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 (0100) 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, onecycle power model, twocycle 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.
Performance on the number line estimation task has the potential to provide crucial information regarding mathematics learning. However, disagreements on what is reflected by performance on the number line estimation task, and what role it might play in mathematics learning remain. Previous findings highlight the relationship between accuracy on the number line task and other mathematical skills (see
The number line task, in its traditional version, consists of a horizontal line with an upper and lower limit. These limits are labelled and indicate the range of the number line (e.g., 010, 0100). The task consists of placing Arabic numbers on the horizontal line based on the number’s value. Accuracy is represented by error measures based on the distance between the estimate and the target number, which are then corrected for the range of the number line (i.e., percent absolute error, PAE). An alternative way to index performance involves mapping the estimates (on the y axis) against the target numbers (on the x axis) to then assess the fit of different mathematical functions to the patterns formed. These patterns are usually referred to as estimation patterns, and what is being assessed is which mathematical function most closely resembles (i.e., explains) the estimates. Estimation patterns can be assessed at a grouplevel, which involves combining the estimates’ placements across participants (but see
The link between estimation patterns and several mathematical functions have been studied (e.g., linear, logarithmic, unbounded, one and twocycle power models). There is evidence to suggest that the patterns followed by children’s estimates may shed some light on what performance on the number line task reflects, as well as the way in which children engage with the task (see
This theoretical account proposes that the mental representation of magnitude shifts from logarithmic to linear (
Numerous studies have provided evidence for estimation patterns evolving from logarithmic to linear. Differences between studies’ methodologies and findings make it difficult to determine when this shift is expected to occur within any given number range. For example, a logarithmic model best fit the estimation patterns of over 80% of four to sixyearold preschoolers on a 01000 number line (
The proportional judgment account proposes that performance on the number line estimation task requires the estimation of the proportion between two numbers (the target number and the number range). This is in line with psychophysical models of relative quantity judgments within a proportion judgment framework (see
As with the studies discussed in the previous section, determining the age at which the transitions between number of benchmarks used are expected to occur for a particular number range is challenging. For example, three and fouryearolds’ estimation patterns on a 010 number line were best explained by an unbounded power model (39.4%), followed by a onecycle power model (18.3%), and finally a twocycle power model (11%). Approximately one third of estimation patterns could not be fitted by any of these three models (
Previous studies have found evidence for and against models arising from both theoretical accounts. These differing findings may occur due to methodological differences and existing challenges for the study of estimation patterns. One of the main challenges is the lack of an absolute reference for model selection. This means that a given mathematical function is the one that best explains the estimation patterns only in comparison with the others being fitted to the same estimation patterns. Another challenge arises when considering the number and types of models that are compared. So far, we have only discussed comparisons between models arising from a single theoretical account. However, findings get more complex when studies compare models that arise from different theoretical accounts. For example, in
A related challenge, which is rarely made explicit, is the slight differences that are sometimes found in model fit between models. This entails that a model is often chosen over another one, without strong evidence for its preference. The fact that this is rarely reported (at least related to individual children’s estimation patterns) makes it harder to know how prevalent it might be. These limitations complicate our understanding of what estimation patterns on the number line task represent and how they relate to other mathematical abilities.
Accuracy on the number line estimation task has been related to concurrent and future mathematical achievement (
We approach the aims of this study based on the representational shift and proportional judgment accounts, first considered separately and then concurrently. The first aim is to determine which mathematical functions (i.e., linear, logarithmic, unbounded power, onecycle power, and twocycle power) best explain children’s estimation patterns on the 0100 number line task. The second aim is to determine if and how children’s mathematical skills are associated with the mathematical function that best explains their estimation patterns.
Only a handful of previous studies have considered both theoretical accounts when analysing individual estimation patterns (
Even fewer studies have looked at the relationship between children’s individual estimation patterns and their mathematical skills (
Singaporean children (
Parents provided written consent, and ethical approval by the Institutional Review Board at Nanyang Technological University was obtained. Testing was conducted in two to three sessions, held on different days. All children were tested in their preschools, in the language of instruction (English), and given a small gift for participating.
Children solved a computerised bounded number line task (0100). Children were presented with a horizontal line with zero at the left and 100 at the right and were shown a number above the middle of the line which they had to place on the line. Children initially completed one practice trial (fifty) to confirm that they had understood the task and were then assessed with 26 numbers (three, four, six, eight, 12, 14, 17, 18, 21, 24, 25, 29, 33, 39, 42, 48, 52, 57, 61, 64, 72, 79, 81, 84, 90, and 96). Numbers on the lower end of the number line were oversampled following
Two mathematics subtests (Numerical Operations and Mathematical Reasoning) from the Wechsler Individual Achievement Test II (WIATII;
In line with the first aim of the study, we fitted children’s individual estimation patterns to the logarithmic, linear, unbounded power model, onecycle power model and twocycle power models (see
Descriptive statistics are presented in
Variable  Min.  Max.  Skewness  Kurtosis  

Mathematical achievement  28.82  5.46  14.00  41.00  0.28  2.63 
Number line estimation (Percent absolute error)  16.15  6.67  4.67  37.89  0.47  2.77 
Within the representational shift account estimation patterns were less frequently best explained by a logarithmic (35.9%, M_{AICc} = 158.19) compared to a linear model (64.1%, M_{AICc} = 145.58; see top panel of
Within the proportional judgment account, 57.6% (M_{AICc} = 148.54) of estimation patterns were best explained by an unbounded power model, 29.7% (M_{AICc} = 149.84) by a onecycle power model, and 12.7% (M_{AICc} = 146.48) by a twocycle power model (see top panel of
Considering that AICc is a relative measure, and that in some cases we had noticed very small AICc differences between models, it was necessary to account for the strength of the evidence for each model.
Within the representational shift account, 10.9% of estimation patterns showed similar support for both models (i.e, ΔAICc < 2), 29.7% (M_{AICc} = 156.46) showed stronger support for the logarithmic model, while 59.4% (M_{AICc} = 144.48) showed stronger support for the linear model (see bottom panel of
We first consider the distribution across both accounts based on the models arising from the previous analyses, which we represent in a twoway table (see
Best fitting model  Logarithmic 
Linear 

Unbounded ( 
51% (81)  49% (78) 
Onecycle ( 
22% (18)  78% (64) 
Twocycle ( 
0% (0)  100% (35) 
Notwithstanding, estimation patterns best explained by the unbounded power model were almost equally divided (
To extend this understanding we directly contrasted the models arising from both theoretical accounts (see top panel in
As before, we also accounted for the strength of the evidence for each model by excluding estimation patterns for which the ΔAICc was less than two (see bottom panel of
Alternative model with similar support  Model that best explains the estimation patterns 


Logarithmic model 
Linear model 
Unbounded power model ( 
Onecycle power model 
Two cycle power model ( 

ΔAICc < 2 with linear model  –  0  9  8  0 
ΔAICc < 2 with logarithmic model  2  –  18  21  8 
ΔAICc < 2 with unbounded power model  30  22  –  20  3 
ΔAICc < 2 with onecycle power model  6  12  10  –  7 
ΔAICc < 2 with twocycle power model  0  3  4  3  – 
To address the second aim of this study we created groups based on the results obtained from the first aim and then compared those groups on their accuracy on mathematical achievement. First, we compared models within theoretical accounts, and then between theoretical accounts. In both cases, analyses were run first including all estimation patterns, and then excluding those that had shown similar support for more than one model (i.e., ΔAICc < 2). Nonparametric tests were used as assumptions of normality were not met. For comparisons within the representational shift account the Wilcoxon rank sum test was used. In all other cases, the Kruskal Wallis test was used, followed up by Dunn’s test with a Holm correction. These analyses were conducted using the Stats (
Within the representational shift account, the linear group outperformed the logarithmic group in mathematical achievement,
When considering all models simultaneously, we found significant differences in mathematical achievement,
The first aim of this study was to determine which mathematical functions best explain children’s estimation patterns on the 0100 number line task. Specifically, we examined estimation patterns based on two dominant perspectives – logarithmic to linear representational shift and proportional judgment. We first examined them separately, and then compared them directly.
Based on the representational shift account (
Based on the proportional judgment account (
Comparing models across theoretical perspectives might be controversial, and it is not without its difficulties (see
When looking at the distribution of models across both theoretical accounts, we found that the percentage of estimation patterns best explained by the linear model increases as the power models become more advanced (i.e., the percentage of children’s estimates best explained by the linear and twocycle power model is higher than the one best explained by the linear and onecycle model, which is higher than the one best explained by the linear and unbounded power models). This trend was opposite for the estimation patterns best explained by the logarithmic model. This finding seems consistent with the idea that the use of additional benchmarks (i.e., more advanced models within the proportional judgement account) is more likely to correspond to more linear estimation patterns within the representational shift account. However, an unexpected finding was that approximately half of the estimation patterns best explained by the unbounded power model were also best explained by the linear model, while the other half was best explained by the logarithmic model.
These findings are unexpected, as we would not anticipate that children who only use the beginning of the number line to guide their estimates (i.e., unbounded power model) can achieve linear estimations. However, there are precedents of estimation patterns being similarly explained by unbounded and linear models. For example, in
First, it is important to note as shown earlier, that other models, beyond the ones that best explain the data might show similar support. For example, from the 78 participants for whom the unbounded power model and the linear model concurred, over 25% showed similar support between the unbounded and one of the other power models, while this was the case for 6% that showed similar support for the logarithmic and linear models.
In addition, almost 20% of the participants for whom the unbounded power model and the linear model concurred showed a less frequent pattern of β > 1, while all participants for whom the unbounded power model and the logarithmic model concurred showed the more frequent pattern β < 1. This might mean that, even though best explained by the unbounded power model, some of these children could be engaging with the task in a different way than other children whose estimates are also best explained by the unbounded power model.
Furthermore, from the estimation patterns in which the linear and unbounded power model had provided the best explanation within each theoretical account, the unbounded power model better explained 62.8% of them. Contrastingly, from the estimation patterns in which the unbounded power model and the logarithmic model had provided the best explanation within each theoretical account, the logarithmic model better explained 75.3% of them.
Finally, as mentioned earlier, estimation patterns best explained by one and twocycle power models were still more likely to also be better explained by linear models, than by logarithmic models. In principle, it seems highly unlikely that a linear estimation pattern would result from the exclusive use of the origin of the number line, though some evidence of this possibility exists (e.g.,
From the comparison of all models to each other, we find that models arising from the proportional judgment perspective best explain most estimation patterns (60.5%, 30.4% after excluding estimation patterns with similar support for more than one model). When considering models individually, the logarithmic, unbounded power, and onecycle power models account for slightly above 23 percent of children’s estimation patterns each (this is around 13 percent for the logarithmic and unbounded power model, and 8 percent for the onecycle power model after excluding estimation patterns showing similar support for more than one model), and the linear and twocycle power model account for approximately 15 and 12 percent of children’s estimation patterns respectively (8 and 5 percent after excluding estimation patterns with similar support for more than one model). These findings are consistent with previous studies in which best fitting models are mostly distributed between models arising from both theoretical accounts (e.g.,
Most previous studies have limited their evaluation of estimation patterns to one theoretical account. This only allows the studies to examine whether or not performance is consistent with this account, but not if there is an alternative account that might be more consistent with performance patterns. Furthermore, most studies report the model with the lowest AIC as the ‘best’ model, ignoring the possibility that other models may explain the data equally well. Taken together our findings are hard to reconcile with the assumptions underlying the chosen theoretical accounts. For example, based on the representational shift account for 10% of participants, either model explained the data equally well. This seems incompatible with the idea that the shape of the estimation patterns can provide a direct reflection of children’s mental representation of magnitude. However, this concurrent fit should be considered in light of overlapping waves theory (
Turning to the proportional judgment account, we found that over a quarter of estimation patterns could be explained equally well by more than one power model. In this regard, the unbounded, onecycle and twocycle power models, do not seem to be reliable indices of the number and types of benchmarks used by children when performing the number line estimation task. This does not imply that children do not use a variety of strategies (or benchmarks) to solve the number line task, but that, at least for our sample, the models arising from this theoretical account do not seem to be very informative regarding children’s strategies. Furthermore, when conducting comparisons between theoretical accounts, we found that over a quarter of estimation patterns best explained by the unbounded power model had similar support for the linear model. Conversely, from the estimation patterns best explained by the linear model, over half show similar support for the unbounded power model. This concurrence between unbounded power and linear models puts into question whether the unbounded power model necessarily represents children who can only rely on the origin of the number line, as they are unlikely to produce linear estimates (though see
The second aim of our study was to determine if the models that best explain children’s estimation patterns were associated with their mathematical performance. When comparisons were conducted within theoretical accounts, children whose estimation patterns were best explained by the linear model had higher mathematical achievement than children whose estimation patterns were best explained by the logarithmic model. This remained the case when excluding children whose estimation patterns showed similar support for both models. This is consistent with the idea that a more linear representation of magnitude is associated with higher levels of concurrent mathematical competence (
Finally, when comparing between theoretical accounts, children whose estimation patterns were best explained by the logarithmic model showed lower mathematical achievement than children whose estimation patterns were best explained by all other models, although the difference with the onecycle power model became nonsignificant when excluding children whose estimation patterns showed similar support for more than one model. Our findings are partially consistent with those of
The finding that children whose estimation patterns were best explained by the unbounded power model outperformed those best explained by the logarithmic model is interesting for several reasons. If we consider both theoretical accounts, both models represent the least advanced stage (i.e., representation, strategy). Moreover, as mentioned prior, it has also been proposed that the unbounded power model under certain circumstances can resemble a logarithmic function (
Our findings, combined with those from other studies (e.g.,
This study is limited to the use of model fitting to distinguish between two theoretical accounts. However, this is not the only way in which these theoretical accounts can be studied or compared. In addition, we only consider the original models arising from the representational shift and proportional judgment accounts. Other models exist within each of these accounts: a mixed logarithmiclinear model (
Additionally, selecting one model that best explains children’s individual estimation patterns might be an artificial differentiation, as a significant percentage of children’s estimation patterns also showed substantial support for at least one other model. We attempted to address this by running the same analyses after excluding these cases. However, this leads to a loss of statistical power, which requires caution for the interpretation of these findings.
A final limitation of this study is that the target numbers used for the number line task were not equally distributed across the number range, showing oversampling of the lower half of the number range. This might have favoured some models (i.e., logarithmic) over others (i.e., cyclical power models).
The current study showed that most children’s estimation patterns can be explained by one of the mathematical functions proposed by the representational shift and proportional judgment accounts, and that which model best explains children’s estimation patterns is associated with their mathematical achievement. However, this study highlighted the obstacles and limitations involved in using estimation patterns as an index of performance, especially given that for a high percentage of children’s estimation patterns at least one other model explained the data equally well as the best model.
Formulas of models to be fitted to the number line data based on
Initial parameter values (slope = 1; intercept = 0)
Initial parameter values (slope = 1; intercept = 0)
Initial parameter values (β = 1; range = 10)
Initial parameter values (β = 1; range = 100)

Otherwise: 


Initial parameter values (β = 1; range = 100)
This manuscript was supported by an International Macquarie University Research Excellence Scholarship "iMQRES" Allocation No. 2020005 to the first author. The original study was funded by the Singapore Ministry of Education (MOE) under the Education Research Funding Programme (OER 16/12RB) and administered by the National Institute of Education (NIE), Nanyang Technological University, Singapore.
The dataset for the project “Number Line Estimation Patterns and Their Relationship with Mathematical Performance” is stored at the Macquarie University Research Data Repository (
Access to the data will be granted upon request. If you are interested in accessing the dataset please contact Rebecca Bull (
The Supplementary Materials contain the following items (for access see
The code used to conduct the analyses
Descriptive statistics and additional analyses for the study
The authors have declared that no competing interests exist.
The authors have no additional (i.e., nonfinancial) support to report.