Unbounded number line estimation as a measure of numerical estimation

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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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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Left digit effects in numerical estimation across development

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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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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The relationship between number line estimation and arithmetic competencies depends on age, operation and skill

PsycEXTRA Dataset ◽

10.1037/e636952013-354 ◽

2013 ◽

Author(s):

Julia Bahnmueller ◽

Stefan Huber ◽

Korbinian Moeller ◽

Hans-Christoph Nuerk

Keyword(s):

Number Line ◽

Number Line Estimation ◽

The Relationship

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Improving Struggling Fifth-Grade Students’ Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures

Council review ◽

10.1177/00144029211008851 ◽

2021 ◽

pp. 001440292110088

Author(s):

Madhavi Jayanthi ◽

Russell Gersten ◽

Robin F. Schumacher ◽

Joseph Dimino ◽

Keith Smolkowski ◽

...

Keyword(s):

Randomized Controlled Trial ◽

Explicit Instruction ◽

Controlled Trial ◽

Number Line ◽

Randomized Controlled ◽

Fifth Grade Students ◽

Linear Representations ◽

Mathematical Explanations ◽

Grade 5 ◽

Number Line Estimation

Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study ( n = 205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students’ mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding ( g = 0.66–0.78), number line estimation ( g = 0.80–1.08), fractions procedures ( g = 1.07), and explanation tasks ( g = 0.68–1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students’ proficiency and understanding of fractions.

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