Number Line Estimation Task Research Articles

You can count on your fingers: Finger-based intervention improves first-graders’ arithmetic learning

The question of whether finger use should be encouraged or discouraged in early mathematics instruction remains a topic of debate. Scientific evidence on this matter is scarce due to the limited number of systematic intervention studies. Accordingly, we conducted an intervention study in which first-graders (Mage = 6.48 years, SD = 0.35) completed a finger-based training (18 sessions of ∼ 30 min each) over the course of the first school year. The training was integrated into standard mathematics instruction in schools and compared with business-as-usual curriculum teaching. At the end of first grade and in a follow-up test 9 months later in second grade, children who received the finger training (n = 119) outperformed the control group (n = 123) in written addition and subtraction. No group differences were observed for number line estimation tasks. These results suggest that finger-based numerical strategies can enhance arithmetic learning, supporting the idea of an embodied representation of numbers, and challenge the prevailing skepticism about finger use in primary mathematics education.

Promoting 5th Grade Students’ Spontaneous Focusing on Quantitative Relations

Spontaneous focusing on quantitative relations (SFOR) is related to the development of rational number knowledge and predicts future mathematical knowledge. However, scarce evidence exists on whether and how SFOR tendency can be supported with classroom instruction. In this quasi-experimental study, 306 5th grade students participated in one of two four-lesson interventions. We investigated the relative effectiveness of both interventions on students’ SFOR tendency and their knowledge of multiplicative relations and fractions. Students in the Spot the Relations intervention (n = 142) completed activities that supported their ability to recognize and describe multiplicative relations in their everyday surroundings. Students in the Fraction Estimation Game intervention (n = 164) trained with digital game-based number line estimation tasks that supported their fraction knowledge. The results revealed that students in the first group showed larger gains in their multiplicative relations knowledge and SFOR tendency, whereas those in the second group showed larger gains in their fraction knowledge from pre- to posttest. Although SFOR tendency was supported by both interventions over time, the effect was larger for Spot the Relations, which used instructional practices that supported students’ multiplicative knowledge and mathematical language and made multiplicative relations relevant and observable parts of the world.

The relationship between numerical magnitude processing and math anxiety, and their joint effect on adult math performance, varied by indicators of numerical tasks

According to the hypothesis of Maloney et al. (Cognition 114(2):293–297, 2010. https://doi.org/10.1016/j.cognition.2009.09.013), math anxiety is related to deficits in numerical magnitude processing, which in turn compromises the development of advanced math skills. Because previous studies on this topic are contradictory, which may be due to methodological differences in the measurement of numerical magnitude processing, we tested Maloney et al.’s hypothesis using different tasks and their indicators: numerical magnitude processing (symbolic and non-symbolic comparison tasks: accuracy, reaction time, numerical ratio, distance and size effects, and Weber fraction; number line estimation task: estimation error), math anxiety (combined scores of learning, testing, math problem solving, and general math anxiety), and math performance. The results of our study conducted on 119 young adults mostly support the hypothesis proposed by Maloney et al. that deficiency in symbolic magnitude processing is related to math anxiety, but the relationship between non-symbolic processes and math anxiety was opposite to the assumptions. Moreover, the results indicate that estimation processes (but not comparison processes) and math anxiety are related to math performance in adults. Finally, high math anxiety moderated the relationship between reaction time in the symbolic comparison task, reaction time in the non-symbolic comparison task, numerical ratio effect in the symbolic comparison task, and math performance. Because the results of the joint effect of numerical magnitude processing and math anxiety on math performance were inconsistent, this part of the hypothesis is called into question.

The line bisection bias as a deficit of proportional reasoning − evidence from number line estimation in neglect

This study aimed to investigate whether neurological patients presenting with a bias in line bisection show specific problems in bisecting a line into two equal parts or their line bisection bias rather reflects a special case of a deficit in proportional reasoning more generally. In the latter case, the bias should also be observed for segmentations into thirds or quarters. To address this question, six neglect patients with a line bisection bias were administered additional tasks involving horizontal lines (e.g., segmentation into thirds and quarters, number line estimation, etc.). Their performance was compared to five neglect patients without a line bisection bias, 10 patients with right hemispheric lesions without neglect, and 32 healthy controls. Most interestingly, results indicated that neglect patients with a line bisection bias also overestimated segments on the left of the line (e.g., one third, one quarter) when dissecting lines into parts smaller than halves. In contrast, such segmentation biases were more nuanced when the required line segmentation was framed as a number line estimation task with either fractions or whole numbers. Taken together, this suggests a generalization of line bisection bias towards a segmentation or proportional processing bias, which is congruent with attentional weighting accounts of line bisection/neglect. As such, patients with a line bisection bias do not seem to have specific problems bisecting a line, but seem to suffer from a more general deficit processing proportions.

Delayed development of basic numerical skills in children with developmental dyscalculia.

Research suggests that children with developmental dyscalculia (DD) have deficits in basic numerical skills. However, there is conflicting evidence on whether basic numerical skills in children with DD are qualitatively different from those in typically developing children (TD) or whether basic numerical skills development in children with DD is simply delayed. In addition, there are also competing hypotheses about deficits in basic numerical skills, assuming (1) a general deficit in representing numerosities (Approximate Number System, ANS), (2) specific deficits in an object-based attentional system (Object Tracking System, OTS), or (3) deficits in accessing numerosities from symbols (Access Deficit, AD). Hence, the purpose of this study was to investigate whether deficits in basic numerical skills in children with DD are more indicative of a developmental delay or a dyscalculia-specific qualitative deviation and whether these deficits result from (selective) impairment of core cognitive systems involved in numerical processing. To address this, we tested 480 children (68 DD and 412 TD) in the 2nd, 3rd, and 4th grades with different paradigms for basic numerical skills (subitizing, counting, magnitude comparison tasks, number sets, and number line estimation tasks). The results revealed that DD children's impairments did not indicate qualitatively different basic numerical skills but instead pointed to a specific developmental delay, with the exception of dot enumeration. This result was corroborated when comparing mathematical profiles of DD children in 4th grade and TD children in 2nd grade, suggesting that DD children were developmentally delayed and not qualitatively different. In addition, specific deficits in core markers of numeracy in children with DD supported the ANS deficit rather than the AD and OTS deficit hypothesis.

Children's number line estimation strategies: evidence from bounded and unbounded number line estimation tasks.

This study investigates the number line estimation (NLE) strategies utilized by children aged 4-7 across both bounded and unbounded NLE tasks. Drawing on prior research, it hypothesized that younger children would predominantly employ benchmark-based strategies, with endpoints as reference points, in bounded tasks, while older children would utilize a wider range of reference points including midpoints and quartiles. For unbounded tasks, it was anticipated that both younger and older children would adopt the scalloped strategy. A total of 181 Chinese children participated, representing three educational backgrounds: Middle Class (kindergarten), Senior Class (kindergarten), and Grade 1 (elementary school). They completed the bounded and unbounded NLE tasks with numbers ranging from 0 to 50. Data analysis focused on estimation accuracy using Percent Absolute Error (PAE) and contour analyses to examine strategy use. Results revealed that all age groups employed benchmark-based strategies using endpoints and midpoints in both bounded and unbounded tasks, and applied the scalloped strategy with units of integers 5 or 10 across all tasks. Findings suggest a coexistence of benchmark-based and scalloped strategies across task types, reflecting children's intuitive estimation strategies. Furthermore, children aged 4 to 7 exhibited consistent strategy utilization, indicating a developmental stage characterized by reliance on specific reference points for estimation. This study contributes to understanding the developmental trajectory of number line estimation strategies in early childhood and emphasizes the importance of task type, learning experiences, and other factors in eliciting different estimation strategies.

Does instructional intervention reduce the left digit effect in number line estimation?

A robust left digit effect arises in number line estimation, whereby the leftmost digits of numerals have an undue influence on placements such that, for example, numbers like 298 are placed far to the left of numbers like 302. Past efforts to motivate more accurate performance using trial-by-trial and summary feedback have not led to a reduction in the left digit effect. In two experiments, we asked whether it is possible to reduce or eliminate the left digit effect in number line estimation through an instructional intervention in which one is explicitly taught about the left digit effect. In Experiment 1 (N = 134), participants completed two blocks (60 trials per block) of a self-paced 0-1,000 number line estimation task and were randomly assigned to either an instruction or a control condition. In Experiment 2 (N = 143), the procedure was enhanced with a learning check, and with additional measures to assess changes in behaviour as a result of instruction. In both experiments, a left digit effect was found in each block of each condition. Although there was evidence that instruction changed behaviour, these changes did not result in any reduction in the left digit effect relative to the control condition. These findings demonstrate that the left digit effect cannot be easily reduced by making people aware of it.

Potential moderators of the left digit effect in numerical estimation

Recent work reveals a left digit effect in number line estimation such that adults' and children's estimates for three-digit numbers with different hundreds-place digits but nearly identical magnitudes are systematically different (e.g., 398 is placed too far to the left of 401 on a 0-1000 line, despite their almost indistinguishable magnitudes; Lai et al., 2018, https://doi.org/10.1111/desc.12657). In two preregistered studies (N = 218), we investigate the scope and malleability of the left digit effect. Experiment 1 used a typical forward-oriented 0-1000 number line estimation task and an atypical reverse-oriented 1000-0 number line estimation task. Experiment 2 used the same forward-oriented typical 0-1000 number line estimation task from Experiment 1, but with trial-by-trial corrective feedback. We observed a large left digit effect, regardless of the orientation of the line in Experiment 1 or the presence of corrective feedback in Experiment 2. Further, analyses using combined data showed that the pattern was present across most stimuli and participants. These findings demonstrate a left digit effect that is robust and widely observed, and that cannot be easily corrected with simple feedback. We discuss the implications of the findings for understanding sources of the effect and efforts to reduce it.

Young children intuitively organize numbers on straight, horizontal lines from left to right before the onset of formal instruction.

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number-space mapping. Three- to six-year-old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non-symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left-to-right-oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space.

Developmental trajectories of number line estimations in math anxiety: Evidence from bounded and unbounded number line estimation

AbstractMathematical anxiety (MA) is a feeling of tension and anxiety that interferes with the manipulation of numbers and the solution of mathematical problems. Past studies have shown that high MA individuals experience difficulties at different levels and domains of mathematics, and that MA results from weakness in spatial abilities, subsequent to deficits in basic numerical abilities. Hence, the main goal of the present study is to examine developmental trajectories of number space associations and their role in MA. In the present study, 2nd, 3rd, and 5th grade children (n = 94) participated in number line estimation tasks, half with low MA (LMA) and half with high MA (HMA). They performed two number line estimation tasks (bounded‐ with numerical values for beginning and end of the number line, or unbounded‐ only the beginning of the number line is marked) in addition to examination of strategy during arithmetical problems. Results indicated that younger children with HMA showed non‐mature numerical estimations compared to participants with LMA. However, for older HMA children, estimations were normalized and group differences were eliminated. Moreover, we found that estimations predicted usage of advanced memory‐based strategies in simple addition operations. These results indicated that (1) non‐mature numerical estimations characterize younger participants with MA, and (2) that numerical estimations are associated with arithmetical performances.

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.

The added value of game elements: better training performance but comparable learning gains

Even though game elements can increase motivation and engagement, they also might distract learners and thereby decrease performance and learning outcomes. In the current study, we investigated the effects of intrinsically integrated game elements on performance and learning outcomes. In a pre-post training study, 85 adult participants were randomly assigned either to the game-based or non-game-based training condition. Participants trained their fraction magnitude understanding with digital number line estimation tasks on five consecutive days (á 15–20 min). The learning outcomes were evaluated using a paper-based number line estimation task. While participants in both the game-based and non-game-based condition improved their fraction magnitude understanding from pretest to posttest, their improvement did not differ significantly. However, during the training, participants in the game-based condition responded more accurately but were slower than those in the non-game-based condition. The current results suggest that game elements might increase participants’ cognitive engagement and change their priorities or strategies (i.e., accuracy over speed) during learning. Nevertheless, better training performance did not lead to superior learning outcomes.

Development of children’s number line estimation in primary school: Regional and curricular influences

Is the development of number line estimation (NLE) similar across regions? Data from Canada (Quebec, n = 67, Mage = 7.9 years; Manitoba, n = 177, Mage = 7.8 years), Chile (n = 81, Mage = 7.9 years), and Northern Ireland (n = 171, Mage = 7.3 years) were analyzed. Twice, approximately one year apart, students completed a 0–1000 NLE task and other mathematical tasks. Using latent profile analysis, students’ estimates were classified as belonging to either a uniform or variable profile. At Time 1, estimation accuracy differed across regions, but at Time 2, patterns of performance were similar. Regional variations in improvements were related to curricular demands. Moreover, mini meta-analyses of the associations between NLE and other mathematical tasks revealed medium effect sizes. Overall, the NLE task can provide insights into concurrent and longitudinal mathematics achievement, but educational experiences should be considered when comparing performance across regions.

Engaging learners with games-Insights from functional near-infrared spectroscopy.

The use of game elements in learning tasks is thought to facilitate emotional and behavioral responses as well as learner engagement. So far, however, little is known about the underlying neural mechanisms of game-based learning. In the current study, we added game elements to a number line estimation task assessing fraction understanding and compared brain activation patterns to a non-game-based task version. Forty-one participants performed both task versions in counterbalanced order while frontal brain activation patterns were assessed using near-infrared spectroscopy (within-subject, cross-sectional study design). Additionally, heart rate, subjective user experience, and task performance were recorded. Task performance, mood, flow experience, as well as heart rate did not differ between task versions. However, the game-based task-version was rated as more attractive, stimulating and novel compared to the non-game-based task version. Additionally, completing the game-based task version was associated with stronger activation in frontal brain areas generally involved in emotional and reward processing as well as attentional processes. These results provide new neurofunctional evidence substantiating that game elements in learning tasks seem to facilitate learning through emotional and cognitive engagement.

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.

Spatial-Numerical Magnitude Estimation Mediates Early Sex Differences in the Use of Advanced Arithmetic Strategies.

An accumulating body of literature points to a link between spatial reasoning and mathematics learning. The present study contributes to this line of research by investigating sex differences both in spatial representations of magnitude and in the use of arithmetic strategies, as well as the relation between the two. To test the hypothesis that sex differences in spatial-numerical magnitude knowledge mediate sex differences in the use of advanced strategies (retrieval and decomposition), two studies were conducted. Study 1 included 96 US first graders (53% girls); Study 2 included 210 Russian first graders (49% girls). All participants completed a number line estimation task (a spatially based measure of numerical magnitude knowledge) and an arithmetic strategy task (a measure of strategy choice). The studies showed parallel results: boys produced more accurate numerical magnitude estimates on the number line estimation task and used advanced strategies more frequently on the arithmetic task. Critically, both studies provide support for the mediation hypothesis (although there were some differences in the pattern obtained for the two strategies). The results are discussed in the context of broader research about the relation between spatial and mathematical skills.

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.

Developmental trajectories of numerical magnitude representations of fractions and decimals.

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0-1 and 0-5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions. Analyses of individual differences revealed positive relations between the accuracy of decimal and fraction magnitude representations at all ages. In Experiment 2, we presented an additional set of 24 fourth graders (14 girls and 10 boys) with the same tasks but with the decimals that were being compared varying in the number of decimal digits. The decimal advantage continued to be present for both magnitude comparison and estimation tasks, indicating that the greater accuracy with decimals was not limited to decimals with equal numbers of decimal digits, though unequal numbers of decimal digits did impact performance with decimals on both magnitude comparison and number line estimation tasks. Implications for understanding numerical development and education are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

A potential dissociation between perception and production version for bounded but not unbounded number line estimation

BackgroundWhat, exactly, do number line estimation (NLE) tasks measure? Different versions of the task were observed to have different effects on performance. MethodWe investigated associations between the production (indicating the location) and perception version (indicating the number) of the bounded and unbounded NLE task and their relationship to arithmetic. ResultsA stronger correlation was observed between the production and perception version of the unbounded than the bounded NLE task, indicating that both versions of the unbounded—but not the bounded—NLE task measure the same construct. Moreover, overall low but significant associations between NLE performance and arithmetic were only observed for the production version of the bounded NLE task. ConclusionThese results substantiate that the production version of bounded NLE seems to rely on proportion judgment strategies, whereas both unbounded versions and the perception version of the bounded NLE task may rely more on magnitude estimation.

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.