Foundations for numeracy

Children have a rich repertoire of conceptual abilities that afford some forms of numerical representation well before entering school, including the ability to represent the approximate numerical magnitudes of sets and the ability to represent a limited number of individual objects in parallel [see 29–31 for reviews]. By the first year of life, infants detect numerical differences between sets of objects as long as the ratio differences are sufficiently large. For example, after watching, and losing interest in, a succession of images of 8 objects, 6-month-old infants regain interest when presented with an image of 16 objects, but their interest remains low when presented with a novel image of 8 or 12 objects [e.g., 32, 33]. By virtue of this approximate number system (ANS), young children and even infants have been shown to add and subtract numerical magnitudes [34]. For example, if 5 rectangles are covered by a screen and then 5 more rectangles move behind the screen, infants look longer if the removal of the screen reveals 5 rectangles (incorrect sum) rather than 10 [34]. However, as the name implies, these numerical representations and associated computations are imprecise, both for infants and for children who are presented with sets too numerous to count [35].

Children also have an object tracking system that allows them to represent and track a limited number of distinct objects in parallel, to distinguish one object from another (numerical identity), to compare sets for numerical equivalence and ordering using one-to-one correspondence, and to perform exact additions and subtractions of small numbers of objects [see 30 for a review]. For example, in a two alternative forced choice search procedure, 12-month-old infants have been shown to reliably approach a bucket with more food items compared to a bucket with fewer food items when the choice involves only a few, individually trackable items (e.g., 1 vs. 2, 2 vs. 3, or 1 vs. 3). However, they fail to reliably choose the bucket with more in cases of 1 vs. 4 or 2 vs. 4, suggesting an upper limit to the number of items that can be reliably tracked in parallel over occlusion [30, 36]. Infants also expect exactly three objects to stand behind a screen if the screen first was lowered over an array of two objects and then another object was placed behind the screen [37]. This object tracking system (OTS) appears to support children’s numerical comparison and addition by facilitating one-to-one matching of individual, trackable items. While it allows more precise comparisons, it is severely limited such that no more than three objects can be tracked at once.

The ANS and the OTS both are present from infancy and are hypothesized to play important roles in numeracy development [e.g., 30]. However, neither system appears to contain the information necessary to represent natural number concepts (i.e., the positive integers), accurately enumerate numbers greater than about 4, or perform exact arithmetic [see 29]. One potential reason for these limitations is that each system requires attention, and the two systems compete for attentional resources: attending to the numerical magnitude of a set reduces processing of the attributes of its individual members, whereas attending to one or more individual members reduces processing of the set’s numerical magnitude [31, 38]. Thus, the systems cannot be combined directly to determine the exact numerical concepts needed for numeracy. It is possible, however, that representations from the ANS and the OTS are combined by the use of numerical language or symbols [29, 39].

In addition to these non-verbal representational systems, children are endowed with a capacity for symbolic thought: a capacity that is seen most prominently in children’s learning of a natural language [e.g., 40] but that also is manifest in their increasing use of spatial symbols to convey complex meanings [see 41]. For example, children begin to learn the meanings of words before the end of the first year [e.g., 42–45]. By 2 years of age, children distinguish singular from plural noun phrases [e.g., 46, 47]. By 3 years, they begin to recite the ordered list of number words used for counting [1–3] and to comprehend and produce number words in quantified noun phrases [48].

In parallel, infants also begin to understand that items such as pictures can be used symbolically. Between the first and second year of life, infants shift from grasping at pictures as if they were objects themselves to pointing at them, suggesting an emerging understanding that the picture is meant to represent the object [49]. By the time children learn number words, they are able to use realistic and simple iconic models or pictures as maps to find hidden items, suggesting a further shift in symbolic thinking [e.g., 49–52]. These developmental shifts are thought to reflect an emerging understanding that physical items can be both objects in their own right and symbolic representations of something that is not currently present to their senses [e.g., 50].

The role of symbols in numeracy development

There is a longstanding and nuanced debate regarding the role of symbols in the development of early numeracy, with a particular focus on the development of natural number concepts [e.g., 4, 29, 53–60]. Although a full discussion of the intricacies of this literature is beyond the scope of this paper, existing theories can be distinguished by whether they posit symbols as necessary for the development of natural number concepts or not. Nativist accounts posit that natural number concepts are innate, and symbols (e.g., number words) only serve to foster processes of enumeration and communication [e.g., 1, 53, 54, 61–64]. In contrast, other theories propose that symbols are critical for the development of natural number concepts [e.g., 29, 55–56, 58, 65, 66]. Some of these theories further posit a privileged role for linguistic symbols in the development of natural number concepts [39, 65, 66].

Training numeracy

Despite some consensus among theories that symbols are important to early numerical development, few studies that we are aware of have tried to directly test the causal role of symbols in the emergence of numeracy through focused experimental interventions. This is not to say that early numeracy training studies with symbolic numbers have not been done. A number of other groups have attempted to improve early numeracy though symbolic number intervention, with some high-profile attempts showing promise [e.g., 12–16, 67, 68, but see 69]. However, most attempts, like many curricular interventions, involve broad training of a rich combination of symbolic and non-symbolic skills. That is, training was multifaceted, often including combinations of non-symbolic number, number words, digits, and/or the mapping between these symbolic and non-symbolic number forms and other magnitudes (especially spatial magnitudes). Given the multifaceted nature of these numeracy intervention studies, it is unclear which aspects of the training produced the changes in children’s numerical abilities.

In one exception, researchers directly contrasted outcomes of symbolic (digit) and non-symbolic numerical training in a small group of preschoolers [70]. Over 10 sessions of training, preschoolers in the experimental groups played computer games where they either associated Arabic digits or dot collections with physical magnitudes. Compared to a non-numerical control group, both the symbolic-digit training group and the non-symbolic training group improved on non-symbolic numerical skills. However, only those in the symbolic training group showed improvement on symbolic number skills, including exact symbolic arithmetic. These results suggest that symbolic number understanding may facilitate exact arithmetic.

Current approach

Here we report an initial, exploratory study using focused experimental training methods combined with a deep assessment of the outcomes to address the role of symbols in early numeracy. More specifically, children were assigned to play one of five variants of a numerical training game for about two weeks. All variants of the game involved tracking and comparing collections of objects on the basis of number and were equated for general gameplay, expectations about the training, and other cognitive and behavioral task demands. We chose non-symbolic numerical comparison as the basis of all training variants because a) it is numerical in nature b) it engages core systems available to all participants and c) non-symbolic numerical magnitudes and symbolic numbers become linked over development [e.g., 71–73]. Five different training variants presented different types of numerical content (see Fig 1A–1E, also see S1 Table in S1 File). Two of the training variants were designed to only emphasize non-symbolic, non-verbal representations of number [see 30, 31, 59]. In one, items were presented sequentially one-by-one to elicit attentional tracking of individual objects (OTS, Fig 1A). In the second, sets of items were presented simultaneously to elicit attention to approximate numerical magnitudes (ANS, Fig 1B). Two other training variants combined non-symbolic representations of number with linguistic number symbols. In one, items were presented sequentially and paired with verbal counting (Fig 1D). In the other, items were presented simultaneously as sets and paired with number word labels embedded within a noun phrase (Fig 1E). The latter variant was designed to associate non-symbolic representations of numerical magnitude with exact cardinal value labels by embedding this information in a richer linguistic context thought to be important for understanding of natural number. A final variant was designed to combine tracking of individual objects with a well-known system of spatial symbols for representing number and performing arithmetic: an abacus. In this condition, sequentially presented items were paired with an animated abacus that tallied items one-by-one (see Fig 1C).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Schematic depiction of the basic training gameplay for the five training variants.

Each column illustrates an example trial for each variant (A-E) where the child is asked to track a “1+2” event in the leftmost basket, and then to compare the sum (3) to the set of 2 items in the rightmost basket. Gameplay progresses from top to bottom (solid arrow). Dotted arrows indicate the animations of objects into the baskets, which could be sequential for each object (columns A, C and D) or simultaneous within each circled set of objects (columns B and E). Both the animation and the feedback period could be accompanied by verbal counting (column D) or numerical labeling (column E), or by a symbolic abacus where items moved in one-to-one correspondence with each added object (column C). Note the minimal differences between conditions and overlapping rows that indicate commonalities across the variants. Additional details of gameplay and feedback can be found in the text.

https://doi.org/10.1371/journal.pone.0259775.g001

Our methodology allowed substantial experimental control over the type of numerical input children received. By systematically combining groups of children trained on different variants and contrasting their outcomes [74], we tested several focused inferences about the role of symbols in numeracy development. Specifically, we contrasted the effects of training with or without symbolic aids, and training with or without verbal support (see S1 Table in S1 File). More broadly, the use of computerized and closely matched training allowed us to equate many aspects of the training procedure across participants, while minimizing the effects of particular teachers on the effectiveness of particular interventions [75].

In order to test the effects of these different types of training on numeracy, we systematically assessed several major components of early numeracy development before and after training rather than taking a single standardized summary outcome score. We assessed children’s symbolic number understanding through counting, their ability to perform non-symbolic numerical comparison, their sensitivity to one-to-one correspondence, and their ability to perform arithmetic transformations. These assessments served to identify the particular areas of numeracy, if any, that were affected by the different types of numerical training.

Procedure overview

Children first were given pre-test assessments and those who had already mastered the critical numeracy skills to be trained were not invited to participate in the training portion of the study (see Participants section below for details). Children who had not yet mastered critical numeracy skills as assessed by the pre-test were pseudo-randomly assigned to play one of five variants of a computer-based numerical comparison training game over two weeks (see Fig 1). Assignment choices were made as the study progressed to keep groups approximately balanced in number of subjects who had completed testing, pre-test abilities, and age, based on pre-test and demographic data. Specifically, children were asked to play their assigned version of the computerized numerical training game twice a day for 9 days over a two-week period (total of 18 sessions). Each session was estimated to take about 15 minutes, although the game was self-paced. After two weeks of training, children were invited back for a post-test session with a majority of the same assessment tasks.

In order to obtain a large sample for this resource-intensive study, two investigators in different countries combined efforts to collect data on the effects of this numerical comparison training on early numeracy in preschool children. This approach yielded some variability in training environment (e.g., preschool vs. home training) and variability in demographics (country, age). All testing and training software and instruction were in English in the U.S. and Italian in Italy. The U.S. children were tested in the lab and in local preschools. The Italian children were tested in local preschools only. Those who were tested in the lab played their training game at home under the supervision of a parent or guardian. All children tested in preschools played their training game in a small group setting under the supervision of trained research assistants from the lab who provided one-on-one support to students as needed. In Italian preschools, all individuals within a single small group played the same training variant and variants differed between groups. In U.S. preschools, individuals within a single small group played different training variants. Regardless, all children tested in preschools in both countries wore headphones and were separated by at least a few feet to eliminate interference in variant-specific and individual player-specific feedback between laptops. All numeracy assessments were conducted one-on-one with children by trained research assistants, graduate students, or postdoctoral researchers. Although it was not always possible, our best efforts were made to have the same assessor conduct both pre- and post- testing sessions with a given child and remain naïve to the type of training the participant had received.

Training

The training game presented all participants with the same basic gameplay scenario and objective: to track the number of items being placed in a basket, remember how many items were in that basket after occlusion, compare the number of items in that basket (occluded) with a second basket of items (visible), and to choose the basket with numerically more items. All five training variants started with a screen showing one exemplar item that served as a prompt to indicate the target items that the child would be working with during that session: fishes, turtles, birds, armadillos, bananas, or nuts. Target items varied from session to session. For trials with turtles as the target items, for example, the computer instructed participants to “Choose the basket with more turtles!”. The screen then updated to reveal two baskets to the left and right of the center of the screen and a set of turtles above the leftmost basket (see Fig 1). The number of items in the first set varied from trial to trial (see details on training variants below). Participants were instructed verbally by the computer to “Get as many turtles as you can. Press the SPACE bar to put the turtles in the basket (see Fig 1).” Participants then pressed the SPACE bar to activate an animation whereby the items moved into the leftmost basket and became occluded (Fig 1). Then a second set of target items appeared above the same leftmost basket and participants continued to press the SPACE bar to move the rest of the items from the second set in that same, leftmost basket (Fig 1). The number of items in this set could also vary from trial to trial (see details on training variants below). After all items from both sets where placed in the leftmost basket and occluded, an animation rotated the rightmost basket to reveal its contents (Fig 1). At that point, the participant, who could only see the contents of the rightmost basket, was instructed by the computer to “Choose the basket with more turtles!”. The participant then pressed the RIGHT or LEFT arrow key to choose the basket they thought had more turtles in it. Trial-by-trial feedback on accuracy was also provided; game difficulty increased across training sessions based on the participant’s individual intersession performance (Fig 1; see below the detailed description of feedback and progression for each variant). For all correct answers, a chime played, the computer said “correct”. For all incorrect answers, a trombone-like sound played, the computer said “look”. Additional feedback was provided depending on which training variant was being played (see details below).

Numeracy assessments

Four main aspects of early numeracy were assessed pre- and post-training: counting proficiency, approximate numerical comparison, one-to-one correspondence, and exact arithmetic with objects (see Fig 2). The tasks were administered in electronic form on a laptop computer in a fixed order for the majority of participants. Computerized versions were used in order to provide more experimental control over the testing, and, as a result make testing more comparable across children, experimenters, and testing sites. Deviations in order, although rare, were required occasionally due to the logistics of testing individual children. These particular assessment tasks have been previously described in several recent publications focusing on relationships between pre-test variables only [72, 75–77]; no published report to date has analyzed post-test data or the effects of training with this dataset.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Depictions of numeracy assessment tasks.

Counting proficiency was assessed using electronic versions of two well-known tasks designed to measure children’s ability precisely produce (Give-N) or enumerate (How Many?) sets of 1–8 items. On a single trial of the Give-N production task, children were asked to bring down a certain number of items (1–8) from a larger collection of 10 items at the top of the screen by pressing the SPACE bar. For the How Many? enumeration task, children were asked to report how many (1–8) items were presented on the screen. Numerical comparison was assessed by showing children a display containing two sets of dots side-by-side and asking them to point to the set with more dots. One-to-one correspondence was assessed by modified non-symbolic and non-verbal versions of our Give-N production and How Many? enumeration tasks, where verbal numbers were replaced by a non-verbal abacus. In the Give-N Abacus task, a visual representation of an abacus with a certain number of highlighted beads was presented at the bottom of the screen as the numerical prompt instead of a number word. Children were asked to match the same number of items as were on the abacus by bringing them to the bottom of the screen through pressing the SPACE bar. For the How many? Abacus task, children were presented with pictures containing 1 to 8 items on the computer screen and were asked to match the number of items on the computer screen by moving over the same number of beads on a physical abacus they were given. Arithmetic transformation was assessed using a non-symbolic subtraction task where 1–6 items entered a can via animation (e.g., 5), some or all of those items came out (e.g., 3), and then children were asked to choose how many items they thought were left in the can by pointing to the options of “zero”, “one”, or “two” items on the right part of the screen. Additional parameters of all tasks can be found in the main text. PLEASE NOTE: Figs are not to visual scale with actual assessment tasks.

https://doi.org/10.1371/journal.pone.0259775.g002

Other variables of interest

Participants

A total of 215 children participated in pre-testing, training, and post-testing and had complete data for the first numeracy assessment (i.e., counting proficiency). Fewer children had complete data for the other assessments (numerical comparison n = 214, one-to-one correspondence n = 170, arithmetic n = 177, see S2 Table in S1 File for more details) due to either an individual child not completing that particular assessment during both testing sessions or a particular task not being run at a particular testing site. Children in the final dataset were, on average, 49.15 months of age (SD = 7.82 months, range = 35.97–76.01 months). Data was collected in the U.S. and Italy. Children in Italy were on average 8.53 months older (n = 79, M = 54.54 months) than preschool children tested in the U.S (n = 136, M = 46.01 months). The range of participant ages was also greater in Italy (range = 40 months; 36–76 months) than the U.S. (range = 21.47 months; 35.97–57.44 months). Children in the U.S. sample also reported less years in preschool (M = .56 years, SD = .61 years) than children in Italy (M = 2.10 years, SD = .84 years), although these estimates are likely biased due to a substantial amount of missing (48/215 missing). Country differences in age, and possibly differences in years in preschool, are not an artifact, but due to the fact that children in this part of Italy tend to start elementary school later than children in U.S.

Analyses

First, we explored the summary characteristics of our data by analyzing the relationships between demographic factors, pre-test scores, and gains after training (post-test scores minus pre-test scores) using basic descriptive statistics, simple correlations, and partial correlations. Second, we analyzed the extent to training in general, regardless of training variant, led to gains in numeracy using t-tests and the extent to which these gains were related to the demographic factors and performance on the training itself using partial correlations. Third, we tested several focused hypotheses regarding the hypothesized mechanisms that may drive numeracy development. Given the resources required to run this study, including the intensity of training and comprehensive scope of pre- and post- training assessment, it was not possible to obtain large enough samples to yield high-powered pairwise statistical comparisons between all five individual training variations as we had originally hoped. However, different variants were able to be conceptually grouped in different ways to explore several hypotheses regarding the role of symbols in numeracy development, as indicated in the introduction [see 74 for similar combinatorial approach]. Sensitivity analyses conducted afterwards suggested that our approach should be able to detect medium effect sizes with a high degree of power (.95) given our statistical design and our obtained sample sizes (see S2 Fig; S2 Text in S1 File).

The first focused hypothesis we tested was that training combining numerical comparisons with symbolic aids of any sort (verbal or visuo-spatial) would lead to more improvement than training with non-symbolic numerical comparisons alone (i.e., without symbolic aids). To test this hypothesis, we compared the outcomes of the variants trained with symbolic aids (verbal-counting; verbal-labeling; abacus-counting, n range = 92–130 depending on assessment task) to those trained with non-symbolic numerical comparison only (sequential non-symbolic and set-based non-symbolic enumeration, n range = 70–85 depending on assessment task).

The second focused hypothesis we tested was that numerical comparisons with verbal number aids specifically lead to more improvement than numerical comparisons without verbal number aids. We tested this hypothesis in several ways. First, we contrasted the outcomes of children who were trained on variants involving verbal numbers (verbal-counting; verbal-labeling, n range = 58–93) with children trained on all the other variants (n range = 103–122). Second, we restricted our analyses to compare outcomes in only those children that received a symbolic variant, contrasting those trained with verbal-symbolic numbers (verbal counting and verbal labeling, n range = 58–93) to those who received non-verbal symbolic abacus training (abacus counting, n range = 34–37). Given the small sample of those trained on the abacus counting variant, we interpret any pairwise comparisons with this single condition with more caution. Finally, we conducted several additional exploratory analyses to better understand the symbolic training advantages that we observed. We also visualize gains for individual training variations and include those visualizations in the supporting information (S3 Fig in S1 File).

We tested for training effects using ANCOVAs with the planned combinations of training variants entered as categorical independent variables and the post-test scores as the dependent measure. Pre-test scores were always entered as a covariate in ANCOVA models to account for any differences in starting point before training, allowing us to make inferences about gains due to training. In cases of significant effects, we determined whether there were any additional pre-training demographic differences between the particular groupings being contrasted that could alternatively explain the difference (see S1 Text in S1 File). Factors on which groupings differed, if any, were then were added to determine if significant results held after controlling for those differences as covariates (i.e., in age, vocabulary, and/or experience with symbolic numbers as tested through count list knowledge). We used pre-post difference scores (post-test minus pre-test scores) for figures in order to more intuitively interpret and compare gains between training groupings and tasks.

Pre-test

Children started with a wide range of numerical abilities before training (counting proficiency M = 59.45%, SD = 20.93%, range 6%-88%; numerical comparison M = 63.55%, SD = 11.08%, range = 39%-94%; arithmetic M = 39.16%, SD = 17.15%, range = 8%-92%; one-to-one correspondence M = 54.63%, SD = 21.20%, range = 13%-100%). Correlations between demographic variables and pre-training abilities revealed that all numeracy measures were related to age, count list knowledge, and vocabulary before training (i.e., pre-test; Table 1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Pre-training correlations between variables.

Pearson’s correlation coefficient (r), p value, and number of participants included in the analysis reported in each cell.

https://doi.org/10.1371/journal.pone.0259775.t001

Training

Children were compliant with the training instructions, completing an average of 211.67 out of 280 planned trials (SD = 67.82, Range = 24–522) and an average of 14.49 (SD = 4.36, Range = 2–25) out of 18 planned sessions of the training game over the course of the two weeks. On average, they also performed fairly well, answering correctly 71.96% (SD = 11.08%, Range = 48%-99%) of the problems they attempted. Finally, children made reasonable progress through the training game, reaching an average level of 10.11 (SD = 2.74, Range = 0–15) out of a possible 15 levels.

Overall outcomes

On average, children showed improvement in all the numerical outcome measures after training except approximate numerical comparison (t(210) = 1.209, p = .228, M gain = 0.82%, SD = 9.81%). That is, children’s gains were significantly different from zero in counting proficiency (M gain = 11.60%, SD = 17.50%, t(214) = 9.716, p < .001), one-to-one correspondence (M gain = 5.98%, SD = 16.38%, t(160) = 4.619, p < .001), arithmetic (M gain = 4.98%, SD = 17.81%, t(172) = 3.679, p < .001), and knower-level (M gain = 1.05 knower levels, SD = 1.75, t(203) = 8.593, p < .001).

An analysis of the relationship between performance on the actual training and post-training numeracy outcomes controlling for pre-test scores using partial correlations revealed a number of effects. Training compliance (total number of trials completed) was related to improvements in counting proficiency (r(212) = .141, p = .039) and corresponding knower-level (r(201) = .192, p = .006). Proportion of correct answers on training was related to post-training improvements on all assessment tasks (numerical comparison r(208) = .325, p < .001; counting proficiency r(211) = .229, p = .001; one-to-one correspondence r(158) = .264, p = .001; arithmetic r(170) = .192, p = .012; knower-level r(201) = .181, p = .010). Highest level of difficulty reached during training was also related to post-training improvements on numerical comparison (r(200) = .197, p = .005) and to counting proficiency (r(203) = .144, p = .040). Together these results link actual performance during training to various improvements in numeracy after training.

An analysis of the relationship between demographic factors (age, vocabulary, count list knowledge) and post-training outcomes accounting for pre-test scores through partial correlations also revealed a number of effects. Age was positively related to gains on all assessments of numeracy except numerical comparison (numerical comparison, r(208) = -.023, p = .735; counting proficiency, r(212) = .280, p < .001; one-to-one correspondence, r(158) = .195, p = .014; arithmetic, r(170) = .185, p = .015; knower-level, r(201) = .225, p = .001). This finding indicates that our training was overall more effective for older compared to younger children within our sample. Vocabulary was also related to gains in counting proficiency (r(207) = .171, p = .013) and corresponding knower-level (r(198) = .199, p = .005). This indicates that children demonstrating better receptive vocabulary at pre-test improved more in counting through training. Knowledge of the count list at pre-test was positively correlated with post-training gains in counting proficiency (r(208) = .280, p < .001), corresponding knower-level (r(199) = .304, p < .001), and one-to-one correspondence (r(158) = .304, p < .001). This finding indicates that the more familiar children were with the count list at pre-test, the greater amount of gain they made in counting proficiency and one-to-one correspondence after training.

The further breakdown by sample showed that Italian children performed better than US children in counting proficiency and arithmetic after training, even accounting for pre-test scores as a covariate (counting proficiency: F(1,212) = 10.272, p = .002, partial eta squared = .046; arithmetic: F(1,170) = 4.265, p = .040, partial eta squared = .024). However, these effects were eliminated when further accounting for age as a covariate (counting proficiency: F(1,211) = 1.079, p = .300, partial eta squared = .005; arithmetic: F(1,169) = 0.927, p = .337, partial eta squared = .005), with age being a significant covariate in the counting proficiency model for counting proficiency (F(1,211) = 8.545, p = .004, partial eta squared = .039). These analyses suggest that differences in the effects of training between the Italian and U.S. children can likely be explained by age differences rather than geographic or cultural differences between the sub-samples. Gains did not differ across the Italian and the US samples for the other aspects of numeracy assessed (numerical comparison: F(1,208) = 1.126, p = .290, partial eta squared = .005; one-to-one: F(1,158) = 1.085, p = .299, partial eta squared = .007; knower-level: F(1,201) = 3.725, p = .055, partial eta squared = .018).

Focused analyses of mechanism

The main purpose of our study was to explore several focused hypotheses regarding the mechanisms that drive early numeracy and their particular effects by contrasting different variants of the training.

Symbolic versus non-symbolic training.

In our first focused analysis we asked whether numerical comparison training with symbolic aids (abacus counting, verbal counting, or verbal labeling, n = 69–85) was more effective than training with non-symbolic numerical comparison alone (set-based or sequential non-symbolic enumeration, n = 92–130). The analyses showed that children trained on numerical comparisons with symbolic aids made larger gains on two of the four outcome measures, one-to-one correspondence (F(1,158) = 7.851, p = .006, partial eta squared = .047) and arithmetic (F(1,170) = 6.260, p = .013, partial eta squared = .036), compared to children who participated in non-symbolic numerical comparison training (Fig 3). These outcomes could not be explained by pre-test differences in the groups or performance differences on the training task itself (see S1 Text in S1 File). There were no differences in improvement between the two training groups on numerical comparison (F(1,208) = 1.982, p = .161, partial eta squared = .009) or counting proficiency (F(1,212) = 0.003, p = .958, partial eta squared < .001). There were also no differences when counting proficiency was coded as a knower-level. (F(1,201) = 0.003, p = .955, partial eta squared < .001). These results show that numerical comparison training with symbolic aids leads to greater increases in several important aspects of numeracy compared to non-symbolic numerical comparison training alone.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Mean gains in numeracy after numerical training with or without symbolic aids.

Gains calculated as post-test score minus pre-test score. Error bars are -/+ 1 SE. * indicates statistically significant differences between numerical comparison training with and without symbolic number aids.

https://doi.org/10.1371/journal.pone.0259775.g003

Verbal versus non-verbal training.

A second focused question concerned whether training variants involving verbal numerical aids conferred a unique advantage over non-verbal training variants. Several analyses were conducted to investigate this possibility. First, we contrasted outcomes of those whose training involved verbal numbers (verbal counting & verbal labeling, n = 58–93) to all others (non-symbolic sequential, non-symbolic sets, & abacus counting, n = 103–122). These comparisons showed that children whose training highlighted verbal numbers did not show greater improvements than those who were trained with nonverbal variants. That is, there were no differences in accuracy at post-test, accounting for pre-test scores, on any of the numeracy assessments between verbal and non-verbal training types (counting proficiency: F(1,212) = 1.545 p = .215, partial eta squared = .007; numerical comparison: F(1,208) = 0.250, p = .617, partial eta squared = .001; arithmetic: F(1,170) = 3.120 p = .079, partial eta squared = .018; one-to-one correspondence: F(1,158) = 0.711, p = .400, partial eta squared = .004). There were also no differences between verbal and nonverbal training when counting proficiency was coded as knower-levels (F(1,201) = 1.850, p = .175, partial eta squared = .009).

Next, we compared outcomes of those who participated in the training with symbolic verbal number aids (verbal counting & verbal labeling, n = 58–93 depending on task) to those who participated in training with the symbolic but non-verbal abacus aid (abacus counting, n = 34–37). There were no clear differences in outcomes on any of the numeracy assessments between verbal and non-verbal symbolic training types either (counting proficiency: F(1,127) = 3.129 p = .079, partial eta squared = .024; numerical comparison: F(1,126) = 0.462, p = .498, partial eta squared = .004; arithmetic: F(1,98) = 0.076, p = .783, partial eta squared = .001; one-to-one correspondence: F(1,89) = 1.724, p = .193, partial eta squared = .019; see Fig 4). There was a small difference between verbal and non-verbal symbolic training when counting proficiency was coded as knower-levels (F(1,121) = 4.298, p = .040, partial eta squared = .034), with non-verbal symbolic abacus training leading to higher knower-levels at post-test than verbal-symbolic training after accounting for pre-test abilities. However, this effect was eliminated when age and/or vocabulary differences between groups (see S1 Text in S1 File) were accounted for as additional covariates (ps > .27).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Mean gains in numeracy after numerical comparison training with symbolic verbal or symbolic non-verbal aids.

Error bars are -/+ 1 SE. Note that gains themselves were significantly different from zero in all areas except numerical comparison, but the differences between verbal and non-verbal symbols on numerical comparison, counting proficiency, one-to-one correspondence, and arithmetic were not significant. A moderate advantage for the non-verbal symbolic aid (abacus) on knower-level.

https://doi.org/10.1371/journal.pone.0259775.g004

Finally, to further explore whether the observed advantages of symbolic training were driven by the children that completed verbal number training or whether the benefits of symbolic training may extend to non-verbal symbolic visual-spatial aids like the abacus, we compared outcomes of those whose training involved verbal-symbolic aids and those in the non-verbal symbolic (abacus) training group separately to the outcomes of those trained on non-symbolic numerical comparison on the two measures where a symbolic training advantage had been observed: arithmetic and one-to-one correspondence. One-to-one correspondence accuracy improved more for those trained with the non-verbal symbolic abacus (n = 34) compared to those trained on non-symbolic numerical comparison alone (n = 69, F(1,100) = 9.035, p = .003, partial eta squared = .083), while there was no clear differences in one-to-one correspondence outcomes between those trained on numerical comparison with verbal-symbolic aids (n = 58) and those trained on non-symbolic numerical comparison alone (F(1,124) = 3.549, p = .062, partial eta squared = .028). There results could not be accounted for by pre-test differences (see S1 Text in S1 File) and results held when controlling for grouping differences in age as an additional covariate (p = .010; p = .063). The reverse seemed to be true for improvements on arithmetic. Improvement on arithmetic was marginally greater for those trained with verbal-symbolic aids (n = 65) compared to those trained on non-symbolic numerical comparison alone (n = 72, F(1,134) = 4.195, p = .042, partial eta squared = .030), but there was no difference in arithmetic outcomes between those who trained with the abacus (n = 36) and those who trained on non-symbolic numerical comparison alone (F(1,105) = 2.665, p = .106, partial eta squared = .025). These exploratory results suggest that gains on one-to-one correspondence may have been driven by those trained to tally with the abacus, whereas gains in arithmetic may have been driven by those whose training involved verbal numbers.

Feasibility of numerical comparison as an intervention tool

We first assessed the overall feasibility of using our particular instantiation of numerical comparison as an intervention tool for enhancing numeracy in preschool aged children. We found that children were compliant and successful on the training game itself. Furthermore, participating in the numerical comparison training was associated with significant gains in three of the four targeted areas of numeracy. That is, on average, children improved in counting proficiency, one-to-one correspondence, and exact arithmetic calculation with objects. Demographic variables of age, vocabulary, and pre-test count-list knowledge, as well as performance during actual training, were all positively related to improvements in numeracy, suggesting our training may have been more effective in older children with more advanced vocabularies and knowledge of the count list, but who had not yet mastered counting.

Although test-retest improvement or ongoing cognitive development may have accounted for some of the observed gains from sessions 1 to 2, several pieces of additional evidence suggest that the numerical training itself also contributed. First, participants made about one “knower-level” gain over the course of two weeks—such a gain normally takes at least several months on a typical developmental timescale [see 3]. Second, these gains were systematic rather than randomly distributed. That is, to make a gain in knower-level, one must improve sequentially in number knowledge performance while retaining performance on previously mastered numbers from pre-test to post-test. As such, systematic gains in knower-level are less likely to result for simple test-retest improvements, where gains would more likely be distributed randomly across trials. Third, individual differences in improvements in numeracy were related to individual differences in training performance, further suggesting that the training itself was driving gains.

The role of symbols in numeracy development

Next, we tested the role of symbols in early numeracy, comparing performance of children that received numerical comparison training with symbolic aid(s) to children who received non-symbolic numerical comparison training alone. Here we provide novel experimental evidence of how symbolic aids may enhance numeracy.

Children who participated in training that combined their non-verbal numerical intuitions with symbolic numerical tools made significantly larger gains on the two non-verbal conceptual assessments of exact number understanding, compared to those children trained on non-symbolic numerical comparison alone. More specifically, children whose numerical comparison training involved using a symbolic abacus, verbal counting, or verbal-numerical set labeling made larger gains on tests of one-to-one correspondence and exact arithmetic with objects than children who were trained with a comparably demanding task that only involved comparing arrays of objects without any additional symbolic-numerical aid. These results are consistent with a recent report [70] and suggest that the unique contribution of symbols to the emergence of numeracy may be to enhance the capacity for thinking about exact equality and correctly computing the effects of numerical transformations of sets.

We also did several follow up analyses to begin to test whether verbal numbers were a more effective symbolic tool compared to non-verbal symbols that rely on visuo-spatial supports in these initial stages of learning. While Western societies rely heavily on verbal-symbolic systems for counting and performing arithmetical computations, other cultures past and present have employed visuo-spatial symbolic supports (e.g., tally systems, abacus) for counting and calculation [see 86 for a review]. Over several analyses we did not see clear evidence that verbal-symbolic aids were significantly more effective than non-verbal symbolic aids, although more work is needed before this conclusion should be fully accepted.

Current theories diverge with regards to the particular combinations of core and cultural numerical abilities that are needed to drive conceptual development in number. These theories differ in nuanced and fundamental mechanistic ways that our current data do not have the precision to address. Furthermore, our analyses contrasting outcomes of verbal versus non-verbal symbolic training involved recombining training variants and, thus, were not orthogonal to other analyses. Nevertheless, our data bring some evidence to bear on one major division between theories: some implicate human language as the critical factor that combines with core number to produce conceptual change [e.g., 29, 39], while other theories propose that cultural-symbolic systems more broadly are the critical factor [87]. Exploratory analyses did not yield evidence of clear differences between those trained with verbal versus non-verbal symbolic aids on any of the numeracy assessments after training. To be clear, this is not evidence that verbal and non-verbal symbolic aids work equally well, nor is it strong evidence in favor of theories that emphasize symbolic systems broadly over language specifically. Nevertheless, these data are some of the first to contrast this distinction in an experimentally controlled manner and, at the least, suggest the possibility that symbols may not have to be verbal-linguistic in order to drive conceptual development in the emergence of numeracy.

To the extent that any differences were observed between the groups trained with verbal and nonverbal symbolic aids, they were isolated to pairwise comparisons on particular assessments. On the one hand, those trained with the non-verbal abacus counting performed better on the one-to-one correspondence task, which itself utilized an abacus, compared to those who received non-symbolic training. On the other hand, those trained with verbal symbolic aids performed better on the exact arithmetic transformation assessment task compared to those who received non-symbolic training. This finding aligns with previous evidence that count words may be essential to exact arithmetic [see 82]. This dissociation hints that verbal and non-verbal symbolic aids make differential contributions to conceptual development in the emergence of numeracy. Further pre-registered replication with larger samples, focused more specifically on this question, would be needed before drawing any strong conclusions.

Given our findings, there are at least three ways in which symbolic number aids might have led to reported improvements in non-verbal exact number tasks. It is possible that children who were trained with symbolic aids understood and/or learned numeracy concepts similarly to children trained without them, but those trained with symbolic aids were more likely to employ symbolic tools to solve these tasks after training even though the assessment tasks themselves did not demand the use of symbols. Such an account would accord with views that symbols are simply tools to extend resource limits for pre-existing human conceptual resources [88, 89]. Another possibility is that training with symbolic aids actually drove novel conceptual development [e.g., 29, 39]. If this is the case, symbolic number systems might provide the basis for children to conceive of exact equality and/or the effects of arithmetic transformation on exact number. Empirical work on the emergence of these concepts suggests that young children as well as people in cultures without formal symbolic number systems already have some conceptual resources for exact numerical representation through one-to-one correspondence [89, 90]. The effects of transformations (arithmetic or spatial/occlusion) on exact number, however, appear to be challenging early in development as well as for adults in cultures without symbolic number systems [89–92]. Thus, a third possibility is that symbols function differently for different aspects of early numeracy. For example, symbolic number may be a tool to extend existing, but limited, capacities for one-to-one correspondence [e.g., 89], paving the way for new conceptual developments regarding the effects of arithmetic transformations on exact number [e.g., 29, 39, 90].

In sum, this study tests the feasibility of using an experimental training framework to contrast specific hypotheses regarding the role of symbols in the emergence of numeracy as well as using numerical comparison in translational applications for improving numeracy. We find evidence that symbols uniquely aid understanding and performance related to one-to-one correspondence and exact arithmetic transformation. Our results also suggest that training with both verbal numbers and a non-verbal symbolic abacus led to gains in numeracy, but the contributions of each type of symbolic training may have been different. Our study provides the methodical and evidentiary groundwork to justify further tests of these and other ideas in a more fine-grained manner.