Left digit effects in numerical estimation across development

Mapping Intimacies ◽

10.31234/osf.io/43dq8 ◽

2021 ◽

Author(s):

Katherine Williams ◽

Alexandra Zax ◽

Andrea Patalano ◽

Hilary Barth

Keyword(s):

Cognitive Development ◽

Numerical Cognition ◽

Number Line ◽

Effect Sizes ◽

Numerical Estimation ◽

Instructional Tool ◽

Learning And Development ◽

Number Line Estimation

Number line estimation (NLE) tasks are widely used to investigate numerical cognition, learning, and development, and as an instructional tool. Interpretation of these tasks generally involves an implicit expectation that responses are driven by the overall magnitudes of target numerals, in the sense that the particular digits conveying those magnitudes are unimportant. However, recent evidence shows that numbers with similar magnitudes but different leftmost digits are estimated very differently. For example, “798” is placed systematically much too far to the left of “801” in a 0-1000 NLE task by children aged 7-11 and adults (Lai et al., 2018). Here we ask whether this left digit effect generalizes to two-digit numerals in a 0-100 NLE task and whether it emerges in younger children. Children aged 5-8 (Study 1, N = 73), adults (Study 2, N = 44), and children aged 9-11 (Study 3, N = 27) completed a standard 0-100 NLE task on a touchscreen tablet. We observed left digit effects for two-digit numerals in children aged 8-11 and adults, with large effect sizes, demonstrating that these effects generalize to smaller numerical ranges. Left digit effects were not apparent in 5- to 7-year-olds, suggesting that these effects do not emerge at younger ages for smaller, more familiar numerical ranges. We discuss developmental emergence of left digit effects in number line estimation and implications within and beyond the field of cognitive development.

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Analogical mapping in numerical development

10.31234/osf.io/t8mcp ◽

2019 ◽

Author(s):

Elisabeth Marchand ◽

David Barner

Keyword(s):

Numerical Cognition ◽

Number Line ◽

Number Word ◽

Numerical Estimation ◽

Analogical Thinking ◽

Number Words ◽

Numerical Development ◽

Analogical Mapping ◽

Number Line Estimation ◽

Abstract Content

This chapter outlines the contribution of analogical thinking in numerical cognition and specifically, to number words learning and numerical estimation. We begin with an overview of number word learning, followed by a description of analogical mapping as defined by Gentner (1983, 2010), and discuss how children might acquire the meaning of counting based on analogical mapping. Next, we review the claim that very similar processes of analogical mapping may support numerical estimation, based on findings from studies of dot-array and number line estimation. These studies suggest that children’s knowledge of how the count list is structured – and in particular the ordering and distance between numbers – affects their ability to make accurate estimates. Finally, we discuss extensions of this idea to other cases where analogy has been proposed as a source of representational change. We conclude that analogical mappings enrich how humans transcend core numerical abilities to represent abstract content.

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Digit identity influences numerical estimation in children and adults

10.31234/osf.io/mgj78 ◽

2017 ◽

Author(s):

Maxine Lai ◽

Alexandra Zax ◽

Hilary Barth

Keyword(s):

Developmental Change ◽

Mental Representations ◽

Number Line ◽

Effect Sizes ◽

Numerical Magnitude ◽

Numerical Estimation ◽

Target Values ◽

Number Line Estimation ◽

Digit Identity ◽

Symbolic Number

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that it is the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, that matter for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7-11 and adults completed 0-1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of “Hundreds” boundaries (e.g. 698 and 701) and “Fifties” boundaries (e.g. 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non-speeded NLE task. Under both speeded and non-speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children’s or adults’ placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.

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Unbounded number line estimation as a measure of numerical estimation

PLoS ONE ◽

10.1371/journal.pone.0213102 ◽

2019 ◽

Vol 14 (3) ◽

pp. e0213102 ◽

Cited By ~ 3

Author(s):

Regina Miriam Reinert ◽

Matthias Hartmann ◽

Stefan Huber ◽

Korbinian Moeller

Keyword(s):

Number Line ◽

Numerical Estimation ◽

Number Line Estimation

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Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks

10.31219/osf.io/vepa2 ◽

2017 ◽

Author(s):

Emily Slusser ◽

Hilary Barth

Keyword(s):

Numerical Cognition ◽

Developmental Change ◽

Mental Representations ◽

Number Line ◽

Bounded Number ◽

Proportion Estimation ◽

Typical Performance ◽

Number Line Estimation ◽

Numerical Magnitudes ◽

Fundamental Shift

How children’s understanding of numerical magnitudes changes over the course of development remains a key question in the study of numerical cognition. In an ongoing debate about the source of developmental change, some argue that children maintain and access different mental representations of number, with evidence coming largely from common number-line estimation tasks. In contrast, others argue that a theoretical framework based on psychophysical models of proportion estimation accounts for typical performance on these tasks. The present study explores children’s (n=71) and adults’ (n=27) performance on two number-line tasks, both the “number to position” or NP task and the inverse “position to number” or PN task. Estimates on both tasks are consistent with the predictions of the proportion estimation account and do not support the hypothesis that a fundamental shift in mental representations underlies developmental change in numerical estimation and, in turn, mathematical ability. Converging evidence across tasks also calls into question the utility of bounded number-line tasks as an evaluation of mental representations of number.

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Linguistic inversion and numerical estimation

Journal of Numerical Cognition ◽

10.5964/jnc.v6i3.273 ◽

2020 ◽

Vol 6 (3) ◽

pp. 263-274

Author(s):

Sophie Savelkouls ◽

Katherine Williams ◽

Hilary Barth

Keyword(s):

Arabic Numeral ◽

Number Line ◽

Read Aloud ◽

Number Word ◽

Numerical Estimation ◽

Digit Number ◽

Number Words ◽

Number Line Estimation ◽

English Speaking ◽

The Right

Number line estimation (NLE) performance is usually believed to depend on the magnitudes of presented numerals, rather than on the particular digits instantiating those magnitudes. Recent research, however, shows that NLE placements differ considerably for target numerals with nearly identical magnitudes, but instantiated with different leftmost digits. Here we investigate whether this left digit effect may be due, in part, to the ordering of digits in number words. In English, the leftmost digit of an Arabic numeral is spoken first (“forty-one”), but Dutch number words are characterized by the inversion property: the rightmost digit of a two-digit number word is spoken first (“eenenveertig” – one and forty in Dutch). Participants (N = 40 Dutch-English bilinguals and N = 20 English-speaking monolinguals) completed a standard 0-100 NLE task. Target numerals were read aloud by an experimenter in either English or Dutch. Preregistered analyses revealed a strong left digit effect in monolingual English speakers’ estimates: e.g., 41 was placed more than two units to the right of 39. No left digit effect was observed among Dutch-English bilingual participants tested in either language. These findings are consistent with the idea that the order in which digits are spoken might influence multi-digit number processing, and suggests linguistic influences on numerical estimation performance.

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Linguistic Inversion and Numerical Estimation

10.31234/osf.io/e3j78 ◽

2020 ◽

Cited By ~ 1

Author(s):

Sophie Savelkouls ◽

Katherine Williams ◽

Hilary Barth

Keyword(s):

Arabic Numeral ◽

Number Line ◽

Read Aloud ◽

Number Word ◽

Numerical Estimation ◽

Digit Number ◽

Number Words ◽

Number Line Estimation ◽

English Speaking ◽

The Right

Number line estimation (NLE) performance is usually believed to depend on the magnitudes of presented numerals, rather than on the particular digits instantiating those magnitudes. Recent research, however, shows that NLE placements differ considerably for target numerals with nearly identical magnitudes, but instantiated with different leftmost digits (Lai, Zax, & Barth, 2018). Here we investigate whether this left digit effect may be due, in part, to the ordering of digits in number words. In English, the leftmost digit of an Arabic numeral is spoken first (“forty-one”), but Dutch number words are characterized by the inversion property: the rightmost digit of a two-digit number word is spoken first (“eenenveertig” - one and forty in Dutch). Participants (N = 40 Dutch-English bilinguals and N = 20 English-speaking monolinguals) completed a standard 0-100 NLE task. Target numerals were read aloud by an experimenter in either English or Dutch. Preregistered analyses revealed a strong left digit effect in monolingual English speakers’ estimates: e.g., 41 was placed more than two units to the right of 39. No left digit effect was observed among Dutch-English bilingual participants tested in either language. These findings are consistent with the idea that the order in which digits are spoken might influence multi-digit number processing, and suggests linguistic influences on numerical estimation performance.

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Sex differences in number line estimation: The role of numerical estimation

British Journal of Psychology ◽

10.1111/bjop.12203 ◽

2016 ◽

Vol 108 (2) ◽

pp. 334-350 ◽

Cited By ~ 7

Author(s):

Regina M. Reinert ◽

Stefan Huber ◽

Hans-Christoph Nuerk ◽

Korbinian Moeller

Keyword(s):

Sex Differences ◽

Number Line ◽

Numerical Estimation ◽

Number Line Estimation

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Integrating numerical cognition research and mathematics education to strengthen the teaching and learning of early number

10.31234/osf.io/ta8gh ◽

2019 ◽

Cited By ~ 1

Author(s):

Zachary Hawes ◽

Rebecca Merkley ◽

Daniel Ansari

Keyword(s):

Mathematics Education ◽

Teaching And Learning ◽

Numerical Cognition ◽

School Context ◽

Intervention Group ◽

Number Line ◽

Control Group ◽

Number Line Estimation ◽

Teachers And Students ◽

And Mathematics

This study reports on the design, implementation, and effects of a 16-week (25-hour) mathematics Professional Development (PD) model for K-3 educators (N=45) and their students (N=180). A central goal of the PD was to better integrate numerical cognition research and mathematics education. The results of the first iteration (Year 1), indicated that compared to a control group, both teachers and students benefitted from the intervention. Teachers demonstrated gains in self-perceived and actual numerical cognition knowledge, while students demonstrated gains in number line estimation, arithmetic, and numeration. In Year 2, teachers in the intervention group demonstrated greater improvements than the control group on a measure of self-perceived numerical cognition knowledge, but no gains in actual numerical cognition knowledge. At the student level, there was some evidence of gains in numeration. Given the mixed findings, questions remain as to why the model may be effective in one school context but not another.

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