Approximate Number System Discrimination Training for 7-8 Year Olds Improves Approximate, but not Exact, Arithmetics, and Only in Children With Low Pre-Training Arithmetic Scores

We investigated whether training the Approximate Number System (ANS) would transfer to improved arithmetic performance in 7-8 year olds compared to a control group. All children participated in Preand Post-Training assessments of exact symbolic arithmetic (additions and subtractions) and approximate symbolic arithmetic abilities (a novel test). During 3 weeks of training (approximately 25 minutes per day, two days per week), we found that children in the ANS Training group had stable individual differences in ANS efficiency and increased in ANS efficiency, both within and across the training days. We also found that individual differences in ANS efficiency were related to symbolic arithmetic performance. Regarding arithmetic performance, both the ANS training group and the control group improved in all tests (exact and approximate arithmetics tests). Thus, the ANS training did not show a specific effect on arithmetic performance. However, considering the initial arithmetic level of children, we found that the trained children showed a higher improvement on the novel approximate arithmetic test compared to the control group, but only for those children with a low pre-training arithmetic score. Nevertheless, this difference within the low pre-training arithmetic score level was not observed in the exact arithmetic test. The limited benefits observed in these results suggest that this type of ANS discrimination training, through quantity comparison tasks, may not have an impact on symbolic arithmetics overall, although we cautiously propose that it could help with approximate arithmetic abilities for children at this age with below-average arithmetic performance.

. Helping those students whose performance is weaker may be particularly challenging, because factors such as motivation (Simzar et al., 2016), math anxiety (Pletzer, Kronbichler, Nuerk, & Kerschbaum, 2015;Z. Wang et al., 2014), lack of a predisposition towards mathematics (Cerda et al., 2015), and teacher effects (Demaray & Elliot, 1998;Tournaki, 2003) come into play. Game-based training of basic math skills may be helpful, as it may be less threatening to those who have difficulty learning mathematics. Indeed, training math-related basic cognitive skills may increase students' confidence in their sense of number, and this might then transfer to their more school-based math abilities (Odic, Hock, & Halberda, 2014;Wang, Odic, Halberda, & Feigenson, 2016; but see Merkley, Matejko, & Ansari, 2017;Wang, Odic, Halberda, & Feigenson, 2017). The search for cognitively motivated, easily implemented, enjoyable and efficient training regimes is currently very active. Here, we implement a one-month intervention to test whether it can improve performance of arithmetic operations in children.
The intuitive sense of number comes from a cognitive system often described as the Approximate Number System (ANS), comprising several components needed to understand and manipulate numerical quantities; among them, a rough ability to estimate magnitudes ("there are many/few marbles in that jar"), magnitude comparisons ("this jar has more marbles than that jar"), and the manipulation of numerical quantities through basic approximate arithmetic operations ("I can remove around half of the marbles from this jar to make it roughly equal to that jar"). Although not necessarily innate (Leibovich, Katzin, Harel, & Henik, 2017), the ANS seems to be present in young infants (Feigenson, Dehaene, & Spelke, 2004;Hyde, 2011;Hyde & Spelke, 2011), in people of all cultures (McCrink, Spelke, Dehaene, & Pica, 2013;Pica, Lemer, Izard, & Dehaene, 2004), and ages , and even in other animals (e.g., Agrillo, Piffer, & Bisazza, 2011;Beran, Evans, & Harris, 2008;Dehaene, Dehaene-Lambertz, & Cohen, 1998;Jones & Brannon, 2012). While all people may have an ANS, the precision of the ANS varies across individuals throughout the lifespan . A more precise ANS allows a person to make more confident distinctions between large close numbers, making more confident numerical distinctions. Interestingly, preschoolers, students across ages, and adults who have a more precise and accurate ANS also tend to do better on tests of formal, written, mathematics ANS Training for 7-8 Year Olds 276 (Amalric & Dehaene, 2016;Y. He et al., 2016;Libertus et al., 2011). Importantly, this relationship holds even when general intelligence, verbal ability, and many other cognitive factors are controlled for .
While some studies suggest that executive functions (EF) might serve as the underlying link between better ANS performance and better school mathematics performance (Gilmore et al., 2013;Li et al., 2018), some recent work has found the opposite to be true (i.e., that ANS precision and number language or number knowledge survive as the only unique predictors of school math ability while EF are not unique (Malone et al., 2020(Malone et al., , 2019Purpura & Logan, 2015). This is an area that will benefit from additional exploration.
Also, early differences in ANS precision, but not other factors such as verbal or performance IQ, executive functions, language knowledge or visuospatial skills, predict later math performance at school (Mazzocco et al., 2011b). Because links between the ANS and formal mathematics abilities are not universally accepted, it remains important to test if ANS precision relates to arithmetic abilities.
The precision of the ANS may improve across life, especially during the school-age years, attaining its highest precision sometime around age 30 years . These improvements justify the hope that interventions aimed at training the ANS might transfer to improvements in some arithmetic skills. Because these basic arithmetic skills can be built into computer tasks Hyde, Khanum, & Spelke, 2014;Park & Brannon, 2014), game-based ANS practice could help to train these early skills, with perhaps a more marked effect in students who are from particularly fragile social environments (Fuhs & McNeil, 2016;Valle-Lisboa et al., 2016;Wilson, Dehaene, Dubois, & Fayol, 2009) or possess low numeracy skills (Liang, Zhang, Long, Deng, & Liu, 2020;Praet & Desoete, 2014;Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009 Wang et al., 2016), potentially transferring to school mathematics performance, these results remain controversial (Szűcs & Myers, 2017). The exact nature of the training and the strength of the results are still unknown. For example, in a brief (5-minute) intervention to improve ANS confidence, children showed immediate gains in symbolic arithmetic (Wang, Odic, Halberda, & Feigenson, 2016) although the causal link was questioned based on concerns about the study design (Merkley, Matejko, et al., 2017). That is, Merkley, Matejko, and Ansari (2017) argued that randomized control trials, with pre-testing of all participants, supported stronger inferences while the Wang et al. (2016) procedure relied on a treatment with post-test design (see also Wang, Odic, Halberda, & Feigenson, 2017). Thus, the possibility of ANS training transferring to symbolic math abilities remains an area of controversy. Even under the assumption that there is some transfer, it is also true that the persistence of these potential gains over an extended period remains unexplored. Other studies showed that training in ANS arithmetic (e.g., estimating the results of adding and subtracting clouds of dots) can lead to improvements in symbolic mathematics in children (Hyde et al., 2014;Obersteiner, Reiss, & Ufer, 2013;Park, Bermudez, Roberts, & Brannon, 2016), in adolescents (Knoll et al., 2016), and in adults (DeWind & Brannon, 2012;Park & Brannon, 2014), but training in simple magnitude comparisons of two collections (e.g., which jar has more marbles) did not produce comparable transfer (Lindskog & Winman, 2016;Park & Brannon, 2014Pinheiro-Chagas et al., 2014).
In the present study, we aimed to add a contribution to the debate surrounding the possible link between the ANS (NB: whose very existence remains controversial) and symbolic arithmetic, using a robust pre-test,

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intervention, post-test design with a well-matched control group. In the school system where we realized our study, the second-grade math program includes arithmetic (additions, subtractions, and some multiplications), basic geometry (spatial orientation, figures, measure of figures), measure of magnitudes (weight, volumes, distances, units of measurement), and basic statistics and probability, among other topics. While the specific program covered by other systems may vary in other countries, it is reasonable to expect that any school system at the target age will include arithmetic, as it gives children the basic steps necessary to succeed in other parts of mathematics. Thus, in our study we focused on arithmetic as the privileged area to test potential effects of ANS training.
In the current study, we focused on one grade level in a controlled environment, while carefully controlling dosing time and conditions of exposure to the ANS training game relative to its control. We studied second-graders, aged 7-to 8-years, for two reasons. First, previous studies have shown that many developmental changes occur around that age (Holloway & Ansari, 2009;Sasanguie, Göbel, Moll, Smets, & Reynvoet, 2013;Siegler & Booth, 2004); and, second, at this grade students know enough arithmetics to be tested with a variety of arithmetic operations, including additions, subtractions, and even some multiplications. The arithmetic operations in second grade (additions, subtractions, and multiplications) represent different levels of difficulty for a child; for example additions are easier than subtractions (e.g., Knops, Dehaene, Berteletti, & Zorzi, 2014), and subtractions are easier than multiplications (Prado et al., 2011). In the present study, due to the approximative nature of the training, we introduced a novel arithmetic operations test to assess performance in approximate symbolic arithmetics, alongside a school-like test of exact symbolic arithmetics.
We asked if three-weeks of ANS training in approximate quantity comparisons could enhance arithmetic skills in children, and if improvements were related to their initial arithmetic abilities. This aspect of our work was motivated by the literature, as several studies suggest that children who are struggling the most may benefit from extended ANS game-based interventions (Langfus et al., 2019;Mazzocco, Feigenson, & Halberda, 2011a;Purpura & Logan, 2015;Valle-Lisboa et al., 2016). In order to understand the sources of children's differences in mathematical achievements, we must understand how the content taught in school relates to basic cognitive abilities. A crucial aspect of this endeavor -one with pivotal societal consequences -is to determine if some training can help struggling students improve. We believe that school-based training is a powerful way to progress in this direction.
In the present study, through our school-based intervention, we aimed to further test the controversial ANS construct and its possible link to symbolic arithmetic. Specifically, we evaluate whether individual differences in ANS efficiency are related to symbolic arithmetic performance; whether training the ANS for an extended period can improve children's ANS efficiency; whether such improvement will transfer to any arithmetic skills; and whether the initial arithmetic level of children before the training helps to determine the efficacy of such an intervention.

Method Participants
Ninety-one second-graders (44 girls; average age = 7 years 10 months, range = 7 years 3 months to 9 years 3 months) were recruited from Hamelin International Laie School (http://www.hamelininternacional-ANS Training for 7-8 Year Olds 278 laie.com/school/) in Barcelona, Catalunya. The school is a non-elite private school with no academic requirements or admissions tests for entering, and does not exclude children along the course of the school years based on academic performance. Tests scores for college entrance exams from the Hamelin International Laie School do not differ from other national schools. Participants came from families of middle to high socioeconomic status. According to teachers' reports, our sample did not include supremely gifted students or students who were greatly below the typical level. Thus, the sample is a fair representation of Catalan school students.
Participants came from four different classrooms (respectively, with n = 24, 24, 23 and 20). We compared a control group, which relied on the standard school activities (Control, n = 44) with an ANS Training group (n = 47). Two teachers of mathematics served two classes each, randomly assigned to Control or ANS Training group so that teacher effects were controlled. Training activity occurred during the computer technology class.
Throughout the study, computer and math classes were kept separate. Teachers and research staff never discussed relationships between computer activity and math classes, so that students were unaware that training was related with math performance.

Materials Arithmetical Competence Assessment
We prepared three problem booklets for pencil-and-paper tests. Two of them contained straightforward arithmetic problems: one additions ( Figure 1a) and one subtractions test booklet (Figure 1b), based on the math fluency subtest of the Woodcock-Johnson Tests of Achievement (Woodcock, McGrew, & Mather, 2001), with additional problems and a 6-minute testing time to better detect individual differences. For these, children had to write in the correct answer to the problems. The third booklet was a novel operations test, which we designed ( Figure 1c) and presented a kind of problem that had not been taught before: each problem was presented along with its result, but the operation sign was omitted from the equation. Children had to mark whether the solution demanded an addition, a subtraction, or a multiplication sign. To correctly answer, students needed not assess whether the given answer was correct (though it always was). Rather, by estimating whether the solution resulted in a slightly larger, a much larger, or a smaller number, they could decide whether the correct operation sign was, respectively, an addition, a multiplication or a subtraction. Our aim was to introduce a novel test relating approximate arithmetic ability and ANS training not confounded with normal arithmetic activities at school during the period of training.
Motivated by findings in the literature (Bonny & Lourenco, 2013;Langfus et al., 2019;Mazzocco et al., 2011a), we wanted some way of characterizing children's initial arithmetic abilities. Because children's school grades at this age were not detailed enough to classify them according to their mathematical abilities, we used the additions and subtractions subtests before the training to identify three initial arithmetic performance groups (Below-Average, Average and Above-Average tertiles) and then use the novel operations subtest to measure pre-and post-training performance in approximate arithmetics.
Test difficulties were created under the supervision of the teachers. Two versions of each subtest were prepared, with different problems, so that pre-and post-training booklets were unique and counterbalanced across the sessions. Difficulty and problem order were randomized. The number of problems for each subtest was large, so that children could not complete all of them during a 6-minute allotted time: (210 additions

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190 subtractions and 117 operations). In the Additions subtest, each addendum could reach at most 18, so that the highest sum was 18 + 18 and the lowest 0 + 0. In the Subtractions subtest, both the minuend and subtrahend ranged between 0 and 18; the result of the subtractions was always positive. Additions and subtractions were presented in 10-sheet booklets, in a column operation algorithm form (Figure 1a and b); and the operations subtest was presented on 3 sheets ( Figure 1c). The Operations test included problems of addition, subtraction and multiplication. For the additions in the Operations test, the maximum value for the first and second addenda was, respectively, 7 and 10 (i.e., 7 ⬜ 10 = 17). For the subtractions in the Operation test, both the minuend and subtrahend ranged between 1 and 11, with the result always positive (i.e., 11 ⬜ 5 = 6). For the multiplications, the timetables of 1 to 5 and 10 were used (i.e., 4 ⬜ 5 = 20), because children had been exposed only to them according to their teachers. The three types of operations and the problem difficulties were presented in random order. The booklet contained a roughly equivalent distribution of problem kinds (42 and 43 additions in the first and second version respectively; 38 and 39 subtractions; and 37 and 35 multiplications).

Computer Activities
The computer classroom had 25 laptops (HP 620, Pentium (R) Dual Core 2.30GHz, 4GB DRAM, 64-bit; Windows 7 Home Premium), with individual headphones. Training and control groups ran different activities.
Controls practiced two commercial programs, Tux Paint, and Microsoft Word. ANS Training children practiced a modified version of Panamath . Written in Java SE6, the program generates trials displaying collections of items inside two rectangles appearing side by side. For example, twelve teddy bears could appear inside the left rectangle and 6 blue dots inside the right rectangle. The number of items within each rectangle was always between 5 and 21. The items were presented in seven different ratios (larger set/smaller set): 3, 2, 1.5, 1.25, 1.17, 1.14, 1.1. For example, in a 3-ratio trial children could see 21 blue vs.
ANS Training for 7-8 Year Olds 280 7 yellow dots. Smaller ratios generate more difficult trials. On each trial, items were displayed for 1382 ms.
To vary the relationship between surface area and number, there were 3 different models controlling object size. Forty-two percent of the trials were size-confounded. In them, items average size was equal for both sets, so that objects occupied cumulative surface areas congruent with number ( Figure 1d). Forty-two percent of the trails were size-controlled, with the object average size smaller for the larger set, so that the ratio of the cumulative area occupied in each set was 1 (Figure 1e). In the remaining 16% of trails, object sizes were stochastically varied, so that children had no consistent size cue for number. In these, the average size of the objects varied randomly between being size-anticorrelated (where the numerically larger set had less total area on the screen), size-controlled, and size-confounded. We called these runs stochastic size control (Figure 1f).
While there are a diversity of views about how to control for non-numerical cues in approximate number stimuli, our method here recapitulates one such approach that has appeared in several papers (Hellgren, Halberda, Forsman, Adén, & Libertus, 2013;Libertus et al., 2013;Libertus, Feigenson, Halberda, & Landau, 2014;Tosto et al., 2014). But, in fact, we expected that our children would rely on continuous cues to some extent and we did not desire to eliminate this possibility in our stimuli because this might be part and parcel of normal ANS functioning (Halberda, 2019).

Procedure
The experiment had three phases: Pre-training, Experimental/Control training, and Post-training. The Pre-training and Post-training assessments were intended to measure the mathematical competence of the participants before and after training. These were conducted in children's classrooms, in the presence of their math teacher.
From the child's perspective, they were independent of the computer activities.
Children had to answer as many questions as possible in 6 minutes per test. The additions subtest was always ran first. Then children had to stop and wait until they were given the subtractions subtest. The final test was the operations subtest; as it contained a novel kind of problems teachers had to briefly explain what children were supposed to do beforehand. Teachers completed 3 examples on the board, in front of the class, one per each problem kind, after which children began the subtest. The Pre-training occurred two days before the first training session (initial assessment); the Post-training one day after training completion (final assessment).
The training phase was administered in six different sessions within a three-week period, at a pace of two sessions per week. This choice was motivated by previous studies (Langfus et al., 2019;Valle-Lisboa et al., 2016). Both the intervention and control groups were trained on the same days (Wednesdays and Fridays), for the same amount of time, and in the same computer classroom.

Control
Control children practiced Tux Paint and Microsoft Word, learning standard activities available in these programs. No activity involved approximate number comparison. Computer classes started with teachers' instructions, after which children worked individually, in the same class as for the same period as the experimental group.

Approximate Number System Training (ANS Training)
Children practiced the Panamath quantity discrimination game. During training, the computer teacher and the experimenter were always present. Children wore headphones and trained simultaneously. By observation, children appeared to like this game. The ANS Training presented children with a sequence of pictures, asking them to make ordinal comparisons of the rapidly flashed collections appearing onscreen. On each trial, they saw two collections of items appearing onscreen side by side, and needed to rapidly estimate which side had more items (Figure 1 d-f), typing their answers on a keyboard ("f" and "j" keys for left or right side respectively).
As each presentation lasted about 1.3 s, children could not count the items, but had to rapidly estimate which of the two sets was larger. Different collections of items were used on each run of trials, so that the game maintained children's interest. For example, one run could present blue vs. yellow dots; another could present cars vs. bears; yet another could display birds vs. dogs. There were 35 trials per run. Children completed approximately 24 runs over the course of three weeks. Feedback was provided after every response, with a high/low pitched beep indicating correct/incorrect answers.
When introducing ANS Training, children were told that they would play a game where they would see some objects -for example, blue and yellow dots -and would have to choose if there were more blue or yellow dots.
They were informed that two different sounds would provide them with feedback. They were also told that the game would vary in difficulty, and that both speed and accuracy were important. Most children completed 24 runs. Six participants completed 21, one 22 and one 20. Each run was composed of 35 trials (or about 6-7 minutes). Each run started with the easiest ratio in the first five trials. Then, every five trials the game increased in difficulty, with the ratios becoming closer to 1, until the seven different ratios were presented in each run.
This confidence-scaffolding procedure was implemented to increase ANS precision and confidence, based on previous studies with brief interventions (Odic, Hock, & Halberda, 2014;Wang et al., 2016). Children completed the ANS Training during their regular computer class time.

ANS Training: Approximate Number Comparison
For our purpose, the primary measure for the effectiveness of the ANS training is post-pretest differences in symbolic math performance. However, before analyzing the effect of training, we assessed whether the training did engage the ANS. We looked for the main signature, ratio-dependent performance that results in a specific curve of percent correct as a function of ratio (Feigenson et al., 2004;Starr, Libertus, & Brannon, 2013a), in which participants' accuracy at determining the bigger of two approximate numerosities decreases as the ratio between the numbers decreases. This ratio-dependence is predicted by Weber's law. Figure 2 presents the data from the ANS Training group, separated by the type of size control of the stimuli. The ratio-dependent performance curve appears for all three size-control trial types: as the numerical ratio between the two collections becomes easier, children's percentage of correct responses improves. Furthermore, notice that even though overall percent correct is somewhat lower, the curve of ratio-dependent performance appears also for the stochastic size-controlled and size-controlled trials. Children chose the numerically greater collection well above chance; this suggests that while they may sometimes have been led astray by the numerically smaller collection having a larger total surface area, they could still succeed on a majority of trials. Some portion of the difference in performance across the stimulus sets could be due to the differing EF demands of the stimuli (Gilmore et al., 2013;Li et al., 2018). And, we note that Inglis, 2015; L. He, Zhou, Zhou, He, & Chen, 2015;Hollingsworth, Simmons, Coates, & Cross, 1991;Izard & Dehaene, 2008;Leibovich & Henik, 2013;Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004).
We note, however, that our main aim was not to identify which dimension is responsible for the ANS signature, but rather to ensure that the ANS is engaged regardless of the cues that empower it. The question of which perceptual features drive the ANS is an important one that continues to inspire debate, but it was not a focus of our present work.
Using the standard psychological model and fitting methods (e.g., Odic et al., 2014) we estimated the observed ANS precision for the whole ANS training group (i.e., Weber fraction or w). We found the best fit model returned  For this reason, efficiency, operationalized as the percentage correct divided by the RT, encompassing both improvements in RT (i.e., getting faster) and in percent correct (i.e., becoming more accurate), seems most appropriate (Bartelet et al., 2014;Libertus et al., 2013;Mazzocco et al., 2011b;Murphy et al., 2007;Park & Brannon, 2014). It increased across the ANS Training task (R 2 = .827, p < .001; Figure 3). While not crucial to our purpose, we observe that while children greatly improved in their response times, they showed a tendency to be less accurate across training; thus, the improvements in ANS efficiency are primarily driven by improvements in RT.  Again, we stress that our main measure of interest is not improvement during training, but the effect of such training on school math; nevertheless, we can show that training affected children's ANS responses, at least as measured by efficiency.
Lastly, it remains an interest whether individual differences in ANS efficiency are stable over time. Here, we could test for this stability, and for improvements in ANS, by looking at regressions across testing days. Figure   5 shows a regression between ANS efficiency in the first day with each of the following testing days. The regressions were all significant (all p < .05) and the overall regression was significant (R 2 = .2, p < .0001). This demonstrates that individual differences in ANS efficiency were indeed stable over time.

ANS Efficiency and Pre-Training Math Performance
While still controversial, some previous research suggests that ANS performance is related to symbolic mathematics performance. Our Pre-Training assessment of Additions, Subtractions and Operations allows us to evaluate this claim for our participants. Indeed, the total number of correct answers in the Pre-Training math tests correlated with the ANS Efficiency before training, R 2 = .086, p < .05; Figure 6. Such a correlation held also when analyzing the Pre-Training Symbolic Math Performance and the total ANS Efficiency across all training sessions. Collapsing across all 3-weeks of ANS-Training Efficiency, we again found that Symbolic Mathematics Performance in Pre-training correlated with ANS Efficiency (R 2 = .146, p < .01). That is, the children with higher symbolic mathematics performance before training also showed a higher ANS efficiency across training.
While previous demonstrations of this relationship between ANS acuity and math ability have relied on brief measures of ANS acuity (Booth & Siegler, 2006;Gilmore et al., 2010;Halberda et al., 2012 Defever, & Reynvoet, 2012) here we extend these results to a much longer temporal interval, showing that this relationship holds even when ANS efficiency is measured across three weeks of training experience. These results suggest that the few studies that failed to find such a relation may be exceptions to the general trend (for discussion see Chen & Li, 2014;Fazio, Bailey, Thompson, & Siegler, 2014;Schneider et al., 2017).

The Effect of ANS Training on Symbolic Math Performance
We now turn to our main interest: the effect of ANS training on symbolic math, or the difference between Pre- as independent variables, we found a main effect of Phase, F(1,89) = 111, p < .001, η 2 = .067, no effect of Training Condition, F(1,89) = 0.78, p = .38, η 2 = .008, and no interaction, F(1,89) = 1.76, p = .19, η 2 = .001; Figure 7. The data were also examined by estimating the Bayes factor, which suggested that the data were 3.33:1 in favor of the null hypothesis, or rather, 3.33 times more likely to occur under the model which only includes the main effect of Phase, rather than the model which includes the main effects of Phase and of Group, and their interaction. In sum, the Bayesian ANOVA reveals that the data provides support for the Phase main effect model, the null hypothesis, over any of the other models. Thus, when inspected globally, no difference was found between the Control and ANS Training children: they all improved in our Symbolic Math Tests during the period of the experiment. We notice that similar results were also found in a sample of Uruguayan seven-year-olds when considering all children as a group (Langfus et al., 2019). Importantly, this previous study (Langfus et al., 2019), along with others (Bonny & Lourenco, 2013;Mazzocco et al., 2011a) have found stronger improvement for children who were initially low-performing in mathematics. These previous studies found these effects by splitting their samples into smaller groups based on initial math performance.

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Motivated by this literature, and our own a priori interests, we planned to take this same approach heresplitting our sample into smaller groups based on pre-training arithmetic scores. Considering the wide differences between children's mathematical abilities, and our interests in the potential effect of training on children with different starting points, we next investigated the relationship between initial math ability and improvement due to ANS training. To assess it, we inspected how the percentage of correct responses varied from pre-to post-test. We regressed number of correct answers on the Symbolic Math Test during pre-training against percentage growth in symbolic math performance from pre-to post-training.
The regression showed that children who gave fewer correct answers on the pre-training Symbolic Math Test showed higher percentage gains in symbolic math ability from pre-to post-test (R 2 = .103, p < .01; Figure 8).
This result suggested that an analysis at the level of tertiles (Below-Average, Average, Above-Average), which has been done in previous publications, may be informative. Converging evidence across several labs suggests that children who are lower-achieving in mathematics may show the strongest (or most easily detected) relationship between the ANS and school mathematics ability (Bonny & Lourenco, 2013;Langfus et al., 2019;Mazzocco et al., 2011a;Purpura & Logan, 2015;Valle-Lisboa et al., 2016). Therefore, inspired by this prior work, we next turned to investigating possible heterogeneity in the gain scores for children according to initial arithmetic performance. We split our groups into tertiles according to their results in the pre-training exact Arithmetic test, comprised of Additions and Subtractions but not the Operations subtest (Figure 9) -our novel measure which we planned to use as an outcome variable. We used the exact Arithmetic test because it includes the problems children are most familiar with and have practiced in school prior to our training.  The split into Below-Average, Average, and Above-Average tertiles regarding their initial arithmetic scores was carried out with a focus on creating roughly equal numbers of participants per group, and resulted in the following cutoffs: below 65 correct responses for the Below-Average tertile, between 66 and 86 for the Average tertile, and above 86 correct responses for the Above-Average tertile (Figure 9; n Below-Control = 15, n Below-ANS = 15, n Average-Control = 16, n Average-ANS = 16, n Above-Control = 13, n Above-ANS = 16). Notice that these tertiles are only indicating performance relative to children's own classmates on our exact Arithmetic test before the training;

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they may not capture a more universal notion of "low-achieving" children (Desimone & Long, 2010), nor "gifted" children (Swanson, 2006). However, given the typical breadth in the composition of our sample, and the fact that our speeded arithmetic tasks are based on the math fluency subtest of the Woodcock-Johnson Tests of ANS Training for 7-8 Year Olds 288 Achievement (Woodcock et al., 2001), our split is likely to capture differences valid beyond our limited sample (see also Langfus et al., 2019;Valle-Lisboa et al., 2016).
Considering that the grouping variable was measured in the pretest, we expected no differences in the number of correct answers for ANS Training and Control children in the Symbolic Math Pre-training Test. A 2 x 3 ANOVA, with Training Condition (ANS Training, Control) and Symbolic Math Level (Below-Average, Average, and Above-Average tertiles) as independent variables, showed that, as expected Symbolic Math Level differed, F(2,85) = 77.3, p < .001, η 2 = .645, but Training Condition did not, F(1,85) = 3.0, p = .087, η 2 = .034. However, an interaction between Training Condition and Symbolic Math Level, F(2,85) = 5.53, p < .01, η 2 = .115, appeared. Bonferroni post hoc tests revealed that the interaction was driven by the Above-Average tertile where the Control group obtained lower scores than the ANS Training group (p < .05; Figure 10). While this difference appears to be a random result of grouping, it is worth bearing in mind that, because of a ceiling effect, it may mask potential ANS training benefits in the Above-Average tertile. We relied on the Operations subtest as our outcome variable, as it assesses how children deal with aspects of mathematical reasoning that they have not been trained to solve, and it was not used as a grouping variable for grouping our children. At the same time, we found it comparable with more traditional arithmetic tests, because it highly correlates with them both before and after training (Additions and Operations: at pre-test r = .68, p < .001, at post-test r = .68, p < .001; Subtractions and Operations: at pre-test r = .64, p < .001, at post-test r = .69, p < .001). In order to compare results across groups, we combined Pre-and Post-training scores F(2,85) = 8.49, p < .001, η 2 = .166; Figure 11. Bonferroni post-hoc tests revealed that the change was higher for the ANS Training group than for the Control group in the Below-Average tertile of children (p < .05), but not for the Average tertile (p = .41). The Above-Average tertile of children showed an effect in the opposite direction, perhaps due to the ceiling effect in the ANS Training group: the Control group scored higher than the ANS Training group (p < .01); however, as we noticed, in this tertile the ANS group was abnormally high, and better than the Control group already at Pre-Training in all tests, including the Operations test, t(27) = 2.09, p < .05.
This makes it hard to interpret the results, probably contaminated by ceiling effects at baseline. Considering the size of the sample (91 participants: Control n = 44 and ANS Training n = 47), we wanted to maintain all participants in order to not lose statistical power. However, with the concern that the observed interaction in the improvement in the Operations test could be driven by the initial differences before the training, specifically in the above-average tertile (Figure 10), and considering that precisely this tertile was the only one not well matched regarding the number of participants between both groups (n Above-Control = 13, n Above-ANS = 16), we proceeded to analyze all participants whose initial arithmetic scores were within 2.5 SDs from the mean, consequently excluding three participants that coincidentally belonged to the Above-Average In Notice that this is specifically relevant for the Below-Average level, in which there is a significant difference of improvement between the ANS Training group and the control group but only in the Operations subtest (approximate symbolic arithmetics) and not in the exact Arithmetic test.
With a concern that our one significant improvement based on training in the below-average tertile might be a groups. This suggests some stability of this effect, but of course, more research is needed.

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In light of the obtained results, we focus our discussion on the Below-Average tertile regarding their initial arithmetic level, where training may be effective, although only in approximate symbolic arithmetics (Operations subtest) and not in exact symbolic arithmetics (exact Arithmetic test: Additions and Subtractions subtests).

Discussion
A considerable amount of research has highlighted the importance of the ANS to math ability, although this role is still controversial and many questions need to be answered before any causal link can be claimed with high confidence. In the present study, we try to make some contributions to this ongoing discussion. Here we found severe restrictions on how and to whom training the ANS resulted in improvements of arithmetic skills.
We extensively trained children on an ANS confidence-scaffolding quantity discrimination task (increasing trial difficulty during sessions). We found that training on speeded, non-symbolic, quantity comparison engages the approximate number system, as shown by the size-ratio signature of the responses, even for size-controlled and stochastic size-controlled trials. Furthermore, ANS training increased ANS efficiency both over the course of the 3-week intervention and within daily sessions. These results confirm and extend the finding that ANS can be partially trained (Knoll et al., 2016;Obersteiner et al., 2013;Park & Brannon, 2014).
In relation to other ongoing debates, we found that initial symbolic math ability related to ANS efficiency ( Figure   6), that individual differences in ANS efficiency were measurable, and that these individual differences were stable throughout training ( Figure 5).
Our main interest, though, was to assess to what extent ANS training would transfer to symbolic mathematics, specifically, arithmetics. Here, our results showed limited benefits. Overall, both Control and ANS Training children improved from Pre-to Post-training. As children's knowledge progresses over the course of a onemonth class activity, this overall improvement was expected. However, we also expected to detect some training-induced improvements in the ANS training group compared to the Control group and, overall, these were not observed. Taking the intervention group as a whole, the ANS training did not help children improve their results in the exact Arithmetics test (Additions and Subtractions tests) nor in the novel Operations test.
This is in fact the null hypothesis of the present study, that is, that the training did not have an overall effect.
The one positive result of note was that, within the children who performed below-average in the pre-training exact Arithmetic test, training the ANS did seem to show a transfer to approximate symbolic arithmetics. This appears to be consistent with patterns seen in other samples of this age (Langfus et al., 2019;Valle-Lisboa et al., 2016). Present results suggest that those who belonged in the ANS training group and had below-average pre-training scores showed a higher improvement in the novel Operations test, which assesses approximate symbolic arithmetics. This improvement, however, was not seen in the exact Arithmetic test (Additions and Subtractions tests).
We would like to stress that the results obtained are positive solely regarding a marginal aspect of arithmetics (symbolic approximate arithmetics), and for a limited number of children (below average in the Arithmetic pre-test). Furthermore, considering the low number of participants per tertile, a word of caution is necessary ANS Training for 7-8 Year Olds 292 when interpreting these results. They should be taken as a suggestion since further research with larger sample sizes in each tertile is needed to support this claim.
This being said, and as an attempt at giving a possible explanation for the obtained results, we cautiously present the following considerations. Different ANS training regimens may train different components of the ANS, such as the approximate sense of cardinality, or of arithmetic transformations, or of quantity comparison, or of a sense of ratios. Other training studies with adults (Park & Brannon, 2013 and preschoolers  improved exact symbolic arithmetic by training approximate arithmetic transformations. In those studies, the cognitive component shared by their ANS training and the symbolic tests was the arithmetic manipulation of numerical quantities. We tentatively suggest that the lack of this cognitive component in our ANS training could be an explaining factor regarding why we obtained null results in the exact symbolic arithmetic. In the present study, we instead focused on training the ANS through approximate quantity comparison, a component also present in our measure, the Operations test. This test requires a previous understanding of how basic arithmetic operations change quantities. It has the sensitivity to assesses approximate symbolic arithmetics since no exact calculation is required; a comparison of both sides of the equal sign (3 ⬜ 5 = 15) allows for a correct answer. Simply stated, "more" for addition, "less" for subtraction, and "much more" for multiplication. For this reason, this novel test may have detected improvements in approximate symbolic arithmetics resulting from practice with the approximate nature of our ANS training procedure. The novel Operations test as a measure of success could, therefore, constitute a strength of our study, allowing us to better detect a positive transfer, if any, from ANS comparison training to approximate symbolic arithmetics.
There are further explanations to be considered. One potential tension in our results is that we found that Pretraining Symbolic Math Test performance is significantly related to ANS performance ( Figure 6) but that ANS training did not lead to greater gains in Symbolic Maths performance (e.g., Figure 7). While not unexpectedbecause the relationship in Figure 6 has developed over the lifetime of the child while the non-result in Figure   7 relies on just 1-month of training -this difference in results leaves open several possibilities: e.g., perhaps 1-month of training isn't enough; or training the ANS through approximate quantity comparison is not the most beneficial type of ANS training; or the relationship in Figure 6 relies on EF while the non-relationship in Figure   7 does not; or that the ANS is important to approximate symbolic arithmetic but not to exact arithmetic per se; etc. In a way, the diverse results in the present study mirror the diversity of interpretations that are current and appropriate in literature, since we are far from understanding the mechanisms that underlie the relation between ANS and symbolic mathematics (Szkudlarek & Brannon, 2017).
We cautiously suggest that ANS quantity comparison training could help children who are struggling with arithmetics to improve some very basic aspects, namely, symbolic approximation. Given that we found some improvements only in approximate symbolic arithmetic, it remains to be studied if these could later positively impact the comprehension and execution of exact symbolic arithmetic. While approximate symbolic arithmetic is not typically taught in schools, exact symbolic arithmetic is, and it is a target outcome of math education.
In the present study, results suggest that the ANS exists, that individual differences in ANS efficiency exist and are stable, and that ANS training can improve ANS efficiency. However, we found that training the ANS, by approximate quantity comparisons, leads to very limited transfer in 7-8 year olds -only in children with low pre-training arithmetic scores and only in approximate symbolic arithmetics.