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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 Pre- and 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.

When children begin their explicit mathematical training, typically their skills vary considerably (

Classroom-based interventions and teaching have not always been found to be effective in helping students who struggle with math (e.g.,

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 (

While some studies suggest that executive functions (EF) might serve as the underlying link between better ANS performance and better school mathematics performance (

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 (

The precision of the ANS may improve across life, especially during the school-age years, attaining its highest precision sometime around age 30 years (

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, 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 (

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 (

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.

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 (

Participants came from four different classrooms (respectively, with

We prepared three problem booklets for pencil-and-paper tests. Two of them contained straightforward arithmetic problems: one

Motivated by findings in the literature (

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 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 (

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,

The experiment had three phases:

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 (

Control children practiced

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 (

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 (

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 (

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.,

It is interesting to assess to what extent, across the training sessions, children improved. However, the measure which best reveals such an improvement is not immediately obvious. Both response time (RT) and accuracy have been shown to be reliable measures of performance in this and similar tasks with children as young as four years of age. However, with few exceptions (e.g. ^{2} = .827,

ANS training was also effective across the three individual daily runs, as revealed by a one-way repeated measures ANOVA with Run Order (first, second, third) as an independent factor and Efficiency as a dependent variable, ^{2} = .072;

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. ^{2} = .2,

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, ^{2} = .086, ^{2} = .146,

We now turn to our main interest: the effect of ANS training on symbolic math, or the difference between Pre- and Post-Training Symbolic Math Performance. A first rough assessment of such an effect can be obtained by collapsing into a single measure all Symbolic Math Tests (Additions, Subtractions, Operations) for all children. In a 2 X 2 ANOVA, with Training Condition (Control, ANS Training) and Phase (Pre-, Post-training) as independent variables, we found a main effect of Phase, ^{2} = .067, no effect of Training Condition, ^{2} = .008, and no interaction, ^{2} = .001;

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 (^{2} = .103,

Percentage improvement in symbolic math for both Control and ANS Training children as a function of Total correct answers in the Symbolic Math Assessment in Pre-training.

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 (

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 (_{Below-Control} = 15, _{Below-ANS} = 15, _{Average-Control} = 16, _{Average-ANS} = 16, _{Above-Control} = 13, _{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; they may not capture a more universal notion of “low-achieving” children (

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, ^{2} = .645, but Training Condition did not, ^{2} = .034. However, an interaction between Training Condition and Symbolic Math Level, ^{2} = .115, appeared. Bonferroni

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 ^{2} = .019, or Training Condition, ^{2} = .002. However, their interaction was significant, ^{2} = .166;

Percentage change in the Operations subtest by Symbolic Math Level (Below-Average, Average, Above-Average) and Training Condition (Control, ANS Training).

Considering the size of the sample (91 participants: Control _{Above-Control} = 13, _{Above-ANS} = 16), we proceeded to analyze all participants whose initial arithmetic scores were within 2.5 ^{2} = .67, but Training Condition did not, ^{2} = .009, and there was no interaction, ^{2} = .06. The three levels for both groups now had an equal number of participants and non-significantly different initial scores across Control and Training groups. Thus, we proceeded to analyze the improvement in Operations subtest. The two-way between-participants ANOVA with Symbolic Math Level (Below-Average, Average, Above-Average) and Training Condition (ANS Training, Control) as factors and percent change on the Operations subtest as the dependent variable revealed no effect of Symbolic Math Level, ^{2} = .019, or Training Condition, ^{2} = .002. However, again, their interaction was significant, ^{2} = .16. Bonferroni _{PreControl} = 40, _{PreANS} = 50, _{PostControl} = 57, _{PostANS} = 57). Therefore, it may be reasonable to consider that the limited time (six minutes) given to the children to do the test possibly caused a ceiling effect of the number of correct problems to be solved. This prevents a detection of potentially greater improvements in the Above-Average tertile in the ANS Training group.

Although our outcome variable was the percentage of change in the Operations subtest, we also analyzed this improvement in the exact Arithmetic test (Addition and Subtraction subtests). Considering that, for the whole sample, the number of correct answers in Operation subtest was highly correlated with the number of correct answers obtained in the Addition and Subtraction subtests pre and post training, it may have been expected for the exact Arithmetic test to show similar results in the percentage of change of the below-average tertile. However, in a 2 X 2 ANOVA, with Symbolic Math Level (Below-Average, Average, Above-Average) and Training Condition (ANS Training, Control) as factors and percent change on the exact Arithmetic test as the dependent variable, showed that Symbolic Math Level differed, ^{2} = .20, but Training Condition did not, ^{2} = .02, and no interaction was found, ^{2} = .02. Thus, no difference was found between the Control and the ANS Training groups regarding the exact Arithmetic test, they all improved on exact additions and subtractions during the period of the experiment. 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 spurious result, we also checked if children showed improvement in all three problem types of the Operations subtest. Focusing on the Below-Average level and the Operations subtest, where the training seemed to be effective, we assessed if any specific equation type (additions, subtractions or multiplications) drove the observed difference between groups. A 2 X 3 mixed ANOVA, with Training Condition (ANS Training, Control) as a between-participant factor and Operation subtest equation type (addition, subtraction, multiplication) as a within-participant factor revealed a main effect of Training Condition, with ANS Training children showing greater percentage change than Control children, ^{2} = .06, but no effect of problem type, ^{2} = .002, and no interaction, ^{2} = .006. Thus, children who performed Below-Average in the pre-training arithmetic scores and had ANS training, improved in all types of equations of the Operations subtest, without any of the equation types solely driving the difference between groups. This suggests some stability of this effect, but of course, more research is needed.

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).

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 (

In relation to other ongoing debates, we found that initial symbolic math ability related to ANS efficiency (

Concerning another controversy, we think it is very likely that our children were relying on some continuous extent cues (

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 one-month 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 (

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 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 (

There are further explanations to be considered. One potential tension in our results is that we found that Pre-training Symbolic Math Test performance is significantly related to ANS performance (

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.

A-test-additions.pdf

B-test-additions.pdf

A-test-subtractions.pdf

B-test-subtractions.pdf

A-test-operators.pdf

B-test-operators.pdf

Instruccions_professores_Catala.pdf

Teachers_instructions_English.pdf

Results_math_tests.csv

The research was supported by a McDonnell Scholar Award to J.H.

Special thanks to Sònia Sas, general director of Hamelin Laie International School at the time of testing, for giving access to the participants and the school facilities essential to conduct this research. We thank the teachers Mar Moreno, Teresa López Guerrero and Catherine Teti for their collaboration with the experimenter during the completion of the research. We thank Luis Morís Fernández for advice in data analysis, and Ariadna Batalla Ferrés and Luca L. Bonatti for useful advice on the manuscript.

The authors have declared that no competing interests exist.