Dissociation between magnitude comparison and relation identification across different formats for rational numbers

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.

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Fraction magnitude: Mapping between symbolic and spatial representations of proportion

Journal of Numerical Cognition ◽

10.5964/jnc.v6i2.285 ◽

2020 ◽

Vol 6 (2) ◽

pp. 204-230

Author(s):

Michelle Ann Hurst ◽

Marisa Massaro ◽

Sara Cordes

Keyword(s):

Rational Numbers ◽

Number Line ◽

Future Research ◽

Spatial Representations ◽

Improve Performance ◽

Comparison Task ◽

Magnitude Comparison ◽

Number Lines ◽

Research And Education ◽

Partitioning Strategy

Fraction notation conveys both part-whole (3/4 is 3 out of 4) and magnitude (3/4 = 0.75) information, yet evidence suggests that both children and adults find accessing magnitude information from fractions particularly difficult. Recent research suggests that using number lines to teach children about fractions can help emphasize fraction magnitude. In three experiments with adults and 9-12-year-old children, we compare the benefits of number lines and pie charts for thinking about rational numbers. In Experiment 1, we first investigate how adults spontaneously visualize symbolic fractions. Then, in two further experiments, we explore whether priming children to use pie charts vs. number lines impacts performance on a subsequent symbolic magnitude task and whether children differentially rely on a partitioning strategy to map rational numbers to number lines vs. pie charts. Our data reveal that adults very infrequently spontaneously visualize fractions along a number line and, contrary to other findings, that practice mapping rational numbers to number lines did not improve performance on a subsequent symbolic magnitude comparison task relative to practice mapping the same magnitudes to pie charts. However, children were more likely to use overt partitioning strategies when working with pie charts compared to number lines, suggesting these representations did lend themselves to different working strategies. We discuss the interpretations and implications of these findings for future research and education. All materials and data are provided as Supplementary Materials.

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Magnitude comparison with different types of rational numbers.

Journal of Experimental Psychology Human Perception & Performance ◽

10.1037/a0032916 ◽

2014 ◽

Vol 40 (1) ◽

pp. 71-82 ◽

Cited By ~ 30

Author(s):

Melissa DeWolf ◽

Margaret A. Grounds ◽

Miriam Bassok ◽

Keith J. Holyoak

Keyword(s):

Rational Numbers ◽

Magnitude Comparison ◽

Different Types

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Supplemental Material for Eye Position Reflects the Spatial Coding of Numbers During Magnitude Comparison

Journal of Experimental Psychology Learning Memory and Cognition ◽

10.1037/xlm0000681.supp ◽

2018 ◽

Keyword(s):

Eye Position ◽

Spatial Coding ◽

Magnitude Comparison

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Eye position reflects the spatial coding of numbers during magnitude comparison.

Journal of Experimental Psychology Learning Memory and Cognition ◽

10.1037/xlm0000681 ◽

2019 ◽

Vol 45 (10) ◽

pp. 1910-1921 ◽

Cited By ~ 2

Author(s):

Samuel Salvaggio ◽

Nicolas Masson ◽

Michael Andres

Keyword(s):

Eye Position ◽

Spatial Coding ◽

Magnitude Comparison

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The Joint Universality for L-Functions of Elliptic Curves

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2004.9.4.15148 ◽

2004 ◽

Vol 9 (4) ◽

pp. 331-348

Author(s):

V. Garbaliauskienė

Keyword(s):

Elliptic Curves ◽

Rational Numbers ◽

Joint Universality ◽

Universality Theorem ◽

Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

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Productly linearly independent sequences

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2015.52.3.1315 ◽

2015 ◽

Vol 52 (3) ◽

pp. 350-370

Author(s):

Jaroslav Hančl ◽

Katarína Korčeková ◽

Lukáš Novotný

Keyword(s):

Rational Numbers ◽

Infinite Products ◽

Linearly Independent ◽

New Concepts ◽

Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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Pembentukan Lapangan Faktor dari Suatu Daerah Integral

Jurnal Natural ◽

10.30862/jn.v8i2.340 ◽

2012 ◽

Vol 8 (2) ◽

Author(s):

Tri Widjajanti ◽

Dahlia Ramlan ◽

Rium Hilum

Keyword(s):

Polynomial Ring ◽

Integral Domain ◽

Rational Numbers ◽

Ring Of Integers ◽

Binary Operations

Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make  a construction such that any integral domain can be  a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that  f : D ® F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.

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