The structure of the notation system in adults’ number line estimation: An eye-tracking study

Abstract

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults’ numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude.

Keywords

Number line estimation rational numbers natural numbers eye tracking adults

In the last two decades, many studies have shown that rational numbers such as fractions (e.g., 3/14) and decimals¹ (e.g., 0.214) are more difficult to understand than natural numbers for not only children but also adults (DeWolf et al., 2015; Ni & Zhou, 2005; Vamvakoussi et al., 2012; Van Hoof et al., 2017). Individuals’ understanding of the numerical magnitude (i.e., the size) of rational numbers has been one important aspect in research on rational numbers (Van Hoof et al., 2017). For instance, when adults compared fraction magnitudes that were inconsistent with natural number reasoning (e.g., 2/3 vs 3/7, where 2/3 is the larger magnitude but has smaller components), they were less accurate and slower to respond than with fraction magnitudes that were consistent with natural number reasoning (e.g., 2/5 vs 5/7; DeWolf & Vosniadou, 2011; Obersteiner et al., 2013; but see also Morales et al., 2020). Similarly, in a study with decimals, adult participants were slower at comparing magnitudes when decimal pairs were inconsistent with natural number reasoning (e.g., 0.27 vs 0.9, where the decimal with more digits is actually smaller in numerical value) compared with when they were consistent (e.g., 0.2 vs 0.87; Varma & Karl, 2013).

Although much research has examined individuals’ understanding of the magnitude of rational numbers in contrast to natural numbers (or natural number reasoning), research is beginning to show that within the category of rational numbers, the magnitude of decimals is understood differently from the magnitude of fractions (DeWolf et al., 2015; Hurst & Cordes, 2016, 2018; Iuculano & Butterworth, 2011; Resnick et al., 2019; Van Hoof et al., 2018; Wang & Siegler, 2013). For instance, DeWolf et al. (2015) found that 12-year-old children’s knowledge about decimals and knowledge about fractions each explained a unique portion of the variance in their algebra performance. As another example, Van Hoof et al. (2018) showed, in a longitudinal study, that each year, children performed better in a subtask assessing the magnitude of decimals than the magnitude of fractions. Finally, Hurst and Cordes (2016) examined adults’ ability to compare magnitudes both across (e.g., decimal vs fraction) and within (e.g., decimal vs decimal) natural numbers, decimals, and fractions. They found that adults were significantly faster to compare magnitudes with decimals and natural numbers in contrast to when they compared magnitudes that involved fractions.

In a study using the number line estimation (NLE) task, Wang and Siegler (2013) measured the estimation accuracy of children in grades 4 and 5 with both decimals and fractions. Children had to indicate the position of a given target number (e.g., 0.214) on a bounded number line ranging from 0 to 1. The researchers found that children estimated the location of decimals more accurately than fractions, which is consistent with the idea that children’s numerical magnitude understanding varies between fractions and decimals. In another NLE study, Iuculano and Butterworth (2011) compared not only decimals and fractions but also natural numbers in a sample of 10-year-old children and adults. The authors found that both children and adults were less accurate and slower to respond when estimating the location of fractions compared with decimals and natural numbers on the number line. In addition, adults were faster than children at estimating decimals and natural numbers, but not fractions. This comparative research between natural and rational numbers as well as between decimals and fractions highlights how difficult working with rational numbers and with fractions in particular can be, even for adults.

One possible explanation for the differences found between natural numbers/decimals on one hand, and fractions on the other hand, could be the structure of the notation system: Natural numbers and decimals are written in a place-value-based notation while fractions are not (see also Tian & Siegler, 2018). For the current article, a place-value-based number notation refers to the Hindu–Arabic numeral system in which a number is written, such that, a digit is multiplied by a power of 10 based on its position, which aids with identifying its value in the number. For natural numbers and decimals, each digit in its particular place is 10 times smaller than the digit preceding it; each digit fits within a system of ones, tens, hundreds, and so on, in the case of natural numbers, and tenths, hundredths, thousandths, and so on in the case of decimals. For clarity, in 241 and 0.241, the digit “1” is 10 times smaller than the digit “4,” which is 10 times smaller than the digit “2.” Fractions, however, convey a multiplicative relationship between two natural numbers (e.g., 1/4) where the fraction’s magnitude is based on the value of the numerator relative to the denominator. Fractions are also not necessarily defined by powers of 10. Altogether, fractions are thus expressed in a non-place-value-based notation.² If children and adults have a more enhanced understanding of the place-value-based compared with the non-place-value-based number structure, then this might explain why the numerical magnitude of decimals has been found to be easier to understand than that of fractions (see Tian & Siegler, 2018 for a review). This reasoning is in line with DeWolf et al. (2015) and Hurst and Cordes (2016) who state that decimals are similar to natural numbers, which are already learned by students at the time of their introduction to decimals and fractions (but see Matthews & Chesney, 2015).

The idea of the structure of the notation system influencing individuals’ understanding of numerical magnitude has been identified in the literature. LeFevre et al. (2013) conducted a longitudinal study in lower elementary school children and found that number system knowledge, which included place-value understanding, significantly predicted children’s natural NLE performance. Moreover, in domains other than estimation, the structure of the notation system has been shown to affect performance in number-related tasks. For instance, children’s place-value understanding has been found to positively relate to arithmetic (Fuson, 1990; Laski et al., 2016; Moeller et al., 2011). Research also suggests that the place-value-transparency of a language’s number system (e.g., Chinese is place-value-transparent while French is not; for details see Ng & Rao, 2010) could be a reason for the improved mathematical performance typically found in Asian students (e.g., Ng & Rao, 2010; Stevenson et al., 1986, 1990). Miura et al. (1994) found that children from China, Japan, and Korea organised place-value-based (i.e., base-10) blocks in tens and ones to display numbers while children from France, Sweden, and the United States did not. Overall, these findings suggest that the structure of the notation system of number plays an important role in working with numbers.

Present study

We aimed to disentangle the extent to which two potential sources of difficulty affect adults’ numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). We created a 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) within-subjects design in which numerical magnitude understanding was assessed by means of the NLE task. The four conditions in the design differed with respect to the type of numbers that had to be positioned on the number line: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based; see Table 1 in the “Method” section). A separated fraction refers to a novel variant of the NLE task in which the target number is the numerator, X, and the endpoint of the number line is the denominator, Y, from a fraction in the form X/Y (e.g., 3 would be the target number and the endpoint of the number line would be 14, from the fraction 3/14). For direct comparability, each target number’s location on the number line was equivalent in all four conditions. The current study therefore extended the design of Iuculano and Butterworth (2011) by including not only natural numbers, decimals, and fractions, but also a new type of NLE condition—a separated fractions condition. This additional condition allowed us to examine the extent to which number type (natural vs rational) and the structure of the notation system (place-value-based vs non-place-value-based) affect adults’ numerical magnitude understanding in the context of an NLE task.

Table 1.

Example of one target number and respective number lines for each of the four conditions.

	Place-value-based	Non-place-value-based
Natural	Natural numbers	Separated fractions
Rational	Decimals	Fractions

In addition to typical performance data, namely, accuracy, which is typically measured in NLE studies by means of the percentage absolute error (PAE) and refers to how inaccurately participants estimate magnitudes on a number line using the percentage of deviation of an estimate from its actual position, and response times, we collected eye-tracking data, as has previously been done in the context of the NLE task (e.g., Sullivan et al., 2011). We used this methodology to investigate adults’ NLE solution process seeing as it has added value in two ways. First, eye tracking provides information over time, also known as process data. These process data are obtained every millisecond and allow trials to be broken down and analysed according to smaller segments, resulting in a more precise picture of what occurred during each trial compared with performance data alone (see MacKay et al., 2020). Specifically, we divided each trial into two phases: the encoding phase and the positioning phase. The encoding phase included data from participants’ first look at the target number while the positioning phase included the rest of the eye-tracking data in the trial (i.e., positioning the target number on the number line and looking back at the target number). Second, eye tracking enables researchers to examine whether, which combination of, and how frequently certain areas of interest (AOIs) in the task are looked at by individuals (e.g., van’t Noordende et al., 2016). This method of breaking down the process data extracted from eye tracking allowed us to understand how participants solve the NLE task for each of the different notations. More specifically, the combined information from the encoding phase and the positioning phase informed us about possible differences between notations with respect to where participants spent most of them time throughout a trial. As such, these eye-tracking data provided us a first glance into how adult participants solve the NLE task for natural numbers, decimals, fractions, and separated fractions as well as into the similarities and differences between each of the notations in terms of eye movement behaviour. Thus, eye tracking offers the possibility to examine adults’ NLE solution process by capturing online, detailed information regarding the precise moment when an individual looked at a certain location.

A note regarding the present study is our definition of numerical magnitude understanding, which will hold for the remainder of this study. Numerical magnitude understanding refers to how well an individual has a grasp of the quantity of a number and reflects how well our cognitive representation of the quantity of numbers is calibrated, which is sometimes referred to as a “mental number line” (Siegler & Opfer, 2003). We assumed participants’ error rates and response times reflected participants’ numerical magnitude understanding whereby lower error rates and shorter response times would be indicative of better numerical magnitude understanding. Moreover, we assumed that the eye-tracking data from each of the phases provided information regarding adults’ numerical magnitude understanding where less time spent processing a certain aspect of the task was associated with better understanding.

Given the well-known findings that participants find the numerical magnitude of rational numbers difficult to understand compared with that of natural numbers, we first hypothesised that participants would estimate natural numbers faster and with lower PAE than rational numbers (Hypothesis 1). Second, based on our review of the literature concerning the structure of the notation system in mathematics, we also hypothesised that participants would estimate place-value-based numbers faster and with lower PAE compared with non-place-value-based numbers (Hypothesis 2). Third, based on our eye-tracking data in the encoding phase, we expected participants would look longer at the target number location for rational numbers compared with natural numbers, seeing as there was difference in the amount of information to be encoded (i.e., rational numbers had more digits than the natural numbers; Hypothesis 3). Fourth, in the positioning phase, specifically regarding eye-movement behaviour on the number line, we expected participants would spend less time looking at the number line for natural numbers compared with rational numbers, given that more cognitive processing (and thus more looking) would be associated with the more difficult number type (i.e., rational numbers; Hypothesis 4a). Similarly, we expected participants would spend less time looking at the number line for the place-value-based numbers compared with non-place-value based numbers seeing as adults’ place-value-based number understanding is stronger and more engrained than non-place-value-based number understanding (Hypothesis 4b). Fifth and finally, in the positioning phase regarding eye-movement behaviour on the target number, we predicted that participants would look more at the target number location for rational numbers compared with natural numbers as adults would likely need more time to process rational numbers, since they are known to be more difficult to understand than natural numbers (Hypothesis 5).

Method

Participants

Fifty-six adult participants were recruited. Four participants were excluded from the experiment because of calibration issues with the eye tracker. The final sample thus consisted of 52 participants (M_age = 20.73 years, SD_age = 6.50 years, Range_age = 18–60 years, N_females = 46, N_right-handed = 47).

Materials

NLE task

Participants completed the number-to-position (NP) variant of the NLE task in which they estimated the location of a given target number on a number line. The number line was a dark grey horizontal line (17°) centred in the screen. The endpoints of the number line were marked by dark grey hatch marks of approximately 0.6° high. The numerical values underneath the endpoints (e.g., 0 on the left, 1,000 on the right) were presented in black Times New Roman font approximately 0.5° high. The target numbers were presented in black Times New Roman font (approx. 1.5° high) within a light grey box (6° wide and 4° high) that appeared randomly in the top left or right corner of the screen at a distance of approximately 3° from the top edge and 1.5° from the left and right edges of the screen.

Participants responded by fixating on the location on the number line where they thought the position of the target number was and clicked a mouse button. At that moment, a red hatch mark (approx. 0.6° high) appeared in the x-position of the participant’s last fixation right before he or she clicked with the mouse. This hatch mark acted as a form of feedback indicating the participant’s final estimate and remained on screen for 500 ms before the next trial began. The hatch mark was constrained to align with the number line (i.e., y-position = 384 pixels or 14° from the top of the screen). This method of responding with the eyes instead of with a mouse cursor has already been successfully adopted in other eye-tracking NLE studies (Di Lonardo et al., 2019; Heine et al., 2010; MacKay et al., 2020; Schneider et al., 2008). It also avoids potential mouse-related issues, in particular, participants’ eye-movement behaviour being disrupted from hand–eye coordination when using the mouse and participants using the mouse as an external place-holder on the number line to aid them in estimating the target number (for more details, see MacKay et al., 2020).

Design and stimuli

A 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) within-subjects design was created such that the NLE task was presented in four conditions: natural numbers, decimals, fractions, and separated fractions.

Fifty-four target numbers were selected for each of the four conditions, such that, they were a large, representative sample of numbers across the number line. We began our stimuli selection with the fractions to ensure we had a balanced combination of fractions that were more familiar (e.g., simple fractions, such as 1/4, 1/3, 1/2) as well as less familiar (e.g., complex fractions, such as 10/13, 11/28, 23/31). Following the fraction stimuli selection, we converted the fractions to decimals, separated fractions, and natural numbers (see the online Supplementary Material A for the full list of simple and complex stimuli across notation). Given the 54 target numbers in each notation, each participant completed a total of 216 experimental trials. To ensure comparability between the four conditions, one target number was selected, such that, the location on the number line was (essentially) equivalent across each of the four conditions. To demonstrate, the target number 214 on a 0–1,000 number line in the natural numbers condition corresponded with 0.214 on a 0–1 number line in the decimals condition, with each decimal being rounded to the third digit after the decimal point. For the two non-place-value-based conditions, 3/14 (which corresponds with 0.214 rounded to three decimal places) on a 0–1 number line in the fractions condition, and 3 on a 0–14 number line for the separated fractions condition were the respective equivalents to the natural numbers and decimals conditions (see Table 1). The complete list of the equivalent target numbers in each condition can be found in the online Supplementary Material B.

Apparatus

A 15.6″ DELL Latitude 5591 Laptop with a screen resolution of 1,024 × 768 and a 60 Hz refresh rate was used to display the stimuli using the Experiment Builder v.2.1.1 software package (SR Research, Ottawa, ON, Canada). Participants were tested in a darkened room with their eyes 60 cm away from the screen in the head-stabilised setting of the eye tracker, meaning that participants’ head movements were restrained using a chin rest attached the table for the duration of the experiment. Each participant’s right eye was recorded using an Eyelink Portable Duo video-based corneal reflection eye tracker at 1,000 Hz (SR Research, Ottawa, ON, Canada). The event detection algorithm was the standard algorithm used by SR Research eye trackers (for a saccade, the velocity threshold was 30°/s; for a blink, the pupil data had to be missing for three samples in a row; all else were fixations). Recording was controlled by a 14″ Lenovo ThinkPad E460 Laptop.

Procedure

Participants were first shown instructions about how to complete the NLE task. Following the instructions, they started by completing a 13-point calibration procedure. A calibration was accepted only if the average error across all 13 points was less than 1.5° and each point’s accuracy was no higher than 2°.

To familiarise themselves with the task and method of responding, participants next completed a block of nine practice trials. More specifically, they were presented with all numbers from 1 to 9 in a randomised order, which they had to estimate the position of on a 0–10 number line. After any questions from the participants were answered, the experimental trials began.

The four conditions of the experimental trials were blocked. The order of the blocks was randomised as well as the order of the 54 trials within each block. In each block, participants completed two practice trials that matched the experimental trials of the respective block (natural numbers: 821 and 133 on a 0–1,000 number line; decimals: 0.821 and 0.133 on a 0–1 number line; fractions: 2/15 and 23/28 on a 0–1 number line; separated fractions: 2 on a 0–15 number line and 23 on a 0–28 number line). Participants had no time restriction while completing the NLE task and were told to complete it as quickly and as accurately as possible.

Each trial began with a drift correction procedure in which a black fixation point (identical to those in the calibration) appeared in the centre of the screen on a white background. Once participants fixated this point, the experimenter accepted the fixation, which allowed the trial to continue. If participants’ fixation deviated more than 1° from the fixation point in any direction, the experimenter initiated another 13-point calibration procedure before continuing with the experiment. Participants were excluded because of calibration issues if they required more than five calibration procedures due to deviation from the fixation point.

Next, the number line, with its labelled endpoints, appeared in the centre of the screen. After a randomly selected amount of time (between 800 and 1,200 ms, in intervals of 20 ms), a grey box appeared randomly in either the top left or top right corner of the screen. This box acted as a cue for participants to make an eye movement towards this box. Once a stable fixation was detected within the box, the target number appeared in it, after which participants positioned the target number on the number line.

If, following a trial, participants spontaneously mentioned that the hatch mark was not in the correct location (i.e., where they were fixating), the experimenter recorded the trial number (to later be removed from the data) and began another 13-point calibration procedure.

Data analysis

The data analyses involved two types of data: performance data and eye-tracking data. Two performance measures were considered. The first was the PAE, which was the absolute value of the difference between the actual position of the presented target number and its estimated position on the number line, divided by the scale of the number line and multiplied by 100. For instance, if the actual position was 380 on a 0–1,000 number line, and the estimated position was 333, the PAE would be 4.7% ([|estimated position–actual position|/scale of the number line] × 100 = [|380–333|/1,000] × 100]. The second was the total response time (RT_tot; in milliseconds), which referred to the response time of the entire trial, starting from the onset of the target number until participant’s response.

For the eye-movement analyses, virtual AOIs were created. For the presented target number, an 8° × 6° AOI (furthermore, target number AOI) was created. Along the number line, five adjacent and equal-sized AOIs (approx. 5° × 5°) were centred around the two endpoints of the number line (0% and 100%) and the three quartile locations on the number line (25%, 50%, and 75%). These number line AOIs were created based on where groups of fixations would be likely to occur on the number line (e.g., M-shaped pattern; Ashcraft & Moore, 2012) with the aim of determining how these fixations differed across the four conditions, within each of the AOIs. Notably, the 100% AOI was of importance to be examined separately, given that the endpoint of the number line changed each trial in the separated fractions condition and not the other three conditions. See Figure 1 for a schematic representation of all AOIs.

For the analyses of the eye-tracking measures, we divided each trial into two phases: the encoding and the positioning phases. The encoding phase started from the onset of the target number until the participant’s eye first left the box in which the target number was presented. The measures examined in this phase were the encoding gaze duration (in milliseconds), which was the total amount of time participants fixated in the target number AOI during the encoding phase, and the encoding fixation frequency (count), which was the total amount of times participants fixated in the target number AOI during the encoding phase. The positioning phase started from the end of the last fixation of the encoding phase (i.e., within the target number AOI) until the participant’s response. In this phase, the same two variables as in the encoding phase were considered for each AOI of the number line: the positioning gaze duration and the positioning fixation frequency. The positioning gaze duration and positioning fixation frequency were analysed across the five AOIs of the number line (0%, 25%, 50%, 75%, and 100%). Seeing as the target number, AOI could also be re-visited in the positioning phase, we examined the probability of revisiting the target AOI, which refers to the percentage of trials that a participant’s eyes returned to the target AOI in the positioning phase.

Before analysing the data, the following trials were removed: (a) spoiled trials during which participants did anything that disrupted a trial (i.e., moved or scratched their heads, squinted, talked, spontaneously told the experimenter that the hatch mark was not in the correct position; 2.07% of total data), (b), trials in which the final response of the participant fell outside of the bounds of the number line itself (i.e., responses that corresponded to less than 0% and greater than 100%) (0.41% of the total data), and (c) trials in which the eye left the target number AOI within 120 ms following target number onset because foveal encoding of the target number may have been hampered in these cases. These trials probably reflect instances in which a saccade (towards the number line) was planned before the onset of the target number (3.13% of the total data). Seeing as the data for all variables (i.e., PAE, RT_tot, and eye-tracking measures) were positively skewed (skewness ranged from 1.78 to 4.29 for all measures), the median was used in all analyses as it is robust against skewness and outlier values (Howitt & Cramer, 2008). It should be noted that the analyses reported in the Results involved averaging these values, thus the means that are reported reflect the means of the medians.

As a final remark, following each analysis in which a significant interaction was found, we conducted post hoc comparisons using paired-sample t-tests and corrected for multiple comparisons by means of Bonferroni corrections (α = 0.05/4 comparisons = 0.0125).

Results

Performance measures

To examine adults’ PAE and response times in the NLE task, two 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) within-subjects ANOVAs were conducted on PAE and RT_tot. For PAE, a significant main effect was found for both number type, F(1, 51) = 9.57, p = .003, $η_{p}^{2} = . 16$ , and structure of the notation system, F(1, 51) = 14.00, p < .001, $η_{p}^{2} = . 22$ , with rational numbers and non-place-value-based numbers eliciting significantly higher PAE than natural numbers and place-value-based numbers, respectively (see Figure 2a and Table 2 for descriptive statistics). No significant interaction between number type and structure of the notation system was found for PAE (F < 1).

Table 2.

Means and standard deviations for participants’ PAE (in %) and RT_tot (in ms) of natural and rational number types across place-value-based and non-place-value-based numbers (N = 52).

	Place-value-based		Non-place-value-based
	M_PAE (SD)	M_RT (SD)	M_PAE (SD)	M_RT (SD)
Natural	Natural numbers		Separated fractions
Natural	3.76 (1.23)	2,526 (1,212)	4.25 (1.22)	2,564 (1,323)
Rational	Decimals		Fractions
Rational	4.13 (1.24)	2,614 (1,312)	4.62 (1.32)	3,205 (1,655)

SD: standard deviation.

For RT_tot, we observed a significant main effect of number type, F(1, 51) = 25.79, p < .001, $η_{p}^{2} = . 34$ , and of structure of the notation system, F(1, 51) = 10.34, p = .002, $η_{p}^{2} = . 17$ . Number type and structure of the notation system significantly interacted, F(1, 51) = 11.79, p = .001, $η_{p}^{2} = . 19$ ; see Figure 2b and Table 2. The post hoc comparisons revealed that participants responded significantly slower in the fractions condition than in both the separated fractions and in the decimals conditions (ps < .001) while there were no differences between the natural numbers and separated fractions conditions (p = .70) or between the natural numbers and decimals conditions (p = .44).

Eye-tracking measures

Encoding phase

To examine participants’ initial encoding of the target number, two 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) within-subjects analyses of variance (ANOVAs) on encoding gaze duration and encoding fixation frequency on the target number AOI revealed significant main effects of number type, encoding gaze duration: F(1, 51) = 113.42, p < .001, $η_{p}^{2} = . 69$ ; encoding fixation frequency: F(1, 51) = 34.60, p < .001, $η_{p}^{2} = . 40$ . No significant main effects of structure of the notation system type were found, encoding gaze duration: F(1, 51) = 0.84, p = .37, $η_{p}^{2} = . 02$ ; encoding fixation frequency: F(1, 51) = 3.76, p = .06, $η_{p}^{2} = . 07$ . Furthermore, there was a significant interaction between number type and structure of the notation system for both encoding gaze duration, F(1, 51) = 66.17, p < .001, $η_{p}^{2} = . 57$ (Table 3 and Figure 3a), and encoding fixation frequency, F(1, 51) = 15.27, p < .001, $η_{p}^{2} = . 23$ (Table 3 and Figure 3b).

Table 3.

Means and standard deviations for participants’ encoding gaze duration (EGD; in milliseconds) and encoding fixation frequency (EFF; count) of natural and rational number types across place-value-based and non-place-value-based numbers (N = 52).

	Place-value-based		Non-place-value-based
	M_EGD (SD)	M_EFF (SD)	M_EGD (SD)	M_EFF (SD)
Natural	Natural numbers		Separated fractions
Natural	446 (125)	2.08 (0.27)	316 (83)	1.94 (0.21)
Rational	Decimals		Fractions
Rational	479 (128)	2.17 (0.39)	579 (210)	2.47 (0.62)

SD: standard deviation.

To disentangle the interaction, the post hoc comparisons revealed significant differences between all conditions (all ps < .003). The separated fractions condition had the lowest encoding gaze duration, followed by the natural numbers condition, the decimals condition, and, finally, the fractions condition. For encoding fixation frequency, the post hoc comparisons revealed a similar pattern: Participants had a significantly higher number of fixations in the fractions compared with the separated fractions (p < .001) and the decimals condition (p = .001) and significantly more in the natural numbers compared with the separated fractions condition (p = .01). Slightly different from the encoding gaze duration pairwise comparisons, no significant difference was found between the natural numbers and decimals conditions for encoding fixation frequency (p = .14).

Positioning phase

To examine adults’ fixations across the number line, two 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) × 5 (AOI: 0%, 25%, 50%, and 75% vs 100%) within-subjects ANOVAs were conducted on positioning gaze duration and positioning fixation frequency (see Table 4 for descriptive statistics, Table 5 for results, Figure 4a and b for visualisations).

Table 4.

Means and standard deviations for participants’ positioning gaze duration (PGD; in ms) and positioning fixation frequency (PFF; count) across number type (natural vs rational), structure of the notation system (place-value-based vs non-place-value-based) and AOI (0%, 25%, 50%, and 75% vs 100%).

	AOI	Place-value-based		Non-place-value-based
	AOI	M_PGD (SD)	M_PFF (SD)	M_PGD (SD)	M_PFF (SD)
Natural	0%	0 (0)	0.00 (0.00)	6 (46)	0.02 (0.14)
	25%	11 (32)	0.12 (0.32)	52 (168)	0.27 (0.48)
	50%	153 (189)	0.49 (0.95)	243 (468)	0.70 (1.04)
	75%	12 (45)	0.07 (0.24)	80 (122)	0.46 (0.53)
	100%	2 (17)	0.02 (0.14)	105 (157)	0.55 (0.73)
Rational	0%	14 (79)	0.06 (0.31)	0 (0)	0.00 (0.00)
	25%	14 (33)	0.14 (0.33)	67 (142)	0.38 (0.58)
	50%	119 (353)	0.37 (0.90)	257 (404)	0.78 (1.01)
	75%	1 (10)	0.02 (0.14)	75 (144)	0.37 (0.50)
	100%	0 (0)	0.00 (0.00)	20 (83)	0.09 (0.37)

AOI: areas of interest; SD: standard deviation.

Table 5.

Results from the 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) × 5 (area of interest [AOI]: 0%, 25%, 50%, and 75% vs 100%) within-subjects ANOVAs on positioning gaze duration and positioning fixation frequency.

Effect	Positioning gaze duration (ms)					Positioning fixation frequency (n)
Effect	F	df_b	df_w	p	$η_{p}^{2}$	F	df_b	df_w	p	$η_{p}^{2}$
Number type	2.09	1	51	.15	.04	3.35	1	51	.07	.06
Structure of notation system	25.92	1	51	< .001***	.34	53.85	1	51	< .001***	.46
AOI	12.12	1.12^a	56.89^a	.001**	.19	14.89	1.49^a	75.95^a	< .001***	.23
Number type × structure of notation system	0.39	1	51	.53	.01	1.38	1	51	.25	.03
Number type × AOI	2.70	1.89^a	96.35^a	.08	.05	1.38	3.39^a	172.67^a	.25	.03
Structure of notation system × AOI	5.64	1.89^a	96.32^a	.01**	.10	7.97	3.17^a	161.86^a	< .001***	.14
Number type × structure of notation system × AOI	3.58	2.13^a	108.76^a	.03*	.07	7.09	3.19^a	162.70^a	< .001***	.12

AOI: areas of interest.

Greenhouse–Geisser corrections used.

p < .05; **p < .01; ***p < .001.

We found a significant main effect of structure of the notation system for both positioning gaze duration and positioning fixation frequency, with participants spending more time looking across the number line for non-place-value-based compared with place-value-based numbers. The main effect of AOI was also found to be significant for both positioning gaze duration and positioning fixation frequency, wherein participants spent the most time looking in the 50% AOI. No main effect of number type was found to be significant in either ANOVA on positioning gaze duration or positioning fixation frequency. We also found a significant two-way interaction between structure of the notation system and AOI which was qualified by a significant three-way interaction for both positioning gaze duration and positioning fixation frequency (see Table 5 for statistical values, see Figure 4 for visualisation). Post hoc comparisons were conducted for both dependent variables by analysing the number type by structure of the notation system interaction within each AOI along the number line. The pattern of results was the same for both positioning gaze duration and positioning fixation frequency and will thus be summarised together. A significant main effect of structure of the notation system was found in the 25% AOI (positioning gaze duration: F(1, 51) = 6.32, p = .02, $η_{p}^{2} = . 11$ ; positioning fixation frequency: F(1, 51) = 11.44, p = .001, $η_{p}^{2} = . 18$ ), the 50% AOI (positioning gaze duration: F(1, 51) = 11.56, p = .001, $η_{p}^{2} = . 19$ ; positioning fixation frequency: F(1, 51) = 12.72, p = .001, $η_{p}^{2} = . 20$ ), the 75% AOI (positioning gaze duration: F(1, 51) = 24.54, p < .001, $η_{p}^{2} = . 33$ ; positioning fixation frequency: F(1, 51) = 7.13, p < .001, $η_{p}^{2} = . 53$ ) and the 100% AOI (positioning gaze duration: F(1, 51) = 17.20, p < .001, $η_{p}^{2} = . 25$ ; positioning fixation frequency: F(1, 51) = 21.92, p < .001, $η_{p}^{2} = . 30$ ), with participants looking longer in the non-place-value-based compared with the place-value-based conditions. No significant main effect of number type was found in any of the AOIs of the number line. Finally, a significant interaction between number type and structure of the notation system was only found in the 100% AOI (positioning gaze duration: F(1, 51) = 20.23, p < .001, $η_{p}^{2} = . 28$ ; positioning fixation frequency: F(1, 51) = 28.31, p < .001, $η_{p}^{2} = . 36$ ). Post hoc comparisons found that, in the 100% AOI (i.e., endpoint), participants looked longer in the separated fractions condition compared with the fractions condition and the natural numbers condition (both ps < .001).

A final analysis in the positioning phase was conducted to examine participants’ eye-movement behaviour associated with returning to the target number AOI. We conducted this analysis to gauge the differences between the notations in terms of participants’ re-encoding of the target number. Thus, we conducted a 2 (number type: natural vs rational) × 2 (structure of the notation system: place-value-based vs non-place-value-based) within-subject ANOVAs on participants’ probability of revisiting the target number AOI. A significant main effect of structure of the notation system, F(1, 51) = 9.59, p = .003, $η_{p}^{2} = . 16$ , and of number type, F(1, 51) = 40.44, p < .001, $η_{p}^{2} = . 44$ , was found for the probability of revisiting the target AOI. These main effects were qualified by a significant interaction effects between number type and structure of the notation system for probability of revisiting the target AOI, F(1, 51) = 28.19, p < .001, $η_{p}^{2} = . 36$ (see Figure 5).

Post hoc comparisons showed that participants had a higher probability of revisiting the target AOI for fractions than for decimals (p < .001) and separated fractions (p < .001). In addition, they had a higher probability of revisiting the target number AOI for natural numbers than separated fractions (p = .003). However, participants showed no differences in the probability of revisiting the target AOI between natural numbers and decimals (p = .43).

Discussion

Previous studies have extensively demonstrated that adults find the magnitude of rational numbers more difficult to understand than natural numbers (Siegler, 2016; Siegler et al., 2011). Within the class of rational numbers, the magnitude of decimals is easier to understand than fractions (DeWolf et al., 2015; Hurst & Cordes, 2016, 2018; Iuculano & Butterworth, 2011; Resnick et al., 2019; Van Hoof et al., 2018). Iuculano and Butterworth (2011) directly compared adults’ numerical magnitude understanding of natural numbers, decimals, and fractions in an NLE task and found that the magnitude of fractions is more difficult to understand compared with the magnitude of natural numbers and decimals. Although the authors did not provide an explanation for this result, one possible explanation could be the structure of the notation system: Natural numbers and decimals are written in a place-value-based notation while fractions are not (Tian & Siegler, 2018)). To clarify, natural numbers and decimals fit within the place-value structure of the Arabic numeral system (i.e., each digit is 10 times smaller than the digit preceding it), whereas fractions are written in a bipartite structure, with a numerator and a denominator which must be related to each other and are not typically related to multiples or powers of 10. Starting from this explanation, the current study aimed to investigate two potential sources of difficulty in adults’ numerical magnitude understanding in the context of an NLE task by means of a 2 × 2 within-subjects design: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This design resulted in four conditions which differed with respect to the notation that had to be positioned on the number line: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to using the performance measures typically used in NLE research (i.e., accuracy and response times), eye tracking was implemented to examine adults’ solution process (i.e., when and where they looked throughout the NLE task) in two consecutive phases of a trial. Specifically, we divided participants’ eye-movement data into an encoding and a positioning phase. The encoding phase included the data from participants’ first investigation of the target number while the positioning phase included the data for the rest of the trial (i.e., positioning the target number on the number line and potentially revisiting the target number). In both phases, we examined gaze duration and fixation frequency. In the positioning phase, we additionally analysed the probability of revisiting the target number. We argue that the implementation of eye tracking and the data from each of the phases provides theoretical insights into adults’ numerical magnitude understanding of the different notations (e.g., less time processing a certain aspect of the task is associated with better understanding of the magnitude), which would not be possible using response time data alone.

The overall pattern of results from the current study illustrates differences in adults’ numerical magnitude understanding across the respective notations. Adults’ numerical magnitude understanding for fractions required the most effort, followed by decimals and separated fractions, and finally, natural numbers. In the following, we have elaborated on the theoretical insights gained about each notation, both on their own and relative to one another. More specifically, we will now provide a summary of results as well as explanations for these differences in numerical magnitude understanding across each notation.

First, taking all findings together, participants found the magnitude of fractions the most difficult notation to understand. We found higher PAE for rational compared with natural numbers. This finding is in line with our first hypothesis (i.e., Hypothesis 1) and follows the same general pattern as previous research (DeWolf & Vosniadou, 2011; Iuculano & Butterworth, 2011; Vamvakoussi et al., 2012). Moreover, as expected based on our second hypothesis (i.e., Hypothesis 2), participants had lower PAE when estimating place-value-based numbers compared with non-place-value-based numbers. It should be noted that, in absolute terms, the PAE for all notations in the current study were quite accurate (i.e., 3.76% for natural numbers; 4.13% for decimals, 4.25% for separated fractions, and 4.62% for fractions). Even the PAE of the fraction notation was quite accurate and similar to the PAE of adults in previous NLE studies with natural numbers (4.49% on an empty 0–1,000 number line, Ashcraft & Moore, 2012; 3.23% on an empty 0–100 number line, Huber, Moeller, & Nuerk, 2014; 1.75% on an empty 0–1,000 number line, Peeters et al., 2017).

In terms of response times, the results partially supported our first and second hypotheses. Participants took longer to estimate fractions than natural numbers, decimals, and separated fractions, suggesting that the response time only increased when a notation was rational and non-place-value-based. So, the response time findings did not fully mirror those of the PAE: The two sources of difficulty had an additive effect on PAE but an interactive effect on response times. However, both the PAE and RT clearly showed that adults found the magnitude of fractions the most difficult to understand.

When looking at the eye-tracking results, the data from the encoding phase did not reveal a difference between natural and rational numbers as we had predicted in our third hypothesis (Hypothesis 3). Rather, we found that participants’ initial gaze duration on the target number AOI differed between all notations: They spent the longest time encoding fractions, followed by natural numbers, then decimals, and finally, separated fractions.

In the positioning phase, we examined participants’ fixations on the number line AOIs (0%, 25%, 50%, 75%, and 100%). Results showed that, for all but the 0% AOI, participants looked longer when estimating non-place-value-based numbers (fractions and separated fractions) compared with when they estimated place-value-based numbers (natural numbers and decimals). These findings partially supported our fourth hypotheses (i.e., Hypotheses 4a and 4b), as we found only differences between place-value-based and non-place-value-based notations but not between natural and rational numbers for both gaze duration and fixation frequency.

Finally, we also examined participants’ revisits to the target number AOI in the positioning phase. Partially in line with our fifth hypothesis (i.e., Hypothesis 5), we observed that not only the number type but also the structure of the notation system influenced the probability of revisiting the target number AOI such that there were more revisits to the target number AOI when participants estimated fractions compared with decimals and separated fractions.

Taking all the evidence from the present study into account, it is clear that adults understood the magnitude of fractions the least well (the rational, non-place-value-based notation) compared with the other notations. We have three interconnected explanations for why this might be. First, compared with the natural numbers, decimals, and separated fractions notations, which all consist of one holistic unit (e.g., 214, 0.214, 3), fractions are the only notation with a bipartite structure (e.g., 3/14). As a result, adults most likely estimated the ratio between the numerator and denominator before positioning the number on the number line (Siegler & Thompson, 2014), which is evidenced by the higher encoding times and the higher total response time for fractions compared with the other notations. Second, this estimation of the fraction’s ratio was, for most of the fractions, by definition, not precise, which explains why participants’ PAE for the fractions notation was higher than for the other notations (see also DeWolf et al., 2014, for similar results in a magnitude comparison task; for further elaboration on these imprecise ratio estimations, see below). Third, when working with fractions, participants might have forgotten or wanted to verify the outcome of their calculation or estimation with the given target number, something which they were likely more inclined to do for this notation compared with the others, which explains why fractions elicited longer re-encoding times than the other notations.

Second, compared with fractions, participants seemed to have a better understanding of the numerical magnitude of decimals and separated fractions, but still had more difficulty with them than natural numbers. The observation that decimals were understood better than fractions as evidenced by the lower PAE, shorter total response time as well as shorter encoding, positioning, and re-encoding times has also been found in previous studies (DeWolf et al., 2015; Hurst & Cordes, 2016, 2018; Iuculano & Butterworth, 2011; Resnick et al., 2019; Van Hoof et al., 2018; Wang & Siegler, 2013). In contrast to previous research, decimals were less accurately estimated than natural numbers. A possible explanation for this unexpected result might be that the decimals in our study had three digits after the decimal point (i.e., 0.214) compared with previous studies which only had two digits after the decimal point (i.e., 0.21; Iuculano & Butterworth, 2011). Moreover, our study had natural numbers consisting of three digits on a 0–1,000 number line (i.e., 214) while other studies have used two on a 0–100 number line (i.e., 21). Arguably, the familiarity of each of these digit lengths, might have had an influence on participants’ estimation accuracy. Perhaps the difference between natural numbers and decimals might not be evident when there are only two digits (after the decimal) involved, since both are fairly familiar, for example, when shopping, we often see prices, such as US$25 and US$0.25. However, the differences between these two notations might emerge when three digits (after the decimal) are involved, such as in our study since, for example, 214 is still fairly familiar (e.g., US$214), while 0.214 is not (e.g., US$0.214). Our response times and eye-tracking results did not reveal any other differences between decimals and natural numbers.

Third, the finding that the magnitude of separated fractions seem to be understood less well than natural numbers is not only a novel result, but also a relevant one regarding the structure of the notation system. Although both notations are natural, it seems as though the non-place-value-based notation of the separated fractions produced a disadvantage compared with the place-value-based notation of the natural numbers. This disadvantage might be explained by participants needing to estimate the position of target numbers on an atypical range of the number line, where the endpoint of the number line is not a round number like 100 or 1,000, as in the separated fractions notation (e.g., 0–14). This is in opposition to estimating target numbers on a typical range of the number line in the natural numbers notation (i.e., 0–1,000), since the NLE task with atypical number lines is more difficult than with typical number lines, as previous research has shown (Di Lonardo et al., 2019; Hurst et al., 2014; Luwel et al., 2018). Moreover, in the positioning phase, we found that in the 100% AOI only, number type and structure of the notation system both influenced participants’ gaze durations and fixation frequencies: They spent more time looking at this AOI when estimating a separated fraction compared with natural numbers, decimals, and fractions. Unique to the separated fractions notation, the endpoint of the number line changed each trial in the separated fractions notation, whereas it remained constant in the other notations. Given that participants needed to update the scale of the number line for each trial, they had more information to encode and retain in that condition compared with the other notations. Furthermore, even though the separated fractions notation is reminiscent of the atypical number line condition from previous research (e.g., Luwel et al., 2018), the separated fractions notation in our study was even more difficult compared with the atypical number lines from previous studies: Participants had to update the scale of the number line every trial, whereas this was not the case in previous atypical number line studies.

To summarise, this study observed that adults understand the magnitude of fractions the least well as they are written in a notation that contains both sources of difficulty (i.e., they are rational and non-place-value-based). When a notation has only one of these sources of difficulty (i.e., rational but also place-value-based, like decimals, or non-place-value-based but also natural, like separated fractions), adults are able to understand its magnitude better than when a notation has both sources of difficulty. Finally, when none of these sources of difficulty are present in a notation (i.e., natural and place-value-based, like natural numbers), adults have the best understanding of its numerical magnitude.

In terms of methodology, it is noteworthy to mention that the current study added a novel methodological element to the NLE literature. Using eye tracking in the current study, we surpassed previous research by not only yielding location-specific information regarding where participants looked on the screen but also, based on our division of a trial into the encoding and positioning phases, when this occurred throughout the trial. By breaking down the process data extracted from eye tracking, we were better able to understand how participants solve the NLE task for each of the different notations. As such, these eye-tracking data provided us a first glance into the details of how adult participants solve the NLE task for natural numbers, decimals, fractions, and separated fractions as well as into the similarities and differences between each of the notations in terms of eye-movement behaviour. For instance, if we examine our encoding phase findings on their own, they revealed how long adult participants initially looked at each notation in our study. Specifically, the encoding phase findings suggested that participants encoded the target number longer when there was more information to encode and when that information was more complex to understand. They spent the most time encoding fractions, which, as mentioned, required estimating the ratio between the numerator and denominator given their bipartite structure. Next, participants spent the second most amount of time encoding natural numbers and decimals. Compared with fractions (e.g., 3/14) which have two values to encode to estimate it accurately, both decimals and natural numbers require encoding one value (e.g., 214 or 0.214), which could explain why participants spent less time encoding these numbers compared with fractions. However, it should be noted that there is an ongoing debate regarding how multi-digit numbers are processed—holistically (e.g., Thomas & Morwitz, 2009) or decomposed (e.g., Huber et al., 2016). The spacing of the digits within our target number AOI does not allow us to provide more specificity in terms of what participants looked at when they fixated within the target number box to be able to draw any conclusions about how multi-digit numbers were processed in the present study. In addition, despite decimals having an additional “0.” at the start of the notation compared with natural numbers, participants might have spent similar time encoding these notations since they resemble each other (i.e., are both in a place-value-based notation structure). Finally, participants spent the least amount of time encoding separated fractions, which included only the single- or double-digit numerator from the equivalent fraction (e.g., 3).

A related second example of the information gained from our two phases of eye-tracking data is the short amount of time spent encoding the separated fraction notation and the longer time spent in the 100% AOI of the number line. Participants needed to encode not only the “numerator” (which was presented in the target number AOI) during the encoding phase but also the “denominator” (which was the endpoint label of the number line) which was only revealed in the positioning phase, to position a separated fraction on the number line. As a result, our eye-tracking data also revealed that this short gaze duration in the encoding phase for the separated fractions notation was compensated by longer gaze durations in the endpoint AOI of the number line, compared with the other notations.

As a final example, our eye-tracking data allowed us to compare two notations with the same components (i.e., the fractions and separated fractions notations) and examine the differences between them based on participants’ pattern of results. As mentioned, both performance and eye-tracking data revealed that participants had a better understanding of the numerical magnitude of separated fractions compared with that of fractions. Our performance data showed that participants had longer response times for fractions than separated fractions. Our eye-tracking data further showed that participants encoded fractions longer than separated fractions, re-encoded fractions longer than separated fractions, yet their eye-movement behaviour on the number line did not substantially differ for fractions and separated fractions. Thus, from our eye-tracking data alone, we are able to extrapolate that it is not the participants determining where the fraction is positioned on the number line that makes fractions less easy to understand compared with separated fractions, but rather determining the ratio of the fraction during encoding and re-encoding.

It should be acknowledged that the design of the present study cannot fully exclude potential confounds that are related to the unique features of the conditions we created, which could have driven differences between them beyond number type and notation system. To exemplify, the digits in the target number varied between the natural numbers (two–three digits), decimals (three–four digits), fractions (two–four digits), and separated fractions (one–two digits). As another example, the range of the number line differed between natural numbers (0–1,000), decimals/fractions (0–1) and separated fractions (0 denominator of fraction). A third and final example is that the range of the number line was constant in the natural numbers, decimals, and fractions notations while this was not the case for the separated fractions. However, it should also be noted that controlling for these potential confounds within this design is inherently difficult given the nature of each of the notations (e.g., each notation has an inherent range of the number line with which it is associated).

A second methodological consideration is related to the stimuli set that was used for the fractions condition of the present study. As discussed above, we argue that participants’ higher error rates for fractions compared with the other notations was due to an imprecise estimation of the ratio between the numerator and denominator of the fractions. However, it should be noted that nine of the 54 fraction stimuli in the current study were fractions that could be relatively easily and precisely converted to decimals (i.e., 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, and 4/5; see Binzak & Hubbard, 2020; Liu, 2017). If participants were relatively easily and precisely converting these nine “easy” fractions, we would expect to see a difference in their PAE of these fractions and the remaining 45 “hard” fractions. To explore this idea, we conducted a paired-samples t-test and found that participants’ PAE for the “easy” fractions was significantly lower than their PAE of the “hard” fractions, t(51) = 2.629, p = .011, d = .365, M_easy = 4.130, SD_easy = 1.544, M_hard = 4.695, SD_hard = 1.484. Moreover, if participants were relatively easily and precisely converting these nine “easy” fractions to decimals, we could expect to see no differences in PAE between these nine “easy” fractions compared with their nine decimal equivalents. This was confirmed by a paired-samples t-test (t(51) = 0.080, p = .936, d = .011, M_decimals = 4.109, SD_decimals = 1.655, M_fractions = 4.130, SD_fractions = 1.544). In contrast, we did find a significant difference in participants’ PAE of the remaining 45 “hard” fractions and their PAE of the 45 decimal equivalents (t(51) = 2.370, p = .022, d = .329, M_decimals = 4.138, SD_decimals = 1.372, M_fractions = 4.695, SD_fractions = 1.484), suggesting that participants imprecisely estimated the ratio of these 45 remaining fraction stimuli. Altogether, these findings suggest that participants are able to estimate the ratio of these nine “easy” fractions relatively easily and precisely, but not for the 45 “hard” fractions. It is recommended that future studies take this distinction between fraction stimuli into account.

Finally, from an educational standpoint, participants estimated separated fractions quite accurately, and notably, tended to estimate them more accurately than fractions, hinting at a potential avenue for future research on the educational use of this type of task. Given that a separated fraction can be considered as a rational number “disguised” in a natural number context, future studies could examine whether the separated fractions task could be used as a tool that would make the transition from natural to rational number numerical magnitude understanding easier for students. For instance, future research could examine the effect on children’s magnitude understanding of fractions of an educator explaining that one way to view the fraction “3/14” is to regard it as a “3” on a 0–14 number line and compare this with a business-as-usual fraction lesson. This could help students learn that “3” would be positioned on a 0–14 number line in the same place as 3/14 would be placed on a 0–1 number line, which might improve children’s fraction magnitude understanding more than a business-as-usual fraction lesson.

Supplemental Material

sj-docx-1-qjp-10.1177_17470218221094577 – Supplemental material for The structure of the notation system in adults’ number line estimation: An eye-tracking study

Supplemental material, sj-docx-1-qjp-10.1177_17470218221094577 for The structure of the notation system in adults’ number line estimation: An eye-tracking study by Kelsey J MacKay, Filip Germeys, Wim Van Dooren, Lieven Verschaffel and Koen Luwel in Quarterly Journal of Experimental Psychology

Footnotes

Acknowledgements

The authors would like to thank all those who were able to participate in this study as well as numerous colleagues for the discussion regarding these findings.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was partially supported by the Internal Funds of KU Leuven (grant number: KA/16/009).

ORCID iD

Kelsey J MacKay

Data accessibility statement

The data from the present experiment are publicly available at the Open Science Framework website: .

Supplementary material

The supplementary material is available at: .

Notes

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