Number Line Estimation Task Research Articles

Number estimation in Down syndrome: Cognition or experience?

BackgroundThe ability to place numbers on a visual “number line” is a hallmark of the understanding of numerical magnitude and it is a strong predictor of mathematical achievement. AimWe examined whether the performance in the number line estimation task is more driven by mental age or experience with numbers in a sample of Italian children with Down syndrome (DS). Method and procedureSixty-three children with DS (Mmonths = 128.62, SD = 30.73) and sixty-three typically developing children (Mmonths = 54.98, SD = 6.34) matched one to one for mental age completed number line estimation tasks and other tests to assess their numerical knowledge. Outcomes and resultsNo significant differences emerged between the two groups in terms of accuracy of positioning numbers on the 1–10 and 1–20 interval. In addition, the accuracy on the 1–10 interval was related to the ability to recognize numbers, while the accuracy on the 1–20 line was related to the ability to compare magnitudes. Conclusion and ImplicationResults suggest that in individuals with DS the linear mapping of numbers is driven by mental age, but the accuracy of positioning numbers is also shaped by the experience with symbolic numbers. Therefore, the improvement of numerical estimation abilities should be a target of intervention programs.

Modeling the left digit effect in adult number line estimation

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0–100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6484 Luxembourgish ninth-graders

The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.

The structure of the notation system in adults' number line estimation: An eye-tracking study.

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults' numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.

Heterogeneity of Dyscalculia Risk Dependent on the Type of Number Line Estimation Task and the Number Magnitude.

An ability that is impaired in developmental dyscalculia (DD) is related to number line estimation (NLE). However, due to variability in NLE task performance, group differences do not exemplify the real difficulty level observed in the DD population. Thirty-two of the fifty-two participants posing dyscalculia risk (DR) (mean age = 9.88) experienced difficulties in mathematics. All the children performed two number-to-position tasks and two tasks requiring a verbal estimation of a number indicated on a line, utilizing the ranges 0–100 and 0–1000. The results showed that the estimation error in the verbal task was greater in the DR group than in the typically developed (TD) group for the 0–1000 range. In the number-to-position task, group differences were found for both ranges and the variability within both groups was smaller than it was in the verbal tasks. Analyses of each of the 26 numerical magnitudes revealed a more comprehensive pattern. The majority of the group effects were related to the 0–1000 line. Therefore, considerable data variability, especially in the DD group, suggests this issue must be analyzed carefully in the case of other mathematical capacities. It also critically questions some well-established phenomena and norms in experimental and diagnostic practices.

Summary accuracy feedback and the left digit effect in number line estimation.

A robust left digit effect arises in number line estimation such that adults’ estimates for numerals with different hundreds place digits but nearly identical magnitudes are systematically different from one another (e.g., 299 is placed too far to the left of 302). In two experiments, we investigate whether brief feedback interventions designed to increase task effort can reduce or eliminate the left digit effect in a self-paced 0–1,000 number line estimation task. Participants were assigned to complete three blocks of 120 trials each where the middle block contained feedback or no feedback. Feedback was in the form of summary accuracy scores (Experiment 1; N = 153) or competitive (summary) accuracy scores (Experiment 2; N = 145). In both experiments, planned analyses revealed large left digit effects in all blocks regardless of feedback condition. Feedback did not lead to a reduction in the left digit effect in either experiment, but improvements in overall accuracy were observed. We conclude that there are no changes in the left digit effect resulting from either summary accuracy feedback or competitive accuracy feedback. Also reported are exploratory analyses of trial characteristics (e.g., whether 299 is presented before or after 302) and the left digit effect.Supplementary InformationThe online version contains supplementary material available at 10.3758/s13421-022-01278-2.

Number sense, trading decisions and mispricing: An experiment

We show that the acuity of the Approximate Number System (ANS), a cognitive system that allows humans and many animal species to evaluate quantities without using exact calculations, is a strong predictor of subjects’ earnings in experimental markets. We measure ANS acuity with a bounded number line estimation (NLE) task and find that subjects who perform better on the NLE task, obtain higher earnings in a continuous double auction experimental market. We underline two channels through which high ANS acuity subjects achieve better performance: they are rewarded for offering liquidity and are faster at exploiting trading opportunities. We also show that, in a given market, the distribution of NLE scores influences mispricing. Our results are unchanged when we control for differences in trading intensity, risk aversion, background education or demographic characteristics.

Conceptual replication and extension of the relation between the number line estimation task and mathematical competence across seven studies

A recent meta-analysis demonstrated the overall correlation between the number line estimation (NLE) task and children’s mathematical competence was r = .44 (positively recoded), and this relation increased with age. The goal of the current study was to conceptually replicate and extend these results by further synthesizing this correlation utilizing studies not present in the meta-analysis. Across seven studies, 954 participants, ranging from 3 to 11 years old (Age M = 6.02 years, SD = 1.57), the overall estimation-competence correlations were similar to those of the meta-analysis and ranged from r = −.40 to −.35. The current conceptual replication demonstrated that the meta-analysis captured a stable overall relation between performance on the NLE task and mathematical competence. However, the current study failed to replicate the same moderation of age group presented in the meta-analysis. Furthermore, the current study extended results by assessing the stability and predictive validity of the NLE task while controlling for covariates. Results suggested that the NLE task demonstrated poor stability and predictive validity in the seven samples present in this study. Thus, although concurrent relations replicated, the differential age moderation, lack of stability, and lack of predictive validity in these studies require a more nuanced approach to understanding the utility of the NLE task. Future research should focus on understanding the connection between children’s developmental progression and NLE measurement before further investigating the predictive and diagnostic importance of the task for broader mathematical competence.

The importance of replicating meta-analyses: Commentary on “Conceptual replication and extension of the relation between the number line estimation task and mathematical competence across seven studies”

The importance of replicating meta-analyses: Commentary on “Conceptual replication and extension of the relation between the number line estimation task and mathematical competence across seven studies”

Left Digit Effects in Numerical Estimation across Development

ABSTRACT 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. 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.

Revisiting the Relationship Between Number-Line Estimation and Basic Addition and Subtraction in Elementary School Children and Adults

This study aims to examine the differences in numerosity estimation on a right-to-left number line between second- to fourth-grade students and undergraduate students, together with whether number-line estimation is related to basic arithmetic tasks (addition and subtraction). Hence, 53 Arabic-speaking children and 63 Arabic-speaking adults performed a paper-based number-line estimation task and also an arithmetic task. Number-line estimation was represented either by the percentage of absolute errors in positing the quantities on the line, by the accuracy, or by the r score for linearity of representation. The results show that, although children's addition (but not subtraction) performance resembles that of adults, the correlations vary between number line and arithmetic scores. Addition scores are positively correlated with error percentages in the children's number-line tasks, while subtraction is negatively correlated with adults' error percentages. These differences are assumed to be attributed to various uses of recall and procedural strategies.

Numerical magnitude understanding in kindergartners: A specific and sensitive predictor of later mathematical difficulties?

Difficulties in mathematics are argued to stem from impairments of a specialized system of numerical magnitude representation. This study investigates whether different measures of numerical magnitude understanding in kindergarten uniquely predict mathematical achievement concurrently and 6 months later, and also examines the relative explanatory of each aspect of magnitude understanding. It also examines the utility of such tasks as sensitive and specific predictors of children at risk of mathematical learning difficulties. Kindergartners (N = 215, M age = 74 months) were administered a number-line estimation task, a nonsymbolic numerical discrimination task, and a nonsymbolic to word mapping task. Multiple measures of math achievement were administered concurrently and upon entering formal education (M age = 80 months). Path analysis revealed that all 3 aspects of magnitude understanding had similar explanatory power in predicting concurrent and prospective math achievement; the only exception was that nonsymbolic numerical discrimination was not a significant predictor of concurrent mathematical reasoning. Of the 3 measures of magnitude understanding, number-line estimation was found to be the most accurate predictor of persistent mathematical difficulties. Approximately half of the children misclassified as having mathematical difficulties showed inconsistent mathematical achievement across the multiple measures. The implications for screening for mathematical difficulties and for designing interventions are discussed. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

Imparare la matematica con Number Worlds: un intervento quinquennale nella scuola primaria

The Number Worlds (NW) program is based on Case’s theory of cognitive development and conceptual learning, and it promotes the learning of mathematical concepts through playful-manipulative activities and respecting children’s level of development. This five years-research is meant to develop an Italian adaptation of the program, compare it with traditional teaching, and determine the impact of working memory (WM) on math learning. 56 primary school children participated in the research. Since grade I, 13 pupils (controls) followed traditional teaching of math; 43 children worked with NW, 3 hours a week for 7 months. The pupils were assessed with two WM tests, the Number Knowledge Test (NKT), and the Number Line Estimation Task (NLET) at the beginning of Grade I and at the end of each Grade. At the end of Grade V, the AC-MT battery was also administered. The experimental group improved more than controls on the NKT and NLET from Grade I to IV. At the end of Grade V, both groups showed a «ceiling effect» in the NKT and NLET performance. However, a difference in favour of the experimental group was found in some AC-MT tasks. WM, together with the curriculum, was predictive of mathematics performance up to class IV.

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

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.

Putting fractions together.

Learning fractions is a critical step in children’s mathematical development. However, many children struggle with learning fractions, especially fraction arithmetic. In this article, we propose a general framework for integrating understanding of individual fractions and fraction arithmetic, and we use the framework to generate interventions intended to improve understanding of both individual fractions and fraction addition. The framework, Putting Fractions Together (PFT), emphasizes that both individual fractions and sums of fractions are composed of unit fractions and can be represented by concatenating them (putting them together). To illustrate, both “3/9” and “2/9 + 1/9” can be represented by concatenating three 1/9s; similarly, 2/9 + 1/8 can be represented by concatenating two 1/9s and one 1/8. Interventions based on the PFT framework were tested in 2 experiments with fourth, fifth, and sixth grade children. The interventions led to improved performance on number line estimation and magnitude comparison tasks involving individual fractions and sums of fractions with equal and unequal denominators. Especially large improvements were observed on relatively difficult unequal-denominator fraction sum problems. The findings suggest that viewing individual nonunit fractions and sums of fractions as concatenations of unit fractions provides a sound conceptual foundation for improving children’s knowledge of both. We discuss implications of the research for teaching and learning fractions, children’s numerical development, and mathematics education in general. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

The left digit effect in a complex judgment task: Evaluating hypothetical college applicants

AbstractA left digit effect has been broadly observed across judgment and decision‐making contexts ranging from product evaluation to medical treatment decisions to number line estimation. For example, $3.00 is judged to be a much greater cost than $2.99, and “801” is estimated strikingly too far to the right of “798” on a number line. Although the consequences of the effects for judgment and decision behavior have been documented, the sources of the effects are not well established. The goal of the current work is to extend investigations of the left digit effect to a new complex judgment activity and to assess whether the magnitude of the effect at the individual level can be predicted from performance on a simpler number skills task on which the left digit effect has also recently been observed. In three experiments (N = 434), adults completed a judgment task in which they rated the strength of hypothetical applicants for college admission and a self‐paced number line estimation task. In all experiments, a small or medium left digit effect was found in the college admissions task, and a large effect was found in number line estimation. Individual‐level variation was observed, but there was no relationship between the magnitudes of the effects in the two tasks. These findings provide evidence of a left digit effect in a novel multiattribute judgment task but offer no evidence that such performance can be predicted from a simple number skills task such as number line estimation.

Linguistic inversion and numerical estimation

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.

Game-based learning environments affect frontal brain activity.

Inclusion of game elements in learning environments to increase motivation and learning outcome is becoming increasingly popular. However, underlying mechanisms of game-based learning have not been studied sufficiently yet. In the present study, we investigated effects of game-based learning environments on a neurofunctional level. In particular, 59 healthy adults completed a game-based version (including game elements such as a narrative and virtual incentives) as well as a non-game-based version of a number line estimation task, to improve fractional knowledge, while their brain activity was monitored using near-infrared spectroscopy. Behavioral performance was comparable across the two versions, although there was a tendency that less errors were made in the game-based version. However, subjective user experience differed significantly between versions. Participants rated the game-based version as more attractive, novel, and stimulating but less efficient than the non-game-based version. Additionally, positive affect was reported to be higher while engaging in the game-based as compared to the non-game-based task version. Corroborating these user reports, we identified increased brain activation in areas associated with emotion and reward processing while playing the game-based version, which might be driven by rewarding elements of the game-based version. Moreover, frontal areas associated with attention were also more activated in the game-based version of the task. Hence, we observed converging evidence on a user experience and neurofunctional level indicating that the game-based version was more rewarding as well as emotionally and attentionally engaging. These results underscore the potential of game-based learning environments to promote more efficient learning by means of attention and reward up-tuning.

Number line estimation and standardized test performance: The left digit effect does not predict SAT math score.

IntroductionRecent work reveals a new source of error in number line estimation (NLE), the left digit effect (Lai, Zax, et al., 2018), whereby numerals with different leftmost digits but similar magnitudes (e.g., 399, 401) are placed farther apart on a number line (e.g., 0 to 1,000) than is warranted. The goals of the present study were to: (1) replicate the left digit effect, and (2) assess whether it is related to mathematical achievement.MethodParticipants were all individuals (adult college students) who completed the NLE task in the laboratory between 2014 and 2019 for whom SAT scores were available (n = 227).ResultsWe replicated the left digit effect but found its size was not correlated with SAT math score, although it was negatively correlated with SAT verbal score for one NLE task version.ConclusionsThese findings provide further evidence that individual digits strongly influence estimation performance and suggest that this effect may have different cognitive contributors, and predict different complex skills, than overall NLE accuracy.

Performance patterns and strategy use in number line estimation among preschool children with different spontaneous focusing on numerosity tendencies

AbstractThis study investigated the influence of spontaneous focusing on numerosity (SFON) tendency on the performance and estimation patterns on number line estimation (NLE). To examine this question, 147 preschoolers (3.51–4.52 years) completed two types of SFON task (referenced SFON task and non‐referenced SFON task) and three conditions of NLE (non‐symbolic area, number‐to‐position, and position‐to‐number). The participants' SFON tendencies were classified as high, average, and poor using latent profile analysis. Children with high SFON tendency performed better and preferred to use more advanced strategy than their poor‐SFON peers on NLE tasks. The proportion of children with irregular NLE pattern increased as SFON tendency decreased. These findings contribute to our understanding of the relationship between performance patterns of different NLEs and SFON tendencies for preschool children.