Spontaneous Focusing on Numerosity Is Linked to Numerosity Discrimination in Children and Adults

Mattan S. Ben-Shachar1*, Svetlana Lisson1* Dalit Shotts-Peretz1, Minna Hannula-Sormunen2, and Andrea Berger 1Department of Psychology, Ben-Gurion University of the Negev, Faculty of Humanities and Social Sciences and Zlotowski Center for Neuroscience, Beer Sheva, Israel 2Department of Teacher Education, University of Turku, Finland

Spontaneous focusing on numerosity (SFON) is the tendency to spontaneously address exact numerosity in the environment without prompting. While previous studies have found children’s SFON to be a stable, domain-specific predictor of mathematical abilities throughout development, it is unclear whether SFON reflects individual differences in quantitative processing. This study examined the relationship between SFON and the acuity of the Approximate Number System (ANS) in children and adults. To measure adults’ SFON, we developed a numerosity bias task (NBT). In children and adults, better ANS acuity was related to higher tendency to spontaneously focus on numerosity. Additionally, in adults, SFON was related to higher mathematical academic achievements. These findings suggest an interplay between SFON and ANS acuity, indicating a mechanism where increased ANS acuity makes numerosity elements in the environment more salient, while early self-initiated numerical practice promotes fine-tuning of the ANS. Possible implications of these reciprocal developmental pathways are discussed. Keywords: spontaneous focusing, numerosity, Weber ratio, mathematical abilities, mathematical achievements

Research on mathematical development during who exhibit SFON tend to identify numerosities and preschool shows that children display considerable incorporate these impressions into decision making or individual differences in the rate of early behaviour, all without any explicit trigger, mathematical skills and knowledge acquisition, and in encouragement, or guidance to do so (Hannula and the fusion of acquired concepts and abilities (Hannula- Lehtinen 2005). Sormunen, McMullen, and Lehtinen 2019). A recent, So far, such individual differences in young growing body of research suggests that children’s children (up to 3 years of age) have been explored SFON, the process of attending to the aspect of the mainly using a number of imitation tasks, developed number of objects or incidents in a self-initiated by Hannula and Lehtinen (2005), that evaluate this manner (without being prompted by others), has an spontaneous tendency. In these tasks, the important role in early numerical development experimenter displays some numerosity properties but (Hannula and Lehtinen 2005; Hannula, Lepola, and at no point are they explicitly marked as important, Lehtinen 2010; Hannula, Räsänen, and Lehtinen 2007; and special care is taken to avoid any wording that Hannula-Sormunen, Lehtinen, and Räsänen 2015; could suggest that the tasks are mathematical or McMullen et al. 2019; Rathé et al. 2016). quantitative in nature. In these tasks, children have Environmental elements of numerosity, even if been found to differ in their tendencies to present, are not always focused on or used in one’s spontaneously focus on numerosity: Some children actions, and studies have found individual differences would imitate the exact actions of the experimenter in the tendency to focus on this aspect of the (e.g., feeding a puppet “sweets”) with no regard to environment without any guidance or numerical numerosity, whereas others would imitate the action, while also attending to the aspect of numerosity (e.g., * These authors contributed equally to this work. feeding the puppet the exact number of sweets as was Corresponding Author: Mattan S. Ben-Shachar Email: [email protected] given by the experimenter, or regarding it in other ways, such as counting the number of distributed context (Hannula and Lehtinen 2005). Individuals sweets, asking if they could give more or fewer sweets

1 Gender, Choices, and STEM 2 than were given by the experimenter, etc.). These The ANS is characterized by an imprecise ability individual differences in the children’s disposition to to distinguish quantities by relying on an estimation focus on environmental numerosites were found to be derived from a distribution of activations on the relatively stable, specifically in preschool ages, as mental number line (Izard and Dehaene 2008; Mou demonstrated by positive correlations for SFON and Van Marle 2014; Stoianov and Zorzi 2012), with scores between 4 and 5 years of age (Hannula and distribution overlap increasing in correspondence with Lehtinen 2005). an increase in numerosity (Izard and Dehaene 2008; A number of studies have demonstrated Stoianov and Zorzi 2012). According to the ANS correlations between SFON and children’s model, the distinction between quantities is made on mathematical abilities (Batchelor, Inglis, and Gilmore the basis of the ratio between them, in accordance 2015; Bojorque et al. 2017; Hannula and Lehtinen with Weber’s law (Dehaene 2003), which states that 2005; Hannula et al. 2010, 2007; McMullen, Hannula- the discrimination threshold between two given Sormunen, and Lehtinen 2015). For example, stimuli (of any given type of sensory modality) children’s mathematical abilities at the age of 3.5 increases by a given factor, as the intensity of the years predicted their SFON tendency at the age of 4 stimulus grows (e.g., Jordan and Brannon 2006). This years, which, in turn, predicted later mathematical discrimination ability improves as we develop, with abilities at the ages of 5 and 6 years which could not the greatest improvement occurring in the first year of be accounted for by insufficient enumeration skills, life (Brannon, Suanda, and Libertus 2007; Halberda et linguistic abilities, or difficulties in comprehending al. 2012; Lipton and Spelke 2004; Xu and Spelke task instructions (Hannula and Lehtinen 2005). Other 2000). For example, day-old infants are able to studies showed that, for kindergarten-aged children, differentiate quantities with a ratio of 1:3 (Izard et al. SFON predicted mathematical abilities over and 2009), at the age of 6 months, infants are able to above other cognitive skills either two or six years discriminate by a ratio threshold of 1:2 (Brannon et al. later -- when the children were either in second or 2007; Xu and Spelke 2000), and, at the age of 9 fifth grade -- but again did not predict their reading months, by a ratio of 2:3 (Lipton and Spelke 2003, skills at this age (Hannula et al. 2010; Hannula- 2004; Xu and Arriaga 2007). By 3-4 years of age, Sormunen et al. 2015; McMullen et al. 2015; Nanu et children are already capable of discriminating ratios of al. 2018). This particular attentional bias is considered 3:4, and adult humans can discriminate quantities with an important precursor to the exact number a ratio of 7:8, with some even succeeding with a ratio recognition process, because using exact number of 9:10 (Halberda and Feigenson 2008). The relative information in action requires analytical, conscious weight of learning experience versus brain maturation processing (Hannula and Lehtinen 2005; Hannula- on these life-long changes in ANS acuity has yet to be Sormunen et al. 2015). Simply put, one must decide to established (Geary 2013). regard specific information (among all of the Although the ANS plays a crucial role in human perceived information) as relevant, in order to act on cognition, individuals differ in their resolution of it. Although the contribution of enhanced mental quantitative processing and numerosity discrimination practice with numerosities, consequent to higher (Halberda and Feigenson 2008). One of the most well- SFON tendency, is well-established (Hannula- known computerized tasks used to measure individual Sormunen et al. 2019; McMullen, Hannula-Sormunen, discrimination ratios is the Panamath dot- and Lehtinen 2014; Rathé et al. 2016), it is unclear discrimination task (Halberda and Feigenson 2008). why numerosity elements in the environment are more During this task, participants are presented with two salient for some children compared to others. dot arrays and are asked to indicate which is more Considering the early manifestation of these numerous, with the task becoming more difficult as individual differences along with the domain- the ratio between both dot arrays decreases (as a result specificity of the SFON tendency, a possible of Weber’s law). Differences in individual Weber underlying influence could be the ANS, which is an ratios (the ratio required for successful discrimination) innate and adaptive system for processing quantities estimated using this task have been found to be a and is a congenital system of numerical representation domain-specific marker for mathematical abilities (Carey 2009; Spelke and Kinzler 2007) that is (Halberda and Feigenson 2008; Halberda, Mazzocco, common to human, and some nonhuman, animal and Feigenson 2008). For example, performance on species (Carey 2009; Feigenson, Dehaene, and Spelke the Panamath task has been found to correlate and 2004; Rugani, Vallortigara, and Regolin 2013). even predict academic mathematical achievements Gender, Choices, and STEM 3 over a 6-month period (Halberda et al. 2008; Libertus, Participants. Feigenson, and Halberda 2013). Moreover, children’s Participants were 51 preschoolers (19 boys) with performances on a symbolic math task have been ages ranging from 3.05 years to 4.9 years (M=3.03 shown to improve following numerosity years, SD=0.39 years). None of the children had a discrimination practice, using a variation on the history of developmental disabilities, learning Panamath task (Wang et al. 2016), suggesting that disabilities, or attention deficits. Participants were ANS precision may have some role in the recruited via the local university’s daycare center development of symbolic math performance, although through an invitation letter distributed by the this evidence is still very limited.1 kindergarten teachers, as well as by word of mouth. The demonstrated predictive relationship between Parents who expressed interest in the study were SFON with later mathematical abilities raises a central contacted by phone and invited to participate in the question regarding the process underlying the study. Prior to the beginning of the experimental individual differences found in children’s SFON session, parents signed informed consent forms. tendency: specifically, whether these individual Although demographic information was not collected, differences in the disposition to attend to surrounding as children were recruited from the University’s numerosities are due to greater resolution of daycare center, a medium-high SES can be assumed. quantitative processing. We hypothesized that SFON Data from five children were discarded, because tendency would be correlated with better numerosity they showed outlier performance in the Panamath task discrimination. Moreover, we hypothesized that such (see details below, and Table 1) and seemed a tendency, i.e., to focus on the numerosity dimension inattentive or disengaged during the task. Thus, data of the environment, should also be present in from a total of 46 children (16 boys) were used in the adulthood. Therefore, in the present study, we tested final data analysis. these hypotheses both in children (Experiment 1) and Apparatus in adults (Experiment 2). SFON imitation tasks. Two sets of imitation tasks were used to assess Experiment 1: SFON and Numerosity individual SFON, in order to create a reliable SFON Discrimination in Children indicator, generalized across task context. Both tasks were carried out in a quiet room either in our lab or in In this experiment, we measured both (a) the the child’s daycare center. Throughout the imitation number of times each child used exact numerosities tasks, the experimenters avoided using any wording during two of the imitation tasks developed by that might suggest that the tasks were mathematical or Hannula and Lehtinen (2005) and (b) the child’s quantitative. The tasks included only small Weber fractions on the Panamath task. We numerosities within a very small number range (1-2), hypothesized that SFON would be negatively related which all children would be able to handle. These to Weber fractions, such that children with higher imitation tasks are especially suitable for the age SFON tendency (i.e., focused on exact numerosities in range of the study’s cohort, following the SFON more trials than average) would show better measurement criteria that tasks must only include discrimination between numerosities (i.e., a smaller small numerosities, such that all children should be Weber fraction). able to recognize them (Rathé et al. 2016). Both the small numerosities and the short instruction pose a minimal confounding effect on working memory and Method number recognition skills. In the first task (Hannula & Lehtinen, 2005), a toy parrot capable of “swallowing” was placed on the 1 When referring to numerosity perception, it should be mentioned table in front of the child, and a bowl of small, that changes in numerosity (as in real life) are always colorful stone “candies” (~3 cm in diameter) was accompanied by changes in other continuous perceptual placed in front of the parrot. The experimenter then properties, such as expanded item size, total surface area, density, introduced the child to Polly (the parrot) and said: and circumference, and hence may be seen as part of one continuous variable (Leibovich et al., 2017). Still, numerosity has “Watch carefully how I feed Polly, and then you do been demonstrated to play a role in nonsymbolic number just like I did.” The experimenter, sitting next to the perception beyond the effects of continuous visual properties (e.g., child, then put two candies, one at a time, into the Cordes and Brannon 2008). However, this debate is beyond the parrot’s mouth, where they disappeared into the scope of the current study. Gender, Choices, and STEM 4 parrot’s stomach. The child was then told: “Now, you – once from each half – a Spearman-Brown do exactly like I did.” After the child completed correction was applied (Webb, Shavelson, and “feeding” the parrot, this procedure was repeated three Haertel 2006). This resulted in a split-half more times with the following number of candies: one reliability coefficient of 0.698, indicating that candy in the second trial, two candies in the third trial, children were consistent in their SFON tendency and one candy in the final, fourth trial. across both tasks. In the second task (Hannula, Mattinen & Panamath task. Lehtinen, 2005), a toy dump truck with an open- Each participant’s threshold of discrimination box bed, a small plastic shovel, and a container (i.e., Weber fraction) was assessed using the full of gravel was placed on the table. The Panamath task (Halberda et al. 2012). This task is a experimenter then said: “Watch carefully how I computerized dot-discrimination task (review and put gravel in the truck, and then you do just like I experience the task at http://panamath.org/). In each did.” The experimenter then scooped two scoops trial, two arrays of dots (one of green dots and one of of gravel, one at a time, into the dump truck, orange dots) are presented side by side, and using the toy shovel. The child was then told: participants must judge which of the two arrays is “Now, you do exactly like I did.” After the child more numerous (see Figure 1). The difficulty of the completed filling the dump truck, this procedure task was adaptive and based on the participants’ performance, with the ratios getting smaller (making was repeated three more times with the following discrimination more difficult), after correct trials, and number of scoops: one scoop in the second trial, larger (making discrimination easier), after incorrect two scoops in the third trial, and one scoop on the trials. Such adaptive tasks are better matched to final, fourth trial. subjects' ability levels and take less time to Individual SFON scores were calculated as 8 administer, while also increasing the reliability of the binary outcome single trials (4 trials in each task), measured outcome (Draheim et al. 2019). Each trial evaluated in real time (during the session) by a began only after the experimenter made sure the child member of the research team. On each trial, a was attending to the screen. child was scored as focusing on numerosity, if she or he imitated the correct numerosity, and/or if she or he was observed presenting any of the following quantifying acts: (a) utterances including number words (e.g., “I’ll give him three candies”); (b) use of fingers to express numbers; (c) counting acts, like whispering number word sequences and indicating acts using their fingers; (d) other comments referring either to quantities or counting (e.g., “I miscounted”); or (e) Figure 1. Sample Panamath trial for children. In this interpretation of the goal of the task as trial, the green array contains 12 dots and the orange quantitative (e.g., “I gave exactly the right array contains five dots. Individuals with Weber number of candies”). An initial, small pilot fraction greater than 1.4 would be able to detect that sample of children was video-recorded, and the the green array is more numerous than the orange videos were sent to Prof. Minna Hannula- array. Sormunen for verification of the SFON Each participant performed 8 minutes’ worth of procedures and the coding. Data collection started trials (one block of roughly 130 trials), and a personal only after a high reliability was reached. Weber fraction was calculated based on performance The correlation between the scores from the throughout the task. Individual Weber fraction scores “parrot” task with those from the “truck” task was reflect the relative size of change in numerosity that r=0.536. Given that split-half correlations an individual requires to be able to discriminate underestimate the overall stability of the total between the numerosities on 75% of the trials (with score due to containing measurement error twice chance at 50%). For example, when presented with one array of eight dots, an individual with a Weber Gender, Choices, and STEM 5 fraction of 0.6 would need the second array to have a categorical outcome variables (such as binary numerosity greater by outcomes), and they also have the added benefit of

N dots ×Weber Fraction=8× 0.6≈ 5 dots (a total of greater statistical power (Jaeger, 2008; Warton & Hui, 8+5=13 dots), to be able to discriminate between the 2011; Wilson et al., 2013). Because age was related to arrays 75% of the time. SFON scores, r ( 44 )=0.44, 95%CI [0.17,0.64 ], As mentioned above, data from five participants p=.002, we examined the relationship between with Weber fractions exceeding their age-appropriate SFON and Weber fractions, while controlling for age 90th percentile were excluded from analysis.2 by including age as a covariate in the repeated- Procedure. measures logistic regression. Tests to see if the data To keep the testing session short, the children met the assumption of collinearity indicated that were tested in two sessions. In the first session, the multicollinearity was not a concern (VI Fage=1.03). children performed the SFON imitation tasks, and, in the second session, they performed the other tasks. All Table 1 the children recruited from the University’s daycare center performed both sessions at the daycare center. Descriptive Statistics for Experiment 1 A small proportion of the children (n = 8) performed the sessions in our lab. Independent sample t-tests Children Adults showed no statistical difference between session N=46 N=77 locations, neither for SFON score, t(44) = 0.29, p = Mean SD Mean SD 0.77, nor for Panamath test score, t(44) = 0.25, p = Age (years) 3.84 0.37 23.70 2.14 0.79. In all of the cases, the children were tested Weber 0.68 0.45 0.18 0.04 individually in a quiet room, where they sat with the Fraction SFON 1 0.33 0.36 0.28 0.33 experimenters. Parents who came to our lab were SFOC -- -- 0.36 0.30 monetarily compensated with the equivalent of about Strategy (%) -- -- 22.07 41.75 $28. The two sessions were about one week apart ( QRS 2 -- -- 131.16 9.41 M=0.35 weeks, SD=0.64 weeks). In the second Raven 3 -- -- 20.25 5.89 1 session, the experimenter sat with the child in a quiet Average proportion of successful trials, out of all single trials (eight trials in children’s tasks; 24 trials in adult’s room, where the child completed the computerized NBT task). Panamath task on a portable PC. The SFON imitation 2 Only 66 participants were able to provide their QRS. tasks always took place first, ensuring that numerical 3 Data from one participant could not be recovered. context could not be inferred prior to the task. A decrease in Weber fractions was found to predict an increase in the probability to incorporate the Results correct numerosities in the imitation task, such that an All statistical analyses were conducted using R individual with a Weber fraction of 0.5 had a (version 3.5.2; R Core Team 2020) in RStudio probability of 26.4% to attend to the numerosity (version 1.1.463; RStudio Team 2018). element, whereas an individual with Weber fraction of The relationship between SFON and Weber 0.75 had a probability of 15.5%, ¿=1.96, fractions was examined using repeated-measures 95%CI [1.04,3.70], z=−2.074, p=.038 (see logistic regression (a generalized linear mixed effects Figure 2). model using the lme4 package; Bates et al. 2015:4; Kuznetsova, Brockhoff, and Christensen 2017), with single-trial SFON scores (1 = focused on number; 0 = didn’t focus on number) as the dependent variable, random intercepts per participant, and Weber fractions as a fixed-effects predictor (see Table 1, for descriptive sample characteristics). This method was chosen because logistic regression models are more suitable than linear regression models, when analyzing

2 Age-percentile data was provided in the Panamath program. Gender, Choices, and STEM 6

were less precise in their imitation of the numerosity used by the experimenter. Overall, in both studies, 2.5- to 3-year-olds showed hallmark ANS representation properties while performing different variations of SFON tasks, suggesting that greater resolution in quantitative discrimination increases the probability of successfully incorporating exact numerosities into decision making and, ultimately, behavior. Another possible explanation for the present findings could be a subsequent influence of SFON tendency on the acuity of numerosity discrimination. Previous studies of SFON have found evidence of a reciprocal relation between SFON and early numerical Figure 2. Partial residual plot of children’s SFON skills (Hannula and Lehtinen 2005; Hannula- scores as a function of their Weber fractions. The Sormunen et al. 2015), indicating that children’s self- black line is the estimated logistic-regression line, initiated practice in focusing on exact numerosity and controlling for age. On the y-axis, SFON scores are the incorporation of it into everyday situations displayed as the proportion of successful trials out of the eight trials per subject. facilitates enhanced mathematical performance and vice versa. Essentially, the idea is that SFON tendency triggers exact number recognition, facilitating Discussion numerical fluency (Hannula et al., 2007, 2010), which In our first experiment, we investigated the possibly includes the acuity of their analog non- relationship between SFON tendency using Hannula symbolic number representation. This relation aligns and Lehtinen’s (2005) imitation tasks, and ANS acuity with studies showing that quantitative discrimination using the Panamath task in preschool children. As could be enhanced by training (Park & Brannon, predicted, we found that ANS acuity, measured by the 2013; Wilson, Revkin, Cohen, Cohen, & Dehaene, individual Weber fractions, was negatively related to 2006). Importantly, these explanations are not SFON, indicating that the children who were better at mutually exclusive, and both directions of influence discrimination between numerosities had a higher could co-exist: Increased salience of numerical SFON tendency, even when controlling for age. information as a consequence of higher ANS acuity One plausible interpretation of the present findings can explain early individual differences in SFON, is that greater resolution of quantitative processing even when environment is controlled (Rathé et al. may facilitate a higher SFON tendency. As discussed 2016). Higher SFON tendency, in turn, may lead to earlier, lower Weber fraction scores reflect an faster, more robust maturation of the ANS via increased saliency of quantitative information enhanced self-initiated numerical practice, providing a (Brannon et al. 2007; Halberda et al. 2012; Lipton and possible, crucial head start during the early stages of Spelke 2004; Xu and Spelke 2000). Previous work has development. In any case, a specific causal demonstrated that the degree to which numerical relationship between SFON and ANS acuity cannot be information is prominent affects the likelihood of determined in the present study, due to its incorporating this information into behavior. For correlational nature. example, Cantlon, Safford, and Brannon (2010) found that children spontaneously matched visual stimuli Experiment 2: SFON and Numerosity based on numerosity, as opposed to surface area, and, Discrimination in Adults more importantly, that their bias toward numerosity- based matching was affected by the numerical ratio As mentioned, the extant research on SFON has between the two visual stimuli. Similarly, when focused mainly on this tendency in children, yet there examining the distribution of responses in a SFON is no reason to believe that individual differences in imitation task, the size of the errors made in each trial this tendency to focus on numerosity should be was proportional to the numerosity presented in that present only in childhood and not in adulthood. trial (Sella, Berteletti, Lucangeli, & Zorzi, 2016). In Because the SFON imitation tasks were developed for other words, as presented numerosity grew, children children and are not suitable for adults, we designed a Gender, Choices, and STEM 7 computerized numerosity bias task (NBT) that would Apparatus allow us to measure adults’ SFON. Although there are NBT. a few works that have measured SFON in adults, the In this computerized forced-choice task, tasks that were used were either designed to measure participants were instructed to learn, for each block, SFON in a parent-child dyadic setting (Chan, Praus- which of two stimuli would earn them points (as Singh, and Mazzocco 2019) or focused on effects of indicated by feedback following their selection). In task properties on eliciting SFON, not on individual each block, the selection of one stimulus would award differences in adults’ SFON (Chan and Mazzocco them points, but the selection of the other stimulus 2017). would result in a deduction of points. Participants The NBT was designed in adherence to the same were instructed to select one of the two stimuli by criteria laid out by Hannula and Lehtinen’s (2005) pressing either a button on the right with their right children’s SFON tasks: (1) The task should have no index finger, to select the right stimulus, or a button explicit request to regard numerosity as relevant, but on the left with their left index finger, to select the left (2) it should include numerosity elements that could stimulus. Responses were collected using a serial be processed, if focused on, along with other elements response box (SRBox). Participants were instructed to that would compete with the numerosity aspect of the respond as quickly as possible. task. Using this task, we investigated adults’ SFON Each block consisted of eight, nine, 10, or 11 trials. tendency. We hypothesized that, similar to findings Each trial began with a fixation cross appearing for among children, adults would demonstrate variance in 500 ms, followed by two stimuli presented on a black their SFON tendency. Moreover, we hypothesized that background and separated by a gray, vertical line. The these individual differences would be associated with two stimuli always differed from each other on two mathematical abilities (specifically, with numerosity dimensions: the number of objects comprising the discrimination) and achievements (as measured by stimulus (one, two, three, or four) and objects’ color mathematical psychometric scores) but not with (red, green, blue, or yellow). For example, the two general intelligence (Hannula and Lehtinen 2005; stimuli could be three red dots and two yellow dots Hannula-Sormunen et al. 2015). Such findings would (see Figure 3). The numbers of objects and colors support the notion that SFON is a perceptual bias that used to comprise the stimuli were randomly selected not only plays a role in childhood but is also apparent and differed between blocks. in adulthood. Moreover, if adult SFON tendencies Unbeknownst to the participants, each block was were found to correlate with other mathematical divided into two phases: a learning phase that abilities in adulthood, but not with general cognitive consisted of all but the last trial, and a test phase that skills (as measured by Raven scores), it would consisted of the last trial. In the learning phase trials, strengthen the status of SFON tendency as a stable the stimuli were presented until a response was made and specific marker for mathematical abilities. or for a maximal duration of 1,500 ms, followed by feedback indicating whether points were won or lost. Method Throughout the learning phase trials (i.e., all but the last trial of each block), color and numerosity were Participants. consistently paired, such that, for example, the three Participants were 84 undergraduate students ( dots were always red, and the two dots were always

M age=23.59 years, SD=2.19; 29 males) who attend yellow. This allowed participants to learn rather our university. All subjects had normal or corrected- quickly, through trial-and-error, which stimulus to-normal vision. Participants received course credit earned them points (e.g., 2-red; see Figure 3A). or the equivalent of about $14. Because the two stimuli differed on two dimensions, Data from seven participants were discarded, the manner in which participants identified the because they showed outlier performance on the rewarding stimulus could be based on either the color Panamath task (see details below). Thus, data from a of the stimulus (“choosing red awards me points”) or total of 77 participants (M =23.70 years, on the number of items comprising the stimulus -- that age is, its numerosity (“choosing three dots awards me SD=2.14; 26 males) were used in the final data points”). Both strategies would lead to the same analysis. performance, because color and numerosity were paired throughout these trials in each block. Gender, Choices, and STEM 8

Figure 3. Example of a number-versus-color block. (A) Example learning trial, with the right stimulus comprised of three red dots and the left stimulus comprised of two yellow dots. Feedback was given according to selection, allowing participants to learn to select three red dots. (B) The last trial in each block was the test trial. In this final trial, color and numerosity were reverse-paired, producing the stimuli 2-red dots and 3-yellow dots. No feedback was given for this trial.

The last trial in each block was designed to test To account for the possibility that our measure whether identification was based on color or might reflect not low vs. high bias towards numerosity numerosity, and this was done by reverse-pairing the but, instead, a high vs. low bias towards color (i.e., two dimensions. For example, if the stimuli in the selecting the stimulus in the test trial that had the same leading trials were 3-red versus 2-yellow, the stimuli numerosity as that of the reward stimuli in the in the final trial would then be 2-red versus 3-yellow previous trials might be because of the obscurity of (see Figure 3B). If the identification was color-based, the color dimension, the other available dimension to the participant should, in this final trial, select the respond to, as opposed to the prominence of the stimulus comprised of the color that previously numerosity dimension), we separately measured awarded points (e.g. 2-red), but, if learning was participants’ spontaneous focusing on color (SFOC). number-based, the participant should select the This was measured using eight additional blocks in stimulus comprised of the numerosity that previously which the stimuli were also comprised of two awarded points (e.g., 3-yellow). This allowed us to dimensions: color and shape (e.g., blue-triangle vs. measure which dimension (numerosity vs. color) was red-square), with the number of objects kept constant more salient to the participant and, thus, to assess the at one. In total, each participant completed 32 blocks: participant’s SFON tendency. In this last trial, the 24 color-versus-numerosity blocks and eight color- stimuli were presented until a response was made or versus-shape blocks. A short break was given after for a maximal duration of 3,000 ms. No feedback was every eight blocks. given for these test trials. After task completion, participants were asked: “What was the strategy you used throughout the Gender, Choices, and STEM 9 task?” Answers were extemporaneously typed by the numbers and mathematical knowledge in solving participants and coded later on by an experimenter as quantitative problems presented in both verbal and a binary variable, with answers containing any visual form, such as tables and graphs. reference to numerosity coded as 1, and answers Of the 77 participants, 11 were unable to provide containing no such reference as 0. their QRS data. Analyses regarding QRS were Throughout the entire task, the side (right or left) performed on the subset of 66 participants for whom of the computer screen that the rewarding stimulus this data was available. appeared on was balanced and randomly selected Raven test. between trials, for both the learning and test trials. General intelligence was assessed using the Moreover, perceptual variables, such as objects’ size Raven Standard Progressive Matrices (Raven and location on the display, were controlled for and 1960), as a possible covariate with SFON. balanced between trials. The average display area that Participants were given 30 minutes to complete was colored was equal between each of the possible 36 matrices. Raven scores were calculated as the number of objects and the different shapes number of correct matrices solved throughout that (approximately 4,700 pixel2), and the location of each object was selected at random, inasmuch as the task. Of the 77 participants, data from one object’s location did not overlap another object, and participant were lost, due to a corrupted computer the objects of each stimulus were confined to one side file. Analyses regarding Raven scores were of the screen. The task was designed using E-Prime performed on the subset of 76 participants, for software (released candidate 2.0.8.9; Psychological whom this data was available. Software Tools Inc. 2010). Procedure. Individual SFON scores were calculated as the When signing up for the study, the study’s binary outcomes of test trials in the first 24 blocks (the description contained no reference to numerosity or color-vs-numerosity blocks) wherein each participant mathematics, to avoid participants arriving at the lab chose according to the numerosity dimension, as and being primed for numerical thinking in any way. opposed to the color dimension, single-trial SFON After giving their written consent, each participant scores (1 = selection based on numerosity; 0 = was seated approximately 70 cm from a computer selection based on color). Similarly, individual SFOC screen and was given the NBT task instructions, scores were calculated as the set of single test trials which contained no explicit request to regard wherein each participant chose according to the color numerosity as relevant information, only that he or she dimension, as opposed to the shape dimension, in the must learn which stimuli are “correct.” Participants last eight color-versus-shape blocks (1 = selection then completed 10 practice trials (that were not used based on color; 0 = selection based on shape). in the final analysis). After ensuring that the task Panamath task. instructions were understood, the experimenter left the Weber fractions were measured using the adult room, and participants completed the rest of the task. version of the Panamath task that we described in After the last trial, participants were asked to describe Experiment 1. Participants performed two practice the strategy they used during the task. trials, in which feedback was given for selecting After completing the NBT task, participants were correctly or incorrectly. Participants then performed 8 asked to report on their past mathematical minutes’ worth of trials (one block of roughly 264 achievements (i.e., QRS). Participants then completed trials). Feedback was not given on these trials. the Raven test and the Panamath task. When all tasks Data from seven participants were excluded from were completed, participants were thanked for their analysis, because their Weber fractions fell above time and were given the equivalent of about $14 or their age-appropriate 90th percentile. course credit. Mathematical achievements. The order of the tasks was deliberate, to ensure that Participants were asked to report the quantitative the subjects did not perform any numerical task prior reasoning score (QRS) that they earned on their higher to the NBT that could serve as an unwanted priming education entrance exam (equivalent to the SAT test). cue. These are standardized scores that range 50 —150, with an average score of 100 points, σ =20. The QRS is derived from the quantitative reasoning section of the test which evaluates individuals’ ability to use Gender, Choices, and STEM 10

Results Individual SFON scores ranged from 0 (responses were never based on the quantitative dimension of the stimuli) to 1 (all responses were based on the quantitative dimension of the stimuli), with a median score of 0.13 (see Table 1, for descriptive sample characteristics). When measuring the reliability of these scores using the Spearman-Brown split-half reliability coefficient (Webb et al. 2006) – comparing the scores that were calculated based on the odd blocks with those based on the even blocks – a reliability of 0.969 was found, indicating that adult participants were consistent in their SFON tendency throughout the task. As expected, SFOC was correlated with SFON, Figure 4. Partial residual plot of adults’ SFON indicating that our SFON measure reflects not only scores as a function of their Weber fractions. The low vs. high bias towards numerosity but also a high black line is the estimated logistic-regression line vs. low bias towards color, r ( 75)=0.48, when controlling for SFOC; the dotted line is the estimated logistic regression line when controlling 95%CI [0.29,0.64 ], p<.001, and, therefore, it was for SFOC and for strategy (see text). On the y-axis, statistically controlled for in all analyses of SFON, by SFON scores are displayed as the proportion of adding it as a predictor. successful trials out of the 24 trials per subject. To test whether individual differences in SFON among adults were related to numerosity A repeated-measures logistic regression analysis discrimination, single-trial SFON scores (1 = selection was conducted, to test the relationship between SFON based on numerosity; 0 = selection based on color) and QRS. Because a gender difference was found in were subjected to a repeated-measures logistic QRS, t ( 64 )=−2.90, p=.005, gender was also regression analysis with random intercepts per included as a predictor in the analysis. As predicted, participant and Weber fractions as a fixed-effects adults with higher QRS also had higher odds of predictor. As hypothesized, a decrease in Weber focusing on the stimuli’s numerosity, such that an fractions was found to predict an increase in the increase in QRS from 120 to 140 (a change of one probability of focusing on the aspect of numerosity in standard deviation) increased the participant’s the NBT task, such that an individual with a Weber probability of attending to numerical information from fraction of 0.25 had a probability of 6.4% to base their 7.9% to 26.0%, ¿=4.08, 95 %CI [1.06,15.67], decision on the numerosity dimension, and an z=2.05, p=.040. individual with a Weber fraction of 0.125 had a Finally, we tested whether SFON was related to probability of 28.8%, ¿=5.91, 95 %CI [1.14,30.73], Raven scores, using a repeated-measures logistic z=−2.11, p=.035 (see Figure 4). When controlling regression. As predicted, Raven scores were not found for strategy, this pattern of results was essentially to have any predictive power, ¿=1.00, unchanged, ¿=4.12, 95%CI [0.84,20.12], 95%CI [0.60,1.67], z=−0.01, p=.999. z=−1.75, p=.070 (see Figure 4). Tests assessing whether the data met the assumption of collinearity indicated that multicollinearity was not a concern in any of the models (all VIF <1.11).

Discussion In the second experiment of our study, we investigated the relationship between SFON tendency (measured using a novel NBT) and ANS acuity (measured with the Panamath task) in adults. Additionally, the associations between SFON and Gender, Choices, and STEM 11 mathematical achievements (measured using QRS) children and adults. In preschool children, SFON was and general intelligence score (measured using the measured using Hannula and Lehtinen’s (2005) Raven Standard Progressive Matrices; (Raven 1960) imitation tasks. In adults, SFON was measured using a were tested, to assess the domain specificity of SFON novel computerized NBT. We found, in both age among adults. As hypothesized, we found that SFON groups, that ANS acuity measured by the individual was negatively related to ANS acuity. Moreover, a Weber fractions was related to SFON, such that higher SFON tendency was shown to be associated individuals with a greater resolution of quantitative with greater mathematical achievements, whereas no processing were more likely to incorporate task evidence for such a relation between SFON and numerosity elements into their behavior or decision general intelligence was found. making without prompting. In line with results among This study provides a novel and reliable method children, individual differences in SFON in adults for measuring individual differences in adults’ SFON. were associated with mathematical abilities: Our findings suggest that SFON could be a stable bias Individuals with a higher SFON tendency were better that continues from childhood into adulthood. This able to discriminate between numerosities. notion is consistent with previous studies that found Additionally, SFON was also associated mathematical stability in SFON throughout childhood (Hannula and achievement, as measured by QRS, suggesting a Lehtinen 2005). Similar findings regarding attentional relation between focusing on nonsymbolic numerosity bias and its stability come from the field of clinical and symbolic mathematical abilities. Together with psychology, where such attentional biases have been the lack of association between SFON scores and found to be stable from a young age (Gupta and Kar general intelligence, these results seem to indicate that 2012). For example, individuals high in anxiety are SFON tendency in adults is a domain-specific marker, abnormally sensitive to threat-related stimuli and, which has already been established in children. during the early automatic stages of processing, tend While our study offers further innovative findings to direct their attention toward threatening information to the growing research on SFON tendency and its (Williams et al., 1988). Here, we provide a novel role in the development of numerical abilities, there insight into the mechanism of SFON, in that we found are several core, open questions remaining, namely, evidence that a higher SFON tendency is associated what is the mechanism underlying individual with high-level mathematical abilities, whereas no differences in SFON tendency and how does this association to general intelligence was found in adults. tendency lead to later mathematical advantages? However, there are some limitations in this Although a growing body of evidence across different experiment’s methodology that should be considered cultures and ages further confirms the domain-specific alongside these results. First, it is unclear to what role of SFON tendency in mathematical achievements extent the SFON tendency measured in the NBT is throughout development, no particular set of factors theoretically identical to the SFON tendency (i.e., executive function, IQ, home and school measured in the imitation tasks, because the tasks environment) has been tested that could sufficiently differ in the way that they are presented to the explain the variability in SFON tendency found in participants (free play vs. learning task) and, thus, children or fully account for its relation to later may inadvertently be measuring different theoretical mathematical achievements (see review in Rathé et al. constructs that are not limited to SFON. However, 2016). As an association between two factors is a measuring strategy post-task and controlling for it prerequisite for causal or mediating relationships statistically during the analyses indicate that the between them, confirmation of a true association observed bias patterns could not be manifested solely between number acuity and SFON tendency would out of a deliberate or premeditated strategy, which is provide a promising starting point to test theories an essential task property to qualify as spontaneous regarding the relationship between these two factors. focusing. Additionally, where children’s SFON was Even more so, considering this relation is found in measured on two imitation tasks, adults’ SFON was adults, suggesting that it’s not manifested simply by measured on a single task. temporal co-development. Earlier works studying the role of ANS in General Discussion and Conclusions mathematical development demonstrate stable, domain-specific support in later on achievements and The present study investigated the relationship skills, but the exact manner of this influence (i.e., as a between SFON tendency and ANS acuity in both mediator between different skills or directly Gender, Choices, and STEM 12 supporting some abilities early on in life) has not yet light on the somewhat surprising results that did not been determined (see meta-analysis by Chen and Li show any relation (Edens and Potter 2013). 2014). The current evidence in this field suggests that Still, our study has several methodological the ANS starts as an innate, core number system that limitations. Crucially, our study design does not allow enables the perception and comparison of quantities, us to determine the causal relationship between SFON on which later, more fine-tuned skills are built. Yet, and ANS acuity. Therefore, we can only speculate the throughout development, it seems that the quality (or exact nature of the relation between them. Both the quantity) of mathematical education influences the theoretical validity of the NBT and the causality in the development of the ANS acuity beyond natural relationship between SFON and ANS acuity should be maturation (Nys et al. 2013). examined in a longitudinal study, which would allow SFON is a behavioral disposition showing natural researchers to examine possible reciprocal relations variation very early on in life. While showing some between these mechanisms over development. stability across different tasks and time, it is also Another limitation is the relatively small number affected, to some extent, by numerical knowledge and of measurements that the SFON measures are based practice (McMullen et al. 2019; Rathé et al. 2016) and on in both tasks (eight trials in the children’s imitation is predictive of later mathematical skills, specifically. tasks, and 24 trials in the adults NBT). However, it is SFON’s relation to later mathematical skills has also this small number of measurements that ensures that been demonstrated as a reciprocal process (Hannula the measurement of SFON is covert and mathematical and Lehtinen 2005; Hannula-Sormunen et al. 2015). context is not deduced (Hannula-Sormunen et al. However, to date, no specific mechanism or property 2015; Rathé et al. 2016). Additionally, despite these that could sufficiently explain the natural variance in few measurements, SFON scores were found to have these dispositions have been found (McMullen et al. acceptable reliability in children and high reliability in 2019). adults. Still, although the results of the logistic Based on the existing literature, we considered regressions were significant, the degree of several possible relations between these factors. significance for the adults and for the children with Evidence of the ANS producing the innate low Weber fractions was just below .05. Further rudimentary ability to perceive environmental research with larger samples is needed to replicate and numerical properties, even in newborns, aligns with confirm our findings. Finally, in both tasks, SFON our speculation that ANS acuity influences SFON tendency on each trial was measured by coding, only tendency, such that higher acuity facilitates higher if numerosity was or was not incorporated into probability noticing and acting on exact numerosities, behavior, dichotomously and not continuously as the as surrounding numerosities may be more salient to extent to or degree by which numerosity was those children in the first place. It can be argued that incorporated into behavior. However, because SFON the opposite relation is also possible: Higher SFON was assessed using multiple measurements per tendency means more frequent practice with exact participant, we were able to assess the degree to which numbers, which has been suggested as a possible individuals tend to incorporate numerosity into their explanation for the impact of SFON on later and more behavior, across many trials and different tasks. complex mathematical skills. Importantly, these In conclusion, the current study presents theories are not mutually exclusive, as evident by the converging evidence for the relationship between reciprocal relations between SFON and other math SFON and numerosity discrimination abilities, based skills. Determining the exact nature of the relation on two populations and using two different tasks. We between ANS and SFON could provide an important designed one of the tasks -- the adult NBT -- to allow advantage for educational programs employing for the investigation of SFON tendency in adults. Guided Focusing On Numerosity (GFON) to support Importantly, our findings offer a possible children’s mathematical education early on (Hannula- developmental mechanism underlying individual Sormunen 2014; McMullen et al. 2019), regardless of differences in SFON tendency and the nature of its the exact, true direction of causality. impact on mathematical abilities later in life. The A more complex relation between ANS acuity and demonstrated association between SFON tendency SFON tendency could also be considered. For and ANS acuity is consistent with the notion that early example, inclusion of ANS acuity as a mediating numerical competence is critically influenced by the factor studying the relation between motivational innate ANS and by the attentional control system, as factors and activity choices to SFON could shed some the interaction between them facilitates core abilities, Gender, Choices, and STEM 13 such as mapping symbols to magnitudes (Ansari 2019; References Geary 2013; Wilkey and Price 2019). 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