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Exact arithmetic abilities require symbolic numerals, which constitute a precise representation of quantities, such as the Arabic digits. Numerical thinking, however, also engages an intuitive non-linguistic number sense, the Approximate Number System (ANS). The ANS allows us to discriminate quantities, approximate arithmetic transformations, and estimate quantities, all without counting individual items. Although the ANS does not require language, estimations made by means of the ANS can be expressed with number words or Arabic digits. A connection between the ANS and school math performance has been established. A child’s accuracy in mapping from approximate quantities to Arabic digits is associated with children’s symbolic math abilities and can also predict their success at learning new arithmetic skills. Here, we explore whether directly training the mapping between estimated quantities and Arabic digits transfers to better math proficiency. The control training was based on discriminating quantity representations, without involving digits. Each of these three-week computer-based trainings were added to the school schedule. We measured improvements in approximate and exact arithmetic after training. Both the experimental and the control group improved in approximate arithmetic performance. However, in exact arithmetic, results show that strengthening the digit-quantity relation improved the 7-year-olds’ competence in symbolic additions and subtractions over and above the improvement measured in the control group. Our results speak to the complexity of the factors involved in developing mathematical abilities, making the case that training the mapping from estimated quantities to digits can be particularly effective in improving children’s mathematical performance.

The construction of mathematical competence is a complex challenge which we still do not fully understand, although much progress has been made in the last 20 years to reveal the complexity of the mental processes involved. The translation of such scientific understanding into practical ways to improve mathematical skills during the crucial ages when basic mathematical abilities are being acquired is still in its infancy. This is not surprising, considering that this acquisition may be affected by many factors, such as gender (

Many studies show that the ANS is present in non-linguistic animals as well as in preverbal infants and adults from different cultures (

When processing non-symbolic representations, two phenomena that are hallmarks of ANS engagement emerge: the distance effect and the size effect (

Many studies have revealed a connection between the ANS and school math performance (

Although the ANS has been related to mathematical achievement across many studies, as reported above, transfer to arithmetic after training has returned inconsistent results, depending on the trained ANS ability. For example, training quantity discrimination leads to small (

Mapping from quantity to numbers, that is, expressing an estimated quantity in mathematical language, can be more or less accurate. Importantly, the accuracy of this mapping relates to children's math abilities, both in number word format (

Mapping between non-symbolic and symbolic numerical representations is an ability that children develop over time (

Mapping can be trained (

As reviewed above, a more accurate mapping between estimated quantities and Arabic digits is related to higher mathematical achievement. Mapping develops over time, and experimental results suggest that its accuracy can be calibrated and trained. However, the development of these findings into concrete training programs to help children has yet to be robustly explored. In the current study, we present a training regime of the estimation of quantities and their mapping to Arabic digits in 7-year-old children during a 3-week period. Our aim is to assess change and transfer from such training to improvements in arithmetic. We named this training regime Numerical Estimation Training (NET). Because the accuracy of the estimations can be improved by calibrating with informative trials (

Our control group replicated the Quantity Discrimination Training introduced by

The study comprised pre- and post-training tests and an intervention of 3 weeks of training. For pre- and post-testing, we used the exact Arithmetic test and the Operations test used in

We also considered the behavioral presence of distance and size effects for both symbolic and non-symbolic representations, as discussed above (

In summary, we replicated both the QDT (here, the training for the control group) and the pre- and post-math tests as in

The main motivation of this study is the suggestion in the literature that in the learning of mathematics, strengthening the mapping between nonsymbolic and symbolic representations of number is a key factor that is often overlooked. We explore the hypothesis that training children to map estimated quantities to their corresponding digits can better calibrate the quantitative meaning of numbers and result in an overall better understanding of the digit-quantity relation, with the potential of manifesting in improvements in their arithmetic abilities.

Ninety-one children from the second grade of primary (38 girls; average age = 7 yrs 9 mos, range = 6 yrs 4 mos - 8 yrs 9 mos) participated in the study. The children mostly came from middle-to-high socioeconomic status families. The study was conducted at the Hamelin International Laie School (

Participants attended four different classrooms, taught by two different teachers. Two classes were randomly assigned to the Numerical Estimation Training group (experimental;

In order to determine participants’ mathematical competence, we administered the three pencil-and-paper tests administered by

Two versions of each test were prepared. Each version contained different problems although, due to the limited amount of possible combinations to form the equations, a few of them were present in both versions. The order of the problems was randomized. In this way, both versions of the booklets could be used for the pre- and post-training tests, and could be counterbalanced across the sessions, so as to control for tests effects. We created a large number of problems for each test, so that the children could not complete all of the problems during the allotted time (6 minutes).

The Additions Test included 210 problems presented in a columnar operation algorithm form, printed on 10 pages. The maximum number that each addendum could reach was 18, with the highest sum being 18 + 18 and the lowest sum being 0 + 0 (

The Subtractions Test included 190 problems presented in a columnar operation algorithm form, printed on 10 pages. Both the minuend and subtrahend ranged between 0 and 18; the results of the subtractions were always positive (

The Operations Test included 117 problems presented in horizontal format, printed on 3 pages (

It is important to note the different nature of the tests. Although all three tests require the previously acquired knowledge of what the basic arithmetic operations do, the Additions and Subtractions Tests also require the ability to make exact calculations, while the Operations Test only requires a comprehension of how the three arithmetic operations roughly change quantities. Stating it in a simple manner, the result is generally “more” for addition, “less” for subtraction, and “much more” for multiplication, allowing for a comparison between both sides of the equal sign.

Twenty-eight computers (model: clon PCs, Intel(R) Core(TM) i3-4170 CPU @ 3.70 GHz, 4GB RAM, 64-bit; monitor: 17” LCD 16/9 from ASUS; operative system: Windows 7 Professional) were used for the training activities. The children wore headphones during training. The stimuli that formed the collections of items for the training part were the same (blue dots, yellow dots, cars, bears, birds, dogs...) in both training groups. In order to maintain children's interest, these items varied along the trials. Both training regimes had the same number of runs and trials: 24 runs with each composed of 35 consecutive trials. In each trial, the stimuli appearing on-screen remained visible too short of a time for children to count the items. This was meant to engage the ANS in the task. Trials were presented in increasing order of difficulty (details below), a procedure which is known to facilitate learning (

Participants in this group were presented with the “Digits” game, written in PsychoPy v1.83.01. The Digits game program generated two types of trials: the

The purpose of the passive learning trials was to provide children with opportunities to directly calibrate their estimation system before the active training trials began (

In the NET, there was a range of distances across trials from the target value to the distractor values. For example, if the target number was 15, a decision between 18, 21 and 15 (i.e., +3, +6, and 0 distances from the correct choice) would be easier than the decision between 15, 13 and 17 (0,-2,+2) because of the differences in distance between the target value (15) and the distractor values. We defined three difficulty levels based on the range that the trials could have. For

All set sizes, from 1 item to 21 items, had to be estimated in each of the three spans. The distances between possible answers were maintained irrespective of the target answer (rather than scaling the distractor answers relative to the correct answer by some ratio). Thus, trial difficulty increased with target answer (

For this group, we used a modified version of the computer game Panamath (

The items were presented in seven different ratios (larger set ÷ smaller set). The ratios could be 3, 2, 1.5, 1.25, 1.17, 1.14, and 1.1. For example, on a 3-ratio trial children might see 21 blue dots on the right side of the screen and 7 yellow dots on the left side. Smaller ratios correspond to more difficult trials. Always, the first five trials of each run presented the easiest ratio. Then, every five trials the game increased in difficulty, with the ratios becoming closer to 1 (without ever reaching 1), until the seven different ratios were presented. This method of presentation uses “confidence hysteresis” and tends to return best possible performance and successful transfer to symbolic mathematics (

To vary the relationship between surface area and number, the Panamath game implemented three different models controlling for object size: size-confounded (42% of the trials in each run;

The experiment was run in three phases:

The pre- and post- assessments were intended to measure the mathematical competence of the participants before and after training. We followed the same procedure as was applied in

The assessments always began with the Additions Test. After completing this 6-minute test, children had to stop answering, return the additions booklet, and wait until they were given the subtractions booklet. The Subtractions Test followed, with the same 6-minute procedure. For the third and final test, the Operations Test, the teacher explained the task to the children in more details, given that this kind of problem was new to them. The teacher briefly described the equation structures and completed 3 examples on the board (one each for addition, subtraction and multiplication) in front of the class. Then children began the 6-minute assessment of the Operations Test.

The Pre- and Post-Training assessments were completed in the students’ regular classrooms, during their regular math class time. The teacher and the experimenter carefully avoided drawing any attention to possible connections between the training schedule and the Pre- and Post- assessments.

Training was administered in six different sessions within a three-week period, at a pace of two sessions per week. Both groups were trained for the same amount of time and with the same number of trials. Each session took approximately 25-30 minutes to complete. The training sessions were integrated into the school schedule of each class, during their computer class-time. Because of this procedure, any effect due to potential deviations of the experiments from class routine was minimized. Training sessions for both conditions were carried out in the same computer classroom and always in the presence of the computer class teacher and the experimenter. Before each training session the experimenter prepared the computer classroom. All material was checked by the experimenter, including the volume of the headphones. At each training session, children worked individually after receiving instructions. They were informed that both speed and accuracy were important. Children were also told that the game would vary in difficulty. They were informed that two different sounds would provide them with feedback about the correctness of their answer, and that this feedback would be provided after every response. Lastly, it was explained to them how to answer. Children of both groups appeared to like the games of their training sessions.

To train children’s ability to map from estimated quantities to digits, children belonging to the Numerical Estimation Training group practiced the Digits game. When introduced to the game, children were told they would play a game where they would first see a collection of objects for a very short time, while an audio recording would tell them how many objects were in the collection. They were told that they would have to pay attention to these trials, but not take any action. They were also informed that they would then see many trials where a collection would be shown for a short period of time, after which they would have to estimate their numerosity by choosing one of three digits that would appear on-screen immediately after the disappearance of the items.

In our control group children practiced the Panamath quantity discrimination game, replicating the procedure of the experimental group in

When introducing this QDT game, children were told that they would play a game where they would see two sets of objects on two sides of the screen – for example, blue dots on the left and yellow dots on the right – and that they would have to choose the set that had more items.

We first present the results of the ANS hallmarks and the efficiency of the training sessions for each group. We then turn to the main results of interest, that is, the effects of the training on symbolic mathematics.

In the NET task, children viewed a quantity of briefly flashed items and had to estimate the amount by choosing the correct match among three possible digits. On average, children responded correctly on 51.3% of trials (

Consistent with predictions of the distance effect, children’s responses were better as the runs were easier (percentage of correct answers on average: _{Easy}_{Medium}_{Hard}^{2} = 0.38. Bonferroni ^{2} = .82,

Next, we analyzed children’s response time. Average RT was 2234.6 ms (

The efficiency, operationalized as the percentage of correct responses divided by RT, increased as children progressed throughout the sessions of the NET (

In the QDT, children briefly viewed two non-symbolic sets of items and had to choose the biggest set. On average, children responded correctly on 72.2% of trials (

In all cases, children's performance exhibits the smooth curve of the Approximate Number System. That is, even if size contributes somewhat to children’s decisions, children's numerical decisions were likely based on the ANS. The curves in

Using the standard psychological model and fitting methods (e.g.,

Children’s average RT across training sessions was 952.8 ms (

We analyzed the efficiency of the QDT, operationalized as the percentage of correct responses divided by the RT. A linear training slope was computed for each child. Children had significantly positive slopes for efficiency across training sessions, indicating efficiency improvement during the training,

In sum, both the NET and the QDT showed signatures of engaging the ANS and had the appropriate level of difficulty, as shown by the fact that children’s task efficiency improved during the 3 weeks of training.

The main analysis of interest concerns the effect of training on children's mathematical performance, in the Additions, Subtractions and Operations Tests. A 2 x 2 mixed ANOVA with Training Condition (NET, QDT) as a between-participant factor, Phase (Pre-, Post-training) as a within-participant factor, and total correct answers as the dependent variable was run for each test. We report the results below.

For the Additions Test, the analysis revealed a main effect of Phase, _{(1,88)} = 10.03, ^{2} = 0.011, no effect of Training Condition, _{(1,88)} = 0.177, ^{2} = 0.001, and a significant Training Condition by Phase interaction, _{(1,88)} = 7.03, ^{2} = 0.008. Bonferroni _{Post}_{Pre}_{Post}_{Pre}

For the Subtractions Test, the ANOVA revealed a main effect of Phase, ^{2} = 0.03, no effect of Training Condition, ^{2} = 0.037, and a significant Training Condition by Phase interaction, ^{2} = 0.006. With respect to the significant interaction, Bonferroni _{Post}_{Pre}_{Post}_{Pre}

For the Operations Test, the ANOVA revealed a main effect of Phase, _{(1,88)} = 29.33, ^{2} = 0.04, no effect of Training Condition, _{(1,88)} = 0.74, ^{2} = 0.007, and a non-significant Training Condition by Phase interaction, _{(1,88)} = 0.10, ^{2} = 0.0001, (

*^{+}^{++}

Many studies have revealed a connection between the ANS and school math performance (

The mapping task between non-symbolic and symbolic number representations focuses on the estimation of quantities (

In the present study, we trained the mapping from estimated quantities to Arabic digits in 7-year-olds during a 3-week period, with the aim of transferring improvements to arithmetic. We introduced a novel training regime, the Numerical Estimation Training (NET), implemented in a computer intervention (“Digits” game). For our control group, we replicated the Quantity Discrimination Training regime (QDT) from

Despite the challenging control training of the QDT, we found that the NET regime transferred stronger improvements to arithmetic performance. Surely, some of the improvements seen in both groups could be due to testing repetition effects or to the standard school activity during the period of the training studies, but these factors affected both groups identically.

In both training regimes, we detected the signatures of ANS engagement, and in both regimes children’s efficiency increased, confirming ANS trainability.

We measured the transfer of the training to arithmetic performance with the Pre-training and Post-training assessments, composed of the Additions Test, the Subtractions Test and the Operations Test. Importantly, the arithmetic knowledge necessary to perform these tests was neither explained nor practiced in any of the training conditions, because we wanted to assess the transfer of the trained aspects of numerical cognition to the performance in formal school arithmetic. Also, it is worth stressing that in the Pre-test phase, both groups started from a very comparable level, because there were no differences in any test at the Pre-training phase. Thus, between-group differences are not likely to have affected the results.

The data suggest that the NET regime improved children’s math abilities more than the QDT regime, at least in the two tests that focus on exact arithmetic: the Additions and the Subtractions Tests. In the Additions Test, the NET group improved the number of correct answers after training while the QDT group did not. This suggests that the NET regime transferred to an improvement in children’s abilities to solve exact additions quickly and correctly.

In the Subtractions Test, both groups significantly increased in their number of correct answers within 6 minutes, although improvement was much higher for the NET group (

In the Operations test we detected a similarly high pre-post test improvement in both groups. This test is sensitive to children’s approximate arithmetic level because it does not require explicit exact calculations but it does require an understanding of how basic arithmetic operations change quantities. In

What could have generated the NET advantage in exact arithmetic improvements? The results suggest that training the mapping from estimated quantities to Arabic digits with the NET has transferred to an improvement in children’s arithmetic abilities. Recruiting the ANS to estimate quantities and mapping these estimations to Arabic digits, along with the calibration provided by the passive trials (

A solid comprehension of the meaning of the numerical symbols used in arithmetic is important for the understanding of the arithmetic calculations themselves. We submit that the educational system may be overestimating 7-8 year olds’ comprehension of this basic aspect of mathematical language. We propose that an appropriate training of mapping from estimated quantities to digits, such as the NET, could be a potent way to foster an overall improvement in children’s arithmetic abilities.

The research was supported by Catalan Government (SGR 2009-1521), MINECO PID2019-108494GB-I00, and a McDonnell Scholar Award to J.H.

The authors have declared that no competing interests exist.

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 conducting this research. We thank the teachers Elvira Cabrespina and Mar Moreno for their collaboration with the experimenter during the completion of the research. We thank Miguel Burgaleta for advice in data analysis, and Cristina I. Galusca and Elena Pagliarini for useful advice on the manuscript.

For this article, a dataset is freely available (

The Supplementary Materials contain the following items (for access see

Math Tests: Two versions of each test were prepared, with different problems, so that pre- and post-training booklets were unique and counterbalanced across the sessions

Instructions to the teachers to guarantee the correct procedure during tests. The version in Catalan (original), and the version in English (translation)

Results of the math tests from the 90 children

Access to training materials (computer games)