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![The hierarchical structure of number representations. This is a also a cognitive taxonomy of numeration systems. At the level of dimensionality, different systems have different dimensionalities. At the level of dimensional representations, the dimensions of different systems are represented by different physical properties. P = Position, Q = Quantity, S = Shape. The two dimensions of the Greek system ([S´S]) are represented in the mind and only separable in the mind. At the level of bases, different systems may have different bases. At the level of symbol representations, different systems use different symbols.](profile/Jiajie-Zhang/publication/2780894/figure/fig3/AS:669538538496001@1536641835335/The-hierarchical-structure-of-number-representations-This-is-a-also-a-cognitive-taxonomy.png)
The hierarchical structure of number representations. This is a also a cognitive taxonomy of numeration systems. At the level of dimensionality, different systems have different dimensionalities. At the level of dimensional representations, the dimensions of different systems are represented by different physical properties. P = Position, Q = Quantity, S = Shape. The two dimensions of the Greek system ([S´S]) are represented in the mind and only separable in the mind. At the level of bases, different systems may have different bases. At the level of symbol representations, different systems use different symbols.
Source publication
This article explores the representational structures of numeration systems and the cognitive factors of the representational effect in numerical tasks, focusing on external representations and their interactions with internal representations. Numeration systems are analyzed at four levels: dimensionally, dimensional representations, bases, and sym...
Contexts in source publication
Context 1
... level has an abstract structure that can be implemented in different ways. The different representations at each level are isomorphic to each other in the sense that they all have the same abstract structure at that particular level (Figure 4). ...
Context 2
... hierarchical structure of number representations in Figure 4 is in fact a cognitive taxonomy of numeration systems. For example, the Egyptian and Cretan systems are in the same group at the level of symbol representations; the Mayan and Babylonian systems are in the same group at the level of bases; the Arabic, Greek, Chinese, Egyptian, Cretan, and Aztec systems are in the same group at the level of dimensional representations; and all the systems in Figure 4 are in the same group at the level of dimensionality. ...
Context 3
... hierarchical structure of number representations in Figure 4 is in fact a cognitive taxonomy of numeration systems. For example, the Egyptian and Cretan systems are in the same group at the level of symbol representations; the Mayan and Babylonian systems are in the same group at the level of bases; the Arabic, Greek, Chinese, Egyptian, Cretan, and Aztec systems are in the same group at the level of dimensional representations; and all the systems in Figure 4 are in the same group at the level of dimensionality. Under this tax- onomy, the lower the level at which two systems are in the same group, the more similar they are. ...
Similar publications
Currently, there is a controversial debate on whether there is an abstract representation of number magnitude, multiple different ones, or multiple different ones that project onto a unitary representation. The current study aimed at evaluating this issue by means of a magnitude comparison task involving Arabic numbers and structured as well as uns...
Citations
... Furthermore, the construction of linear representations of numerical magnitudes allows children to learn place value and perform mental calculations (Jordan et al. 2008). Place value refers to the value of a digit defined by its position within the sequence of digits (Zhang and Norman 1995). Successful mastery of the place value structure of the Arabic number system is needed to solve multidigit magnitude comparison tasks and arithmetic computations such as additions and subtractions (Moeller et al. 2011). ...
A study was conducted to analyze the factorial structure and measurement invariance of the Curriculum-Based Measurement (CBM) Indicadores de Progreso de Aprendizaje en Matemáticas (IPAM [Indicators of Basic Early Math Skills]), in 2nd grade Spanish students. The model proposed is a one-factor model in which the five IPAM tasks (i.e., number comparison, missing number, single-digit computation, multi-digit computation, and place value) serve as observable indicators for a single underlying factor (i.e., number sense). The IPAM, composed of three parallel forms (i.e., A, B, and C), was administered to 252 Spanish second graders, three times throughout the school year (i.e., fall, winter, and spring). Consequently, the goodness of fit of the proposed model was analyzed for each measurement time. Furthermore, longitudinal measurement invariance was explored to analyze whether the measurement model would remain stable throughout the three-time points of measurement. Discriminant, predictive, and concurrent validity were also tested. The results support that the number sense latent factor explains the common variance of the observable indicators throughout the school year. Each latent factor was found highly related to the next. Moreover, the IPAM showed adequate indices for discriminant, concurrent and predictive validity. We conclude that the IPAM is an appropriate measure to assess number sense competence in second grade.
... Second, many societies have systems of numerical symbols to represent quantities (e.g., number words, Arabic numerals, Chinese ideographs, etc.). These systems of number symbols are cultural inventions that rely on arbitrary (i.e., noniconic) conventions to determine the form and rules of mathematical manipulation (Zhang & Norman, 1995). The development of children's symbolic number abilities is a protracted process that takes place over many years (Wynn, 1992). ...
Research has shown that two different, though related, ways of representing magnitude play foundational roles in the development of numerical and mathematical skills: a nonverbal approximate number system and an exact symbolic number system. While there have been numerous studies suggesting that the two systems are important predictors of math achievement, there has been substantial debate regarding whether and how these basic numerical competencies may be developmentally interrelated. Specifically, the causal direction of their relation has been the subject of debate: whether children's approximate number abilities predict later symbolic number abilities (the mapping account) or the other way around (the refinement account). Our sample included 622 kindergarten children (mean age = 62 months, SD = 3.5, 279 females, 75 born outside Canada), whose dot comparison, number comparison, and mixed comparison skills were assessed over three time-points and math achievement assessed over four time-points. We contrasted multiple theoretical predictions of the interrelations between the variables of interest posited by these two developmental accounts using longitudinal random intercept cross-lagged models. Results were most consistent with the refinement account, suggesting that earlier symbolic number ability is consistently the strongest predictor of approximate number ability, mixed-comparison ability, and arithmetic skills. Notably, our results demonstrated that, when individual models are examined in isolation, model fit was adequate or near adequate for all models tested. This highlights the need for future research to contrast competing accounts, as our results suggest the examination of any one account in isolation may not reveal the best fitting developmental model. (PsycInfo Database Record (c) 2021 APA, all rights reserved).
... A haiku easily contains words that establish a season as well as the strong feeling of the Japanese as regards the season. Haiku-dubbed the shortest verse form in the world-is often divided into 17 morae or a Japanese unit of syllable weight: five for the first line, seven for the second line, and five for the last line (Wilkinson, 1971;Zaidan, 1943;Zhang, 1999). ...
Purpose: The article presents both numeric implementation in Japan and the Arabic gematrical calculation (hisababajadun) in Indonesia. It is common for the people in Japan and Indonesia to believe in the power of numerals and to assume whether or not certain digits induce bad luck. Methodology: The research applies a qualitative descriptive method through a contrastive approach. Main Findings: Data analysis is based on both Koizumi's approach (1995) to the meaning of numbers in the Japanese language and Al Bani's view on hisababajadun. Our study shows that the numeric calculation in Japan uses luckiness as its ground while the hisababajadun builds its mechanism on birth date calculation according to Quranic Neuro-Hypnosis. Implications/ Applications: The research may enrich existing theoretical references on the belief toward numbers in both Japan and Indonesia. While in practice, it may as well be a reference for the application of numeric calculation methods in both cultures. Novelty/Originality of this study: This study is unique and novel as it enhances the understanding of the importance of numerals in two distinct cultures i.e., Japan and Indonesia. Moreover, this study sheds light on some crucial aspects of numeral calculation including how these numerals are used, what beliefs are attached to them and how and where these numerals are used for solving life problems.
... Second, many societies have systems of numerical symbols to represent quantities (e.g., number words, Arabic numerals, Chinese ideographs, etc.). These systems of number symbols are cultural inventions that rely on arbitrary (i.e., noniconic) conventions to determine the form and rules of mathematical manipulation (Zhang & Norman, 1995). The development of children's symbolic number abilities is a protracted process that takes place over many years (Wynn, 1992). ...
Research has shown that two different, though related ways of representing magnitude play foundational roles in the learning of classroom math abilities: a non-verbal, approximate number system (ANS) and an exact, symbolic number system (SNS). While there have been a multitude of studies suggesting that the ANS and SNS are important predictors of math achievement (MA), there has recently been substantial debate regarding whether and how these basic numerical competencies may be developmentally interrelated. Specifically, there has been discussion on whether children’s ANS abilities predict later SNS abilities (the mapping account) or children’s SNS abilities predict later ANS abilities (the parallel development account). We modelled and contrasted multiple theoretical predictions posited by these two developmental accounts using multiple longitudinal path models. Our sample included 622 kindergarten children (M = 62 months, SD = 3.5), whose ANS, SNS and mixed-comparison skills were assessed over three time points and MA assessed over four time points. Results were most consistent with the parallel development account, suggesting that earlier SNS abilities are consistently the strongest predictor of ANS abilities, mixed-comparison abilities, and MA. Notably, our results demonstrated that, when individual models are examined in isolation, model fit either reached or approached adequate fit for all models tested. This highlights the need for future research to contrast competing accounts, as our results suggest that the examination of any one account in isolation may lead to misleading theoretical conclusions. In sum, our results are consistent with the view that children’s SNS abilities predicts later ANS abilities.
... That is, perhaps there are, on average, three strokes per character, independent of writing system size, because all the strokes can be simultaneously processed, whereas processing times increase substantially for greater than around three objects. It has been thought that this may underly why number systems tend to represent '1' by one stroke, '2' by two strokes, and '3' by three strokes, but this stops for greater numbers (Ifrah 1985; Zhang & Norman 1995; Dehaene 1997). The combinatorial degree value of 3/2, and the connected rate at which the number of stroke types increases with writing system size (namely as the 3/2 power), would be a consequence of the redundancy and subitizing limit. ...
A writing system is a visual notation system wherein a repertoire of marks, or strokes, is used to build a repertoire of characters. Are there any commonalities across writing systems concerning the rules governing how strokes combine into characters; commonalities that might help us identify selection pressures on the development of written language? In an effort to answer this question we examined how strokes combine to make characters in more than 100 writing systems over human history, ranging from about 10 to 200 characters, and including numerals, abjads, abugidas, alphabets and syllabaries from five major taxa: Ancient Near–Eastern, European, Middle Eastern, South Asian, Southeast Asian. We discovered underlying similarities in two fundamental respects.
(i) The number of strokes per characters is approximately three, independent of the number of characters in the writing system; numeral systems are the exception, having on average only two strokes per character.
(ii) Characters are ca . 50% redundant, independent of writing system size; intuitively, this means that a character's identity can be determined even when half of its strokes are removed.
Because writing systems are under selective pressure to have characters that are easy for the visual system to recognize and for the motor system to write, these fundamental commonalities may be a fingerprint of mechanisms underlying the visuo–motor system.
