Number Line Task Research Articles

Real-world estimation taps into basic numeric abilities.

Accurately estimating and assessing real-world quantities (e.g., how long it will take to get to the train station; the calorie content of a meal) is a central skill for adaptive cognition. To date, theoretical and empirical work on the mental resources recruited by real-world estimation has focused primarily on the role of domain knowledge (e.g., knowledge of the metric and distributional properties of objects in a domain). Here we examined the role of basic numeric abilities - specifically, symbolic-number mapping - in real-world estimation. In Experiment 1 ( ) and Experiment 2 ( ), participants first completed a country-population estimation task (a task domain commonly used to study real-world estimation) and then completed a number-line task (an approach commonly used to measure symbolic-number mapping). In both experiments, participants with better performance in the number-line task made more accurate estimates in the estimation task. Moreover, Experiment 2 showed that performance in the number-line task predicts estimation accuracy independently of domain knowledge. Further, in Experiment 2 the association between estimation accuracy and symbolic-number mapping did not depend on whether the number-line task involved small numbers (up to 1000) or large numbers that matched the range of the numbers in the estimation task (up to 100,000,000). Our results show for the first time that basic numeric abilities contribute to the estimation of real-world quantities. We discuss implications for theories of real-world estimation and for interventions aiming to improve people's ability to estimate real-world quantities.

Varieties of Number-Line Estimation: Systematic Review, Models, and Data

The psychophysical function that best fits human data from number-line estimation is the subject of a lively, on-going debate with important theoretical and practical implications. We comprehensively reviewed articles which tested competing psychophysical functions and found systematic variablility in task design. To test whether one function could account for data across diverse tasks, we examined 158 children’s and adults’ estimates using two 2 × 2 designs, crossing symbol (symbolic, non-symbolic) and boundedness (bounded, unbounded) on free number-line tasks (Experiment 1) and crossing the same factors on anchored tasks (Experiment 2). This yielded eight varieties of number-line estimation: four old varieties for testing replicability and four new varieties for testing generalizability. Across the eight varieties, 88.84 % of participants provided estimates better fit by a mixed log-linear model than competing models, with weights of the logarithmic component (λ) decreasing with age in each task. Unlike parameters of competing models, λ on any given task significantly predicted λ on the other 7 tasks, as well as predicting arithmetic skills. Results suggest that representations of numerical magnitude play the largest part in number-line estimation, and the “logarithmic-to-linear shift” provides the most accurate and generalizable description of how number-line estimation develops. (196 words)

Using diffusion models for symbolic numeracy tasks to examine aging effects.

We present a model-based analysis of aging effects in three symbolic numeracy tasks using three groups of subjects (young adults, 60- to 69-year-olds, and 70- to 90-year-olds). The tasks are number discrimination (is this number greater or less than 50), number memory (was this number in the list of numbers just displayed), and number line (point to where this number is on this number line). The first two tasks were fit by the standard two-choice diffusion model and the last one by the spatially continuous diffusion model (Ratcliff, 2018). Results showed good fits of the models to accuracy (choices) and response time distributions. In the tasks, nondecision time (the time to encode a stimulus and make a response) increased with age, but the amount of evidence needed for a decision (boundary settings) increased in the number discrimination and number memory tasks, but not the number line task. The number discrimination task produced conflicting accuracy and response time results as a function of age, but the model-based analyses resolved these differences. In the number memory task, drift rates (evidence used to drive the decision process) were lower for the older adults than for young adults, but for the other two (easier) tasks, there was no change in drift rate with age. The analyses extracted differences among individuals in model components, some of which were systematic across tasks. In particular, drift rates were correlated across tasks, which shows consistent individual differences across tasks, results that could not have been obtained without model-based analyses. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Prior proficiency level matters: The differential impact of erroneous examples in online homework related to decimal number line tasks

AbstractBackgroundThe present study delves into the debate surrounding the effectiveness of homework, focusing specifically on the use of online homework in teaching decimal numbers to 5th and 6th grade primary school students. It highlights the challenges students face in representing and identifying decimal numbers, a concept that is often difficult for 5th and 6th graders to grasp.ObjectivesThe aim to evaluate the effectiveness of online homework in enhancing understanding of decimal numbers and compare the impact of instructional methodologies based on correct and incorrect examples.MethodsThe research employs an experimental design, comparing the effects of online homework with correct and incorrect examples, assessing how these methodologies influence students' ability to understand and use decimal numbers, especially on a number line.ResultsThe results support the overall effectiveness of online homework in improving students' performance in decimal numbers, regardless of their initial level of understanding. Specifically, the study reveals that the type of examples used in homework plays a key role in the effectiveness of learning. Incorrect examples are more beneficial for students with higher prior knowledge, while correct examples are more appropriate for students with lower prior levels.ConclusionsThese outcomes highlight the potential value of online homework in primary education and underline the importance of tailoring educational activities to the individual needs of pupils. The study contributes to the broader conversation about the role of homework in education, suggesting practical ways to optimise online learning experiences for young learners.

The left digit effect in an unbounded number line task.

The left digit effect in number line estimation refers to the phenomenon where numerals with similar magnitudes but different leftmost digits (e.g., 19 and 22) are estimated to be farther apart on a number line than is warranted. The effect has been studied using a bounded number line task, a task in which a line is bounded by two endpoints (e.g., 0 and 100), and where one must indicate the correct location of a target numeral on the line. The goal of the present work is to investigate the left digit effect in an unbounded number line task, a task that involves using the size of one unit to determine a target numeral's location, and that elicits strategies different from those used in the bounded number line task. In a preregistered study, participants (N = 58 college students) completed four blocks of 38 trials each of an unbounded number line task, with target numerals ranging between 0 and 100. We found a medium and statistically reliable left digit effect (d = 0.70). The study offers further evidence that the effect is not driven by response strategies specific to the bounded number line task. We discuss other possible sources of the effect including conversion of symbols to magnitudes in these and other contexts.

Number line estimation patterns and their relationship with mathematical performance

There is ongoing debate regarding what performance on the number line estimation task represents and its role in mathematics learning. The patterns followed by children’s estimates on the number line task could provide insight into this. This study investigates children’s estimation patterns on the number line task and assesses whether mathematics achievement is associated with these estimation patterns. Singaporean children (n = 324, Age M = 6.2 years, Age SD = 0.3 years) in their second year of kindergarten were assessed on the number line task (0-100) and their mathematical performance (Numerical Operations and Mathematical Reasoning subtests from WIAT II). The results show that most children’s number line estimation patterns can be explained by at least one mathematical model (i.e., linear, logarithmic, unbounded power model, one-cycle power model, two-cycle power model). But the findings also highlight the high percentage of participants for which more than one model shows similar support. Children’s mathematical achievement differed based on the models that best explained children’s estimation patterns. Children whose estimation patterns corresponded to a more advanced model tended to show higher mathematical achievement. Limitations of drawing conclusions regarding what performance on the number line task represents based on models that best explain the estimation patterns are discussed.

Assessing the impact of LEGO® construction training on spatial and mathematical skills.

Lego construction ability is associated with a variety of spatial skills and mathematical outcomes. However, it is unknown whether these relations are causal. We aimed to establish the causal impact of Lego construction training on: Lego construction ability; a broad range of spatial skills; and on mathematical outcomes in 7-9-year-olds. We also aimed to identify how this causal impact differs for digital versus physical Lego construction training. One-hundred and ninety-eight children took part in a six-week training programme, delivered twice weekly as a school lunch time club. They completed either physical Lego training (N=59), digital Lego training (N=64), or an active control condition (crafts; N=75). All children completed baseline and follow-up measures of spatial skills (disembedding, visuo-spatial working memory, spatial scaling, mental rotation, and performance on a spatial-numerical task, the number line task), mathematical outcomes (geometry, arithmetic, and overall mathematical skills) and Lego construction ability. Exploratory analyses revealed evidence for near transfer (Lego construction ability) and some evidence for far transfer (arithmetic) of Lego training, but overall transfer was limited. Despite this, we identified key areas for further development (explicit focus on spatial strategies, training for teachers, and embedding the programme within a mathematical context). The findings of this study can be used to inform future development of Lego construction training programmes to support mathematics learning.

Adults systematically underestimate decimals and whole number exposure induces further magnitude-based underestimation.

Decimal numbers are generally assumed to be a straightforward extension of the base-ten system for whole numbers given their shared place value structure. However, in decimal notation, unlike whole numbers, the same magnitude can be expressed in multiple ways (e.g., 0.8, 0.80, 0.800, etc.). Here, we used a number line task with carefully selected stimuli to investigate how equivalent decimals (e.g., 0.8 and 0.80 on a 0-1 number line) and proportionally equivalent whole numbers (e.g., 80 on a 0-100 number line) are estimated. We find that young adults (n = 88, Mage = 20.22 years, SD = 1.65, 57 female) have a linear response pattern for both decimals and whole numbers, but those double-digit decimals (e.g., 0.08, 0.82, 0.80) are systematically underestimated relative to proportionally equivalent whole numbers (e.g., 8, 82, 80). Moreover, decimal string length worsens the underestimation, such that single-digit decimals (e.g., 0.8) are perceived as smaller than their equivalent double-digit decimals (e.g., 0.80). Finally, we find that exposing participants to whole number stimuli before decimal stimuli induces magnitude-based underestimation, that is, greater underestimation for larger decimals. Together, these results suggest a small but persistent underestimation bias for decimals less than one, and further that decimal magnitude estimation is fragile and subject to greater underestimation when exposed to whole numbers. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

The relationship between number line estimation and mathematical reasoning: a quantile regression approach

Number line estimation has been found to be strongly related to mathematical reasoning concurrently and longitudinally. However, the relationship between number line estimation and mathematical reasoning might differ according to children’s level of performance. This study investigates whether findings from previous studies that show number line estimation significantly predicting mathematical reasoning replicate, and whether this relationship holds across several points of the mathematical reasoning distribution. Participants include 324 Singaporean children (162 girls, Mage = 74.1 months, SDage = 4.0) in their second year of kindergarten who were assessed on the number line estimation task (0–10 and 0–100) and mathematical reasoning skills. The results replicate previous findings showing that higher accuracy on the number line estimation task is predictive of higher mathematical reasoning, for both the 0–10 and 0–100 number line. Quantile regressions show that performance on the number line task similarly predict mathematical reasoning across the performance distribution. Possible differences between the 0–10 and 0–100 number line’s predictive capacity are discussed.

Symbolic and non-symbolic predictors of number line task in Italian kindergarteners.

The number line estimation task (NLE) is often used as a predictor for broader measures of mathematical achievement. In spite of its popularity, it is still not clear whether the task is based on symbolic or non-symbolic numerical competence. In particular, there is only a very limited amount of studies investigating the relationship between NLE performance and symbolic vs. non-symbolic math skills in children who have not yet begun formal schooling. This study investigates the strength of the association between NLE performance and symbolic and non-symbolic tasks in young kindergarteners. Ninety two 5-year-old children completed the NLE task (range 0-100) and a battery of early numerical competence tests including symbolic-lexical tasks, symbolic semantic tasks, and non-symbolic semantic tasks. The relationship between symbolic and non-symbolic early numerical competence and NLE performance was analyzed using a regression model based on the Bayesian Information Criterion (BIC). Results show that only symbolic semantic tasks are significant predictors of NLE performance. These results suggest that symbolic numerical knowledge is involved in number line processing among young children, whilst non-symbolic knowledge is not. This finding brings new data to the debate on the relationship between non-symbolic numeral knowledge and symbolic number processing and supports the evidence of a primary role of symbolic number processing already in young kindergarteners.

The real preschoolers of Orange County: Early number learning in a diverse group of children

The authors assessed a battery of number skills in a sample of over 500 preschoolers, including both monolingual and bilingual/multilingual learners from households at a range of socio-economic levels. Receptive vocabulary was measured in English for all children, and also in Spanish for those who spoke it. The first goal of the study was to describe entailment relations among numeracy skills by analyzing patterns of co-occurrence. Findings indicated that transitive and intransitive counting skills are jointly present when children show understanding of cardinality and that cardinality and knowledge of written number symbols are jointly present when children successfully use number lines. The study’s second goal was to describe relations between symbolic numeracy and language context (i.e., monolingual vs. bilingual contexts), separating these from well-documented socio-economic influences such as household income and parental education: Language context had only a modest effect on numeracy, with no differences detectable on most tasks. However, a difference did appear on the scaffolded number-line task, where bilingual learners performed slightly better than monolinguals. The third goal of the study was to find out whether symbolic number knowledge for one subset of children (Spanish/English bilingual learners from low-income households) differed when tested in their home language (Spanish) vs. their language of preschool instruction (English): Findings indicated that children performed as well or better in English than in Spanish for all measures, even when their receptive vocabulary scores in Spanish were higher than in English.

Children grow upwards, and so does the number line: Evidence from a directional number line paradigm.

Technological advancements give researchers the opportunity to explore the internal metric that allows to mentally place numbers in a spatial and ordered way to establish relationships between quantities. In this study, we implement the cMNL, an embodied number line paradigm to investigate the configuration of children's number space mappings under multiple conditions. A sample of 185 primary school children aged 8-10years old completed digitally an embodied number line task encompassing directionality and modality as variables. Contrary to the premise of a fixed internal number line moving from left to right in many Western scripts, our results suggest that children's number-space mapping is more robust along a vertical axis. In addition, children's embodied number line estimation differed depending on input modality. The findings provide insight into the variability in children's number line estimation, and the usability of digital assessment in understanding the mechanisms of the developing number-space system.

Contributions of linguistic, quantitative, and spatial attention skills to young children's math versus reading: Same, different, or both?

AbstractThe study used Bayesian and Frequentist methods to investigate whether the roles of linguistic, quantitative, and spatial attention skills are distinct in children's acquisition of reading and math. A sample of 175 Chinese kindergarteners was tested with measures of linguistic skills (phonological awareness and phonological memory), quantitative knowledge (number line task, symbolic digit comparison, and non‐symbolic number estimation), spatial attention skills (visual span, mental rotation, and visual search), word reading, and calculation. After statistically controlling for age and nonverbal intelligence, phonological awareness and digit comparison performance explained unique variance in both math and reading. Moreover, number line estimation was specifically important for math, while phonological memory was specifically essential for reading. These findings highlight the possibility of developing early screening tools with different cognitive measures for children at risk of learning disabilities in reading and/or math.

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children.

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3-5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1-9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

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.

초등 예비교사의 수직선을 활용한 분수 과제 변형 사례 연구

With the importance and students’ difficulties with the number line in learning fractions, this study aims to investigate prospective teachers’ textbook fraction task modification using the number line and identify their strengths and weaknesses. The analysis of 58 prospective teachers’ modification tasks about fraction concept and addition and subtraction of fraction indicated that the prospective teachers chose a wide range of topics in the textbooks and modified the external contexts and types of the number lines. Moreover, the prospective teachers showed strengths in making efforts to help students understand the number line such as designing additional number line activities and anticipating students’ strategies about the modified number line tasks. However, some limitations and mathematical errors were found in modifying the tasks in a conceptual way and establishing specific lesson plan for the modified tasks.

Whole Number Bias of Students in Fraction Number Line Tasks

Whole Number Bias of Students in Fraction Number Line Tasks

Tracking priors and their replacement: Mental dynamics of decision making in the number-line task

Several theories of decision making assume that optimal decisions are reached by computing a prior distribution over possible responses, and then updating it according to the evidence received. We show how this prior replacement, with its two processing stages, can be captured with a simple behavioral method: tracking the finger movement as participants point to a response location. On each trial, participants saw a number and pointed to its location on a number line. In two experiments, we manipulated either the prior, via the distribution of target numbers, or the initial finger direction, via explicit instruction. In both experiments, when a trial started the participants pointed towards the instructed direction, and in the last part of the trial they pointed towards the target. Critically, between these two stages there was a third, interim stage in which the participants pointed towards the prior before deviating towards the target. Transient pointing towards the prior was observed even when it induced a brief deviation away from the target. This pattern fits a model wherein decisions are first driven by prior knowledge, followed by the accumulation of trial-specific evidence. We propose that the number-to-position mapping task with finger tracking is a powerful paradigm to investigate fine-grained aspects of priors in a simple decision-making scenario.

Large number placements on a bounded number line: The importance of explanations

AbstractUnderstanding large number data is essential for making sense of real‐world problems. For the research reported here, our intent was to find connections between quantitative cognitive science studies and our prior qualitative study about participants' understanding of the relative size of large numbers. While all 23 cognitive science research studies that we reviewed focused only on participants' placements of large numbers on number lines, our study focused on preservice teachers' (PSTs; n = 128) strategies – placements and explanations for their placements – of one billion on a number line bounded by zero and one trillion. Our intent was to answer the following question: What are the connections between three plausible placements substantiated by cognitive scientists – linear; log‐linear; and segmented linear – of large numbers on number line tasks and the nine strategies identified in our qualitative study? We highlight exemplars from the nine PSTs' strategies to show connections between the three plausible placements. In doing so, we illustrate that asking participants to place large numbers on number lines only gives a partial picture of their understanding. Their explanations expand the placements by revealing inconsistencies among the placements and explanations. Implications for future research are also discussed.

A number-line task with a Bayesian active learning algorithm provides insights into the development of non-symbolic number estimation.

To characterize numerical representations, the number-line task asks participants to estimate the location of a given number on a line flanked with zero and an upper-bound number. An open question is whether estimates for symbolic numbers (e.g., Arabic numerals) and non-symbolic numbers (e.g., number of dots) rely on common processes with a common developmental pathway. To address this question, we explored whether well-established findings in symbolic number-line estimation generalize to non-symbolic number-line estimation. For exhaustive investigations without sacrificing data quality, we applied a novel Bayesian active learning algorithm, dubbed Gaussian process active learning (GPAL), that adaptively optimizes experimental designs. The results showed that the non-symbolic number estimation in participants of diverse ages (5-73 years old, n = 238) exhibited three characteristic features of symbolic number estimation.