Introduction

The number-line task is one of the most commonly used tasks to study numerical cognition. In a conventional task, participants estimate the location of a symbolic number (e.g., Arabic numeral of 8) along a line flanked by two numbers (e.g., 0 and 50). Performance on number-line tasks with symbolic numbers has been viewed as depicting the approximate numerical representations that are also shared with non-symbolic numbers (e.g., number of dots; Dehaene et al., 2008; Siegler & Opfer, 2003). In this view, the meanings of symbolic numbers are learned by mapping to their non-symbolic referents (e.g., mapping numeral 8 to 8 dots), forming an association between symbolic and non-symbolic number representations (Dehaene, 2011). Therefore, performance on symbolic number-line tasks is thought to predict how non-symbolic numbers would be estimated on number lines.

Against the shared-representation account, however, some suggest independent representations for symbolic and non-symbolic numbers (Carey, 2004; Carey & Barner, 2019; Lyons et al., 2012; Rips et al., 2008). According to this perspective, symbolic-number learning does not require mapping to the cardinality of a set from early development (Carey & Barner, 2019; Rips et al., 2008). Alternatively, symbolic numbers may be mapped to their non-symbolic referents early but become “estranged” from them late in development (Lyons et al., 2012). These accounts would not predict major findings in symbolic number-line studies to be observed in number-line tasks with non-symbolic numbers. Despite its theoretical importance, whether the characteristic features of symbolic number estimation are also evident in non-symbolic number estimation has not been fully investigated.

One of the characteristic features of symbolic number-line estimation is the “log-to-linear shift” (Siegler et al., 2009). This shift refers to the observation that younger and/or less educated populations produce logarithmic estimates in situations where older and/or more educated populations produce more linear estimates (Berteletti et al., 2010; Opfer & Siegler, 2007; Siegler & Booth, 2004; Siegler & Opfer, 2003; Thompson & Opfer, 2008). A prominent interpretation of the log-to-linear shift is that it reflects adaptive changes in numerical representations. The default representation, shared across human and non-human species, is logarithmically scaled. Here, the difference between 1 and 10 is subjectively larger than the difference between 101 and 110. With age and schooling, however, the formal properties of the decimal system are gradually learned and can be applied to new situations where the difference between 1 and 10 is equal to the difference between 101 and 110 (Dehaene, 2011). In this account, non-symbolic estimation would also be expected to show log-to-linear shifts over development (Kim & Opfer, 2018; Yuan et al., 2020).

A second feature of symbolic number-line estimation is that log and linear representations coexist (Siegler & Opfer, 2003; Thompson & Opfer, 2010). The coexistence of different representations is typically explored by manipulating the number range in number-line tasks (e.g., estimating 20 on a 0–100 vs. 200 on a 0–1,000 number line). A common finding is that people are more likely to rely on a log representation for a larger number range (Opfer et al., 2019; Siegler & Booth, 2004; Siegler & Opfer, 2003). If the same representations were used for non-symbolic numbers, non-symbolic number estimates would also be more logarithmic when to-be-estimated numbers are large (e.g., 200 dots) rather than small (e.g., 20 dots). This question, however, has not been systematically explored.

A third feature of symbolic number-line estimation is that the linearity of estimates predicts proficiency with numbers in other contexts (Booth & Siegler, 2008; Siegler & Booth, 2004). The linearity of symbolic estimates predicts number memory (Opfer et al., 2019), counting (Östergren & Träff, 2013), math learning (Booth & Siegler, 2008; Siegler & Ramani, 2008), math scores (Fazio et al., 2014; Kim & Opfer, 2017), dyscalculia (Geary et al., 2008), and a genetic disorder, like Williams syndrome (Opfer & Martens, 2012). Whether the linearity of non-symbolic number estimates is associated with non-symbolic math is still an open question.

Despite previous attempts to address these questions using symbolic and non-symbolic number-line tasks, the findings in the literature are not consistent. For example, whereas symbolic and non-symbolic estimation appears to share the log-to-linear developmental trajectory in some studies (Kim & Opfer, 2018; Sasanguie et al., 2012; Sella et al., 2015), other studies report that non-symbolic estimation develops differently from symbolic estimation (Kolkman et al., 2013; Sasanguie et al., 2016). Table 1 summarizes previous research examining the development of symbolic and non-symbolic number estimation. As shown in the table, research parameters, including participants’ ages, number scales, and applied models, were different across studies, which could have caused the discrepancy in the results. This state of affairs calls for a rigorous examination of the relations between the two types of number estimation.

Table 1 Summary of number-line studies with symbolic and non-symbolic number
Full size table

The purpose of the present study, using a non-symbolic version of the number-line task, is to systematically investigate whether the preceding three features of symbolic number estimation also characterize non-symbolic number estimation. Exploring these features in number-line tasks is inherently challenging because of the potential range dependency of number-line estimates. If the number range tested is too small or too large, developmental differences and the association between estimates and math skills may appear absent (Clarke et al., 2018). Determining appropriate number range can be more challenging in a non-symbolic number-line task. For symbolic number, there is a certain range of numbers that children more frequently encounter in a particular grade in education (e.g., one-digit numbers in kindergarten, two- and three-digit numbers in third grade). Symbolic number-line tasks with familiar number ranges might show individual variations that correlate with symbolic math competence. However, it is difficult to specify the range of non-symbolic numbers children come across in everyday life. Non-symbolic numbers vary greatly for every individual in size, from a relatively small number (e.g., number of cookies in a jar) to a very large number (e.g., number of people in a large stadium).

The number-line range is determined by the upper-bound number, which is often determined by an educated guess of investigators. When it is unclear which upper-bound number should be used, a simple but costly approach would be to examine estimates on multiple number-line tasks with different upper-bound numbers. The same approach has been used to explore the coexistence of log and linear representations, but with only a few different number ranges (e.g., 0–10 and 0–100; Berteletti et al., 2010). To fully examine the changes in estimates across number ranges, it would be desirable to use as many upper-bound numbers as possible. However, this approach would require many more trials than the typical task with a single, fixed scale (total trials = number of trials in each range × number of upper bounds).

The present study aimed to address the design bottleneck in number-line tasks by applying a novel multi-scale non-symbolic number-line task (Fig. 1) that can be as brief as a fixed-scale number-line task but without sacrificing data quality. This highly efficient task is developed based on an algorithm that combines optimal experimental design in statistics (Atkinson & Donev, 1992) with active learning in machine learning (Settles, 2012). It provides a computational means to characterize non-symbolic number estimation over a much wider range of number scales for an individual, thereby quantifying developmental changes in estimates, as efficiently and accurately as possible. Specifically, in an experiment using a Bayesian active learning algorithm, dubbed Gaussian process active learning (GPAL), that our lab has developed (Chang et al., 2021), we delineate number-line estimation functions across four or more different number ranges, and compare the functions among individuals.

Fig. 1
figure 1

Non-symbolic number-line task. Note. A given number (in the red box) is estimated on a line with zero and an upper-bound number (in the blue box)

Full size image

We hypothesize that if non-symbolic numbers share representations with symbolic numbers, non-symbolic number-line estimation would exhibit the three features of symbolic number-line estimation described above: Non-symbolic estimates would change from log to linear with age, become more logarithmic in the same individuals as upper bounds increase, and show associations with non-symbolic math proficiency measured in a non-estimation context (approximate addition).

Method

Experiment

The present study applied GPAL to the non-symbolic version of the number-line task in Fig. 1 in order to infer psychophysical functions underlying non-symbolic number estimates in the two-dimensional design space. That is, the given number (design variable 1) and the upper-bound number (design variable 2; scale) were varied from one trial to another in an adaptive and optimal manner prescribed by the GPAL algorithm. This is unlike typical number-line tasks that manipulate only the given number during the task.

Participants

Seventy-three children aged between 5 and 13 years (47 males, Mage: 8.69, SDage: 2.09) were tested at a local science museum. 165 adults were recruited from a local university (n = 31 (16 males), Mage: 19.44, SDage: 0.99) and from Amazon Mechanical Turk (MTurk; n = 134 (69 males), Mage: 39.25, SDage: 12.91). All participants were recruited from the USA. Experiments in the current study were approved by the Institutional Review Board (IRB) in the local university.

Bayesian power analysis

We determined the number of participants based on a Bayesian power analysis (Kruschke, 2010). The effect of interest was whether the logarithmicity component (λ) in a mixed log-linear model (MLLM; described in a later section) changes with the upper-bound number across age groups. Statistical power was calculated as the estimated probability that new data replicate the difference in the values of μλ (a hyperparameter that determines the prior mean of λ) in a 0–50 number line (μλ,50) and a 0–100 number line (μλ,100), which was found in our pilot experiment (0.074 effect size). The effect was considered replicated if Bayesian 99% highest posterior density interval (HPDI) of μλ,100–μλ,50 did not include zero. We set a desired level of statistical power at 0.8. Estimated power was 0.8 for 73 participants (children) and 0.88 for 165 participants (adults). When 95% HPDIs were used instead of 99% HPDIs, the power level was 1 for both children and adults.

Stimuli and procedure

Each participant completed a non-symbolic number-line task (Fig. 1) and a non-symbolic math proficiency task – i.e., approximate addition. The number-line task was always given first. In this task, a group of dots was presented above a line for 2,000 ms every trial. Participants were asked to decide the location of the given number of dots on a line. The response was made by mouse-clicking the assumed location of the number on the line. The number of dots to estimate was chosen by GPAL every trial, among the integers between 5 and an upper bound. Small numbers (0–4) that are subitizable (Feigenson et al., 2004) were excluded from the design.

Besides given numbers, upper-bound numbers also varied trial by trial. The number of dots at the right end of a line (i.e., the upper bound) in Fig. 1 was chosen by GPAL every trial, whereas the number of dots at the left end (i.e., the lower bound) was always zero. The task for children had fewer total trials and fewer possible upper bounds than those for adults, to make the task suitable for children. The possible upper-bound values were 50, 100, 200, and 400 for children, and 50, 100, 150, 200, 250, 300, 350, 400, 450, and 500 for adults. The task consisted of 50 trials for children, and 90 trials for adults. The GP posterior was reset after the first half (block) of the trials, in order to test the reliability of GPAL by comparing the function estimates in the first block (i.e., test) and the second block (i.e., retest), each with a half of the trials (25 trials for children, 45 trials for adults). After each trial, participants had to press the spacebar to proceed to the next trial. There were five practice trials before the test block. We controlled for the size and the cumulative area of the dots in the given-number box. On half of the trials, the size of the dots was held the same in both the given-number and the upper-bound boxes, with the cumulative area increasing with the given number. On the other half, the cumulative area of the dots in the given-number box was equal to that in the upper-bound box, with the dot size decreasing with the given number. Also, the dots of given- and upper-bound numbers were randomly spread in the boxes, such that the convex hull of the dots would not vary with numbers. According to Yuan et al. (2020), estimation of non-symbolic numbers is correlated with symbolic number estimation when perceptual cues, such as the convex hull, are not readily available. The random distribution of dots, therefore, was expected to elicit the use of representations and skills shared with symbolic numbers.

To test numerical performance outside estimation contexts, participants were also given a non-symbolic addition task upon completion of the number-line task. All adult participants completed both tasks. For children, we asked if they were willing to do another task when they completed the number-line task, to prevent loss of data quality due to fatigue. Fifty-one children chose to complete the approximate addition task. In this task, participants viewed two arrays of dots going into a box, consecutively for 1,500 ms per array, on the left side of the screen. Then, there was another array of dots presented on the right side of the screen for 1,500 ms. Participants were instructed to choose which side had more dots using the f or j key press. Each array had from 5 to 50 dots, and the sum of two arrays of dots in the gray box was less than 51. The array size was drawn from five ratio bins (1.05, 1.14, 1.2, 1.5, 2.0). There were 30 trials for children (six trials for each ratio bin), and 50 trials (ten trials for each ratio bin) for adults. There were two practice trials without feedback.

In what follows, we provide a brief overview of GPAL with which the experiment was conducted.

Gaussian Process Active Learning (GPAL)

GPAL is an algorithm-based experimental method for adaptively optimizing experimental designs in order to infer an unknown function with the fewest possible number of observations (Chang et al., 2021). GPAL comprises the following three iterative steps that are performed on each trial: (1) Design optimization step in which the optimal design is identified based on the current state of knowledge about the model being inferred; (2) Experiment step in which stimuli are presented with the optimized design configuration and a response is observed; and (3) Model inference step in which the observed response is used to infer an updated model, which in turn becomes a new model for the next iteration.

GPAL has been developed as a nonparametric extension of a parametric Bayesian active learning algorithm for optimal experimental design, dubbed Adaptive Design Optimization (ADO; Cavagnaro et al., 2010). A key difference between GPAL and ADO is in the model inference step. ADO requires the assumption of a parameterized model that specifies the functional form that generates responses in an experiment. For example, in the number-line task, the model might assume that number estimates follow log, linear, or cyclic power functions. In contrast, GPAL is “model-free” in that it does not make a priori assumptions about the possible model form, and instead, directly infers the form based on observed responses.

To infer the underlying functional form, GPAL utilizes a nonparametric Bayesian method known as a Gaussian process (GP) (e.g., Cox et al., 2012; Griffiths et al., 2009; Rasmussen & Williams, 2006; Schulz et al., 2018). Being nonparametric, GP is capable of modeling a virtually limitless range of functional forms without being subject to the constraints imposed by a parametric model family as in ADO. Formally, a GP is defined as a stochastic random process that forms a Gaussian distribution of functions over the function space (Rasmussen & Williams, 2006):

$$f(x)\sim GP\left(m(x),k\left(x,{x}^{\prime}\right)\right),$$
(1)

where f (x) is the underlying function to be inferred from observed data, and x and x′ are two different points in the design space. In the above equation, the mean function m and the kernel function k, the latter of which governs the smoothness of the function f (x), are defined as statistical expectations with respect to the distribution of functions:

$${\displaystyle \begin{array}{c}m(x)=\mathbbm{E}\left[f(x)\right]\\ {}k\left(x,x^{\prime}\right)=\mathbbm{E}\left[\left(f(x)-m(x)\right)\left(f\left({x}^{\prime}\right)-m\left({x}^{\prime}\right)\right)\right].\end{array}}$$
(2)

In the present study, we use the squared exponential kernel function.

Once the model is inferred by GP, it is then used in the design optimization step to identify the optimal design. This optimization process is known as active learning in machine learning (Settles, 2012). The active learning in GPAL relies upon the (Bayesian updated) distribution of the function f (x) in Eq. (1). In particular, we implement the uncertainty sampling scheme of Lewis and Catlett (1994) in which GPAL queries the design point x with the highest variance of f (x). As such, an optimal design is the one that leads to the largest reduction in uncertainty about the unknown function f (x). Shown in Fig. 2 is an example of the model inference and design optimization steps in GPAL.

Fig. 2
figure 2

An illustrated scheme of the Gaussian Process Active Learning (GPAL) algorithm. Note. The left panel summarizes the current state of knowledge about the underlying model after five observations. In this graph, the black dots are observed data, and the black curve is the inferred model as the GP mean function. The blue area indicates the two standard deviations from the GP mean and represents the uncertainty about the underlying model. The grey curves are a few functions sampled from the GP. GPAL selects the optimal design for the next trial as the point with the largest variance, indicated by the dashed vertical black line. Shown on the right panel is a new observation (red dot) made with the optimized design, which is then used to update the GP. The iterative process repeats itself for the next trial. For additional technical details, readers are directed to Chang et al. (2021)

Full size image

Models of numerical estimation

We used formal models of numerical estimation in literature to assess the characteristics of GPAL-inferred functions (i.e., GP-estimated posterior mean function). The three features of number-line estimates could be interpreted in a log-linear framework, but GP-estimated functions per se do not provide a quantifiable measure of logarithmicity. To quantify the degree of log compression in GPAL-inferred functions, we used a mixed log-linear model (MLLM; Anobile et al., 2012; Opfer et al., 2016; see below for details).

However, interpreting the results in a log-linear framework might not be justifiable if GPAL produces functions better explained by alternative models, such as cyclic power models (CPMs; Hollands & Dyre, 2000). The CPMs are derived from the proportional reasoning account, assuming log or linear patterns in estimates do not reflect numerical representations, but proportional reasoning skills required for number-line estimation (Barth & Paladino, 2011; Cohen & Sarnecka, 2014; Rouder & Geary, 2014; Slusser et al., 2013). Both the MLLM and CPMs can describe log and linear functions, but some versions of the CPM also predict distinctive patterns that the MLLM would not describe. Given that GPAL is free of assumptions about the shapes of underlying functions, whether logarithmic or cyclic, comparing the two models with the functions inferred by GPAL will allow us to determine the functional form of numerical estimation in an unbiased manner without favoring one model or another (see Appendix for GPAL simulations).Footnote 1

Mixed log-linear model (MLLM)

The MLLM is a commonly used model to describe number-line estimates. This model is defined as:

$$y=a\left(\lambda \frac{U}{\mathit{\ln}(U)}\mathit{\ln}(x)+\left(1-\lambda \right)x\right)+b,$$
(3)

where y is the estimate of a given number x in a number line with the upper bound of U. The parameter λ, as a measure of logarithmicity, is the relative weight of the logarithmic function to the linear function in estimation. The estimate y becomes completely linear with λ = 0, and completely logarithmic with λ = 1. a and b are two scaling parameters.

The model parameters were estimated by fitting a hierarchical Bayesian model (Lee & Wagenmakers, 2014) using JAGS in MATLAB (Plummer, 2003; Steyvers, 2011). The MCMC sampling was iterated 150,000 times, with the first 50,000 samples being discarded as burn-in samples. The model parameters for participant i were defined as follows: λi ∼ Beta (μληλ, (1−μλ)ηλ), ai ∼ Beta (1, 1), and bi ∼ Uniform (0, U), where μλ and ηλ are hyperparameters following μλ ∼ Beta (1, 1), and ηλ ∼ Gamma (1, 20).

Cyclic power model (CPM)

Two versions of the CPM, one- and two-cycle power models (1CPM and 2CPM), were compared with the MLLM. Another version of the CPM, zero-cycle power model (0CPM), was not tested because its prediction is qualitatively similar to that of MLLM, likely making the models indistinguishable. The MLLM would not predict the power function with the value of the exponent (i.e., β) larger than 1, but such large β values in the 0CPM were rarely reported (Slusser et al., 2013; Spence, 1990). The 1CPM assumes that there are two reference points, at the two ends of the number line. The 2CPM assumes the midpoint as an additional reference point in addition to the two ends. We used the models provided by Hollands and Dyre (2000), which extended the model in Spence (1990). The model equations are as follows:

$$1 CPM:y=U\left(\frac{x^{\beta }}{x^{\beta }+{\left(U-x\right)}^{\beta }}\right)$$
(4)
$$2 CPM:y=\left\{\begin{array}{c}\left(U/2\right)\left(\frac{x^{\beta }}{x^{\beta }+{\left(U/2-x\right)}^{\beta }}\right),\mathrm{if}\ x\le U/2.\\ {}U/2+\left(U/2\right)\left(\lambda \frac{x^{\beta }}{x^{\beta }+{\left(U/2-x\right)}^{\beta }}\right),\mathrm{if}\ x>U/2.\end{array}\right.$$
(5)

In the equations above, y is the number estimate, x is the given number, U is the upper bound of the number line, and β is the exponent of the power function. CPMs predict linear functions with β = 1, and predict more cyclic biases as β deviates more from 1. For the 2CPM, we assumed the use of a reference point in the middle (U/2). The models were fitted using the same procedure as the MLLM. The model parameters in the hierarchical Bayesian model for participant i were defined as follows:

  • βi ∼ Gamma (μβ /ηβ, ηβ), where μβ and ηβ are hyperparameters following μβ ∼ Gamma (1, 1), and ηβ ∼ Gamma (2, 1).

Model comparison analysis

The MLLM and CPMs were fitted to the Gaussian process (GP) posterior mean functions estimated by the GPAL algorithm from the experimental data. Model fits were compared by the deviance information criterion (DIC; Spiegelhalter et al., 2002), which is a generalization of the Akaike information criterion (AIC; Akaike, 1998) for hierarchical Bayesian models. The DIC evaluates model fit while being penalized by model complexity (e.g., number of parameters). The model comparison based on the DIC supports the model with the smallest DIC value. The model comparison results are described and discussed in the following sections.

Results and discussion

Studies that use number-line tasks typically use the medians of the raw data to recover an underlying function. This process was not necessary in the present study because GP automatically interpolated raw data to form an estimate of the underlying function. Therefore, instead of median points, we used the posterior mean of the estimated GP functions (see Appendix for outlier detection method). The GP posterior means averaged over participants are shown in Fig. 3.

Fig. 3
figure 3

GP-estimated functions for children and adults. Note. The large plots on the left side are the mean functions collapsed across test and retest blocks. The small plots on the right side show the functions from the test and retest blocks separately

Full size image

Despite the small number of trials in each block (45 for adults; 25 for children), the functions inferred by GPAL were consistent between the first and the second blocks (i.e., test and retest). Reliability of GPAL was measured using the concordance correlation coefficient (CCC; Lawrence & Lin, 1989), a measure of agreement between the first and second measurements. The mean CCC between the posterior mean functions from test and retest (Fig. 3) over participants was 0.89 for children (SD: 0.11) and 0.90 for adults (SD: 0.15). The strong correlation demonstrated high reliability of GP predictions.

Next, we compared the MLLM and CPMs using GP-estimated posterior mean functions (see Appendix for the results of the same analysis with the raw data) to determine how we should interpret the shapes of the functions. GP functions would be better explained by the MLLM if they were from numerical representation or by the CPMs if they were from proportional reasoning skills. Table 2 shows that the DIC values were the lowest for the MLLM regardless of the upper bound, for both children and adults. A better fit of the MLLM over the CPMs suggests that non-symbolic number estimates may reflect log-linear representations for number. Therefore, we interpreted the results in a log-linear framework. Results are discussed in relation to the three questions raised in the Introduction.

Table 2 DIC values for the hierarchical Bayesian models MLLM, 1CPM, and 2CPM
Full size table

Characteristic features of number-line estimation

Log-to-linear developmental improvement

To measure the degree of logarithmic compression in the GP functions, the posterior mean of the GP was fitted by the hierarchical Bayesian MLLM (see Appendix for the results with the raw data). The developmental changes in logarithmicity were explored by dividing participants into three age groups, younger children (kindergarteners – 2nd graders; n = 32), older children (3rd – 7th graders; n = 39), and adults (n = 159). Figure 4 shows the posterior means of the hyperparameter μλ, which represents the mean of the prior distribution for λ, estimated from the data combined over test and retest blocks for each age group across upper bounds. Larger values of the posterior mean indicate more logarithmic estimation.

Fig. 4
figure 4

The logarithmicity measure (μλ) of Gaussian process (GP) estimates for the age groups plotted against upper bounds. Note. Error bars indicate Bayesian 95% highest posterior density interval (HPDI)

Full size image

Overall, the values of μλ were the largest for younger children and decreased with age. Younger children’s estimation was highly logarithmic, whereas adults’ estimation was much more linear. The estimates of older children had intermediate levels of logarithmicity. The developmental decrease in μλ from the novel multi-scale task is consistent with previous research showing that the logarithmicity components (λ) decreases with age in a conventional fixed number-line task with symbolic numbers (Kim & Opfer, 2017, 2020; Opfer et al., 2016; Siegler & Booth, 2004).

Larger number ranges led to more logarithmic estimation. The posterior mean μλ value in every age group was the smallest at the smallest upper bound of 50 (Fig. 4). In 0–50 number lines, younger children showed somewhat logarithmic estimation, with the posterior mean of μλ being 0.12 (95% highest posterior density interval (HPDI): [0.02,0.22]; see Appendix for interpretation of HPDI intervals). Older children and adults were highly linear in estimation and differed little among one another. The posterior mean of μλ was 0.02 (95% HPDI: [0.01,0.04]) for older children, and 0.06 (95% HPDI: [0.04,0.08]) for adults. The age-related differences were much larger in 0–400 number lines. With the upper bound of 400, the posterior mean of μλ was 0.69 (95% HPDI: [0.60,0.78]) for younger children, 0.45 (95% HPDI: [0.37,0.54]) for older children, and 0.20 (95% HPDI: [0.16,0.24]) for adults, presenting the most salient log-to-linear transition in development.

Coexistence of log-linear representations

The changes in μλ across the upper-bound numbers were more apparent in younger children, but estimates of older children and adults also became more logarithmic on large-scaled number lines. Even adults who were least affected by the number range showed somewhat logarithmic estimation with a large upper-bound number. Specifically, adults’ estimation on 0–400 number lines was nearly as logarithmic as younger children’s estimation on 0–50 number lines. In line with logarithmic estimates of symbolic numbers in adults (Landy et al., 2013), this outcome suggests that Western, educated adults still rely on logarithmic representations when estimating large non-symbolic numbers.

Logarithmicity and approximate arithmetic

The degree of logarithmicity correlated with participants’ performance in the approximate addition task. To assess the correlation, we obtained individuals’ Weber fraction (w, Pica et al., 2004) in the approximate addition task. Weber fraction is an accuracy measure with numerical ratios taken into account. For example, if w is .5, participants would reliably compare 15 versus 10 dots (i.e., 50% difference) once they correctly perform arithmetic operations.

Next, we computed Bayesian partial correlations between λ and w values, while controlling for the effects of education levels (see Appendix for education level coding). Overall, there were positive correlations between λ and w across upper bounds. The posterior mean of the correlation coefficient (r) was 0.10 for 0–50 number lines, 0.15 for 0–100 number lines, 0.18 for 0–200 number lines, and 0.19 for 0–400 number lines. The 95% HPDIs of r included zero only for 0–50 number lines (95% HPDI = [-.04,0.24]). These results show considerable correlation between the two non-symbolic number tasks, suggesting that the associations between number-line estimation and math skills are found with non-symbolic numbers, not only with symbolic numbers.

Characteristics of GPAL-selected designs

A novelty of the present experiment was that the data collection was controlled by an active learning algorithm. A combination of an upper bound and a given number was selected by GPAL trial-by-trial to optimize function estimation. The resulting frequencies of the designs selected by GPAL are shown in Fig. 5. The distribution of given numbers selected by GPAL was markedly different from typical fixed designs used in number-line tasks, where given numbers are evenly distributed over the design space (Slusser et al., 2013) or concentrated in the first half of the range (Siegler & Opfer, 2003). Rather, GPAL most frequently chose the designs at the edges of the triangular design space. This indicates that estimates of the smallest number (i.e., 5) and of the upper-bound number of each number line had the highest uncertainty (i.e., posterior variance) most of the time.

Fig. 5
figure 5

Design selection frequencies in the number-line task with multiple scales Note. On every trial, Gaussian process active learning (GPAL) selected a given number (design variable 1) and an upper-bound number (design variable 2)

Full size image

A comparison of the frequencies with which designs were selected for children and adults in Fig. 5 suggests that design selection in GPAL is sensitive to the shape of estimated function. The designs for adults were more extreme than those for children, presumably because the function was more linear for adults. Samples at the two ends of the lines are highly informative for estimating nearly straight lines in linear functions. In contrast, if the function is logarithmic, a broader sample of given numbers (i.e., designs) is required because the endpoints alone are insufficient to estimate the shape of a non-linear curve. This idea is shown in the two side plots with red circles in Fig. 5.

General discussion

The present study examined children’s and adults’ estimates of non-symbolic numbers across multiple number ranges by adaptively optimizing the choice of design variables in a number-line task. This Bayesian active learning algorithm provided several insights into the development of non-symbolic number estimation.

The primary insight concerned the similarity of non-symbolic and symbolic number-line estimation. Our results showed all three features of symbolic number-line tasks. Specifically, we found that: (1) non-symbolic number estimation also shows a “log-to-linear shift”; (2) logarithmic and linear patterns of non-symbolic number estimates co-exist in the same individuals, with logarithmicity increasing with numeric range; and (3) logarithmicity of non-symbolic number estimates is positively correlated with the quality of estimates for arithmetic sums. The model comparison using model-neutral designs provided by GPAL supported the interpretation of these findings in the log-linear framework. All findings about non-symbolic number estimation (but the log-to-linear shift) were novel and predicted by the idea that symbolic and non-symbolic numbers are estimated through shared numerical representations with a common developmental pathway. Together, the results suggest that estimation of non-symbolic and symbolic numbers relies on common representations (Dehaene, 2011; Dehaene et al., 2008; Siegler & Opfer, 2003), rather than qualitatively distinct representations (Carey, 2004; Carey & Barner, 2019; Rips et al., 2008).

Another insight, which manifested in several unexpected ways, was that size matters. One way that it mattered was in differentiating age groups: Sensitivity to the upper bound decreased with age. Specifically, with age, the upper bound had a smaller effect on the logarithmicity of estimates, and so a large number scale (e.g., 0–400) best distinguished among age groups. This finding highlights the importance of using large numbers for studying numeric development, but it also shows the utility of GPAL in finding these numbers, which were not known a priori.

Size also mattered when addressing the association between number representations and arithmetic ability, where conclusions in the literature often conflict (Fazio et al., 2014; Halberda et al., 2008). In the case of symbolic numbers and symbolic arithmetic, there is a robust correlation (Booth & Siegler, 2008; Fazio et al., 2014; Kim & Opfer, 2017), and between non-symbolic number estimation and symbolic arithmetic, there is a weak correlation (Fazio et al., 2014). Here we looked at non-symbolic estimation and non-symbolic arithmetic, and found that logarithmicity of estimates correlated with the Weber fraction in the approximate addition task, with the strength of the correlation tending to increase with upper bound. These results support the idea that the approximate number system supports arithmetic intuitions (Halberda et al., 2008), but the influence of the approximate number system is reduced by crossing format (from non-symbolic to symbolic) and limiting the range of numbers tested.

Still another way that size matters is how it reveals the process of developmental change. Past studies that explored the range-dependency of the numerical estimation typically used number-line tasks with a few different scales. For example, Berteletti et al. (2010) presented 0–10 and 0–100 number lines to the same participants. Thompson and Opfer (2010) used 0–1,000, 0–10,000, and 0–100,000 number lines in a between-subject design. These studies showed that log compression increased with increasing number range in children, but the changes in logarithmicity were abrupt rather than gradual, possibly because there were only a few selected upper-bound numbers that differed by orders of magnitude. In contrast, when the upper-bound was included as a design variable that could be comprehensively controlled, we found a gradual change from more logarithmic to more linear estimation in all age groups. Thus, GPAL revealed an unexpected feature of developmental change that will be important to follow in future studies.

GPAL was particularly useful in the current study because we explored participant behaviors in a large design space in an efficient and model-free manner. Alternative methods such as using a fixed design or model-based experimental design optimization could be employed, but they would reduce the efficiency in data collection and also the informativeness of the data. That is, number-line tasks with fixed designs would require many more trials or multiple sessions to compare computational models and estimate a logarithmicity measure across upper bounds as in the current study. A model-based design optimization method (e.g., ADO; Cavagnaro et al., 2010) could facilitate parameter estimation (e.g., estimation of the logarithmicity parameter of the MLLM) across upper bounds, but the model would predefine the relation between the upper bound and logarithmicity, which may mischaracterize representation of number. When this happens (i.e., model misspecification), a design optimization algorithm might bias the design selection so the conclusion of the study. GPAL overcomes both of these limitations, namely, inefficient data collection and model misspecification.

In conclusion, the present study explored non-symbolic number estimation in children and adults using a novel active learning algorithm (GPAL). The three characteristic features of symbolic number-line estimation were observed in the study. The results, taken together, suggest that non-symbolic and symbolic numbers may be represented through common processes that share a developmental trajectory.