Math anxiety results in a drop in performance on various math-related tasks, including the symbolic number ordering task in which participants decide whether a triplet of digits is presented in order (e.g. 3-5-7) or not (e.g. 3-7-5). We investigated whether the strategy repertoire and reaction times during a symbolic ordering task were affected by math anxiety. In study 1, participants performed an untimed symbolic number ordering task and indicated the strategy they used on a trial-by-trial basis. The (...) use of the memory retrieval strategy, based on the immediate recognition of the triplet, decreased with high math anxiety, but disappeared when controlling for general anxiety. In the study 2, participants completed a timed version of the number order task. High math-anxious participants used the decomposition strategy (e.g. 5 is larger than 3 and 7 is larger than 5 to decide whether 3-5-7 is in the correct order) more often, and were slower in responding when both memory- and other decomposition strategies were used. Altogether, both studies demonstrate that high-math anxious participants are not only slower to decide whether a number triplet is in the correct order, but also rely more on procedural strategies. (shrink)

The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial—accessing underlying magnitude representation of symbols (i.e., symbol‐magnitude associations), processing relative order of symbols (i.e., symbol‐symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots‐number word matching task—thought to be a measure of symbol‐magnitude associations (numerical magnitude processing)—a numeral‐ordering task that (...) focuses on symbol‐symbol associations (numerical order processing), and a digit‐number word matching task targeting symbolic processing per se. Results showed that both numerical magnitude and order processing were uniquely related to arithmetic achievement, beyond the effects of domain‐general factors (intellectual ability, working memory, inhibitory control, and non‐numerical ordering). Importantly, results were different when a general measure of mathematics achievement was considered. Those mechanisms of symbolic number processing did not contribute to math achievement. Furthermore, a path analysis revealed that numerical magnitude and order processing might draw on a common mechanism. Each process explained a portion of the relation of the other with arithmetic (but not with a general measure of math achievement). These findings are consistent with the notion that adults’ arithmetic skills build upon symbol‐magnitude associations, and they highlight the effects that different math measures have in the study of numerical cognition. (shrink)

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties can easily (...) be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space. (shrink)

Recent years have witnessed an enormous increase in behavioral and neuroimaging studies of numerical cognition. Particular interest has been devoted toward unraveling properties of the representational medium on which numbers are thought to be represented. We have argued that a correct inference concerning these properties requires distinguishing between different input modalities and different decision/output structures. To back up this claim, we have trained computational models with either symbolic or nonsymbolic input and with different task requirements, and showed that (...) this allowed for an integration of the existing data in a consistent manner. In later studies, predictions from the models were derived and tested with behavioral and neuroimaging methods. Here we present an integrative review of this work. (shrink)

Developmental dyscalculia (DD) is a pervasive difficulty affecting number processing and arithmetic. It is encountered in around 6% of school-aged children. While previous studies have mainly focused on general cognitive functions, the present paper aims to further investigate the hypothesis of a specific numerical deficit in dyscalculia. The performance of 10- and 11-year-old children with DD characterised by a weakness in arithmetic facts retrieval and age-matched control children was compared on various number comparison tasks. Participants were asked to compare a (...) quantity presented in either a symbolic (Arabic numerals, number words, canonical dots patterns) or a nonsymbolic format (noncanonical dots patterns, and random sticks patterns) to the reference quantity 5. DD children showed a greater numerical distance effect than control children, irrespective of the number format. This favours a deficit in the specialised cognitive system underlying the processing of number magnitude in children with DD. Results are discussed in terms of access and representation deficit hypotheses. (shrink)

The book of Jonah uses four indications of time: Jonah spends 3 days and three nights in the fish; the city of Nineveh takes 3 days to cross; Jonah enters the city to the extent or distance of one day’s travel; and Jonah proclaims to Nineveh that she has 40 days to repent. In this article it will be pointed out that each of the four instances where time is mentioned in the book of Jonah has a symbolic meaning (...) and intends to mean more than is evident on the surface of the text.Contribution: This article contributes to the ongoing critical scholarship of the book of Jonah. In this article the interpretation of time in relation to the number of days mentioned in the book of Jonah is revisited and it is argued that the days should be understood symbolically and not literally. An evaluation is also given of what their meanings are. (shrink)

A dual-code model of number processing needs to take into account the difference between a number symbol and its meaning. The transition of automatic non-abstract number representations into intentional abstract representations could be conceptualized as a translation of perceptual asemantic representations of numerals into semantic representations of the associated magnitude information. The controversy about the nature of number representations should be thus related to theories on embodied grounding of symbols.

The article is devoted to the four principle realization in the creative work of two poets, namely the English literature classic T.S. Eliot (1888-1965) and a representative of the Russian poetry Elena Swarz (1948-2010). Number four is a structural principle in their works composition and crucifix of Christ. On the example of the two poets the author shows that on the one hand, the symbol of four is a cosmic transversalism of being and the cross image, on the other. Thus, (...) two contradictory meanings are combined here: harmony and drama as truly Christian forms of dialectics. (shrink)

Mathematics often seems incomprehensible, a melee of strange symbols thrown down on a page. But while formulae, theorems, and proofs can involve highly complex concepts, the math becomes transparent when viewed as part of a bigger picture. What Is a Number? provides that picture. Robert Tubbs examines how mathematical concepts like number, geometric truth, infinity, and proof have been employed by artists, theologians, philosophers, writers, and cosmologists from ancient times to the modern era. Looking at a broad range of topics (...) -- from Pythagoras's exploration of the connection between harmonious sounds and mathematical ratios to the understanding of time in both Western and pre-Columbian thought -- Tubbs ties together seemingly disparate ideas to demonstrate the relationship between the sometimes elusive thought of artists and philosophers and the concrete logic of mathematicians. He complements his textual arguments with diagrams and illustrations. This historic and thematic study refutes the received wisdom that mathematical concepts are esoteric and divorced from other intellectual pursuits -- revealing them instead as dynamic and intrinsic to almost every human endeavor. (shrink)

In mathematics education, students' difficulties with negative numbers are well known. To explain these difficulties, researchers traditionally refer to obstacles raised by the concept of NEGATIVE NUMBERS itself throughout its historical evolution. In order to improve our understanding, I propose to take into consideration another point of view, based on Vygotsky's principles, which define a strong relationship between signs such as language or symbols and cognitive development. I show how it is of great interest to consider students' difficulties (...) with negatives in the light of the use of symbols, in particular the minus sign. I describe the results of an empirical study about students' strategies in solving arithmetical equations with negatives. My analysis shows that the problems raised by the presence of negatives in the equations are attributable to students' very restricted and often inadequate use of the minus sign, which prevents them from finding a negative solution or making sense of it. (shrink)

In this paper, we recall and study the new type of algebraic structures called Symbolic Plithogenic Algebraic Structures. Their operations are given under the Absorbance Law and the Prevalence Order.

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that were (...) exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

Twenty classes in ten schools with 627 sixth?grade students in five cities in Taiwan participated in this study. The research provides information on the performance differences among written computation, pictorial representation, symbolic representation and number sense. The results of One?way ANOVA analysis indicate that significant difference was found among WCT, PRT, SRT and NST tests, with F=536.327, p=0.000. The a posteriori comparisons show for each pair (WCT vs PRT, WCT vs SRT, WCT vs NST, PRT vs SRT and SRT (...) vs NST) significant difference at the 0.001 level (p=0.000), except for the pair comparison between PRT and NST. This implies that these Taiwanese students were highly skilled in written computation but their written skills were not equally transferred to use of non?computational paths that depended on symbolic representation, pictorial representation and number sense to solve similar problems. (shrink)

Clarke and Beck's defense of the theoretical construct “approximate number system” is flawed in serious ways – from biological misconceptions to mathematical naïveté. The authors misunderstand behavioral/psychological technical concepts, such as numerosity and quantical cognition, which they disdain as “exotic.” Additionally, their characterization of rational numbers is blind to the essential role of symbolic reference in the emergence of number.