Number Line Estimation Research Articles

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)

"It's Different for Girls!" The Role of Anxiety, Physiological Arousal, and Subject Preferences in Primary School Children's Math and Mental Rotation Performance.

(1) Background: This study examines the role of subjective anxiety (mathematics and spatial anxiety), along with physiological responses, in mathematics or math and mental rotation performance in 131 German primary school students (65 girls, 66 boys; Mean age = 8.73 years). (2) Method: Students' preference for math vs. German and their subjective anxiety were assessed using standardized questionnaires. Emotional reactivity was measured using the Galvanic Skin Response (GSR). Math performance was evaluated via percentage scored and completion times on number line estimation, word problems, and missing terms tasks. Spatial skills were assessed using a novel mental rotation task (nMRT) incorporating gender-congruent and -neutral stimuli. (3) Results: Girls outperformed boys on percentage scored on the math task but took longer to complete this. No gender differences were found in performance on the nMRT. Girls demonstrated higher math anxiety and were less likely to prefer math over German. Math anxiety predicted math scores and accuracy on the nMRT while gender predicted math performance and mental rotation response time. Subject preference was associated with longer completion times and emotional reactivity with longer response times. Girls' preference for math and lower emotional reactivity was linked to shorter completion times, while lower math anxiety predicted higher scores. In contrast, these factors did not affect boys' math performance. Additionally, subjective anxiety, emotional reactivity, or subject preference did not impact spatial performance for either gender. (4) Conclusions: Supporting mathematical self-efficacy and emotional regulation, especially in girls, is crucial for enhancing STEM outcomes in primary education. Gender-fair assessment in mental rotation reveals equitable spatial performance and reduces the impact of anxiety.

Rapid automatised naming is related to reading and arithmetic for different reasons in Chinese: Evidence from Hong Kong third graders

BackgroundRapid automatised naming (RAN) has been found to predict children's reading and arithmetic abilities. However, the underlying mechanisms for its involvement in the two abilities are not clear. This study examines how RAN shared variances with domain‐general and domain‐specific abilities in predicting reading and arithmetic in Chinese children.MethodsOne hundred and sixty‐four children (mean age = 8 years 0 months, SD = 4 months) were administered with RAN tasks, word reading and arithmetic tasks and measures of working memory, processing speed, morphological awareness, phonological awareness and number line estimation.ResultsRAN mainly shared variance with morphological awareness in predicting word reading, while it shared variance with processing speed and number line estimation in predicting arithmetic calculation.ConclusionsThe findings indicated that RAN was related to reading and arithmetic for different reasons. The RAN–reading relationship partly reflected the semantic facilitation of the orthography–phonology links for both RAN stimuli and Chinese characters, while the RAN–arithmetic relationship partly reflected the shared process of retrieving semantic information from long‐term memory embedded in the two tasks.

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.

Improving Low Mathematics Achievers’ Number Sense via Number Line Training with Board Games

This study investigated the effect of a number line training via lineer board games with specific contents, namely numerical and figural, on students’ number line estimation skills, arithmetic performance, and mathematics achievement. A total of 30 first graders (15 in experimental and 15 in control group) participated in the study. There were 2 speech and language disorder (SLD) students, one in each, in the control and experimental groups. Number line training with a board game, Sahibingo® was implemented in the experimental group, while a non-numerical board game, Animalbingo, was used in the control group. Training lasted for 30 minutes a day, one day a week, and 4 weeks (2 hours in total). Results indicated that students, including the SLD, in the experimental group improved their number line estimation skills in 0-10 and 0-20 number range, but not in the 0-100 range. Improvements did not reflect in arithmetic performance and mathematics achievement scores. Although there are some improvements in small range number line estimations possibly because of the familiarity there is no improvement in the larger range and in terms of mathematics achievement scores, possibly because of the shorter period of training. It can be concluded that low mathematics achievers’ number sense can be improved via lineer board games with numerical content. Future research may investigate the effect of longer periods and may include other students with special needs.

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.

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.

Unpacking the Connections Between Fractions and Algebra: The Importance of Fraction Schemes and Units Coordination

ABSTRACT Researchers from multiple disciplines have found that fractions and algebra knowledge are linked. One major strand of research has identified children’s units coordination structures as crucial for success with fractions and algebra via multiplicative reasoning, whereas a second strand of research points to magnitude knowledge as a central conceptual structure supporting fractions and algebra knowledge. In an online study with 59 eighth graders, we compared the associations of these two foundational factors with fractions and algebra knowledge by assessing students’ units coordination skills, magnitude knowledge, and several general cognitive skills alongside multiple aspects of fractions and algebra knowledge. In regression models controlling for math anxiety and general cognitive skills, units coordination and fraction schemes were better predictors of students’ algebra knowledge than were fraction magnitude or fraction arithmetic skills. We replicated findings from previous studies that showed units coordination and fraction schemes were closely related and showed for the first time that units coordination was significantly associated with fraction number line estimation and fraction arithmetic scores, even after accounting for general cognitive skills. We emphasize the need for interdisciplinary teams to continue investigating relations among these constructs to further our understanding and ultimately support student learning of fractions and algebra.

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.

Examining the malleability of young children's flexible attention to numerical and spatial magnitudes

Math ability is critical to children's future school and career success, and prior studies show that flexible attention to magnitudes (FAM) predicts children's future math abilities over and above other early math skills. FAM broadly refers to the ability to switch flexibly between attending to different dimensions of magnitude (e.g., size and numerosity). In the current study, we created an intervention using number books to test, for the first time, the malleability of children's FAM ability. A randomized experiment was conducted with 116 preschool-age children (Mage = 55.6 months; 54.3 % female) to test our hypothesis that FAM ability is malleable and can be trained using number books. The intervention included four conditions to which children were randomly assigned: size-to-number, mixed, conventional counting, and non-numerical. The experimental conditions were size-to-number and mixed, while the conventional counting and non-numerical conditions served as controls. Consistent with our hypothesis, children significantly improved in FAM task performance from pre- to post-intervention within the experimental groups, but not within the control groups. Furthermore, children in the two experimental conditions combined significantly improved number line estimation scores from pre- to post-intervention, unlike the combined two control conditions. Implications of these findings on the development of future FAM interventions are discussed.

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.

Integrated knowledge of rational number notations predicts children’s math achievement and understanding of numerical magnitudes

We propose that integrated number sense, the ability to fluidly translate and compare magnitudes within and across notations, is central to understanding of rational numbers. Consistent with this hypothesis, two studies of 6th through 8th grade students (N = 264 and N = 46) indicated that accuracy comparing magnitudes within and across notations predicted overall math achievement and fraction number line and arithmetic estimation accuracy. Cross-notation magnitude comparison accuracy (i.e., fraction vs. decimal, percentage vs. fraction, and percentage vs. decimal) accounted for variance in math outcomes beyond that explained by magnitude representations within individual notations. The findings also revealed a percentages-are-larger bias, in which percentages are perceived as larger than equivalent fractions and decimals in cross-notation comparisons. There were no differences by grade level, suggesting that such number sense does not necessarily develop with age. Explicitly promoting integrated number sense may help improve mathematics instruction.

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.

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).

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.