Number Line Estimation Research Articles

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

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.

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.

Estimation in the Mathematics Curricula of Denmark, Norway and Sweden: Inadequate Conceptualisations of an Essential Competence

ABSTRACT Acknowledging evidence that the ability to estimate has major consequences for both later mathematics learning and real-world functionality, this paper examines the national mathematics curriculum for compulsory school for each of Denmark, Norway and Sweden for the estimation-related opportunities it offers children. Framed against four conceptually and procedurally different forms of estimation (computational, measurement, quantity and number line), each of which is implicated differently in the later learning of mathematics, analyses indicated that none of the four forms of estimation were addressed explicitly in the Norwegian curriculum. Expectations of computational and measurement estimation were present in both the Danish and the Swedish curricula, although neither referred to either quantity or number line estimation. Even when estimation-related learning outcomes were articulated, there was no evidence of the processes by which they might be realised. Finally, there were no acknowledgements that estimation may contribute to the learning of other mathematical topics.

Learning Improper Fractions with the Number Line and the Area Model

ABSTRACT Number lines and area models are both used pervasively in teaching fractions. Prior studies found that second and third graders demonstrated better magnitude knowledge of proper fractions after a 15-minute training using the number line as compared to using the area model. The current study aimed to extend these findings to improper fractions. We randomly assigned fourth and fifth graders to a number line training, an area model training, or a non-numerical control condition. The number line and area model trainings involved both proper and improper fractions and were closely modeled on the training procedures in prior studies. Fraction training with the area model produced improvements in children’s area model estimation of proper and improper fractions. However, contrary to our expectations, training with the number line did not improve number line estimation, and neither training led to improvements in transfer tasks assessing fraction magnitude knowledge. These findings suggest that children can develop the skill to represent improper fractions on area models with brief training. Nevertheless, it is unclear whether this skill enhances a comprehensive understanding of fraction magnitudes.

Can feedback, correct, and incorrect worked examples improve numerical magnitude estimation precision?

We investigated whether three interventions – studying incorrect worked examples, studying correct worked examples, or receiving feedback – improved children’s 0–1,000 (Experiment 1) and adults’ 1 thousand–1 billion (Experiment 2) number-line estimation precision relative to a no intervention control group. At pretest, participants estimated numbers on number lines (without feedback or examples) and judged their confidence in each estimate. They were then assigned to an intervention group, completed intervention trials, and then estimated numbers and judged their confidence at posttest. Children’s and adults’ confidence judgments did not increase from pretest to posttest. However, consistent with prior research, feedback improved children’s and adults’ estimation precision from pretest-to-posttest. Importantly, studying correct worked examples also improved precision, but studying incorrect worked examples did not. Although not all children benefited from training, those who did were able to transfer their knowledge to a number-categorization task. Adults in the feedback group were also able to transfer their learning. These outcomes highlight benefits of studying worked examples for improving children’s and adults’ estimation precision, but to obtain benefits, corrective feedback must be included.

Number Line Estimations, Place Value Understanding and Mathematics Achievement

Current study investigated the relationships among mathematics achievement, place value understanding and number line estimations. We used a curriculum-based math achievement test (MAT, Mathematics Achievement Test), a place value test (PVT), and a mental number line estimation test (MNL) as the data collection tools. A total of 355 fourth graders participated in the study. They were recruited from schools located in middle-low socioeconomic areas of central Anatolia. Correlations were statistically significant among all these tests scores (MAT, PVT, MNL). The highest correlation was obtained between PVT and MAT scores. PVT have also stronger correlations with MNL (0-1000). The average scores of MAT and MNL tests were correlated inversely as expected. The correlations between mathematics achievement scores and MNL (0-1000) was highest. The three MNL tests alltogether have explained 40% of the variance in MAT. Similar to PVT, MNL (0-1000) alone accounted for 36% of the variance in MAT. Meanwhile, MNL (0-1000) (large numbers) contributed MAT more. PVT explained 70% of the variance in MAT. In all tests, PVT contributed more to MAT. Generally, boys did better in all MNL tests but only the result of MNL (0-1000) was significant favoring boys. We can conclude, based on these results that both approximate number acuity and place value understanding contribute to the general math achievement as well as each other. While the place value concept seems to be a representational tool that enhence the acuity of approximate numbers, the relative magnitudes of numbers support place value understanding conceptually.

Estimation in the primary mathematics curricula of the United Kingdom: Ambivalent expectations of an essential competence

ABSTRACT In this paper, we examine the national curricula for primary mathematics for each of the four constituent nations of the United Kingdom (England, Northern Ireland, Scotland and Wales) for the estimation-related opportunities they offer children. Framed against four conceptually and procedurally different forms of estimation (computational, measurement, quantity and number line), the analyses indicate that computational estimation and measurement estimation were addressed in all four curricula, albeit from a skills-acquisition perspective, with only the Scottish offering any meaningful justification for their inclusion. The process of rounding, absent in the Northern Ireland curriculum, was presented as an explicit learning objective in the English, Scottish and Welsh curricula, although it was only the Scottish that made explicit the connections between rounding and computational estimation. In all curricula, both quantity estimation and number line estimation were effectively absent, as was any explicit acknowledgement that learning to estimate, irrespective of its form, has a developmental role in the learning of other mathematical topics.

Estimativa numérica e desempenho aritmético

O presente estudo relacionou a precisão da estimativa numérica em dois testes de desempenho em estimativa numérica: o Teste de Estimativa Numérica de Quantidades (TENQ) e o Teste de Estimativa na Reta Numérica (TERN), com o desempenho em um teste padronizado de aritmética (TDE), em estudantes do 5o e 6o ano escolar, de uma escola pública e outra particular de Porto Alegre/RS. O principal objetivo desse estudo foi verificar se o desempenho em diferentes tarefas de estimativa numérica está relacionado ao desempenho aritmético desses estudantes. Os dados indicam que o desempenho aritmético apresentou correlação de moderada a forte com o desempenho em estimativa, em ambas as tarefas, quando considerado o intervalo numérico de 0 a 100. Esse resultado indicou que conhecimentos aritméticos possam estar relacionados, de alguma forma, à habilidade em realizar estimativas.

The relationship between mental rotation and arithmetic: do number line estimation, working memory, or place-value concept matter?

Mental rotation is positively related to arithmetic ability; however, the mechanism underlying this relationship remains unclear. The possible roles of working memory, place-value concept, and number line estimation in the correlation between mental rotation and whole-number computation were investigated. One hundred and fifty-five first-grade students were tested to determine their mental rotation ability, arithmetic ability, and non-verbal intelligence. One year later, their working memory, place-value concept, number line estimation, and overall arithmetic ability were assessed. After controlling for age, gender, and prior arithmetic ability, we found that mental rotation uniquely predicted arithmetic ability after one year. Further mediation analyses demonstrated that number line estimation significantly mediated the relationship between mental rotation and arithmetic ability. In contrast, neither working memory nor place-value concept significantly mediated the relationship between mental rotation and arithmetic ability. This study highlights that mental number line estimation is the most important element explaining the influence of a dynamic spatial skill, that is, mental rotation, on arithmetic ability among young Chinese children.

The Role of Non-symbolic and Symbolic Skills in the Development of Early Numerical Cognition from Preschool to Kindergarten Age

ABSTRACT The development of (early) numerical cognition builds on children’s ability to understand and manipulate quantities and numbers. However, previous research did not find conclusive evidence on the role of symbolic and non-symbolic skills in the development of (early) numerical cognition. The aim of the current study was to clarify the relation between different types of non-symbolic quantity skills, symbolic numerical skills and early numerical cognition. A sample of 43 children was tested at the age of 3.5 years and at the age of 5 years. At 3.5 years, non-symbolic number line estimation, non-symbolic quantity comparison and symbolic enumerating skills were measured. At 5 years, early numerical cognition, defined as symbolic number line estimation and counting, were measured It was found that non-symbolic number line estimation at 3.5 years could predict both symbolic number line estimation and counting at 5 years. Enumerating at 3.5 years could only predict counting at 5 years. This suggests that both non-symbolic and symbolic skills play a role in the development of early numerical cognition, although enumerating skills do not transfer to all types of early numerical cognition. Furthermore, not all non-symbolic skills seem to play an important role in the development of early numerical cognition. The results suggest that non-symbolic quantity comparison does not contribute much to the development of early numerical cognition. Associations between non-symbolic quantities and space, operationalized here as non-symbolic number line estimation, seem central to the development of early math from preschool to kindergarten age.

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.

Children’s Math Anxiety Predicts Their Math Achievement Over and Above a Key Foundational Math Skill

ABSTRACT Math anxiety negatively predicts young children’s math achievement. While some researchers have suggested that math anxiety may stem from poor math ability, others have argued that math anxiety occurs at all levels of math ability. An important question is whether math anxiety predicts math achievement over and above foundational math skills. We sought to address this issue by examining whether math anxiety predicts future math achievement, controlling for number line estimation, a foundational math skill that predicts future math achievement. We found that 1st graders’ math anxiety predicts their math achievement in 1st through 3rd grade, controlling for their number line estimation at the beginning of 1st grade. This finding suggests that math anxiety contributes to future math learning over and above an important foundational math skill. Additionally, we examined whether there are age differences in the relation of first-grade math anxiety to number line estimation at various scales in order to test the hypothesis that the negative effect of early math anxiety on math performance depends on task difficulty. In support of this hypothesis, early math anxiety more closely related to the 0–100 number line task in 1st graders but to the 0–1000 number line task in 3rd graders, suggesting that math anxiety most strongly relates to math skills that are at the cusp of children’s understanding at particular grade levels. Together, these findings underscore the importance of fostering young children’s positive emotions toward math in addition to providing them with a strong cognitive foundation for math learning.

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.

Gender differences in confidence during number-line estimation

Prior research has found gender differences in spatial tasks in which men perform better, and are more confident, than women. Do gender differences also occur in people’s confidence as they perform number-line estimation, a common spatial-numeric task predictive of math achievement? To investigate this question, we analyzed outcomes from six studies (N = 758 girls/women and boys/men with over 20,000 observations; grades 1–5 and adults) that involved a similar method: Participants estimated where a provided number (e.g., ¾, 37) was located on a bounded number line (e.g., 0–1; 0–100), then judged their confidence in that estimate. Boys/men were more precise (g = .52) and more confident (g = .30) in their estimates than were girls/women. Linear mixed model analyses of the trial-level data revealed that girls’/women’s estimates had about 31% more error than did boys’/men’s estimates, and even when controlling for precision, girls/women were about 7% less confident in their estimates than were boys/men. These outcomes should encourage researchers to consider gender differences for studies on math cognition and provide pathways for future research to address potential mechanisms underlying the present gender gaps.

To Add or Not to Add Game Elements? Exploring the Effects of Different Cognitive Task Designs Using Eye Tracking

Research on instructional design provides inconsistent results on the use of game elements in cognitive tasks or learning. Cognitive load theory suggests that game elements increase extraneous cognitive load and, thus, may distract the users. In contrast, from an emotional design perspective, the use of game elements is argued to increase performance by providing a more interesting and motivating task environment. To contribute to this debate, the current study investigated the effect of game elements on behavioral performance, attention, and motivation. We designed two versions of the number line estimation task-one with game elements and one without. Participants completed both versions of the task while their eye-fixation behavior was recorded. Results indicated that participants paid attention to game elements, that is, they fixated them, although they were not necessary to complete the task. However, no difference in estimation accuracy was observed between the two task versions. Moreover, the task version with game elements was rated to be more attractive, stimulating, and novel, and participants reported experiencing greater flow. In sum, these data indicate that game elements seem to capture attention but also increase motivational aspects of learning tasks rather than decreasing performance.

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.

Effect of Finger Gnosis on Young Chinese Children’s Addition Skills

Evidence has revealed an association between finger gnosis and arithmetic skills in young Western children, however, it is unknown whether such an association can be generalized to Chinese children and what mechanism may underlie this relationship. This study examines whether finger gnosis is associated with addition skills in young Chinese children and, if so, what numerical skills could explain this correlation. A total of 102 Chinese children aged 5–6 years were asked to complete finger gnosis and addition tasks in Study 1. Results showed that finger gnosis was significantly associated with addition performance. However, no significant correlation was found between finger gnosis and the use of finger counting in solving addition problems. Moreover, girls’ finger gnosis was better than boys’, and children with musical training demonstrated better finger gnosis than those without. In Study 2, 16 children with high finger gnosis and 20 children with low finger gnosis were selected from the children in Study 1 and asked to perform enumeration, order judgment, number sense, and number line estimation. Children with high finger gnosis performed better in number line estimation than their counterparts with low finger gnosis. Moreover, the number line estimation fully mediated the relationship between finger gnosis and addition performance. Together, these studies provide evidence of a correlation between finger gnosis and addition skills. They also highlight the importance of number line estimation in bridging this association.

Comparing eye fixation and mouse cursor response modes in number line estimation

ABSTRACT Recently, responding with eye fixations in the number line estimation (NLE) task was introduced. To determine if this new mode of responding replicates typical mouse cursor NLE results, 33 adults completed a 0 –1000 NLE task and responded with their eyes (eye condition) or the mouse (mouse condition), while their eyes were tracked. In the mouse condition, participants had lower error rates, higher response times, and looked at the three quartiles of the number line and back to the presented target number for a longer time compared to the eye condition. Despite these differences, both conditions elicited an “M-shaped pattern” in response times and error rates, thus replicating the typical NLE results in both response modes. Overall, a similar (strategic) process occurs for both response modes, suggesting that both are appropriate to use in the NLE task. Researchers should, however, proceed with caution, as there are mode-related constraints on response times and error rates.

Spontaneous focusing on multiplicative relations and fraction magnitude knowledge

ABSTRACT To test the hypothesis that a higher tendency to spontaneously focus on multiplicative relations (SFOR) leads to improvements in rational number knowledge via more exact estimation of fractional quantities, we presented sixth graders (n = 112) with fraction number line estimations and a novel task in which numerical information embedded in narratives could be estimated as fractions. Consistent with our main hypothesis, we found that SFOR tendency predicted both forms of fraction estimation. However, the relation between SFOR and fraction magnitude comparisons was mediated by fraction estimation, both on the number line and on whole number relations embedded in narrative vignettes. Thus, a higher tendency to recognize multiplicative relations in non-explicitly mathematical situations may contribute to increases in the precision with which students encode fractional relations in everyday contexts, both as spatial and numerical relations.