Over the past two decades, the number line estimation (NLE) task has arguably become one of the most well-known methods of measuring numerical competence in adults and children alike. A simple Google Scholar search using the term “Number Line Estimation” yielded over 3300 hits, and the seminal NLE papers have each been cited over 1100 times (Booth & Siegler, 2006, 1102 citations; Siegler & Booth, 2004, 1566 citations). Though each of these citations certainly does not indicate that the task itself was used in the study, it is clear, at a minimum, that results from NLE tasks have been widely used to shape our understanding of numerical cognition.

The core NLE task involves presenting a series of blank number lines with the endpoints of the line marked by specific numbers; for each item, a third number that is numerically somewhere between the two endpoint numbers is given, and the participant’s task is to estimate where that number would fall on the number line (e.g., Siegler & Booth, 2004). For example, on a 0-100 number line task, the number “0” would be marked at the left endpoint and “100” marked at the right endpoint, and an individual item might ask participants to demonstrate where they believe individual numbers (e.g., 23, 47, 2, 96, 14, etc.) belong on the line. Scoring of this task typically involves either computing a participant’s estimation accuracy across all of their items

(Siegler & Booth, 2004) or examining a participants’ pattern of estimates across all of their items by fitting different types of models to their estimates (e.g., linear, logarithmic, exponential (Booth & Siegler, 2006); power models (Barth & Paladino, 2011; Ruiz et al., 2023); mixed log linear models (Qin et al., 2024)).

The use of the NLE task has been widespread across both contexts and cultures. For example, the task has been administered in laboratory or one-on-one settings (Praet & Desoete, 2014; Siegler & Booth, 2004; Wall et al., 2016), online using platforms such as MTurk (Landy et al., 2017) and Qualtrics (Fitzsimmons & Thompson, 2022), in an FMRI scanner (Vogel et al., 2013), and in both K-12 (Barbieri et al., 2021; Jung et al., 2020) and college classrooms (Steinke, 2017). NLE tasks have also been employed in studies across the globe, including in Israel (Ashkenazi & Cohen, 2023), Singapore (Ruiz et al., 2024), Chile (Xu et al., 2023), Luxembourg (Nuraydin et al., 2023), Zambia (Sudo et al., 2022), China (Li et al., 2024), Germany (Jung et al., 2020), Turkey (Sarı & Olkun, 2021), and with individuals from Indigenous tribes (Dehaene et al., 2008).

The characteristics of the number line task have varied widely across studies. For example, many studies use a paper-and-pencil administration (e.g., Booth & Siegler, 2006; Chan & Mazzocco, 2024), while others use computerized versions of the task (e.g., Booth & Siegler, 2008; Fazio et al., 2014; Gunderson & Hildebrand, 2021). Most studies use the number-to-position version of the task described above, though others have given a mark on a number line and asked the participant to estimate what number goes there (Position-to-Number, e.g., Peeters et al., 2017; Siegler & Booth, 2004). And while most studies present each item individually as described above (i.e., participants only see one number line at a time), others have provided a page containing several number lines that are visible at once (Barbieri et al., 2023; Young & Booth, 2015) or have asked participants to place multiple numbers on the same number line (Siegler & Booth, 2004; Steinke, 2017).

Perhaps the greatest variability across studies using number line estimation tasks is the numerical scale and the type of numbers to be placed. While many studies have continued to employ the 0 – 100 or 0 – 1000 whole-number scales introduced in the original studies (e.g., Hoard et al., 2008; Praet & Desoete, 2014; Sarı & Olkun, 2021; Sullivan et al., 2011), others have used much smaller (e.g., 0 – 10, Dietrich et al., 2016; 0 – 20, Cornu et al., 2017; 0 – 8, Yu et al., 2022) or larger scales (e.g., 0 – 10,000, Jung et al., 2020; 0 – 1,000,000, Slusser et al., 2013), irregular scales such as 0 – 62,571 (Booth et al., 2014) or 1,000 – 1,000,000,000 (Landy et al., 2017), and scales involving negative numbers (-1000 – 0, Brez et al., 2016; -1,000 – 10,000, Young & Booth, 2015); some have even explored the use of unbounded number lines (e.g., Link et al., 2014; Reinert et al., 2019). The number line task has also been used to measure participants’ understanding of different types of numbers, including fractions (e.g., Booth & Newton, 2012; Namkung & Fuchs, 2016), decimals (DeWolf et al., 2015; Schneider et al., 2009), and percentages (Schiller et al., 2024).

Despite the wide variability in the particulars of NLE task administration, there is a large consensus that individuals’ NLE performance is related to (and often predictive of) their general mathematical competence. For example, NLE has been shown to predict arithmetic performance (e.g., Dietrich et al., 2016; Gunderson & Hildebrand, 2021), problem-solving skills (e.g., Zhu et al., 2017), and mathematical reasoning (e.g., Ruiz et al., 2024). In a meta-analysis of 41 papers, Schneider and colleagues (2018) reported strong associations between NLE and counting, computation skills, and school mathematics achievement. Ellis and colleagues (2021) replicated these overall findings, emphasizing the importance of performance on the NLE task, but found performance to be particularly predictive of mathematical competence in younger children. NLE performance even correlates with brain activation while solving arithmetic problems (Berteletti et al., 2015), and poor NLE performance has been linked to mathematical learning disabilities (Geary et al., 2012). In a recent systematic review of 33 manuscripts that examine predictors of algebra performance, NLE tasks were one of the most studied student-level factors, with fraction NLE tasks consistently demonstrating their predictive utility, but whole number line tasks were also often used (Silla et al., under review).

The prevalence of its employment and the usefulness of the task for predicting more complex mathematical competencies make it likely that the NLE will continue to be an essential measure in psychology and education research. Given its prominence as a predictor of school mathematics achievement and the fact that NLE data collection often takes place in schools during precious classroom time—either with full-class administration during class or by pulling individual students out of class—it is necessary to ensure that the task is as efficient as possible. However, the number of target items has varied quite a bit from study to study, ranging from a minimum of 6 items to 44 or more items (Schneider et al., 2018). Additionally, although studies with older children and adolescents often use paper-based tasks that can be administered reasonably quickly in whole-group settings (e.g., Barbieri et al., 2021), this is not the case for younger children. In studies with young children, number line tasks are often administered one-on-one with an experimenter using an iPad (e.g., Geary et al., 2008). Given that the 0 – 100 scale, in particular, is among the most widely used scales (696 Google Scholar hits, compared with 465 for 0 – 1000 and 542 for 0 – 1) and is useful for young children (who likely do not yet have fully developed attentional control; Tremolada et al., 2019), we aim to determine how few target items could be used to get an accurate measure of children’s numerical magnitude understanding to ensure responsible use of teachers’ and students’ time.

Over the past two decades, the number line estimation task has been widely used as a measure of children’s numerical competence (e.g., Li et al., 2024; Ruiz et al., 2023). However, the number of trials given within the task has varied considerably, involving up to 44 or even more trials given to individual students. The present study used data from two previously published studies involving number line estimation on a 0 – 100 scale (Booth & Siegler, 2006; Siegler & Booth, 2004) to determine potential reduced sets of trials that could be used to produce similar results while taking up less time in data collection.

The current findings demonstrate that our reduced whole number line (0 – 100) scale successfully replicates the patterns of developmental findings from both prior foundational work (Booth & Siegler, 2006; Siegler & Booth, 2004) and recent assertions (Qin et al., 2024). However, we find inconsistencies specifically in relation to patterns of results for second graders, discussed further below. That is, the reduced scale (15 target items) has comparable psychometric properties compared to the original scale (23 target items). Further, as in seminal pieces by Siegler and Booth (Booth & Siegler, 2006; Siegler & Booth, 2004), the logarithmic-to-linear shift is supported with the reduced scale, with kindergarteners’ estimates better fit by a logarithmic than a linear function and third graders’ estimates better fit by a linear than a logarithmic function. Analysis of individual student distributions also supports this claim. The previous work demonstrated developmental change in how children represent numerical magnitudes on a mental number line. They found that young children (around 5) hold a logarithmic representation of numbers where smaller numbers (e.g., 1, 2, 3) are spaced farther apart on their mental number line, while larger numbers (e.g., 30, 40, 50) are compressed closer together. That is, they overestimate the size of smaller numbers relative to larger numbers. This reflects a non-linear representation of numerical magnitude in which equal intervals represent exponential rather than additive changes. Children’s numerical representations become more linear as they develop (~6 – 8 years old). That is, they shift towards holding a linear mental number line in which equal intervals represent the same magnitude as demonstrated by their placement of target numbers spaced equally and appropriately on a number line; a similar finding from Qin and colleagues (2024) regarding increased linearity in student estimates with age is also replicated with our reduced scale.

As in the prior work, first-graders were no more linear in their estimates than logarithmic. However, there were some slight deviations in the findings of the current work and prior work with second graders’ estimates. In the seminal work, second graders’ estimates were significantly more linear than logarithmic. In the current analysis with the reduced scale, the second graders’ estimates did not significantly differ in terms of their linearity or logarithmic nature. An inspection of the R² values shows that second graders’ estimates seemed to be more linear (Rlin2 = .95) than logarithmic (Rlog2 = .90), but these differences are not significantly different. This discrepancy does not appear to be an issue with power, as the analysis included second graders across three original studies. However, it is possible that the sensitivity of the reduced scale for detecting pattern shifts in number line estimation varies by grade. That is, our reduced scale may be most accurate with respect to linear and logarithmic patterns for the grades chronologically furthest from the developmental transition (kindergartners and third graders) and the grades around the transition (e.g., 2nd grade) may need a more expanded measure to more precisely and fully capture the nuanced change from logarithmic to linear representations.

Next we consider the finding that the best option of the reduced 0 – 100 scales comprised 15 target items, with 12 of those items in the first half of the range of the scale. That is, the scale included the following targets: 3, 4, 6, 8, 12, 17, 21, 24, 25, 29, 33, 39, 52, 57, and 81. This is likely due to the composition of the sample in relation to their developmental stage of numerical magnitude understanding. That is, the majority of the sample held a logarithmic representation of numerical magnitude, and so more target items in the first half of the scale is likely better able to capture the shape of the curve in that part of the scale. As mentioned previously, it is likely that the 0 – 100 scale employed in the present study is not the optimal way to capture the numerical competence of the older students in this study (second and third grade students), and that their performance on a 0 – 1000 scale may be more useful. However, when it is desirable to assess the 0 – 100 number line estimation skills of these older students who have more accurate or linear representations of the scale, researchers may need even fewer target items to capture their magnitude understanding in this range. Future work with older samples may wish to investigate further reductions. Following this logic, it is possible that reduction procedures for larger scales (e.g., 0 – 1000, 0 – 10,000) may follow the same patterns with older children. For example, the logarithmic to linear shift that occurs within the 0 – 1,000 scale happens at the age of 8 – 10 (Booth & Siegler, 2006; Siegler & Booth, 2004). The logarithmic to linear shift that occurs within the 0 – 10,000 scale happens around the age of 10 – 12 (Siegler et al., 2011). It is possible that reduced scales for these ranges would rely more heavily on targets in the 0 – 500 range (for the 0 – 1000 scale) and in the 0 – 5000 range (for the 0 – 10000 scale) if studying 8 – 10 year olds and 10 – 12 year olds, respectively. Alternatively, it is possible that there is already bias reflected in the design of the original scale due to the intentional oversampling of the first half of the scale. That is, because there are more target items in the first half of the number line, it may be easier to replicate a logarithmic curve found with a reduced set of target items sampled from this same distribution. To determine whether this possibility is a better explanation for the perceived accuracy of the reduced scale, a set of targets that are equally representative of the entirety of the scale may be necessary. Though this is a possibility, and more research with a wider variety of targets may serve to support or challenge prior work, we consider the findings of Gashaj et al. (2016), who used an approximately equal number of target items above and below 50 on a 0 – 100 scale and found that kindergarteners’ numerical magnitude on a 0 – 100 number line was more logarithmic than linear. Still, further research using a more even distribution of target items across scales at different grades or ages may clarify whether potential biases exist in the original scale.

We also employed rigorous psychometric methods to evaluate and refine the scale, addressing limitations in the original scale's development. We utilized the graded response model (GRM), which is specifically designed for analyzing ordered polytomous data such as Likert-style items (Samejima, 2016). This approach allowed us to examine item-level properties and overall scale performance with greater precision than classical test theory methods alone.

Our analyses also included exploratory factor analysis (EFA) with scree plot examination, providing insights into the scale's dimensionality (Cattell, 1966). We also conducted parallel analysis, which is considered one of the most accurate methods for determining factor retention (Horn, 1965). The use of multiple criteria for assessing dimensionality, including eigenvalue ratios and explained variance, strengthened our conclusions about the scale's structure. We also used both Cronbach's alpha and McDonald's omega to check for reliability, which gave us a more complete picture of internal consistency (Dunn et al., 2014). These methodological strengths, combined with our systematic approach to scale reduction, provide a solid foundation for confidence in our findings. The refined scale not only maintains the psychometric integrity of the original but also potentially improves its unidimensionality and efficiency, addressing the limitations of the original scale's atheoretical development.

Despite these strengths, the current study has the following limitations that can be addressed in future research. First, we note some methodological limitations. Our method for factor analysis was exploratory in nature. In future studies, confirmatory factor analysis (CFA) could be used to confirm the structure of the scale even more. It would be possible to test the hypothesized factor structure that came out of our exploratory analyses more thoroughly with CFA. This could make the construct validity of the refined scale stronger. Moreover, the assumptions of unidimensionality and local independence, which are fundamental to the graded response model used in this study, warrant further examination. Future research should investigate how well these assumptions hold across different contexts and populations, as violations of these assumptions could impact the accuracy of item parameter estimates and overall scale performance.

Further, while we believe that the reduced scale sufficiently captures variance in students’ number line estimation skills, it is, of course, not certain that they would have made the same estimations on the target items if they had only been given those items. That is, it is possible that the students’ estimations on the retained target items were influenced by their experience considering the other, non-retained target items. Future work would need to directly compare children’s estimations when given only the reduced scale vs. the full scale to determine whether this is of consequence on estimates. Additionally, while our sample (N = 234) was drawn from three studies by the same group—which may limit external generalizability—we note that the present work primarily targets scale reduction and item functioning (discrimination, thresholds) rather than population-level generalization; nonetheless, future validation in independent, diverse samples is warranted. We also note here that we had far fewer third graders in our analytic sample than earlier grades. Future work in this area should attempt to recruit a more even distribution of participants per grade.

We also consider the age of the data itself. That is, the data used in the current study originates from studies conducted as early as 2004. Some of the trends we see in the logarithmic to linear shift may happen at a different age or grade level, as schools have taken steps to focus more on inclusion of number line activities in their standard mathematical practices in second grade (e.g., National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA & CCSSO], 2010). More current number line work with the 0 – 100 scale has replicated the same pattern but has argued that the pattern is more reflective of skill at mapping numbers to space (e.g., Cohen & Quinlan, 2018; Haman & Patro, 2022; Sasanguie et al., 2016). Indeed, most recent work using the 0 – 100 NLE task emphasizes results regarding accuracy (e.g., PAE) rather than linearity (e.g., Ruiz et al., 2024); or evaluate linearity without consideration of the logarithmic function (e.g., Li et al., 2024). Thus, the usefulness of the NLE task for comparing patterns of estimation may be less important now. Researchers who maintain the goal of investigating transitions in estimation patterns may need to confirm that the reduced scale we derived in the current study yields the same or similar findings with children currently in third grade or below.

It is also important to note that although discrimination might differ by grade, our per-grade subsamples are too small to support stable multi-group GRM estimation; small group sizes in polytomous IRT can lead to imprecise discrimination and threshold estimates, so we report a single-group calibration and reserve formal grade-level invariance tests for future work with larger, grade-balanced samples.

Finally, it is worth noting that the reduced scale is comprised almost entirely of items in the lower half of the numerical scale. When the original task was designed, numbers were oversampled from the lower third of the scale in order to capture a potential logarithmic-to-linear shift (e.g., Siegler & Booth, 2004). Oversampling in the case of the 0 – 100 number line meant including 10 items between 0 – 30 and 14 items between 31 – 100. In contrast, our reduced scale retained 8 items between 0 – 30 and only 5 items between 31 – 100. Thus, oversampling of the low end of the scale is indeed important and may in fact be even more critical than originally anticipated. That is, accurately capturing variance in young students’ number line estimations appears to require an even greater proportion of numbers at the low end and many fewer items in the top two-thirds of the scale.