Number Line Estimation Performance Research Articles

Task features change the relation between math anxiety and number line estimation performance with rational numbers: Two large-scale online studies.

Math performance is negatively related to math anxiety (MA), though MA may impact certain math skills more than others. We investigated whether the relation between MA and math performance is affected by task features, such as number type (e.g., fractions, whole numbers, percentages), number format (symbolic vs. nonsymbolic), and ratio component size (small vs. large). Across two large-scale studies (combined n = 3,822), the MA-performance relation was strongest for large whole numbers and fractions, and stronger for symbolic than nonsymbolic fractions. The MA-performance relation was also stronger for smaller relative to larger components, and MA relating to specific number types may be a better predictor of performance than general MA for certain tasks. The relation between MA and estimation performance changes depending on task features, which suggests that MA may relate to certain math skills more than others, which may have implications for how people reason with numerical information and may inform future interventions. (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.

The role of math anxiety in the relationship between approximate number system and math performance in young children

AbstractAlthough interest in mathematics anxiety in young children (6–11 years) has increased recently, the results of previous studies on the role of mathematics anxiety in the relationship between number sense and math performance are inconsistent. The study's main objective was to establish whether math anxiety is a mediator or a moderator in the relationship between number line estimation (NLE) and mathematical outcomes. A cross‐sectional, correlational design was used. A total of 234 Turkish public primary school children (third and fourth graders) participated in the study. The following hypotheses were tested: (1) math anxiety mediates the relationship between the approximate number system (ANS) (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency); (2) math anxiety moderates the relationship between ANS (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency). The results mostly suggest no mediation effect of math anxiety between NLE and math performance in young children. The results also suggest that increases in difficulty level in NLE (increases of errors in NLE) is related to more anxiety regarding mathematics. Furthermore, moderation analyses indicated that the acuity of mental representation of numbers in high math‐anxious children is a key factor contributing to math performance. Developmental aspects of the obtained results are also discussed in the paper.

Relational language influences young children's number relationskills.

Relational language is thought to influence mathematical skills. This study examines the association between relational language and number relation skills-knowledge of cardinal, ordinal, and spatial principles-among 104 U.S. kindergartners (5.9years; 44% boys; 37% White, 25% Black, 14% Asian, 24% other) in the 2017-2018 academic year. Controlling for general verbal knowledge, executive function, and counting and number identification skills, relational language predicted later number relation skills, specifically number line estimation, β=.30. Relational language did not differentially predict number line estimation performance in children with low or high number relation skills, likely due to the restricted ranges of data within subgroups. Number relation skills, specifically number line estimation and number ordering, may be a pathway between relational language and mathematical skills.

Numerical estimation strategies are correlated with math ability in school-aged children

Various laboratory-based numerical cognition measures are often found to be linked with standardized math scores, yet the exact nature of these links over development remains in question. In this study, we investigated how 0–100 number line estimation performance and dot numerosity discrimination explained variance in 6- to 8-year-olds’ standardized math ability scores, and we assessed links between tasks. Our primary goal was to assess the link between estimation performance and math ability from a theoretical perspective that models number line estimation as an instance of proportion estimation. As age increased, children’s proportion-based strategies for number line estimation became more sophisticated and estimation bias decreased. Children who spontaneously inferred and used a central reference point to guide their proportion estimates scored higher on the math ability measure (TEMA-3), demonstrating a relation between more advanced proportion estimation strategies and higher math abilities. Individual differences in estimation bias also made a lesser contribution to explaining variance in children’s math ability. Dot discrimination accuracy uniquely contributed to children’s math ability. Number line estimation performance and dot discrimination accuracy were not associated, consistent with prior research suggesting that number line estimation does not directly map onto mental numerical magnitude representations. Educational implications and future directions are discussed.

Operational momentum during children’s approximate arithmetic relates to symbolic math skills and space–magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space–magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space–magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Children’s neural activity during number line estimations assessed by functional near-infrared spectroscopy (fNIRS)

Number line estimation (NLE) is an educational task in which children estimate the location of a value (e.g., 25) on a blank line that represents a numerical range (e.g., 0–100). NLE performance is a strong predictor of success in mathematics, and error patterns on this task help provide a glimpse into how children may represent number internally. However, a missing and fundamental element of this puzzle is the identification of neural correlates of NLE in children. That is, understanding possible neural signatures related to NLE performance will provide valuable insight into the cognitive processes that underlie children’s development of NLE ability. Using functional near-infrared spectroscopy (fNIRS), we provide the first investigation of concurrent behavioral and cortical signatures of NLE performance in children. Specifically, our results highlight significant fronto-parietal changes in cortical activation in response to increases in NLE scale (e.g., 0–100 vs. 0–100,000). Furthermore, our results demonstrate that NLE performance feedback (auditory, visual, or audiovisual), as well as children’s grade (2nd vs. 3rd) influence cortical responding during an NLE task.

The strategy matters: Bounded and unbounded number line estimation in secondary school children

Changes in number line estimation (NLE) performance are frequently used as an indicator for the development of the number magnitude representation. For this purpose, two different task versions have been applied: a traditional bounded and a relatively new unbounded NLE task. Previous studies mainly assessed primary school children or adults showing that these tasks differ in terms of i) estimation accuracy and solution strategies employed as well as ii) with respect to their relation to other basic numerical/arithmetic skills. So far, data from secondary school children are scarce for bounded NLE, and even no data is available for unbounded NLE. Thus, we assessed bounded and unbounded NLE in grade levels 5–7 to evaluate a) developmental as well as strategic influences, and b) the relation of bounded and unbounded NLE performance with basic arithmetic skills. Our results show that children employed the use of different solution strategies for bounded (i.e., proportion-judgement) and unbounded (i.e., magnitude-estimation based) NLE. Moreover, only for bounded NLE, estimation accuracy increased with age. Furthermore, estimation performance for bounded but not unbounded NLE was strongly associated with basic arithmetic (addition, subtraction, multiplication, and division). Our findings indicate that the differential results for bounded and unbounded NLE obtained in primary school children seem to generalize to older secondary school children. Presented results substantially contribute to the knowledge about the (consecutive) development of skills pertaining to bounded and unbounded number line estimation.

Developmental change in number line estimation: A strategy-based perspective.

The current study investigated developmental changes in children's benchmark-based strategy use in number line estimation. Third and fifth graders solved a 0-1,000 number line estimation task in one of three conditions. In the control condition, only the origin and endpoint were specified, the midpoint condition included an additional benchmark at 50%, and the quartile condition contained three additional benchmarks at 25%, 50%, and 75%. Trial-by-trial verbal strategy reports revealed that fifth graders, in comparison to third graders, spontaneously applied quartile-based strategies more frequently while they used strategies based on the origin, midpoint, and endpoint about equally often. Finally, children's mathematics achievement was positively related to the variety and frequency of benchmark-based strategies as well as their number line estimation performance. We conclude that developmental changes in number line estimation performance can at least partially be attributed by a refinement in children's benchmark-based strategy use. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Individual Differences in Implicit and Explicit Spatial Processing of Fractions.

Recent studies have explored the foundations of mathematical skills by linking basic numerical processes to formal tests of mathematics achievement. Of particular interest is the relationship between spatial-numerical associations—specifically, the Spatial Numerical Association of Response Codes (SNARC) effect—and various measures of math ability. Thus far, studies investigating this relationship have yielded inconsistent results. Here, we investigate how individual implicit and explicit spatial representations of fractions relate to fraction knowledge and other formal measures of math achievement. Adult participants (n = 105) compared the magnitude of single digit, irreducible fractions to ½, a task that has previously produced a reliable SNARC effect. We observed a significant group-level SNARC effect based on overall fraction magnitude, with notable individual variability. While individual SNARC effects were correlated with performance on a fraction number-line estimation (NLE) task, only NLE significantly predicted scores on a fractions test and basic standardized math test, even after controlling for IQ, mean accuracy, and mean reaction time. This suggests that–for fractions–working with an explicit number line is a stronger predictor of math ability than implicit number line processing. Neither individual SNARC effects nor NLE performance were significant predictors of algebra scores; thus, the mental number line may not be as readily recruited during higher-order mathematical concepts, but rather may be a foundation for thinking about simpler problems involving rational magnitudes. These results not only characterize the variability in adults’ mental representations of fractions, but also detail the relative contributions of implicit (SNARC) and explicit (NLE) spatial representations of fractions to formal math skills.

Benchmark-based strategy use in atypical number lines.

Previous studies have indicated that the presence of atypical end points (e.g., 1,639 and 2,897) on a number line has a negative effect on number line estimation (NLE) performance (Booth & Newton, 2012; Hurst, Leigh Monahan, Heller, & Cordes, 2014). In the present study, we investigated whether this effect could be attributed to a disruption in the ease with which benchmarks on the number line can be determined and whether this possible disruption changes with age. Hence, we asked 5th graders and adults to perform a NLE task in a typical and an atypical condition with a number line ranging from 0 to 1,000 and from 367 to 1,367, respectively. Results showed that participants' overall estimates were less accurate in the atypical condition compared to the typical condition. A similar pattern of results was observed for the estimates around the self-created benchmarks at 50% in children and at 25%, 50%, and 75% in adults. Moreover, in both age groups, we found a shift from a larger proportion of estimation patterns' being best fit by more complex power models in the typical condition toward a larger proportion of estimation patterns best described by simpler power models in the atypical condition. Of importance, we also found evidence that adults' estimation patterns in the typical condition are best described by a 4-cycle power model, hereby extending previous modeling results. Overall, our findings indicate that hindering the application of benchmark-based strategies negatively affects children's and adults' NLE performance. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

Evaluating the Effect of Labeled Benchmarks on Children's Number Line Estimation Performance and Strategy Use.

Some authors argue that age-related improvements in number line estimation (NLE) performance result from changes in strategy use. More specifically, children’s strategy use develops from only using the origin of the number line, to using the origin and the endpoint, to eventually also relying on the midpoint of the number line. Recently, Peeters et al. (unpublished) investigated whether the provision of additional unlabeled benchmarks at 25, 50, and 75% of the number line, positively affects third and fifth graders’ NLE performance and benchmark-based strategy use. It was found that only the older children benefitted from the presence of these benchmarks at the quartiles of the number line (i.e., 25 and 75%), as they made more use of these benchmarks, leading to more accurate estimates. A possible explanation for this lack of improvement in third graders might be their inability to correctly link the presented benchmarks with their corresponding numerical values. In the present study, we investigated whether labeling these benchmarks with their corresponding numerical values, would have a positive effect on younger children’s NLE performance and quartile-based strategy use as well. Third and sixth graders were assigned to one of three conditions: (a) a control condition with an empty number line bounded by 0 at the origin and 1,000 at the endpoint, (b) an unlabeled condition with three additional external benchmarks without numerical labels at 25, 50, and 75% of the number line, and (c) a labeled condition in which these benchmarks were labeled with 250, 500, and 750, respectively. Results indicated that labeling the benchmarks has a positive effect on third graders’ NLE performance and quartile-based strategy use, whereas sixth graders already benefited from the mere provision of unlabeled benchmarks. These findings imply that children’s benchmark-based strategy use can be stimulated by adding additional externally provided benchmarks on the number line, but that, depending on children’s age and familiarity with the number range, these additional external benchmarks might need to be labeled.

Children can accurately monitor and control their number-line estimation performance.

Accurate monitoring and control are essential for effective self-regulated learning. These metacognitive abilities may be particularly important for developing math skills, such as when children are deciding whether a math task is difficult or whether they made a mistake on a particular item. The present experiments investigate children's ability to monitor and control their math performance. Experiment 1 assessed task- and item-level monitoring while children performed a number line estimation task. Children in 1st, 2nd, and 4th grade (N = 59) estimated the location of numbers on small- and large-scale number lines and judged their confidence in each estimate. Consistent with their performance, children were more confident in their small-scale estimates than their large-scale estimates. Experiments 2 (N = 54) and 3 (N = 85) replicated this finding in new samples of 1st, 2nd, and 4th graders and assessed task- and item-level control. When asked which estimates they wanted the experimenter to evaluate for a reward, children tended to select estimates associated with lower error and higher confidence. Thus, children can accurately monitor their performance during number line estimation and use their monitoring to control their subsequent performance. (PsycINFO Database Record

When prior knowledge interferes, inhibitory control matters for learning: The case of numerical magnitude representations.

The present study tested the interference hypothesis—that learning and using more advanced representations and strategies requires the inhibition of prior, less advanced ones. Specifically, it examined the relation between inhibitory control and number line estimation performance. Experiment 1 compared the accuracy of adults’ (N = 53) estimates on 2 number line tasks, 1 with standard (power of 10) endpoints (0–1,000) and the other with nonstandard endpoints (364–1,364). Inhibition, as measured by Stroop task performance, predicted the accuracy of estimation on the nonstandard number line task, above and beyond estimation on the standard task. In Experiment 2, changes in kindergartners’ (N = 42) 0–100 number line estimation were elicited through randomized training conditions, which involved playing a numerical board game. Stroop task performance was related to the rate of improvement in estimation, controlling for pretest number line task performance. The results provide a potential explanation for the relation between inhibitory control and mathematics achievement: Individuals with better inhibitory control may be better able to suppress the activation of prior knowledge and may be less vulnerable to interference from such knowledge. Potential implications for instructional design are discussed. (PsycINFO Database Record (c) 2016 APA, all rights reserved)

Children’s use of number line estimation strategies

This study tested whether second graders use benchmark-based strategies when solving a number line estimation (NLE) task. Participants were assigned to one of three conditions based on the availability of benchmarks provided on the number line. In the bounded condition, number lines were only bounded at both sides by 0 and 200, while the midpoint condition included an additional benchmark at the midpoint and children in the quartile condition were provided with a benchmark at every quartile. First, the inclusion of a midpoint resulted in more accurate estimates around the middle of the number line in the midpoint condition compared to the bounded and, surprisingly, also the quartile condition. Furthermore, the two additional benchmarks in the quartile condition did not yield better estimations around the first and third quartile, because children frequently relied on an erroneous representation of these benchmarks, leading to systematic estimation errors. Second, verbal strategy reports revealed that children in the midpoint condition relied more frequently on the benchmark at the midpoint of the number line compared to the bounded condition, confirming the accuracy data. Finally, the frequency of use of benchmark-based strategies correlated positively with mathematics achievement and tended to correlate positively also with estimation accuracy. In sum, this study is one of the first to provide systematic evidence for children’s use of benchmark-based estimation strategies in NLE with natural numbers and its relationship with children’s NLE performance.

How number line estimation skills relate to neural activations in single digit subtraction problems

The Number Line (NL) task requires judging the relative numerical magnitude of a number and estimating its value spatially on a continuous line. Children's skill on this task has been shown to correlate with and predict future mathematical competence. Neurofunctionally, this task has been shown to rely on brain regions involved in numerical processing. However, there is no direct evidence that performance on the NL task is related to brain areas recruited during arithmetical processing and that these areas are domain-specific to numerical processing. In this study, we test whether 8- to 14-year-old's behavioral performance on the NL task is related to fMRI activation during small and large single-digit subtraction problems. Domain-specific areas for numerical processing were independently localized through a numerosity judgment task. Results show a direct relation between NL estimation performance and the amount of the activation in key areas for arithmetical processing. Better NL estimators showed a larger problem size effect than poorer NL estimators in numerical magnitude (i.e., intraparietal sulcus) and visuospatial areas (i.e., posterior superior parietal lobules), marked by less activation for small problems. In addition, the direction of the activation with problem size within the IPS was associated with differences in accuracies for small subtraction problems. This study is the first to show that performance in the NL task, i.e. estimating the spatial position of a number on an interval, correlates with brain activity observed during single-digit subtraction problem in regions thought to be involved in numerical magnitude and spatial processes.

On the Relation between the Mental Number Line and Arithmetic Competencies

In this study, we aimed at investigating whether it is indeed the spatial magnitude representation that links number line estimation performance to other basic numerical and arithmetic competencies. Therefore, estimations of 45 fourth-graders in both a bounded and a new unbounded number line estimation task (with only a start-point and a unit given) were correlated with their performance in a variety of tasks including addition, subtraction, and number magnitude comparison. Assuming that both number line tasks assess the same underlying mental number line representation, unbounded number line estimation should also be associated with other basic numerical and arithmetic competencies. However, results indicated that children's estimation performance in the bounded but not the unbounded number line estimation task was correlated significantly with numerical and arithmetic competencies. We conclude that unbounded and bounded number line estimation tasks do not assess the same underlying spatial-numerical representation. Rather, the observed association between bounded number line estimation and numerical/arithmetic competencies may be driven by additional numerical processes (e.g., proportion judgement, addition/subtraction) recruited to solve the task.

Predictors for Mathematics Achievement? Evidence From a Longitudinal Study

Numerical processing has been extensively studied by examining the performance on basic number processing tasks, such as number priming, number comparison, and number line estimation. These tasks assess the innate “number sense,” which is assumed to be the breeding ground for later mathematics development. Indeed, several studies have associated children's performance in these tasks with individual differences in mathematical achievement. To date, however, most of these studies have cross‐sectional designs. Moreover, the few longitudinal studies either use complex tasks (e.g., story problems) or investigate only one of these basic number processing tasks at a time. In this study, we examine the association between the performance of children on several basic number processing tasks and their individual math achievement scores on a curriculum‐based test measured 1 year later. Regression analyses showed that most of the variance in children's math achievement was predicted by nonsymbolic number line estimation performance (i.e., estimating large quantities of dots) and, to a lesser extent, the speed of comparing symbolic numbers. This knowledge about the predictive value of the performance of 5‐ to 7‐year‐olds on these markers of number processing can help with the early identification of at‐risk children. In addition, this information can guide appropriate educational interventions.

P.4.b.002 The latent structure underlying the comorbidity of panic, agoraphobia and specific phobias

Various laboratory-based numerical cognition measures are often found to be linked with standardized math scores, yet the exact nature of these links over development remains in question. In this study, we investigated how 0–100 number line estimation performance and dot numerosity discrimination explained variance in 6- to 8-year-olds’ standardized math ability scores, and we assessed links between tasks. Our primary goal was to assess the link between estimation performance and math ability from a theoretical perspective that models number line estimation as an instance of proportion estimation. As age increased, children’s proportion-based strategies for number line estimation became more sophisticated and estimation bias decreased. Children who spontaneously inferred and used a central reference point to guide their proportion estimates scored higher on the math ability measure (TEMA-3), demonstrating a relation between more advanced proportion estimation strategies and higher math abilities. Individual differences in estimation bias also made a lesser contribution to explaining variance in children’s math ability. Dot discrimination accuracy uniquely contributed to children’s math ability. Number line estimation performance and dot discrimination accuracy were not associated, consistent with prior research suggesting that number line estimation does not directly map onto mental numerical magnitude representations. Educational implications and future directions are discussed.