The aim of this investigation was to explore the developmental transitions of the mental representations associated with numerical estimation (logarithmic-linear shift). This was achieved by focusing on the estimation behaviors of individual target digits within a familiar number range (0-20), adapting and extending the methods of Siegler and colleagues [2, 5, 10] and building on the ideas of Barth and Paladino [26], Berteletti et al. [7], Ebersbach et al. [8] and Thompson and Siegler [11]. The statistical analysis of individual numbers inferred that there is merit in future in depth analyses of strategies application in conjunction with regression modeling.

On a developmental front, Year 1 children did not demonstrate a dominance of any representation in either the individual regression models or group level residuals. In fact, looking at the individual regression models alone, the variance explained by any model (linear, logarithmic or power), was significantly lower in Year 1 than in Years 2 and 3. When examining the group level model, children in Years 2 and 3 demonstrated the lowest residuals from a linear model, in comparison to a logarithmic model. The complexity of this transition to a linear mental representation is indicated by the categorization of residuals into linear, logarithmic and ambiguous (Table 4). For selected target digits, there were a high percentage of Year 1 participants who demonstrated linear-like behaviors however this was not always the case (e.g. target digits 7 and 8, Table 4). In contrast, Years 2 and 3 demonstrated a consistently high percentage of participants with trials that were best represented by a linear model. This matches the trend observed across the three groups of preschool children in the Berteletti et al. study [7]. In the familiar number scale tasks used in that study (1-10 and 1-20) the youngest group (mean age 4 years) did not demonstrate a bias towards either representation, meanwhile the middle and oldest groups (4.5-6.5 years) indicated a significantly lower linear residual. Furthermore, the present findings are in line with the 0-100 number line developmental findings of Siegler and colleagues [2, 5, 10] except in Year 1 with no significant bias.

Extending the analysis to include 1- and 2-cycle power functions [26], in this case, did not create any further clarity in terms of R^{2} values. Perhaps it was the reduced number range (0-20) and minor differences in task instructions that limited the potential for power models to represent the data, as Barth and Paladino [26] focused on a 0-100 number line and indicated 50 as being 'halfway' at the beginning of the experiment. The individual data in the present study was typically best fit by a 1-cycle model (Table 1), which aligns with the fact that participants were not directed towards the 'halfway' point when task instructions were given. Further exploration into the use of proportional power models is required, particularly in relation to the appropriateness of using such an approach, as highlighted by Opfer et al. [27].

Extending the work of Siegler and Opfer [10] the results from the individual target digits are a unique contribution to the body of literature, as this begins to explore the possibility that the development of mental representations could be marked by the use of the external 'anchor points' as described by Ebersbach et al. [8]. In the case of this research application of external anchor points should be observed for target digits 2 and 17. According to existing research [2, 5, 7, 8, 10], a low linear residual would indicate a more accurate mental representation. The number scale 0-20 is familiar to students in Years 1-3, however, the strategies involved in positioning numbers accurately on an externally represented number line may not be fully established. For Year 1 the linear and logarithmic residuals for both of these numbers were similar (Table 2). Year 2, showed a significantly lower linear residual for target number 2, and perhaps as a function of the magnitude effect or incomplete transition, only a marginal preference for a linear model for target number 17 (Table 2). Finally then, for Year 3, the difference between linear and logarithmic residuals for target numbers 2 and 17 was significant with a linear model providing the lowest residual (Table 2). This indicates a developmental transition, but also highlights that the greatest disparity between the logarithmic and linear models is likely to occur in close proximity to external anchor points. It is this external anchor point reasoning that we use to speculate that both target digits 2 and 17 have lower linear residuals in comparison the logarithmic model residuals.

Parallels can be drawn between the observations of the present study and the developmental progression seen with arithmetic strategies in the classroom. Existing research (e.g. [34–36]) purports the importance of sequences and counting in the early stages of development, but also identifies that the application of the base-ten structure in constructing novel relationships among numbers up until approximately 9 years of age. As a further example, Dutch mathematics education programs teach mental arithmetic strategies that employs decomposition as the basis of instruction. From Grade 2, Dutch children are encouraged to use mental jumps and decomposition, often beginning on a number line, in order to encourage flexible mental strategies [37]. Given this information and linking back to the present data, it is proposed that the Year 1 children did not demonstrate any clear anchor point application because they were limited to counting strategies and were unable to link the numerical value to the spatial cues provided by the number line. Subsequently, in Years 2 and 3 we do see evidence of the continued development of more flexible strategies, and use of anchor points, that utilize decomposition and part-whole relations.

A question that comes out of this discussion is, given the flexibility of strategy application, is it in fact meaningful to try and model the mental representation of numbers using a fixed linear/logarithmic model? What the previous paragraph has posited is that specific numbers could exhibit unique behaviors as a function of the familiarity with the number range, proximity to either external or mental anchor points, as well as knowledge of arithmetic strategy. While Ebersbach et al. [8] focused on the role of external anchor points, the mental anchor points in particular would relate to more advanced strategy application, such as knowledge of proportions (e.g. half, quarter etc.) and ability to mentally partition the external number. This potential for individual difference represents a limitation of the linear/logarithmic modeling approach. In the followings we discuss what a more detailed approach could add to the current knowledge base.

Siegler and Booth [5], in their first experiment with 0-100 scale, had error rates of 27% for Kindergartners, 18% for Grade 1 and 15% for Grade 2. The later Booth and Siegler [2] study demonstrated a more obvious plateau with Kindergartners: 24%, Grade 1: 12%, Grade 2: 10%, and Grade 3: 9%. This could be a function of the 0-100 number scale being in the unfamiliar range. In these two studies [2, 5], the youngest groups were always significantly different to the subsequent year levels; however, this was not the case in the present research, as average percent absolute error was lower and there was no significant group difference.

Overall, the results from the percent absolute error data indicated that the most accurately estimated numbers on the 0-20 number line were digits 2 and 4. This could be for two reasons; first, referencing the lower values (2 and 4) to the external lower anchor point of zero or their knowledge of one. Second, it could be that number magnitudes up to 4 have a stronger representation, because these quantities are understood from infancy [38–40] in what Feigenson et al. [15] describes as the small number system. This could also be linked to how frequently these numbers are encountered in a young child's everyday language (e.g. [13, 25]). This also fits with theories of enumeration and subitizing abilities being present prior to verbal counting (e.g. [41–43]). The fact that these enumeration and subitizing abilities are limited to numerosities of four (e.g. [44, 45]), and that they are present from infancy, could explain the strength of the representation of digits 2 and 4, and thus producing more accurate estimations. This is further evidenced in the categorization of trials (Table 4), where all year groups demonstrated a high percentage of linear classifications of digits 2 and 4. Then for the subsequent numbers 7 and 8, Year 1 stood out with a high percentage of logarithmic classifications, which were not evident in Years 2 and 3 (Table 4). It is our interpretation that for Year 1 digits 7 and 8 are likely to be less frequently encountered and could contribute to the variation.

On the developmental front, the Year 1 children again showed separation from the older year groups in the estimation accuracy of individual target digits. For the positioning of numbers 11 and 13, Year 1 children produced estimates that were significantly less accurate than Years 2 and 3 (Figure 3). This gained further support when investigating the standard deviations for each target digit, with the greatest variation evident for digits 11 and 13 for Year 1 participants (Figure 4). The idea of applying external anchor points to aid in the estimation of numbers has been purported, although only briefly in existing studies [5, 8]. However, it could be argued that a further number estimation strategy could be to apply mental anchor points and divide/partition the external number line into segments which would increase the accuracy of positioning. The Year 2 and 3 children of this study were probably exhibiting these behaviors. A potential mental partition would be that of halfway, and number 11 was the closest target digit to the mental anchor point of 10. It is proposed that, in Year 1, as both a result of both mental representation (no clear logarithmic-linear preference) and educational experience, children lacked the requisite representations/skills to apply mental anchor point strategies and accurately estimate these central numbers (i.e. 11 and 13). The core learning concepts for Year 1, as prescribed by the National Framework in England focus on counting and skills related to addition and subtraction, not division or the part-whole relation. The early understanding of division principles often stems from the relationships between doubling and halving, which is promoted in Year 2. This knowledge of 'half' is required for the application of mental anchor point strategies, which as a function of educational experience the Year 1 children do not typically possess. The fact that the formal teaching of this concept in Year 2, coincides with increased accuracy of estimation for central digits (Figure 3), lends further support to this argument. Further detailed analyses would be required to strengthen these proposals and will be the focus of subsequent studies.

Examining percent absolute error for specific target numbers allowed the discussion to go beyond the limitations of structured modeling to further explore the potential of strategy application. The merits of both external and mental anchor points, as estimation strategies were supported. In line with the initial hypothesis, Year 2 appeared to represent a transitional phase, with the apparent onset of part-whole strategies to aid the creation of mental anchor points. It was hypothesized that the developmental shift from logarithmic to linear mental representation would likely coincide with evidence of strategy use; the present study supports this.

The development of the anchor point strategy application could be described as follows; the first would be to utilize the external anchor points to assist positioning of the numbers. Children in the first year of school may only use the left most point, rather than having the strategic knowledge to employ both extremes. The developmental progression, along with the logarithmic-linear shift, extends to include the use of both anchor points after Year 1. This is followed by some level of mental partitioning, which again advances in complexity and is the result of educational experience, which became evident in Year 2 (Figure 3). It would be an interesting investigation to more closely map the development of these individual strategies and educational experience with the logarithmic and linear modeling scenarios. This extension would ideally include saturation of all target digits within the prescribed number range and a direct means of determining the strategy used to solve the problem, whether requesting verbal reports from participants or applying the use of eye-tracking technology as introduced by Schneider et al. [28]. Furthermore, it would be worthwhile to investigate whether the type and flexibility of strategies used during these first years of school could predict later mathematical achievement. It is the combination of these proposals that could facilitate the most meaningful insights for informing education.