Learning to Count: Structured Practice with Spatial Cues Supports the Development of Counting Sequence Knowledge in 3-Year-Old English-Speaking Children

Research Findings: Children who speak English are slower to learn the counting sequence between 11 and 20 compared to children who speak Asian languages. In the present research, we examined whether providing children with spatially relevant information during counting would facilitate their acquisition of the counting sequence. Three-year-olds (n = 54) who played a 1–20 number board game in which numbers were grouped by decade into 2 rows learned significantly more of the counting sequence than children who played a linear version of the game or those who were in the control group. Both the row and linear versions of the game helped children improve their performance on an object counting task. Children's performance on a number line task did not show an effect of either game intervention. Practice or Policy: These results suggest that counting practice that includes spatially informative cues can facilitate young English-speaking children's learning of the challenging number sequence from 11 to 20.

Before they start formal schooling, children acquire a variety of numeracy skills that are essential for developing future mathematical concepts (Duncan et al., [14]; Li & Geary, [23]). In particular, researchers have consistently found that early counting skills are predictive of later mathematical performance (Aunio, Aubrey, Godfrey, Pan, & Liu, [1]; Jordan, Kaplan, Ramineni, & Locuniak, [18]; Ramani & Siegler, [32]). We were interested in children's acquisition of the counting sequence between 10 and 20. For English-speaking children (and those who speak other Indo-European languages), learning the sequence of number names from 11 to 19 is particularly difficult because these number names do not follow a consistent pattern (Aunio et al., [1]; Cankaya & LeFevre, [3]; Dowker, Bala, & Lloyd, [12]; Dowker & Roberts, [13]; LeFevre, Clarke, & Stringer, [20]; Miller, Smith, Zhu, & Zhang, [25]). In contrast, number names in many Asian languages follow a predictable sequence that combines the word for ten with the unit word (e.g., 12 is ten-two; Chan, [6]; Miller et al., [25]). Asian parents and teachers also support children's learning by providing highly structured practice with numeracy concepts (Chang, Sandhofer, Adelchanow, & Rottman, [7]; Ng & Rao, [28]; Pan, Gauvain, Liu, & Cheng, [29]; Zhou et al., [44]). Accordingly, Asian children acquire counting knowledge much earlier than children who speak English, and this knowledge may also facilitate their understanding of the base-10 structure of the number system (Göbel, Shaki, & Fischer, [16]; Miura, Kim, & Okamoto, [26]). In the present research, we used a board game intervention that divided the numbers from 1 to 20 into two rows showing decades 1–10 and 11–20. We hypothesized that experience with the rows board game would support English-speaking children's acquisition of the counting sequence between 11 and 20 by providing spatial cues to support the counting process and emphasize the structure of the number system.

Spatial processes are linked to children's mathematical development in a variety of ways (Mix & Cheng, [27]). In particular, spatial abilities predicted children's counting sequence knowledge in kindergarten (Zhang et al., [43]), suggesting that spatial cues may be useful in supporting acquisition of counting knowledge. Numerical board games are one context in which spatial cues may help support numeracy knowledge (Ramani & Siegler, [31]; Ramani, Siegler, & Hitti, [33]; Siegler & Ramani, [37], [38]; Whyte & Bull, [40]). Most often, numerical board games used in research with young children have a linearly arranged board extending from 1 to 10 with equally sized spaces numbered from left to right. According to Ramani and Siegler ([32]), playing such numerical board games exposes children to a series of repetitive cues corresponding to the magnitude and sequencing of numbers. In several studies, after playing a linear (1–10) game, kindergarten children showed improved performance on numeracy measures such as counting, number line performance, and number naming (Ramani & Siegler, [31]; Ramani et al., [33]; Siegler & Ramani, [37], [38]; Whyte & Bull, [40]).

The specific spatial orientation or layout of numerical board games influences children's learning from these games (Laski & Siegler, [19]; Siegler & Ramani, [38]). For example, Siegler and Ramani ([38]) found that 1–10 board games arranged linearly were more effective at improving the numerical understanding of young children than circular games. In research with kindergarten children using a 0–100 board game arranged as a 10 × 10 matrix, Laski and Siegler ([19]) found that children who played the game by counting on from their current board position (i.e., moving five squares from 34 by counting "35, 36, 37, 38, 39") improved considerably more on several numeracy measures than children who simply counted from one on each turn (i.e., counting the squares as "1, 2, 3, 4, 5" from the original position). In particular, children who counted on were better able to locate positions of numbers within the 10 × 10 matrix. To explain these findings, Laski and Siegler proposed the cognitive alignment hypothesis, which predicts maximal learning from a number game when the structure of the game and the activities that the children participate in are closely aligned with the nature or spatial organization of the mental representation that children are expected to acquire.

In the present research, our interest was in facilitating English-speaking children's acquisition of the counting words from 11 to 20. We hypothesized that providing spatial cues to highlight the structure of the number sequence from 1 to 20 would help children consolidate sequence knowledge. This prediction is consistent with the cognitive alignment hypothesis in that children's knowledge of the counting sequence ultimately should reflect the base-10 structure of the formal number system. To test this hypothesis, children who participated in the number game conditions played either a row number game arranged in two sections (i.e., one extending from 1 to 10 and the other extending from 11 to 20, as in the middle panel of Figure 1) or a linear number game arranged in a single section (top panel of Figure 1). Children in control conditions either did not play a game or played one with colors but no numbers (bottom panel of Figure 1). The linear board game was an extended version of the 1–10 board game used in several previous studies (Ramani & Siegler, [31]; Siegler & Ramani, [38]; Whyte & Bull, [40]). Both number game conditions provided children with experience moving through the counting sequence between 1 and 20; however, the cognitive alignment hypothesis predicts that the row game, which aligns the teens digits with the corresponding ones digits and emphasizes the parallel structures of the 1–10 and 11–20 sequences, should facilitate children's learning of the counting sequence between 11 and 20 more than the linear game.

Graph: Figure 1. Game boards. The top panel shows the linear game, the middle panel shows the row game, and the bottom panel shows the color game. A version of this figure was published in Cankaya et al. ([4]) without the row game.

In summary, the purpose of this study was to determine whether spatial support for the base-10 structure of the counting string in a 1–20 numerical board game would help English-speaking children learn the counting sequence from 11 to 20 compared to children who played a linear version of the board game. Three-year-olds were of particular interest as they were not yet proficient verbal counters and were not learning numeracy skills in a formal setting. In other work using numerical board games with kindergarten or preschool children, the board game experience was related to improvements in a range of numerical skills (Ramani & Siegler, 2008; Whyte & Bull, [40]). First, our main hypothesis was that children who played the rows game would improve more on their counting sequence knowledge between 11 and 20 than children in the other conditions. Second, we hypothesized that numeracy skills that involved counting knowledge in the 1–10 range (i.e., object counting, numeration) may show improvements when children played either counting game because of increased counting practice. Third, children were not expected to improve differentially across conditions on a nonsymbolic arithmetic task because performance is not strongly linked to verbal counting skills (Canobi & Bethune, [5]; Jordan, Huttenlocher, & Levine, [17]). Fourth, we included number line tasks in the current research because in previous research, older children showed improvements after playing linear 1–10 games on a number line task that required them to play numbers on a line marked with 0 and 10 at the endpoints (Ramani & Siegler, [32]; Whyte & Bull, [40]). However, these tasks proved to be very difficult for these young children, so their performance did not improve over the training sessions.

Children participated in one of four training conditions over 4 weeks: a number board game arranged in two rows (1–10 and 11–20), a linear board game arranged in a single row (1–20), or one of two control conditions in which they either played a color board game or simply met briefly with the experimenter. Children were pre- and posttested on a variety of numeracy measures. In all of the conditions, children were asked to count as high as possible each week, and number naming and number line performance were assessed at the midpoint (i.e., after two training sessions). This design allowed us to examine changes in performance during the training as well as between pre- and posttest sessions.

Over two consecutive years, 3- and 4-year-old children were recruited from day cares located in a large Canadian city. A total of 94 children participated in a pretest session (48 girls). Of these, 78 children went on to participate in the intervention: 54 three-year-olds and 24 four-year-olds. Children were not included in the intervention if they were not cooperative in the pretesting tasks (n = 4), if they did not want to play the games (n = 2), or if they had already mastered the targeted verbal counting skills (i.e., counted higher than 30 with no errors; n = 10). Only the data from the 54 three-year-old children were analyzed for this article. The 24 four-year-olds who participated in the intervention had better numeracy skills than younger children (e.g., half could count to 16 or higher), and some had received instruction in preschool or kindergarten. Thus, for the purposes of the present analyses, 3-year-olds were more suitable candidates for testing the main hypothesis. (Note that Year 1 children from the linear and control conditions [n = 23] were also included in analyses described by Cankaya, LeFevre, & Dunbar, 2014). In summary, the present analyses combine data from 54 three-year-old children who participated in two consecutive years (Year 1, n = 34; and Year 2, n = 20). The methodologies used in the 2 years differed slightly, as described in "Procedure."

The 54 three-year-olds included 27 girls and 27 boys. All but one child had been born in Canada. All participants spoke English at day care, and all testing and training sessions were conducted in English. Twenty-three children also spoke an additional language at home. Twelve parents reported the use of another language sometimes (four French, one German, two Mandarin, one Polish, three Spanish, and one Urdu), and nine parents reported the use of another language at home often or always (four Arabic, four Mandarin, and one Somali). Parents' education was measured on a 5-point scale, where 1 = less than high school, 2 = high school graduate, 3 = college graduate, 4 = university graduate, and 5 = postgraduate degree. Parents' education level ranged from 1 to 4 for this group (Mdn = 3).

Children completed nine measures at pre- and posttest. Numeracy measures included verbal counting, object counting, number naming, two versions of a number line task (marked endpoints of 0–10 and 0–20), and a nonsymbolic arithmetic task. Control measures included spatial span and receptive vocabulary. Because the number naming task was administered differently in the two studies it could not be compared directly across studies and so is not discussed further. Approximately half of the children also completed a rapid naming task as part of a pilot test for this measure. Rapid naming task performance is not discussed here (see Sowinski, [39], for more information about this measure).

This task measures knowledge of verbal number names and of the counting sequence. The experimenter asked the children to count as high as they could. In some instances, when the child was hesitant to count, the experimenter used a puppet to engage the child in the activity. The researcher explained that the puppet did not know how to count and wanted to learn to do so from the child. Three different strategies were used to encourage a child to continue counting. For instance, if the child stopped counting at 6, the researcher first said, "What comes after 6?" If the child did not continue counting, the experimenter repeated the last three numbers counted. Finally, the researcher repeated the last number counted while ending on an expectant tone (e.g., "4, 5, 6?" or "6?"). The verbal counting task was administered six times, once each at pre- and posttesting and at the beginning of each intervention session.

Children's counting performance was scored in two ways. The strict scoring method credits the child with the highest number counted without any errors. A more lenient scoring is to use the highest number the child counted to correctly while allowing one error (Miller et al., [25]; Rasmussen, Ho, Nicoladis, Leung, & Bisanz, [35]). For example, if a child counted 1, 2, 3, 4, 6, 7, 12, 20, according to the strict scoring his or her highest count would be 4, whereas in the lenient scoring the child was credited with a highest count of 7, allowing for one error (i.e., skipping 5). We added an additional criterion to the lenient scoring, however. If the child skipped more than two numbers, then he or she was not credited with the higher number. For example, a count of 1, 2, 3, 4, 8 would be scored as a highest count of 4 because the child skipped 5, 6, and 7, whereas a count of 1, 2, 3, 4, 7 would be scored as a highest count of 7. Note that the majority of children (78%) either had the same score with the strict and lenient scoring or received a lenient score that was within 3 of the strict score. The relatively small number of children who had much higher counts with the lenient than the strict scoring were those who skipped a single number but continued to count higher (e.g., 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12—skipped 6). In summary, the lenient scoring technique was used to ensure that the child's counting ability was not underestimated because of other factors such as distraction but also was not overestimated for children who simply stated a higher number name.

To empirically evaluate whether the strict and lenient scoring provided equally consistent data, we analyzed the first two counting attempts (i.e., in the pretest and the first game session, where the counting occurred before any game playing) to compare the test–retest reliability of the two scoring schemes. The test–retest correlation (n = 51) was significantly higher for the lenient scoring (r = .798) than for the strict scoring (r = .580), z = −2.89, p = .0038, with the confidence interval of the difference of [−0.41, −0.07] (see Diedenhofen & Musch, [11], for a description of the online software used for this comparison). Thus, the lenient scoring appears to have provided a more stable estimate of a child's counting ability than the strict scoring. Nevertheless, both are discussed in relation to the central hypothesis.

In this task children are asked to give the experimenter a specified number of small toy animals. This task is often used to assess children's knowledge of cardinality; however, in this study we were most interested in seeing whether children could use the counting string successfully across a range of stimuli. Stimuli were presented in consecutive order and potentially included the numbers 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, and 17 (following Miller et al., [25]), although the task was terminated after three incorrect trials, so most children were only asked about the smaller numbers. For each trial, after the child finished presenting the requested stimuli, the experimenter would ask in a neutral tone, "Is that X sheep?" The experimenter ended each trial after the child responded affirmatively or if no answer was given in 30 s. Performance was scored as the number of trials correct (maximum = 12).

KeyMath–Revised Numeration Subtest Form A and Form B were used to provide an overall assessment of children's numeracy knowledge (Connolly, [9]). The KeyMath test examines a child's ability to count objects, recognize Arabic digits, and understand the ordinal position of items. The task is standardized for individuals from kindergarten to Grade 10. Only raw scores were used for the analysis, because the participants were prekindergartners and no standardized scores were available for this group. The task was terminated when the child answered three questions in a row incorrectly. Maximum score was 10.

The purpose of this task was to examine the child's ability to represent and manipulate the number of items in a set using concrete materials (Canobi & Bethune, [5]; Jordan et al., [17]). There were three parts to the task: (a) matching, (b) addition, and (c) subtraction. The materials used in this task were a placemat to represent a farm, a farmer figurine, and multiple animal figurines. The experimenter explained to the child that the farmer had a farm with some animals in it and that the child must also have the same number of animals in his or her farm. In the warm-up trial, first children were shown animals in the farmer's farm, next the animals in the farmer's farm were hidden behind a toy door, and last the children were asked to represent the same number of animals in their own farm. The matching trials had quantities of 2, 3, and 5. If a child successfully completed at least one of the matching trials, he or she was then given addition trials. On the addition trials, the experimenter showed the children the animals, closed the door, and then added more animals. The experimenter asked the child to show how many animals were behind the door. The stimuli for the addition trials were 1 + 2, 3 + 1, 2 + 3, and 4 + 2. On the subtraction trials, the experimenter started with the larger amount, closed the door, and then took away some animals. The subtraction stimuli were 3 – 1, 4 – 3, 5 – 2, and 6 – 4. The subtraction trials were administered if the child successfully completed at least one addition trial. The experimenter terminated the task if the child made three consecutive errors. Performance was scored as the total number of trials solved correctly (maximum of 10).

This task assesses a child's understanding of the ordinal relations among numbers in a specified range (Petitto, [30]). In older children, it is correlated with spatial abilities (LeFevre et al., [22]). Two versions of the number line task were administered using an iPad. The first number line extended from 0 to 10, and the other extended from 0 to 20. Other than the iPad administration, the procedures were similar to those used in other studies (e.g., Ramani & Siegler, [31]; Siegler & Ramani, [38]). First the experimenter showed each child the number line. Next the child was presented with each number between the endpoints, one at a time, in a random order and asked to touch where the respective number should be placed on the line. Percent absolute error was calculated as (number indicated – location touched) × 100 and then averaged across locations.

A spatial span task was included in this study as a control measure because visual-spatial working memory is a correlate of early mathematical development (Li & Geary, [23]). This task was implemented as an iPad application but was similar to that used in other studies (e.g., LeFevre et al., 2010; Rasmussen, Ho, & Bisanz, [34]). In this task, a set of green circles appeared on a white screen. For each trial, a series of circles lit up in a predetermined sequence. The child was asked to recall the order in which the circles lit up by touching the circles in the same order. The stimuli consisted of a warm-up trial of two sequences of length two to familiarize the child with the task. Following the warm-up trial, three new sequences of length two were presented. If the child correctly recalled at least one of the sequences, he or she was shown three sequences of length three, and so on, for a maximum sequence length of five. All details of this task were described to the participants before the warm-up trials and, if necessary, repeated once again before the first trial. All data were automatically recorded by the application. Performance was scored as the total number of correct trials.

Peabody Picture Vocabulary Test–Third Edition, Form B, was used to assess receptive vocabulary in participants (Dunn & Dunn, [15]). The task involved the child viewing four pictures presented by the experimenter. Children were asked a question, such as "Which one is the dog?" The child indicated his or her answer by choosing and pointing to the appropriate picture out of the four options. The task began with the first trial and ended after Trial 60. If the child made eight or more errors in a set, the test was terminated. The scores were standardized by age.

Three different games were used. Each game had colored squares organized either in a single line, in two rows, or in a line without any Arabic digits (see Figure 1).

The data collection process extended over a 6-week period. Weeks 1 and 6 were the pre- and posttest sessions, respectively. The gameplay occurred from Weeks 2 through 5. All testing sessions took place in a quiet area of the child's day care.

Once parental consent was provided, a female researcher pretested each child individually. Pretesting was divided into two sessions of approximately 20 min each in order to prevent fatigue in the child. The child's two pretesting sessions were completed on the same day. The first pretesting session included, sequentially, the object counting task, nonsymbolic arithmetic, verbal counting, number naming, and the KeyMath Numeration Subtest. The second pretesting session included the rapid automatic naming task, the number line estimation tasks, the spatial span task, and the vocabulary task.

Following pretesting, children were randomly assigned to either the control, linear, or row conditions. At the beginning of each session, children were asked to count verbally as high as possible. In the control condition of Year 1, children played the color board game, whereas in the control condition in Year 2, children did not play a game. They met with the experimenter each week and counted verbally, however, so that we could assess possible improvements in verbal counting and equate familiarity with the experimenter across groups. They also participated in the midpoint testing session (described in "Midpoint Testing [Week 4]").

In Year 1, children in all three groups played the board game with the same female researcher. In Year 2, children in the linear and row experimental conditions were paired with a peer based on similar age and counting ability, supervised by the same female researcher. The same pair of children played the game once a week for 15 min over the 4-week period. In the game conditions, the children played the game as many times as possible within the 15-min timeframe, with an average of two games played per session. By the end of the 4-week period, children in the linear and row conditions had been exposed to the game for approximately 1 hr.

During the first week of the game intervention, the researcher showed the children how to play the game: "We are going to play a game. These animals are stuck on this island. There is no food or shade here. We need to help them cross this bridge to get to this island where there is food and shade." In the number game conditions, they were told, "There are different numbers on the bridge from 1 to 20," whereas in the color version they were told, "There are different colors on the bridge: red, green, yellow, and blue." Then the experimenter said, "This is a magical bridge. In this game you have to say the numbers/colors out loud when it is your turn, to find the food for your animal." In order to demonstrate the rules of the game, the researcher spun the spinner and moved the required number of spaces while saying each number/color out loud. To emphasize the importance of saying the numbers on each square out loud, the researcher spun the spinner a second time and moved the appropriate number of spaces. For instance, the researcher demonstrated that if the toy was on the number 3 and the child spun a 2, he or she would move the toy animal two spaces on the board while saying "4, 5" out loud. In the row condition, children were given the same instructions as in the linear game; however, they were told that once they crossed the bridge extending from 1 to 10 they had to take a rope swing up to the next bridge that extended from 11 to 20. If at any point during the gameplay the child did not know the name of the number or color, or if he or she responded incorrectly, the experimenter would prompt the correct response and ask the child to repeat it. The researcher recorded each time she corrected the child and the total time it took to play each game.

At the beginning of the third game-playing session, children in all three experimental conditions were tested individually in a midpoint testing session. The midpoint testing included the verbal counting, number naming, and number line estimation tasks. After the midpoint testing was completed, children in the game conditions played the game once.

One week after the last intervention session, children were posttested individually on all of the numeracy measures. The vocabulary and the spatial span tasks were not included in the posttesting sessions.

Initial analyses were conducted to determine whether children's performance between pre- and posttest differed across the 2 years of testing. To summarize, in Year 1 children played the games alone with the experimenter and the control condition was a color game, whereas in Year 2 children played the game in groups of two and those in the control condition met briefly with the experimenter each week but did not play a game. There were no effects of year or interactions between year and other variables in any analyses. Thus, in all analyses presented in this article, data were collapsed across the year of testing and a single control group was used.

Children were randomly assigned to conditions, and thus we did not expect to find differences in performance across conditions at pretest. To check for possible differences, however, we analyzed performance on each pretest measure in a 3 (condition: linear, row, control) one-way analysis of variance (ANOVA). Means are shown in Table 1. There were no significant differences at pretest across the game conditions for any of the pretest measures (Fs <1), including age, vocabulary, and spatial span. The number of children who spoke another language at home did not vary significantly across intervention groups, χ2(2, _I_N_i_ = 54) = 1.63, p = .446. The distribution of gender did not vary across intervention groups, χ2(2, _I_N_i_ = 54) = 1.27, p = .529. In summary, there were no statistically significant preexisting differences across groups before the intervention.

Table 1. Mean Scores and Standard Deviations at Pre- and Posttest.

2 Note. At pretest, for number line to 10, one child was missing data, and for number line to 20, 10 children were missing data because they refused to complete the task. Three children were missing data on the spatial span task. PAE = percent absolute error.

Finally, because the game-playing sessions were limited in time to approximately 15 min, and children varied in how quickly they counted and moved their game pieces, we compared the number of times each child played the number game between the row and linear groups to ensure that the game conditions were comparable in terms of children's experiences. These means were not significantly different, with children in the row condition playing the game an average of 6.4 times and children in the linear condition playing the game an average of 6.0 times, t(49) = 1.01, p = .317. In both groups, the range was from four to eight complete game sessions over the course of the experiment, so the experiences of the children were relatively uniform. The number of times the children played the game was not significantly correlated with performance in either group, presumably because of the limited variability. Similarly, controlling for the number of times the game was played did not influence analyses of performance.

At pretest, children's numeracy skills were modest. On average, children counted to 10 without making an error. They were correct on the first two or three items on the numeration test, which involved counting and recognizing small numbers. They were able to give the experimenter four or five items in the object counting task. Most were able to solve one or two nonsymbolic arithmetic problems. Consistent with the generally modest performance, the children had a large amount of error on the number line tasks and appeared to have relatively little understanding of the ordinal relations among the numbers. Hence, performance on the number line task was much worse than in previous studies with older children.

Two different analyses are included in the following sections to assess the effects of the number game experiences on children's early numeracy performance. When appropriate, repeated measures ANOVAs comparing performance across the testing sessions or between pre- and posttest sessions are presented to evaluate changes in performance across training conditions. In addition, posttest scores controlling for pretest performance using analysis of covariance (ANCOVA) are included along with measures of effect size for differences in adjusted posttest scores between training conditions. The latter analyses provide direct tests of the size of the differences in children's performance at posttest, whereas the former evaluate changes in performance from pre- to posttest and thus provide somewhat different information. In most of the current analyses, the conclusions about differences across game conditions are the same regardless of the specific analysis that was used.

The central hypothesis in the present research was that playing the row number game would help children learn the counting sequence between 11 and 19. Children were asked to count as high as possible at the start of each session, and thus six counting attempts were recorded: 50 of the 54 children had complete data across the six attempts. The children's highest counts, allowing for one error (i.e., the lenient scoring), were analyzed in a 6 (session) × 3 (group: control, linear, row) ANOVA, with repeated measures on the first factor. The main effect of session was significant, F(5, 47) = 10.73, p < .001, ηp2 = .186. The interaction of session and group approached significance, F(5, 47) = 1.80, p = .061, ηp2 = .071. The means are shown in Figure 2. To test whether the changes in children's counting varied across game conditions, we focused on the linear effect of session (i.e., improvements over time) and interactions between session and game condition. The linear effect of session was significant, F(1, 47) = 27.39, p < .001, ηp2 = .368, and the interaction of game condition and session was significant for the linear contrast, F(2, 47) = 4.11, p = .023, ηp2 = .149, indicating that there were different linear effects among the three groups. As is shown in Figure 2, there were no statistically significant differences across groups in the three conditions at pretest or before the first or second game sessions (Weeks 1–3). However, at the start of the third intervention session (i.e., Week 4, after children had played the game for two sessions), children in the row condition counted higher than those in the linear game and control conditions. This advantage grew over the next two sessions, such that at posttest more children counted to 20 or higher in the row condition (50%) than in the linear condition (25%) or the control condition (17%), χ2(1, _I_N_i_ = 54) = 5.88, p = .052. In summary, although all children improved their counting over sessions, the children who played the row game showed the most improvement.

Graph: Figure 2. Mean highest count (allowing for one error) across testing sessions by intervention condition. Error bars are the 95% confidence interval (95% confidence interval = 1.69) for the linear interaction of condition and session (Masson & Loftus, 2003).

As an alternative to the use of repeated measures analyses to evaluate improvements in counting performance, we analyzed the difference in adjusted posttest counts between the row and linear groups in an ANCOVA, controlling for pretest counting performance. The posttest means were used to calculate effect size using Hedges's g, an index that has been recommended for comparing interventions in education research with children. The posttest adjusted means for the lenient scoring were 15.7 (SD = 7.96) for the linear group and 20.8 (SD = 7.99) for the row group, F(1, 33) = 3.57, p = .068, Hedges's g = +0.64, indicating a moderate effect size. The posttest adjusted \ means for the strict scoring gave very similar results, with average highest counts of 12.5 (SD = 8.38) for the linear group and 17.2 (SD = 6.16) for the row group, F(1, 33) = 5.02, p = .032, Hedges's g = +0.62. Thus, the various ways of comparing the children's counting performance in the linear and row conditions indicated that children in the row group learned more of the counting string in the teens than children in the linear group.

Children's object counting performance was analyzed in a 2 (session: pretest, posttest) × 3 (game condition: control, linear, row) mixed ANOVA. Children's performance improved from pre- to posttest (2.3 vs. 3.5), F(1, 51) = 14.19, p < .001, ηp2 = .218. There was an interaction between session and game condition, F(2, 51) = 3.91, p = .026, ηp2 = .133. Bonferroni post hoc tests were used to compare pre- and posttest performance in each condition. As shown in Table 1, children in the control condition did not improve across sessions (3.1 vs. 3.0, p = .918), whereas children in the linear game (1.2 vs. 3.0, p = .002) and the row game (2.6 vs. 4.4, p = .002) conditions showed significant improvements in performance. Note that for the object counting task, an average of 1 on the task indicates that the child successfully gave the experimenter three objects, and an average of 2 indicates that the child gave the experimenter four objects. The data were also analyzed using pretest scores as the covariate to calculate posttest effect sizes across conditions, F(2, 50) = 3.02, p = .058. Children in the rows condition had somewhat higher posttest adjusted \ scores (M = 4.2, SD = 2.2) than children in the control condition (M = 2.4, SD = 2.3; p = .068 [Bonferroni adjustment], Hedges's g = +.78). Similarly, children in the linear condition (M = 3.8, SD = 2.3) had somewhat (but not significantly) higher scores than those in the control condition (M = 2.4, SD = 2.3; p = .235, Hedges's g = +.63). The difference between children in the linear and rows groups was not significant (p = 1.00, Hedges's g = +.15). Thus, there was some evidence that both the number game groups showed moderate gains in their object counting performance after the intervention.

The object counting task requires children to understand cardinality (i.e., that the number word denotes the quantity of the set) and to use counting to determine quantities. The game conditions may have helped children by providing practice assigning number words to objects (squares on the game boards). Although children only counted one or two squares during each of their turns, that experience may have helped children to improve their understanding of the link between the number indicated and the quantity of the set (in this case, the number of moves they could make and the final location of the game piece).

Children's number correct on the KeyMath numeration measure was analyzed in a 3 (game condition: control, linear, row) ANCOVA controlling for pretest numeration number correct. ANCOVA was used because children received one of two different versions of the measure (A or B) at pretest and posttest, so a difference score was not appropriate. The covariate was significant, F(1, 50) = 10.86, p = .002. The effect of game condition was not significant, F(2,50) = 2.27, p = .114, ηp2 = .083. Nevertheless, further analyses were done to calculate effect sizes. Using the Bonferroni adjustment for multiple comparisons, we found that children in the row condition had numerically, but not statistically significantly, higher adjusted scores at posttest than those in the control condition (3.19 vs. 2.24, respectively, p = .157, Hedges's g = +.67) and the linear condition (3.19 vs. 2.38, respectively, p = .269, Hedges's g = +.58), whereas children in the linear condition did not score significantly higher than those in the control condition (2.38 vs. 2.24, respectively, p = 1.00, Hedges's g = +.15). Thus, there was a trend for children in the rows condition to show better performance after training than children in the linear and control groups, with moderate effect sizes. Only a few of the items on this measure required children to count, so we did not expect large improvements in performance on this task.

Children's performance on the nonsymbolic arithmetic task was analyzed in a 2 (session: pretest, posttest) × 3 (game condition:control, linear, row) mixed ANOVA. Children's performance improved from pre- to posttest (1.4 vs. 2.2, respectively),F(1, 51) = 11.74, p = .001, ηp2 = .187. However, the effect of game condition was not significant, F <1. The interaction of session and game condition was not significant, F(1, 51) = 1.79, p = .177, ηp2 = .187. Similarly, the analysis of posttest adjusted means from the ANCOVA was not significant, F(2, 50) = 1.93, p = .156, ηp2 = .072, with both the rows and control groups having somewhat higher adjusted posttest scores (2.49 and 2.56, respectively) than the linear group (1.61), but none of the comparisons approached significance (ps >.255, Bonferroni corrected).

Forty-five children completed the 0–10 number line task at all three sessions. Their percent absolute error on the number line task was analyzed in 3 (session: pretest, midpoint test, posttest) × 3 (game condition: control, linear, row) mixed ANOVAs. No effects were significant, Fs <1. Similarly, for the 34 children who completed the 0–20 number line task at all three sessions, there were no significant effects of session or game condition, Fs <1. Although the children enjoyed participating in this task, their poor performance suggests that they did not have sufficient knowledge of the ordinal relations among numbers to do the task.

In the present research, we found that participation in a 1–20 number game that used spatial cues to highlight the relations between the numbers from 1 to 10 and from 11 to 20 facilitated 3-year-old children's learning of the counting sequence compared to participation in a linear 1–20 board game or in a control condition. The results supported the cognitive alignment hypothesis, which predicts that activities in which numerical information is closely aligned to the desired mental representation will support children's learning (Laski & Siegler, [19]; Whyte & Bull, [40]). Specifically, children learned more about verbal counting when the number board game was organized spatially to emphasize the links between the structure of the counting sequence and the intended mental representation of the number system. More generally, adding structure through spatial grouping to the irregular English number word sequence may help children grasp the decade structure of the base-10 system more easily. Yang and Cobb ([42]) analyzed number activities at home and school for Taiwanese and American parents and found that Taiwanese teachers and parents were more likely to encourage children to view numbers and collections beyond 10 as composites of tens and units, whereas American parents and teachers focused on one-to-one matching of objects to number names, even for numbers in the teens. In the present research, the rows board game was similarly designed to emphasize the base-10 structure of the number system because the numbers arranged by decade (two rows, 1–10 and 11–20) and the colors of the squares matched based on the ones column. The arrangement of numbers by decade may have assisted children in recognizing that 11 is 10 plus one more square. Furthermore, the break between the two sets of numbers may also have helped children to notice and attend to patterns of number names between decades.

Furthermore, children in both number game groups also showed improvements in their object counting performance, most likely due to the practice provided in matching number words to locations on the game board. Children in the row condition also showed somewhat better performance on a general early numeracy measure that involved counting small quantities than children in the linear and control conditions. In English, the counting sequence between 11 and 20 is difficult to learn because it does not follow a predictable linguistic pattern. The row game condition provided spatial support for children's learning of the linguistic counting sequence.

In contrast to the improvements shown on the linguistic measures of number knowledge, improvement in the nonsymbolic arithmetic task did not depend on game condition. Children in all three groups improved over the 6-week period, suggesting that exposure to numeracy activities may have facilitated posttest performance on this measure. Furthermore, children did not show any improvement on the number line tasks, even though previous researchers have shown that the use of numerical board games can be an effective way to improve children's number line placements (e.g., Ramani & Siegler, [31]; Siegler & Ramani, [38]; Whyte & Bull, [40]). There are three important differences between the present research and other studies that may account for why improvements in number line performance were not observed in the present research. First, children in the present research were younger. The average age of children in the present research was 3.3 years compared to 3.8 years in Whyte and Bull ([40]) and average ages of 4.5 and older in other studies (e.g., Laski & Siegler, [19]; Ramani & Siegler, [31]; Siegler & Ramani, [37], [38]). Second, in previous studies children were in preschool or kindergarten, whereas the children in the present research were not receiving any formal instruction. Third, the number game went from 1 to 20 rather than from 1 to 10, as in most previous studies (except for Laski & Siegler, [19], with Grade 1 children, which used a 0–100 number game). Thus, children in the present research received less experience with the 1–10 component of the number line during training. To the extent that number line performance develops slowly and is related to children's understanding of the number system in the specified range (Dackermann, Huber, Bahnmueller, Nuerk, & Moeller, [10]), the children in the present research may have needed more practice to show increased understanding. Other researchers also have questioned the extent of transfer from similar number games to other numeracy tasks (see Baroody & Purpura, [2], for a more general critique of these studies).

The modest improvement in verbal counting for children who played the linear game condition indicates that the linear game did not help these children to develop their number sequence knowledge as much as the rows game, although the counting practice did lead to some improvement in the object counting task. Turkish children, in contrast, showed improvement in the linear version of the game compared to English-speaking children (Cankaya et al., [4]). The Turkish number language is like Chinese, in that it constructs the number names between 11 and 19 as combinations of the word for 10 plus the unit number (e.g., ten-three). Hence, simply practicing the counting sequence appeared to be helpful for Turkish children, whereas English-speaking children needed more explicit support, as was provided in the rows version of the game. Similarly, Dowker and colleagues (Dowker et al., [12]; Dowker & Roberts, [13]) have shown that 6-year-old children learning mathematics in Welsh, which also constructs number names as combinations of 10s and units, show some specific advantages in linguistic number tasks compared to their peers who are learning in English.

There are several limitations of the present research. First, there were differences in the control groups between Year 1 and Year 2 of the study. One control group played a color game, whereas the other simply met with the experimenter and counted. Second, another difference between Years 1 and 2 is that children participated in groups of two in Year 2 compared to participating alone in Year 1 (Ramani et al., 2012). Variations across testing years presumably added additional variability to the results. Note, however, that these differences across subgroups maintained the familiarity of children with the experimenters and, in the second case, preserved the amount of gameplay in the number conditions across years. Nevertheless, there may have been unintended effects on children's motivation or attention to the games. Third, the design of the study, in which all children counted six times over the experiment, may have reduced the specific impact of the training conditions. Future studies in which counting practice is contrasted with other kinds of practice (e.g., learning the alphabet) would be useful in understanding the impact of different experiences on children's acquisition of the counting sequence.

In conclusion, the present research suggests that a simple number board game can help English-speaking 3-year-old children learn the counting string between 11 and 20. As predicted by Laski and Siegler ([19]), the most learning occurred when the game structure corresponded to the mental representation that the intervention was designed to support. Specifically, the rows game in which the first 20 numbers were organized into two groups of 10 provided the greatest benefit to children's knowledge of the counting sequence. This result suggests that adding spatial structure to an activity involving counting may be helpful for English-speaking children whose number language does not correspond closely to the structure of the symbolic number system. Knowledge of how such structure can facilitate children's counting may help early childhood educators to maximize opportunities to stress the structure of the counting sequence. Mathematics educators have long argued in favor of the value of playing number-related games with children (e.g., Rutherford, [36]) and the use of artifacts such as hundreds charts to emphasize base-10 number patterns (Clements & Sarama, 2007; Wynroth, [41]). The present research provides specific support for the use of structured counting in a game context to facilitate children's acquisition of the counting sequence. More generally, the present results add to the growing body of evidence that simple and enjoyable numerical board games can provide specific learning opportunities to young children in home and early education contexts.