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The knowledge of various types of numbers, such as integers and fractions, has been shown to be crucial for students' success in mathematics. However, the significance of negative number knowledge has not been explored enough. This study aims to investigate the role of negative number knowledge in mathematics achievement. A group of seventh graders (n = 192) in Hong Kong were assessed on their understanding of positive and negative integers, mathematics achievement, fluid intelligence, working memory, and inhibition. Using hierarchical linear models, the study found that negative number understanding significantly predicted mathematics achievement, even after controlling for school, general cognitive skills, and positive number understanding. The findings confirm the importance of negative number understanding in early adolescents' mathematics learning. Educators should pay more attention to this aspect of numerical magnitude. Educational Impact and Implications Statement: The current study found that the understanding of negative numbers significantly predicted early adolescents' mathematics achievement after controlling for general cognitive factors and the corresponding positive number understanding. Given the relatively poor understanding of negative numbers among early adolescents, educators should devote more attention to this construct, as a better understanding of negative numbers may enable adolescents to master the abstract concepts in more advanced mathematics topics. [ABSTRACT FROM AUTHOR]

In this study, we examine different approaches to the presentation of Y coordinates in mobile auditory graphs, including the representation of negative numbers. These studies involved both normally sighted and visually impaired users, as there are applications where normally sighted users might employ auditory graphs, such as the unseen monitoring of stocks, or fuel consumption in a car. Multi-reference sonification schemes are investigated as a means of improving the performance of mobile non-visual point estimation tasks. The results demonstrated that both populations are able to carry out point estimation tasks with a good level of performance when presented with auditory graphs using multiple reference tones. Additionally, visually impaired participants performed better on graphs represented in this format than normally sighted participants. This work also implements the component representation approach for negative numbers to represent the mapping by using the same positive mapping reference for the digit and adding a sign before the digit which leads to a better accuracy of the polarity sign. This work contributes to the areas of the design process of mobile auditory devices in human-computer interaction and proposed a methodological framework related to improving auditory graph performance in graph reproduction. [ABSTRACT FROM AUTHOR]

In this study, we examine different approaches to the presentation of Y coordinates in mobile auditory graphs, including the representation of negative numbers. These studies involved both normally sighted and visually impaired users, as there are applications where normally sighted users might employ auditory graphs, such as the unseen monitoring of stocks, or fuel consumption in a car. Multi-reference sonification schemes are investigated as a means of improving the performance of mobile non-visual point estimation tasks. The results demonstrated that both populations are able to carry out point estimation tasks with a good level of performance when presented with auditory graphs using multiple reference tones. Additionally, visually impaired participants performed better on graphs represented in this format than normally sighted participants. This work also implements the component representation approach for negative numbers to represent the mapping by using the same positive mapping reference for the digit and adding a sign before the digit which leads to a better accuracy of the polarity sign. This work contributes to the areas of the design process of mobile auditory devices in human-computer interaction and proposed a methodological framework related to improving auditory graph performance in graph reproduction.

Mastery of mathematics depends on the people’s ability to manipulate and abstract values such as negative numbers. Knowledge of arithmetic principles does not necessarily generalize from positive number arithmetic to arithmetic involving negative numbers (Prather & Alibali, 2008, https://doi.org/10.1080/03640210701864147). In this study, we evaluate the relationship between participant’s knowledge of the Relation to Operands arithmetic principle in both positive and negative numbers and their spontaneous on numerical relations. Additionally, we tested if the feedback that directs attention to relations affects participants’ attention to relation and their arithmetic principle knowledge. This study contributes to our understanding of the specific skills and cognitive processes that are associated with understanding high-level mathematics.

While some researchers place negative numbers on a so-called extended mental number line to the left of positive numbers, others claim that negative numbers do not have mental representations but are processed through positive numbers combined with transformation rules. We measured spatial associations of negative numbers with a modified implicit association task that avoids spatial confounds present in most previous studies. In two lab-based magnitude classification experiments (each including 24 participants) and two online replications (with 74 and 77 participants, respectively), positive and negative numbers were combined with two spatial contexts: either directional symbols (left- or right-pointing arrows) or rectangles of varying sizes. In all experiments, we found a robust distance effect for negative numbers. However, there were no consistent associations of negative numbers with directional or size contexts. In the context of directional symbols, holistic processing was prevalent only in the small negative number range (-9, -8, -7, -6) when ensured by the stimulus set, supporting an extended mental number line. In the context of rectangles, however, large negative numbers from -4 to -1 were perceived as small, thus supporting rule-based processing. For negative number processing in the context of size, we further suggest the Semantic-Perceptual Size Congruity Cuing model (SPeSiCC model). We show that associations of size with negative numbers underly more complex processing mechanisms than mere recruitment of a transformation rule. In general, we conclude that associations of negative numbers with space and size are situated in the context, as they depend on the presented number range and differ for spatial direction and size.

The world of numbers keeps us intrigued for one reason or the other. From negative numbers to imaginary numbers and from various constants to peculiar ratios, they never cease to surprise us. And our curiosity and imagination keeps on inspiring us to look for these patterns in nature. Consequently leading the way for discoveries that are important to meeting not only our material goals but also spiritual ones at the same time. However, due to an extraordinary reliance of modern applied science on numbers, we have generated more scholarly interest, especially in the last few centuries, around the application of numbers to the material realm as compared to the spiritual. Amidst our attention to the peculiar or special numbers and ratios, there are some numbers -- or rather concepts -- that still look in our face, inviting us to explore a whole new world of possibilities all over again. Though these concepts laid the foundation of modern day mathematics as we use it, yet at times we take them for granted and are not usually interested in revisiting our understanding about their nature, their past and future possibilities. In this chapter, we revisit three of these foundational concepts -- Zero, One and Infinity (ZOI) from an integrative spiritual perspective. [ABSTRACT FROM AUTHOR]

Copyright of Educational Academic Research is the property of Ataturk University Coordinatorship of Scientific Journals and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The teaching of negative numbers is dealt with herein from the point of view of expanding the number classes from those containing only positive numbers to those that also include the negatives (and zero). The expansion involves generalizing the concept of “number” and making modifications in the definitions of the arithmetical operations. Each of these actions generates cognitive obstacles, both historically and pedagogically. In contrast to the historical situation, the concept of negative number is not problematic for the contemporary student because he/she encounters them with certain frequency in his/her daily life. The addition of negative numbers, especially in the case in which it becomes equivalent to subtraction in the natural numbers with the subtrahend greater than the minuend, is subject to a didactical obstacle, which can be overcome by using the number line as a visual/conceptual support. Also, the multiplication of two negative numbers, resulting in a positive number, constitutes a genuine epistemological obstacle due to the fact that the usual metaphors used to explain the negative numbers do not clarify this operation. The article thus proposes to consider double negatives as inverses of inverses and delineates a sequence of activities based on this metaphor to overcome the mentioned epistemological obstacle.

Mastery of mathematics depends on the people’s ability to manipulate and abstract values such as negative numbers. Knowledge of arithmetic principles does not necessarily generalize from positive number arithmetic to arithmetic involving negative numbers (Prather & Alibali, 2008, https://doi.org/10.1080/03640210701864147). In this study, we evaluate the relationship between participant’s knowledge of the Relation to Operands arithmetic principle in both positive and negative numbers and their spontaneous on numerical relations. Additionally, we tested if the feedback that directs attention to relations affects participants’ attention to relation and their arithmetic principle knowledge. This study contributes to our understanding of the specific skills and cognitive processes that are associated with understanding high-level mathematics. [ABSTRACT FROM AUTHOR]

While some researchers place negative numbers on a so-called extended mental number line to the left of positive numbers, others claim that negative numbers do not have mental representations but are processed through positive numbers combined with transformation rules. We measured spatial associations of negative numbers with a modified implicit association task that avoids spatial confounds present in most previous studies. In two lab-based magnitude classification experiments (each including 24 participants) and two online replications (with 74 and 77 participants, respectively), positive and negative numbers were combined with two spatial contexts: either directional symbols (left- or right-pointing arrows) or rectangles of varying sizes. In all experiments, we found a robust distance effect for negative numbers. However, there were no consistent associations of negative numbers with directional or size contexts. In the context of directional symbols, holistic processing was prevalent only in the small negative number range (-9, -8, -7, -6) when ensured by the stimulus set, supporting an extended mental number line. In the context of rectangles, however, large negative numbers from -4 to -1 were perceived as small, thus supporting rule-based processing. For negative number processing in the context of size, we further suggest the Semantic-Perceptual Size Congruity Cuing model (SPeSiCC model). We show that associations of size with negative numbers underly more complex processing mechanisms than mere recruitment of a transformation rule. In general, we conclude that associations of negative numbers with space and size are situated in the context, as they depend on the presented number range and differ for spatial direction and size. [ABSTRACT FROM AUTHOR]

To date, of the many studies on early algebraic thinking, none, to our knowledge, has examined the relationships between algebraic thinking and negative numbers. Students encounter persistent difficulties in dealing with these numbers, and we believe that these could be addressed through the development of algebraic thinking. We are particularly interested in relational thinking, a form of algebraic thinking involved in generalised arithmetic, characterised by the ability to identify the structure of an expression as well as the relationships between numbers. The idea of the 'subtractive number' has been highlighted in this context. The aim of the study was to investigate the role of relational thinking in dealing with negative numbers. We submitted a paper-and-pencil test to 166 grade 6 students in order to analyse their skills in operations with integers, as well as their relational thinking in questions relating to the compensation strategy in subtraction. We then examined the extent to which the students who answered the compensation questions correctly performed the operations with integers better than those who answered them incorrectly. Our results showed that students' ability to see the subtraction operation as a 'transformation' involving a unary use of the minus sign appears to be a factor in their success in operations with negatives. Few students demonstrated this ability, yet it can be seen as an essential stage on which to base the progressive development of relational thinking. [ABSTRACT FROM AUTHOR]

Abstract This research is based on several inquiries from mathematics educators on the origins of negative numbers and their acceptance in the community. This study used library research, or more precisely, bibliographical studies, as a method. The research examined how the Asian and European communities accept negative numbers. The study's findings indicated that the Asian and European communities accept negative numbers differently. In comparison to the Asian population, the European populace takes a long time to accept negative figures. It resulted in numerous contrasts between the Asian and European Communities, one of which is in terms of religiosity. This study established that religiosity has an effect on the Asian and European communities' acceptance of negative numbers. Abstrak Penelitian ini didasarkan pada beberapa pertanyaan dari pendidik matematika tentang asal-usul bilangan negatif dan penerimaannya di masyarakat. Penelitian ini menggunakan metode penelitian kepustakaan atau lebih tepatnya studi kepustakaan. Penelitian tersebut mengkaji bagaimana masyarakat Asia dan Eropa menerima angka negatif. Temuan studi menunjukkan bahwa komunitas Asia dan Eropa menerima angka negatif secara berbeda. Dibandingkan dengan penduduk Asia, penduduk Eropa membutuhkan waktu lama untuk menerima angka negatif. Hal ini mengakibatkan banyak perbedaan antara Komunitas Asia dan Eropa, salah satunya dalam hal religiositas. Studi ini menetapkan bahwa religiusitas berpengaruh pada penerimaan masyarakat Asia dan Eropa terhadap angka negatif. How to Cite: Maryati, T. K., Aisah, K., Miftah, R. (2021). How the Asian and European Communities Accept Negative Numbers. TARBIYA: Journal of Education in Muslim Society, 8(2), 167-178. doi:10.15408/tjems.v8i2.25042.

Whether negative numbers have a fixed spatial–numerical association of response codes effect (SNARC effect), and (if they have) whether the spatial representation of negative numbers is associated with negative numbers' absolute or signed values remains controversial. In this study, through three experiments, the coding level of the magnitude and the spatial-direction is manipulated. In the first experiment, participants are required to code the magnitude and spatial-direction explicitly by using a magnitude classification task. In the second experiment, participants are forced to code the magnitude implicitly as well as to code the spatial-direction explicitly by utilizing a cuing task. In the third experiment, participants are obliged to code the magnitude explicitly as well as to code the spatial-direction implicitly by adopting a magnitude and arrow-direction classification tasks with Go/No-Go responses. The results show that (1) the absolute value of negative numbers associates with space when the magnitude of negative numbers is explicitly coded, no matter employing the explicit or implicit spatial-direction; (2) the signed value of negative numbers associates with space under the condition of implicit magnitude as well as explicit spatial-direction. In conclusion, the current study indicates that the SNARC effect of negative numbers is variable in different conditions, and the type of SNARC effect about negative numbers is modulated by the joint coding level of the magnitude and spatial-direction. [ABSTRACT FROM AUTHOR]

The notion of negative numbers is one of the most fundamental in mathematics. For many years, they have caused confusion and they have been an object of controversy among great researchers, until the development of symbolic algebra led to their acceptance in the form they have today. When students are first introduced to the concept of negative numbers and the operations between them, various difficulties occur and misconceptions arise, as expected. Their comprehension is one of the most challenging tasks of teaching mathematics due to their complexity and abstract nature. For this purpose, various models have been devised and employed. In the present study we expose some of the epistemological obstacles commonly observed in understanding negative numbers and then exhibit and compare two different models frequently used for introducing the four basic operations between them.

Whether negative numbers have a fixed spatial–numerical association of response codes effect (SNARC effect), and (if they have) whether the spatial representation of negative numbers is associated with negative numbers’ absolute or signed values remains controversial. In this study, through three experiments, the coding level of the magnitude and the spatial-direction is manipulated. In the first experiment, participants are required to code the magnitude and spatial-direction explicitly by using a magnitude classification task. In the second experiment, participants are forced to code the magnitude implicitly as well as to code the spatial-direction explicitly by utilizing a cuing task. In the third experiment, participants are obliged to code the magnitude explicitly as well as to code the spatial-direction implicitly by adopting a magnitude and arrow-direction classification tasks with Go/No-Go responses. The results show that (1) the absolute value of negative numbers associates with space when the magnitude of negative numbers is explicitly coded, no matter employing the explicit or implicit spatial-direction; (2) the signed value of negative numbers associates with space under the condition of implicit magnitude as well as explicit spatial-direction. In conclusion, the current study indicates that the SNARC effect of negative numbers is variable in different conditions, and the type of SNARC effect about negative numbers is modulated by the joint coding level of the magnitude and spatial-direction.

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value. We completely classify the cases of rank one, torsion groups for which the density vanishes and the the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we propose some partial results. Finally, we present some computations comparing the approximated density computed with primes up to 1010 and those predicted by the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

While the idea of the naturalness of the positive integers is ancient, the idea of the naturals as the foundation of our number system is not. This latter idea, along with other factors, eventually led to the abstract definitions of natural numbers at the end of the 19th century. But, what led to such an idea that was already present among Germanic mathematicians in the first third of the 19th century? This article examines the tensions around the notion of number among the Germanic mathematicians of the second half of the 18th century with the aim of contributing to a better understanding of some of the factors that explain the emergence of such a different approach to naturals.

Copyright of JMD: Journal für Mathematik-Didaktik is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Visualising positive and negative numbers on a number line is helpful for exploring problems involving operations with positive and negative numbers. This is because number lines lend themselves to exploring problems involving continuous linear contexts such as travelling distances and temperature. Teachers in a professional development program explored how to make sense of operations with positive and negative integers by visualising a context on a number line. [ABSTRACT FROM AUTHOR]

Numerical magnitude knowledge is related to current math abilities and predictive of future math learning. However, this relationship has previously been shown only for knowledge of positive magnitudes; the present study is the first, to our knowledge, to examine the connection between magnitude knowledge and math ability with negative integers. In the current study, 8th Grade Algebra I students (n = 154) were given assessments of their algebra and negative sign knowledge and 2 number line tasks, an all negative (−10,000 to 0) and a bidirectional scale (−1,000 to 1,000) as part of a pretest and posttest surrounding their unit on 2-step equation solving. Results show that students' negative sign and numerical magnitude knowledge are associated with their current, or pretest, algebra performance. However, only negative magnitude knowledge from the bidirectional number line was predictive of improvement in students' overall algebra knowledge. These findings illustrate differences between knowledge of positive and negative magnitudes and provide support for the integrated theory of numerical development. This study examined whether prior relationships among numerical magnitude knowledge and mathematics knowledge and learning apply to students' understanding of negative magnitudes. Our results revealed that students' negative sign knowledge and magnitude knowledge, of both positive and negative numbers, were related to their current overall algebra knowledge. However, only their negative number magnitude knowledge from a bidirectional number line predicted improvement of students' overall algebra knowledge at the end of their mathematics unit on 2-step equation solving. These findings demonstrate that it is important to provide students with experiences with an extended number system that includes both negative and positive numbers. [ABSTRACT FROM AUTHOR]

We describe a game that uses inexpensive and accessible materials to encourage students to make important discoveries about integer operations. Go-High/Go-Low gives students studying negative numbers a fun way to not only practice addition, subtraction, and ordering numbers, but also to discover the surprising idea that subtraction can actually lead to a greater number, depending on the number subtracted. [ABSTRACT FROM AUTHOR]

Abstract Experiments with multiple objectives form a staple diet of modern scientific research. Deriving optimal designs with multiple objectives is a long-standing challenging problem with few tools available. The few existing approaches cannot provide a satisfied solution in general: either the computation is very expensive or a satisfied solution is not guaranteed. A novel algorithm is proposed to address this literature gap. We prove convergence of this algorithm, and show in various examples that the new algorithm can derive the true solutions with high speed. Highlights • A novel algorithm is proposed to address multiple objectives optimal designs. • The convergence of this algorithm is proved. • The new algorithm is shown to enjoy high speed in various examples. [ABSTRACT FROM AUTHOR]

In recent years, an increasing interest has been shown in pure discount bonds. A number of papers have looked at corporate capital structure in an option pricing framework, assuming the issuance of pure discount bonds by corporations. Merton has presented a theoretical discussion of the impact of default risk upon pure discount bonds. Racette and Lewellen have argued that long-term pure discount bond issues could present special tax advantages to corporations. Fisher and Weil have suggested that long-term pure discount bonds would be very attractive to holders of bond portfolios in order to immunize these portfolios against changes in interest rates. A similar position is taken by Caks. In spite of the claimed advantages of pure discount bonds from the viewpoint of both potential bondholders and stockholders, in the United States long-term pure discount bonds are not issued by corporations or governments. This paper will provide an explanation of this non-issuance by showing that the tax treatment of original issue pure discount bonds is highly disadvantageous to potential bondholders and firms. U.S. tax law requires the original purchaser of a long-term pure discount bond to amortize as regular income over the bond's life the difference between the purchase price and the face value (to be received at maturity). One purpose of this paper is to show that this tax law implies that, in equilibrium, pure discount bond issue prices would have to be negative for a wide spectrum of tax rates. Corporations would never issue bonds at negative prices, since the corporation would never receive any cash inflows. Secondly, it will be shown that pure discount bonds initially issued at positive prices can subsequently have negative prices if interest rates were to rise. Consequently, such bonds would be very risky to potential purchasers. [ABSTRACT FROM AUTHOR]

Our topic is the understanding of the nature of negative numbers - the entities to which expressions such as ‘-1’ refer. Following Frege, we view positive whole numbers as providing the answer to the question „how many?“ In this light, how are we to view negative numbers? Both positive and negative numbers can be ordered through the relation of larger or smaller. It is then true of all negative numbers that they are entities which are (somehow) smaller than zero. For many, this has been understood as an ontological paradox: how can something be „less than nothing?“ Some propose to avoid the paradox by treating negative numbers as mere façons de parler. In this paper, we propose a more realist account, taking as our starting point the thesis that there is at least one familiar type of object, the magnitude of which can be expressed with negative numbers, namely, debt. How can the sense of an expression be ontologically paradoxical, yet the expression itself still plausibly refer to a social object such as a debt? Or, put differently, how is it possible to be, at the same time, a realist in financial theory and a nominalist in mathematical theory? The paper first shows that the paradox arises when the two distinct ways in which negative numbers are connected to real objects are run together. The first of the two refers to debt only, whereas the second could refer to debt, as well as to physical objects. Finally, we claim that a debt is at once a specifically social object and part of reality as described by physics.

Understanding negative numbers and doing operations on them is one of the fundamental subjects in mathematics that students face many problems when they encounter them at school. One of the problems that leads to very serious learning difficulties in mathematics is those misconceptions students may have from previous inadequate teaching, informal thinking, or poor remembrance. Recognition of misconception and the origins that create them in the fields and at different levels of education, can improve learning. Misunderstanding in this study is incomplete or incorrect interpretation of a concept that cause systematic errors in the performance. This study investigates the student’s understanding and misconception of negative numbers. Descriptive statistics, survey, is used as a method of this study. The population of the study is all second level students of guidance school in the academic year 1391-92 and all the seventh level students of Darmian town in the academic year 1392-93. 443 students in second level of guidance school and 55 students in seventh level were chosen as a sample of the study based on cluster random sampling. A self-designed questionnaire by the researcher is used as the instrument for the study, the questionnaire was based on “Bofferding” and “Kilhamn” questionnaire. Results showed that most of the students don’t have a true understanding of negative numbers in school and face problems in applying their knowledge to solve mathematical problems. Also the results of the current study showed that some of the students consider "-" as a reducing operator and some consider "+" as an increasing operator.

Numerous studies have demonstrated the relevance of magnitude estimation skills for mathematical proficiency, but little research has explored magnitude estimation with negative numbers. In two experiments the current study examined middle school students’ magnitude knowledge of negative numbers with number line tasks. In Experiment 1, both 6th (n = 132) and 7th grade students (n = 218) produced linear representations on a -10,000 to 0 scale, but the 7th grade students’ estimates were more accurate and linear. In Experiment 2, the 7th grade students also completed a -1,000 to 1,000 number line task; these results also indicated that students are linear for both negative and positive estimates. When comparing the estimates of negative and positive numbers, analyses illustrated that estimates of negative numbers are less accurate than those of positive numbers, but using a midpoint strategy improved negative estimates. These findings suggest that negative number magnitude knowledge follows a similar pattern to positive numbers, but the estimation performance of negatives lags behind that of positives.

Abstract Whether negative numbers are represented componentially as two separate components (a digit and a sign) or represented holistically as a whole is still under debate. The present study investigated the representation of negative numbers via duration comparison tasks that might eliminate the possible influences of the strategies participants usually use when processing numbers. In the duration comparison task, participants are required to compare the durations of two numbers that were presented sequentially, thus the numerical value is irrelevant to the task. Our results showed that negative numbers were processed holistically when positive numbers and negative numbers were presented in separate blocks (Exp. 1). But negative numbers showed a componential representation when positive numbers were intermixed with negative numbers within the same block (Exp. 2). This inconsistency might be due to the spatial separation of the polarity sign and digit components in negative numbers which may remind participants that negative numbers are formed by two components when there are positive numbers as a comparison in the same block. Therefore, when the spatial separation of the polarity sign and digit components in negative numbers was eliminated by using colors to mark the polarity (Exp. 3), the negative numbers were processed holistically even when positive numbers and negative numbers were intermixed. These results suggest that negative numbers are represented holistically and may be prone to be represented componentially when intermixed with positive numbers within the same block. Highlights • The representation of negative numbers was investigated via duration comparison tasks. • The current results mainly support the holistic representation of negative numbers. • The context effect in the duration comparison task was possibly due to a representation superiority of positive numbers. [ABSTRACT FROM AUTHOR]