Negative Numbers Are Generated in the Mind

2008 ◽  
Vol 55 (3) ◽  
pp. 157-163 ◽  
Author(s):  
Dana Ganor-Stern ◽  
Joseph Tzelgov
Keyword(s):  
Absolute Magnitude ◽  
Number Line ◽  
Numerical Comparison ◽  
Comparison Task ◽  
Negative Numbers ◽  
The Mind ◽  
Holistic Representation

Abstract. The goal of the present study was to disentangle two possible representations of negative numbers - the holistic representation, where absolute magnitude is integrated with polarity; and the components representation, where absolute magnitude is stored separately from polarity. Participants' performance was examined in two tasks involving numbers from -100 to 100. In the numerical comparison task, participants had to decide which number of a pair was numerically larger/smaller. In the number line task, participants were presented with a spatial number line on which they had to place a number. The results of both tasks support the components representation of negative numbers. The findings suggest that processing of negative numbers does not involve retrieval of their meaning from memory, but rather the integration of the polarity sign with the digits' magnitudes.

Exploring the Boundaries of the Number–Temporal Order Association

2015 ◽  
Vol 62 (3) ◽  
pp. 198-205 ◽  
Author(s):  
Dana Ganor-Stern
Keyword(s):  
Embodied Cognition ◽  
Mental Representation ◽  
Temporal Order ◽  
Past Research ◽  
Numerical Comparison ◽  
Negative Number ◽  
Comparison Task ◽  
Negative Numbers ◽  
The Past

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

scholarly journals Neural coding partially accounts for the relation between children’s number line estimation and numerical comparison performance

2018 ◽  
Author(s):  
Richard Prather
Keyword(s):  
Neural Coding ◽  
Standardized Test ◽  
Number System ◽  
Number Line ◽  
Standardized Test Scores ◽  
Numerical Comparison ◽  
Comparison Task ◽  
Mathematics Interventions ◽  
Number Line Estimation ◽  
Number Perception

Numerical comparison is a primary measure of the acuity of children’s approximate number system (ANS). ANS acuity is associated with key developmental outcomes such as symbolic number skill, standardized test scores and even employment outcomes(Halberda, Mazzocco, & Feigenson, 2008; Parsons & Bynner, 1997). We examine the relation between children’s performance on the numerical comparison task and the number line estimation task. It is important to characterize the relation between tasks in order to develop mathematics interventions that lead to transfer across tasks. We find that number line performance is significantly predicted by non-symbolic comparison performance for participants ranging in age from 5 to 8 years. We also evaluate, using a computational model, if the relation between the two tasks can be adequately explained based on known neural correlates of number perception. Data from humans and non-human primates characterizes neural activity corresponding to the perception of numerosities. Results of behavioral experimentation and computational modeling suggest that though neural coding of number predicts a correlation in participants’ performance on the two tasks, it cannot account for all of the variability in the human data. This is interpreted as consistent with accounts of number line estimation in which number line estimation does not rely solely on participants’ numerical perception.

Holistic Representation of Unit Fractions

2011 ◽  
Vol 58 (3) ◽  
pp. 201-206 ◽  
Author(s):  
Dana Ganor-Stern ◽  
Irina Karasik-Rivkin ◽  
Joseph Tzelgov
Keyword(s):  
Distance Effect ◽  
Congruity Effect ◽  
Complex Numbers ◽  
Numerical Comparison ◽  
Comparison Task ◽  
Semantic Congruity ◽  
Stimulus Set ◽  
Unit Fractions ◽  
Semantic Congruity Effect ◽  
Holistic Representation

The present study examined the processing of unit fractions and the extent to which it is affected by context. Using a numerical comparison task we found evidence for a holistic representation of unit fractions when the immediate context of the fractions was emphasized, that is when the stimuli set included in addition to the unit fractions also the numbers 0 and 1. The holistic representation was indicated by the semantic congruity effect for comparisons of pairs of fractions and by the distance effect in comparisons of a fraction and 0 and 1. Consistent with previous results (Bonato, Fabbri, Umilta, & Zorzi, 2007) there was no evidence for a holistic representation of unit fractions when the stimulus set included only fractions. These findings suggest that fraction processing is context-dependent. Finally, the present results are discussed in the context of processing other complex numbers beyond the first decade.

Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

2009 ◽  
Vol 62 (8) ◽  
pp. 1598-1616 ◽  
Author(s):  
Gaëlle Meert ◽  
Jacques Grégoire ◽  
Marie-Pascale Noël
Keyword(s):  
Mental Representation ◽  
Priming Effect ◽  
Rational Numbers ◽  
Numerical Comparison ◽  
Natural Numbers ◽  
Comparison Task ◽  
Magnitude Comparison ◽  
Distance Effects ◽  
Holistic Representation ◽  
Common Components

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.

A teaching aid for signed numbers

1966 ◽  
Vol 13 (7) ◽  
pp. 589-591
Author(s):  
Edna M. Pratt
Keyword(s):  
Mathematical Proof ◽  
Number Line ◽  
Negative Numbers ◽  
Inductive Approach ◽  
Hard Time ◽  
Teaching Aid ◽  
Model Approach

Each year the time comes when it is necessary to illustrate the multiplication of signed numbers. There are many approaches to helping the student develop insight and skill. Some of the more common are multiplying two negative numbers using the number line, 1 * an inductive approach,2 the modern mathematical proof approach,3 the statement approach,4 or the model approach,5 but some young mathematicians still have a hard time understanding these ideas. The following is a very effective device which may be used to help students discover and then reinforce the operation of multiplication of signed numbers.

scholarly journals Active Learning dalam Pembelajaran Matematika SD Melalui Permainan Engklek Mamun

2020 ◽  
Vol 5 (2) ◽  
pp. 233-238
Author(s):  
Ulfatun Khasanah
Keyword(s):  
Active Learning ◽  
Mathematics Learning ◽  
Number Line ◽  
Negative Number ◽  
Negative Numbers

The weak ability of the students on the summation material and reduction in the background of this research. Mathematics is abstract, so in mathematics learning the necessary medium/intermediary that serves to confine so that facts clearer and more easily accepted by students. Media used is the game usually Mamun, which is a game conducted in groups in the form of advanced crank if the number of positive and crank backward if the number is negative. Numbers is a row of students holding positive and negative number flags. Each student holds a number. Students who do not hold the number do the forward or backward crank according to the prescribed number. The method is qualitative. The result is that through the game Engklek Mamun students can do the counting operation summation and reduction on the number line.

scholarly journals Numerical Comparisons and Sex-Related Brain Connectivity

2021 ◽  
Vol 13 (2) ◽  
pp. 62
Author(s):  
Fabiola R. Gómez-Velázquez ◽  
Andrés A. González-Garrido ◽  
Ricardo A. Salido-Ruiz ◽  
Sulema Torres-Ramos ◽  
Aurora Espinoza-Valdez ◽  
...  
Keyword(s):  
Brain Connectivity ◽  
Cognitive Strategies ◽  
Numerical Comparison ◽  
Math Learning ◽  
Comparison Task ◽  
Behavioral Measures ◽  
Functional Brain ◽  
Progressive Development ◽  
Interhemispheric Connectivity ◽  
Number Comparison

Despite the recent literature on sex-related anatomic, maturational and functional brain differences, the study of significant individual developments in math learning and achievement has scarcely approached this perspective. We aimed to compare the influence of sex in functional brain connectivity and behavioral measures in a numerical comparison task. Therefore, a group of school children with ages from 8 to 11 years old was evaluated during a number comparison task. Even though the behavioral performance was similar across the sexes, males distinctly showed a significant correlation between their math WRAT-4 scores and the number of correct responses in the experimental task and working memory scores. Besides, the analysis of the concurrent EEG during task performance showed that males comparatively had a greater brain left intra-hemispheric connectivity, as well as greater interhemispheric connectivity, particularly in Theta and Alpha bands during task performing -as compared to resting-. In contrast, females showed a significantly different decrement of brain connectivity in the Alpha band from resting to task performing. Present results are interpreted as probably reflecting sex-related maturational dissimilarities in neurodevelopment, along with the progressive development of more efficient cognitive strategies, processes running not necessarily parallel in both sexes. 

scholarly journals Numerical cognition in action: Reaching behavior reveals numerical distance effects in 5- to 6-year-olds

2018 ◽  
Vol 4 (2) ◽  
pp. 286-296 ◽  
Author(s):  
Christopher D. Erb ◽  
Jeff Moher ◽  
Joo-Hyun Song ◽  
David M. Sobel
Keyword(s):  
Spatial Representation ◽  
Numerical Cognition ◽  
Spatial Information ◽  
Numerical Comparison ◽  
Digital Display ◽  
Numerical Distance ◽  
Tight Coupling ◽  
Comparison Task ◽  
The Right ◽  
Trial Participants

This study investigates how children’s numerical cognition is reflected in their unfolding actions. Five- and 6-year-olds (N = 34) completed a numerical comparison task by reaching to touch one of three rectangles arranged horizontally on a digital display. A number from 1 to 9 appeared in the center rectangle on each trial. Participants were instructed to touch the left rectangle for numbers 1-4, the center rectangle for 5, and the right rectangle for 6-9. Reach trajectories were more curved toward the center rectangle for numbers closer to 5 (e.g., 4) than numbers further from 5 (e.g., 1). This finding indicates that a tight coupling exists between numerical and spatial information in children’s cognition and action as early as the preschool years. In addition to shedding new light on the spatial representation of numbers during childhood, our results highlight the promise of incorporating measures of manual dynamics into developmental research.

The fractions SNARC revisited: Processing fractions on a consistent mental number line

2018 ◽  
Vol 71 (8) ◽  
pp. 1761-1770 ◽  
Author(s):  
Elizabeth Y Toomarian ◽  
Edward M Hubbard
Keyword(s):  
Number Line ◽  
Mental Number Line ◽  
Snarc Effect ◽  
Spatial Representations ◽  
Comparison Task ◽  
Single Digit ◽  
Magnitude Processing ◽  
Number Comparison ◽  
Within Subjects ◽  
Solid Foundation

The ability to understand fractions is key to establishing a solid foundation in mathematics, yet children and adults struggle to comprehend them. Previous studies have suggested that these struggles emerge because people fail to process fraction magnitude holistically on the mental number line (MNL), focusing instead on fraction components. Subsequent studies have produced evidence for default holistic processing but examined only magnitude processing, not spatial representations. We explored the spatial representations of fractions on the MNL in a series of three experiments. Experiment 1 replicated Bonato et al.; 30 naïve undergraduates compared unit fractions (1/1-1/9) to 1/5, resulting in a reverse SNARC (Spatial-Numerical Association of Response Codes) effect. Experiment 2 countered potential strategic biases induced by the limited set of fractions used by Bonato et al. by expanding the stimulus set to include all irreducible, single-digit proper fractions and asked participants to compare them against 1/2. We observed a classic SNARC effect, completely reversing the pattern from Experiment 1. Together, Experiments 1 and 2 demonstrate that stimulus properties dramatically impact spatial representations of fractions. In Experiment 3, we demonstrated within-subjects reliability of the SNARC effect across both a fractions and whole number comparison task. Our results suggest that adults can indeed process fraction magnitudes holistically, and that their spatial representations occur on a consistent MNL for both whole numbers and fractions.

Number line multiplication for negative numbers

1966 ◽  
Vol 13 (3) ◽  
pp. 213-217
Author(s):  
Lewis H. Coon
Keyword(s):  
Elementary School ◽  
School Child ◽  
Number Line ◽  
Elementary School Child ◽  
Intermediate Grades ◽  
Negative Numbers ◽  
Vector Representations

The number line, a traditional tool in arithmetic classes, continues to find greater acceptance in some of the new mathematics programs. The logic behind some of the applications of “cricket- Jumps,” vectors, or hops1 to and fro on the line seems to give an elementary school child a model suited to his learning level. Thus the jumps of the earlier grades are replaced by moves in the intermediate grades, which are replaced by vector representations in the upper grades. Figure 1 shows the addition of 3 to 2, written 2+3.

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