals, and the evolution of a convenient symbolism. The idea might, indeed, be traced to the very beginning of the place-value of the Arabic notation. The symbolism has not been standardized today, as witness the English 3.14, the French 3,14 and our 3.14. Yet, strangely enough, the theory of operations with decimals appeared in complete form in the first work devoted wholly to the subject. La Disme (1585) by Simon Stevin of Bruges.