scholarly journals Number line tasks and their relation to arithmetics in second to fourth graders

2021 ◽  
Vol 7 (1) ◽  
pp. 20-41
Author(s):  
Carrie Georges ◽  
Christine Schiltz
Keyword(s):  
Fourth Graders ◽  
Proportional Reasoning ◽  
Number Line ◽  
Numerical Magnitude ◽  
Cognitive Mechanisms ◽  
Bounded Number ◽  
Bias Model ◽  
Number Lines ◽  
Power Models ◽  
Log Linear

Considering the importance of mathematical knowledge for STEM careers, we aimed to better understand the cognitive mechanisms underlying the commonly observed relation between number line estimations (NLEs) and arithmetics. We used a within-subject design to model NLEs in an unbounded and bounded task and to assess their relations to arithmetics in second to fourth grades. Our results mostly agree with previous findings, indicating that unbounded and bounded NLEs likely index different cognitive constructs at this age. Bounded NLEs were best described by cyclic power models including the subtraction bias model, likely indicating proportional reasoning. Conversely, mixed log-linear and single scalloped power models provided better fits for unbounded NLEs, suggesting direct estimation. Moreover, only bounded but not unbounded NLEs related to addition and subtraction skills. This thus suggests that proportional reasoning probably accounts for the relation between NLEs and arithmetics, at least in second to fourth graders. This was further confirmed by moderation analysis, showing that relations between bounded NLEs and subtraction skills were only observed in children whose estimates were best described by the cyclic power models. Depending on the aim of future studies, our results suggest measuring estimations on unbounded number lines if one is interested in directly assessing numerical magnitude representations. Conversely, if one aims to predict arithmetic skills, one should assess bounded NLEs, probably indexing proportional reasoning, at least in second to fourth graders. The present outcomes also further highlight the potential usefulness of training the positioning of target numbers on bounded number lines for arithmetic development.

Promoting Discourse: Fractions on Number Lines

2021 ◽  
Vol 114 (4) ◽  
pp. 284-289
Author(s):  
Susan Ahrendt ◽  
Debra Monson ◽  
Kathleen Cramer
Keyword(s):  
Fourth Graders ◽  
Number Line ◽  
Number Lines ◽  
Mathematical Discussions ◽  
Five Practices

Examine fourth graders’ thinking about the unit, partitioning, order, and equivalence on the number line and consider ways to orchestrate mathematical discussions through the Five Practices.

EXPRESS: Spatial-numerical associations from a novel paradigm support the Mental Number Line account

2021 ◽  
pp. 174702182110087
Author(s):  
Lauren Aulet ◽  
Sami R Yousif ◽  
Stella Lourenco
Keyword(s):  
Memory Task ◽  
Verbal Working Memory ◽  
Number Line ◽  
Mental Number Line ◽  
Task Demands ◽  
Numerical Magnitude ◽  
The Novel ◽  
Alternative Account ◽  
Subject Design ◽  
Multiple Tasks

Multiple tasks have been used to demonstrate the relation between numbers and space. The classic interpretation of these directional spatial-numerical associations (d-SNAs) is that they are the product of a mental number line (MNL), in which numerical magnitude is intrinsically associated with spatial position. The alternative account is that d-SNAs reflect task demands, such as explicit numerical judgments and/or categorical responses. In the novel ‘Where was The Number?’ task, no explicit numerical judgments were made. Participants were simply required to reproduce the location of a numeral within a rectangular space. Using a between-subject design, we found that numbers, but not letters, biased participants’ responses along the horizontal dimension, such that larger numbers were placed more rightward than smaller numbers, even when participants completed a concurrent verbal working memory task. These findings are consistent with the MNL account, such that numbers specifically are inherently left-to-right oriented in Western participants.

Spatial-numerical associations from a novel paradigm support the mental number line account

2021 ◽  
Author(s):  
Lauren S Aulet ◽  
Sami Ryan Yousif ◽  
Stella F. Lourenco
Keyword(s):  
Memory Task ◽  
Verbal Working Memory ◽  
Number Line ◽  
Mental Number Line ◽  
Task Demands ◽  
Numerical Magnitude ◽  
The Novel ◽  
Alternative Account ◽  
Subject Design ◽  
Multiple Tasks

Multiple tasks have been used to demonstrate the relation between numbers and space. The classic interpretation of these directional spatial-numerical associations (d-SNAs) is that they are the product of a mental number line (MNL), in which numerical magnitude is intrinsically associated with spatial position. The alternative account is that d-SNAs reflect task demands, such as explicit numerical judgments and/or categorical responses. In the novel ‘Where was The Number?’ task, no explicit numerical judgments were made. Participants were simply required to reproduce the location of a numeral within a rectangular space. Using a between-subject design, we found that numbers, but not letters, biased participants’ responses along the horizontal dimension, such that larger numbers were placed more rightward than smaller numbers, even when participants completed a concurrent verbal working memory task. These findings are consistent with the MNL account, such that numbers specifically are inherently left-to-right oriented in Western participants.

Fixated in unfamiliar territory: Mapping estimates across typical and atypical number lines

2019 ◽  
Vol 73 (2) ◽  
pp. 279-294
Author(s):  
Sabrina Michelle Di Lonardo ◽  
Matthew G Huebner ◽  
Katherine Newman ◽  
Jo-Anne LeFevre
Keyword(s):  
Numerical Range ◽  
Number Line ◽  
Strategy Use ◽  
Reverse Direction ◽  
Tracking Data ◽  
Spatial Processes ◽  
Number Lines ◽  
Number Line Estimation ◽  
The Right ◽  
Typical Range

Adults ( N = 72) estimated the location of target numbers on number lines that varied in numerical range (i.e., typical range 0–10,000 or atypical range 0–7,000) and spatial orientation (i.e., the 0 endpoint on the left [traditional] or on the right [reversed]). Eye-tracking data were used to assess strategy use. Participants made meaningful first fixations on the line, with fixations occurring around the origin for low target numbers and around the midpoint and endpoint for high target numbers. On traditional direction number lines, participants used left-to-right scanning and showed a leftward bias; these effects were reduced for the reverse direction number lines. Participants made fixations around the midpoint for both ranges but were less accurate when estimating target numbers around the midpoint on the 7,000-range number line. Thus, participants are using the internal benchmark (i.e., midpoint) to guide estimates on atypical range number lines, but they have difficulty calculating the midpoint, leading to less accurate estimates. In summary, both range and direction influenced strategy use and accuracy, suggesting that both numerical and spatial processes influence number line estimation.

Order and Magnitude Share a Common Representation in Parietal Cortex

2009 ◽  
Vol 21 (11) ◽  
pp. 2114-2120 ◽  
Author(s):  
Michael S. Franklin ◽  
John Jonides
Keyword(s):  
Distance Effect ◽  
Number Line ◽  
Judgment Task ◽  
Mental Number Line ◽  
Numerical Magnitude ◽  
Intraparietal Sulcus ◽  
Ordinal Information ◽  
Common Representation ◽  
Distance Effects

The role of the intraparietal sulcus (IPS) in the representation of numerical magnitude is well established. Recently, there has also been speculation that the IPS is involved in the representation of ordinal information as well. These claims, however, overlook the fact that all neuroimaging paradigms in which participants make judgments about either magnitude and/or order result in a behavioral distance effect (i.e., the comparison is easier when the stimuli span a greater distance). This leaves open two possibilities: It may be that activation of the IPS is due to the mechanism that yields distance effects, or it may be that the IPS is involved in the representation of information about both magnitude and order. The current study used fMRI to compare a magnitude task in which participants show distance effects to an order-judgment task that yields reverse-distance effects. The results reveal activation of the IPS for both the magnitude and order tasks that is based on participants' strategies as opposed to the actual distance between the numbers. This leads to the conclusion that the IPS represents a mental number line, and that accessing this line can lead to distance effects when participants compare magnitudes and to reverse-distance effects when participants check for order.

scholarly journals The Force of Numbers: Investigating Manual Signatures of Embodied Number Processing

2021 ◽  
Vol 14 ◽  
Author(s):  
Alex Miklashevsky ◽  
Oliver Lindemann ◽  
Martin H. Fischer
Keyword(s):  
Embodied Cognition ◽  
Grip Force ◽  
Numerical Cognition ◽  
Stimulus Presentation ◽  
Number Line ◽  
Number Processing ◽  
New Technique ◽  
Numerical Magnitude ◽  
Left Hand ◽  
Number Magnitude

The study has two objectives: (1) to introduce grip force recording as a new technique for studying embodied numerical processing; and (2) to demonstrate how three competing accounts of numerical magnitude representation can be tested by using this new technique: the Mental Number Line (MNL), A Theory of Magnitude (ATOM) and Embodied Cognition (finger counting-based) account. While 26 healthy adults processed visually presented single digits in a go/no-go n-back paradigm, their passive holding forces for two small sensors were recorded in both hands. Spontaneous and unconscious grip force changes related to number magnitude occurred in the left hand already 100–140 ms after stimulus presentation and continued systematically. Our results support a two-step model of number processing where an initial stage is related to the automatic activation of all stimulus properties whereas a later stage consists of deeper conscious processing of the stimulus. This interpretation generalizes previous work with linguistic stimuli and elaborates the timeline of embodied cognition. We hope that the use of grip force recording will advance the field of numerical cognition research.

scholarly journals Cognitive mediators of US—China differences in early symbolic arithmetic

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0255283
Author(s):  
John E. Opfer ◽  
Dan Kim ◽  
Lisa K. Fazio ◽  
Xinlin Zhou ◽  
Robert S. Siegler
Keyword(s):  
Standardized Tests ◽  
Number Line ◽  
Chinese Children ◽  
Numerical Magnitude ◽  
Magnitude Comparison ◽  
Cognitive Mediators ◽  
Number Line Estimation ◽  
Mathematics Knowledge ◽  
Us Children ◽  
Symbolic Number

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

Key Mathematical Competencies From Arithmetic to Algebra

2020 ◽  
Author(s):  
Kristie J. Newton ◽  
Christina A. Barbieri ◽  
Julie L. Booth
Keyword(s):  
Mathematics Learning ◽  
Numerical Magnitude ◽  
Major Area ◽  
Equal Sign ◽  
Negative Numbers ◽  
Mathematical Ideas ◽  
Whole Numbers ◽  
Mathematical Competencies ◽  
Number Lines ◽  
Over Time

Mathematics learning encompasses a broad range of processes and skills that change over time. Magnitude and equivalence are two fundamental mathematical ideas that students encounter early and often in their mathematics learning. Numerical magnitude knowledge is knowledge of the relative sizes of numbers, including whole numbers, fractions, and negative numbers, within a given scale. Understanding mathematical equivalence means understanding that two or more specific quantities with the same value can be represented in a variety of ways and remain equal and interchangeable. A major area of research on equivalence is knowledge of the equal sign. Both equal sign knowledge and magnitude knowledge are foundational in that they predict later learning in mathematics, including algebra. Implications for practice include the use of number lines and more variation in the way that arithmetic problems are formatted.

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