The authors assessed a battery of number skills in a sample of over 500 preschoolers, including both monolingual and bilingual/multilingual learners from households at a range of socio-economic levels. Receptive vocabulary was measured in English for all children, and also in Spanish for those who spoke it. The first goal of the study was to describe entailment relations among numeracy skills by analyzing patterns of co-occurrence. Findings indicated that transitive and intransitive counting skills are jointly present when children show understanding of cardinality and that cardinality and knowledge of written number symbols are jointly present when children successfully use number lines. The study’s second goal was to describe relations between symbolic numeracy and language context (i.e., monolingual vs. bilingual contexts), separating these from well-documented socio-economic influences such as household income and parental education: Language context had only a modest effect on numeracy, with no differences detectable on most tasks. However, a difference did appear on the scaffolded number-line task, where bilingual learners performed slightly better than monolinguals. The third goal of the study was to find out whether symbolic number knowledge for one subset of children (Spanish/English bilingual learners from low-income households) differed when tested in their home language (Spanish) vs. their language of preschool instruction (English): Findings indicated that children performed as well or better in English than in Spanish for all measures, even when their receptive vocabulary scores in Spanish were higher than in English.
Children’s earliest number skills—skills such as counting out loud, labeling quantities, and recognizing written numerals—undergird later mathematics learning and predict later mathematics performance in both primary and secondary school ( Children this young are more properly called “dual-language learners” or “multiple-language learners” than bilinguals or multilinguals because they are still learning their languages. But because the longer terms are both unwieldy and unfamiliar to many readers, we will use the term ‘bilingual’ here to mean children who are exposed to and learning in more than one language before age 5.
When children learn the counting system of their language, they acquire a system for representing exact numbers (
At the same time as they are learning to count, children are also learning the cardinal meanings of number words. Children learn the meanings of the first three or four number words one at a time and in order, starting with one and two (
A particularly important step seems to be when children figure out how counting is connected to exact set sizes via the cardinality principle (i.e., that the last word of a count is the exact, cardinal number of the whole set). Children who understand the cardinality principle identify numerosity (as opposed to some other property of sets, such as total summed area) as the dimension of experience that number words pick out (
Distinct from both counting and understanding the meanings of spoken number words, children in literate societies also learn to recognize and name written numerals, and this understanding relates to their later mathematical achievement (
Another potential influence on early numeracy is the child’s nonsymbolic, approximate number sense (
The present study sought to document the performance of a diverse sample of preschool-age children on a range of early numeracy tasks. Although there is a substantial literature on each area (counting, cardinality, nonsymbolic number discrimination, etc.), the variety of tasks used and the small sample sizes in most studies make it difficult to assemble an overall picture of preschool numeracy development or to compare developmental data across studies.
In addition to number tasks, the present study also included measures of children’s vocabulary in order to better understand the interplay of language and numeracy in early development. We were particularly interested in whether bilingual learners acquire number concepts on the same timeline as monolingual learners and whether a child’s number knowledge in the language of their preschool instruction differs from number knowledge in their home language. On the one hand, one might expect exposure to two languages to be helpful: Switching back and forth between languages might help children learn to control their attention and ignore irrelevant information, both of which are important skills for math learning (
On the other hand, there are reasons to expect numeracy development to take longer in bilingual learners. A child learning two languages has, in a way, twice as much information to acquire as a child learning only one.
Finally, the present study included a large and economically diverse sample of preschoolers. Basic research in cognitive development, as in other areas of psychology, has often relied on convenience samples of participants from middle-class, primarily white and English-monolingual households (
The study had three goals. The first goal was to explore entailment relations among early numeracy skills by identifying which skills (empirically, not theoretically) seem to form the foundation for other skills, such that they are present by the time the other skills are present. Our measures included symbolic skills (those using spoken and written numerals) as well as a nonsymbolic number comparison (ANS acuity) task where children had to decide which of two dot clouds had “more” dots. The study’s second goal was to examine the relation between symbolic number-skill development and language use, comparing bilingual to monolingual learners and separating bilingualism from well-documented socio-economic influences such as household income and parental education. The study’s third goal was to find out, for bilingual learners, whether and how children’s symbolic number knowledge differed when tested in their home language vs. the language of their preschool instruction.
A total of 566 children were recruited for the study. Of these, 4 had developmental or language delays (as reported by their parent or guardian on the study consent form) and 3 had consent forms where the question about developmental and language delays was left blank. These 7 children were allowed to participate in the study (interact with the researchers and play the games), but their data were excluded from the analysis. Another 13 records were duplicate enrollments where the same child had signed up for the study more than once. These were merged with the original records to create one record per child. An additional 22 children signed up for the study but did not participate, either because they were not at school on any of the testing days or because they did not want to play. Finally, 11 children were either younger than 3 years or older than 6 years at the time of the first testing session. These 11 children were also excluded from the analysis. The resulting sample of 513 children (246 girls, 267 boys) contributed data for the analysis.
These children ranged in age from 3 years, 0 months to 5 years, 11 months (
Parents were asked three questions about the child’s home language use: (1) In what language(s) do you and other caregivers speak to your child at home? (2) In what language(s) does your child speak to you and other caregivers at home? (3) Please specify the other language here. Questions 1 and 2 were answered on a 5-point Likert scale with the following values: English only, Mostly English but sometimes another language, Both languages about equally, Mostly another language but some English, Only another language. Based on the parent’s responses to these questions, each child participant was assigned a score representing the amount of time that he or she spent interacting in English
The demographic questionnaire also asked parents to “list the caregivers who spend the most time with the child in a typical week (for example: mother, father, grandmother, babysitter, etc.) and mark that person’s highest level of education completed.” Responses were given on a Likert scale from “1st-11th grade (please write the highest grade completed” to “Master’s or doctoral degree.” Each participant was assigned an educational attainment score that reflected the highest level of education attained by any of the child’s primary caregivers.
Household income was assessed with the question, “What was the total combined income of all members of the household this past year? (Please include money from jobs, welfare, social security payments, pensions, dividends, etc.)” Respondents were free to write in an exact number or choose an income category.
The demographic questionnaire asked parents to identify their child’s ethnicity (Hispanic or non-Hispanic). Fifty-four percent of children were identified as Hispanic (
Research sites included four Head Start programs and seven private pre-kindergarten programs. Head Start is a federally funded program to provide early childhood education to children from low-income households. Accordingly, over 90% of children attending Head Start had annual household incomes below $30,000, whereas over 90% of children attending private pre-kindergarten programs had annual household incomes of over $75,000. English was the language of instruction at all of the programs. Although the Head Start teachers were fluent in Spanish, they used Spanish mainly for classroom management and English for curriculum instruction, as their goal was to prepare children for kindergarten programs that would take place entirely in English (see also
Undergraduate research assistants visited preschools during the morning drop-off and afternoon pick-up hours to invite parents and children to participate in the study. At preschools where many children were Spanish/English bilingual learners, recruitment was done by Spanish/English bilingual research assistants. Consent was obtained in either Spanish or English, as parents preferred. If the parent was unable to fill out the form (e.g., many parents were carrying infants, diaper bags, backpacks, etc.) a research assistant read the questions and recorded the parent’s answers. When the person dropping off or picking up the child was not the parent, they were given a consent form to take home for the parent to sign and return later. Children were given a small prize (e.g., a rubber duck or a slinky) at the time of sign-up; no prizes were given during testing.
We followed an inclusive policy in recruiting and testing, meaning that all families who wanted to sign up were allowed to sign up, and all of their children who had parents’ permission and wanted to participate were allowed to do so. Before starting data analysis, we excluded children who had a previously diagnosed language or developmental delay, whose parents had not provided the information needed to place the child in a language or SES group, or who were not between the ages of 36 and 71 months, inclusive.
Testing took place at school, during school hours. Researchers first visited each classroom on two occasions, for at least an hour each time, to build rapport with the children. Testing was done at a table in a quiet room or in a hallway near the classroom. Researchers worked in pairs, with one researcher administering the task while a second researcher recorded the data using a video camera and/or pen and paper, depending on the task. Children who seemed bored or unhappy during testing were asked, “Do you want to go back to your classroom?” and returned to the classroom if they wished. All research activities were approved by the IRB of UC-Irvine [Human Subjects Protocol No. 2005-4735].
For English monolinguals, and for those bilingual learners whose home language was not Spanish, all assessments were conducted in English. For Spanish/English bilingual learners, assessments of vocabulary and symbolic number knowledge (spoken and written) were conducted twice—once in Spanish and once in English, in counterbalanced order. The assessment of nonsymbolic approximate number system (ANS) acuity was done in English, Spanish, or a combination of both, as the child preferred. Testing with Spanish/English bilingual children was carried out by fully bilingual research assistants who were native speakers of both Spanish and English. While at the schools, each of these research assistants spoke only in one language (Spanish or English) in order to give children the impression that they (the research assistants) were monolingual speakers of the language. This was done to encourage the children to cooperate with testing in the assigned language with each research assistant.
All children were given the PPVT–III (L.M.
Spanish-speaking bilingual learners were given the TVIP (
The purpose of this task was to assess children’s knowledge of the verbal counting list. Children were asked to count out loud from one to ten. The experimenter introduced the task by saying, “Let’s count to ten. Ready? One, two, three, four, five, six, seven, eight, nine, ten. Now you count.” After the child counted once, the experimenter said, “Thanks. Can you count again?” If the child stopped counting, the experimenter prompted them to continue by repeating the last two numbers the child had said (e.g., “two, three . . . ?”) or by asking “Do you know what comes after [three]?” repeating the last number the child had said. Test-retest reliability for the two verbal counts was .75.
Scores reflected the highest number the child counted to without error (maximum possible score: 10). For example, a count of “One, two, three, five, six, ten,” would be scored as 3. Each child’s highest score of the two counts (or four counts--two in each language for Spanish/English bilingual learners) was used. For the entailment analysis and to enable comparison with the transitive counting task, scores of 6 were coded as proficient; scores of 5 or below were coded as not proficient. (Although children were invited to count to ten, we treated counts of six and above as successes in order to facilitate comparison with the transitive counting task, where children were never asked to count rows of more than six objects.)
In this task, children were asked to count rows of three and six items (
This task, also called Give-a-Number (
In this task, researchers asked children to give some number of objects between one and six to a stuffed animal. Children were asked for each number three times in a preset, randomized order for a total of 18 trials. The experimenter set up the task by placing the following items on a table: a stuffed animal (e.g., a dinosaur), a tub with small, plastic objects in it (e.g., yellow bananas), and a lid from the top of the tub, to be used as a plate (
In this task, we asked children to identify the written numerals 1 through 10. Materials included a small picture frame cut from foam core and shaped like a house, with a small square of velcro in the center.
There was also a set of laminated cards showing the written numerals zero through ten, with a square of velcro on the back of each card. The cards were placed face up on the table in an unordered jumble (
This task was developed as a simpler alternative to the commonly-used continuous number line task (e.g.,
Children were asked for the numbers one through four and six through nine in a preset, randomized order (maximum possible score: 8). They were given generalized positive feedback (e.g., “Thank you!”) on every trial, regardless of their responses. After each trial, the experimenter removed the number from where the child had placed it on the number line (prior to asking for the next number). Children’s responses were coded as correct or incorrect. In the current sample, reliability was α = .90. For the entailment analysis, scores of 8 were coded as proficient; scores of 7 or below were coded as not proficient.
In this task, the experimenter placed a strip of paper on the table in front of the child. On the paper was printed a horizontal line, with the numeral 0 at the left end and the numeral 10 at the right end (
For each trial of the task, children were shown a laminated card, 21.5cm long by 12.5cm high, with two clouds of black dots printed side by side (
First, children were given training trials to make sure that they understood the task (
After answering correctly on eight training trials in a row, children were given test trials. These consisted of nine blocks with ratios at 1:2 (= .50), 7:12 (= .58), 2:3 (= .66), 17:24 (= .71), 3:4 (= .75), 4:5 (= .80), 5:6 (= .83), 7:8 (= .87), and 9:10 (= .90). There were eight trials per block, and all trials within each block contained the same ratio. Feedback was given after every trial (e.g., “That’s right— this side has more.” or “Uh oh, this side has more dots, you see?”). Trials were presented too rapidly for children to count the dots, and no children were observed attempting to count. Children’s performance was scored as the average of the number of correct trials in each block (maximum possible score: 8). In the current sample, reliability was α = .62.
Of the 513 children who contributed data to the analysis, 77% (
The majority of parents (92%,
Descriptive statistics for each numeracy skill are presented in
Task | Minimum | Maximum | |||
---|---|---|---|---|---|
Vocabulary | 502 | 89.71 | 19.49 | 40.00 | 130.00 |
Nonsymbolic number comparison | 358 | 5.31 | 0.67 | 3.33 | 7.11 |
Intransitive counting | 510 | 9.17 | 2.07 | 1.00 | 10.00 |
Transitive counting | 512 | 5.68 | 0.97 | 1.00 | 6.00 |
Give-N | 510 | 3.43 | 1.70 | 0.00 | 5.00 |
Number identification | 490 | 6.35 | 3.64 | 0.00 | 10.00 |
Scaffolded number line | 477 | 3.56 | 2.89 | 0.00 | 8.00 |
Continuous number line | 495 | 0.47 | 0.47 | -0.98 | 1.00 |
The changes that we made to the ANS testing procedure ensured that for those children who did complete the task (that is, who reached criterion on the training trials and then went on to complete at least one block of test trials), we can be confident that what we measured really was their ANS acuity. However, these changes also meant that a lot of children simply could not complete the task. Nearly one-third of the children we tested (
The entailment relations among the early symbolic numeracy skills are illustrated in a parallel plot (
We examined the relation between symbolic number skills (those using spoken or written numerals) and language use to look for differences in patterns of development for bilingual vs. monolingual learners, separating these from other well-documented socio-economic influences such as household income and parental education. For each symbolic number skill, we present the results of a traditional frequentist multiple linear regression with bilingualism and household income as predictors, controlling for child age. For this analysis, language context was dichotomized as bilingual vs. monolingual, with households using English at least 95% of the time considered monolingual, and all others considered bilingual. (All children in the study attended English-speaking preschool programs, so there were no monolingual learners of languages other than English.) We calculated the multiple linear regression models using the
Bilingualism was not a significant predictor of children’s intransitive counting skills after controlling for parental education, household income, and child age (
Task | Intercept-only model | Age, education, income model | Age, education, income, bilingualism model |
---|---|---|---|
Intransitive counting | 1 to 571167 | * | 1 to 2.89 |
Transitive counting | 1 to 502 | * | 1 to 3.48 |
Give-N | 1 to 4.66 | * | 1 to 6.46 |
Numeral identification | 1 to 2.04 | * | 1 to 5.50 |
Scaffolded number line | 1 to 1047 | 1 to 1.19 | * |
Continuous number line | 1 to 1026 | * | 1 to 1.45 |
There was not a significant effect of bilingualism on children’s transitive counting skills after controlling for parental education, household income, and child age,
There was not a significant effect of bilingualism on children’s Give-N performance after controlling for parental education, household income, and child age,
There was not a significant effect of bilingualism on children’s numeral identification skills after controlling for parental education, household income, and child age,
Bilingualism was a significant predictor of children’s scaffolded number line performance after controlling for parental education, household income, and child age,
Bilingualism was not a significant predictor of children’s continuous number line performance after controlling for parental education, household income, and age,
In our third research question, we were interested in whether and how bilingual children’s symbolic number knowledge differed when tested in their home language vs. the language of their preschool instruction. To address this question, we considered only the Spanish/English bilingual learners in our sample (
The PPVT and TVIP were used to measure children’s receptive vocabulary in English and Spanish, respectively. On the PPVT, the mean standard score was 78.50 (
This task measured children’s ability to correctly count out loud to ten; thus, the maximum score possible was 10. The average counting out loud score was 6.22 (
This task measured children’s ability to correctly count six objects; thus, the maximum score possible was 6. In Spanish, the mean score was 4.80 (
This task measured children’s knowledge of the exact meanings of the numbers “one” through “six,” to determine their knower level in each of their languages; the highest knower level possible was CP-knower. When assessed in Spanish, there were 28 pre-knowers (12%), 50 one-knowers (21%), 68 two-knowers (28%), 47 three-knowers (19%), 19 four-knowers (8%), and 30 CP-knowers (12%). When assessed in English, there were 18 pre-knowers (7%), 62 one-knowers (25%), 51 two-knowers (20%), 44 three-knowers (18%), 16 four-knowers (6%), and 60 CP-knowers (24%). Children performed significantly better in English than Spanish,
This task measured children’s ability to correctly identify the written numerals one through ten; thus, the maximum score possible was 10. The average numeral identification score was 3.41 (
This task measured children’s ability to correctly place the numbers one through nine (excluding five) on a number line; thus, the maximum score possible was 8. In Spanish, the mean score was 1.84 (
This task measured children’s ability to correctly place the numbers one through nine (excluding five) on a number line, with scores calculated as the correlation between each placement mark and the correct location on the line. Thus, the maximum possible score was 1 (perfect correlation). In Spanish, the mean score was 0.20 (
This study investigated early childhood number skills in a linguistically and economically diverse sample of preschoolers in Orange County. The study had three goals: To describe entailment relations among a set of early numeracy skills; to examine the relation between symbolic number-skill development and bilingualism/monolingualism when controlling for known effects of socioeconomic status; and to find out whether and how bilingual learners’ symbolic number knowledge differed when tested in their home language vs. the language of preschool instruction. To these ends, children were tested in English (and also in Spanish, where appropriate) on a battery of tasks related to symbolic number skills, nonsymbolic number comparison, and receptive vocabulary.
The first set of questions related to entailment relations among early numeracy skills. Logic dictates that certain skills build on others. For example, intransitive counting (reciting the number list) is part of transitive counting (reciting the number list and pointing to objects); and both of these provide a foundation for children to use counting to construct sets on the Give-N task. But not all relations among tasks are so easy to predict. We were interested both in symbolic skills (those using spoken and written numerals) and in nonsymbolic number comparison, which we tested by asking children to decide which of two cards had “more dots.”
Among the symbolic skills, there was strong evidence that numeracy skills build on each other, in mostly unsurprising ways. Children learn to count, then learn how counting represents number, as demonstrated by the fact that transitive and intransitive counting skills were jointly entailed by the Give-N task. Similarly, it is not surprising that children who can place number symbols correctly on a number line can also identify those written symbols. These findings replicate previous work showing that children learn the counting procedure before they master the cardinality principle (
Perhaps more surprising was the link between the Give-N task (where children used counting to construct sets of a given size) and the Numeral ID task (where children identified the written number symbol matching a given number word). We had no
Unlike the clear entailments found among the symbolic number tasks, we found little correlation between children’s performance on a nonsymbolic number comparison task (testing approximate number system, or ANS, acuity) and their performance on any of the symbolic number tasks. Methods for assessing ANS acuity in preschoolers must be chosen carefully, as many children may not interpret the phrase “more dots” to refer to numerosity until they have a grasp of the cardinality principle (
A second question that we sought to answer with this study was how the child’s language environment (specifically comparing monolingual to bilingual or multilingual environments) relates to the early development of number skills. We wanted to use this large and diverse sample to separate bilingualism from well-documented socio-economic influences such as household income and parental education. In terms of demographics, we found abundant and unsurprising evidence that economic hardship is bad for early learning: Children whose families were under the greatest economic stress showed the most delayed acquisition of early numeracy. This is obviously of concern, given that children who enter kindergarten with poorer understanding of numbers often continue to struggle throughout the early years of elementary school (
A third goal of the present study, concerning just the Spanish/English bilingual learners in our sample, was to compare children’s performance on symbolic number tasks when tested in Spanish versus when tested in English. English was the language of instruction in their preschool programs; Spanish was the language used all or part of the time at home. Assessments of the children’s receptive vocabulary in each language showed that most had larger vocabularies overall in Spanish than in English. However, when it came to counting, constructing sets on the Give-N task, and identifying written numerals on the Numeral ID task, most children performed as well or better in English--the language of instruction at their preschools--than they did in Spanish. We take this as evidence that preschool programs had provided these children with important numeracy-building experiences in English that went beyond the experiences that their home environments provided in Spanish. Even though the at-school experiences took place in English, which was not the most familiar language for most of these children, those experiences were still key to their developing numeracy skills. Unlike older bilingual learners for whom the challenge may be “translating” prior knowledge from one language into another, preschoolers are likely to be learning skills for the first time. And because children under five essentially become native speakers of whatever language they are exposed to, experiences with counting and number in any language are valuable. On a policy level, these findings would seem to argue for providing preschool services to as many children as possible (even if such services can only be provided in English) rather than providing fewer children with bilingual services, which may be more expensive to implement. Limited resources for bilingual services should be allocated first to older students, who have more prior knowledge in their home language, and who face longer delays in reaching the level of English proficiency needed for instruction in English at higher grades.
The fact that the Spanish/English bilingual learners in our sample performed as well or better in English than in Spanish also has methodological implications for researchers. Testing children twice (once in each language) requires nearly twice the effort and expense of testing them only once. The present study, for example, employed native Spanish-speaking research assistants who spoke to the children only in Spanish and pretended not to understand English at all, as well as English-speaking research assistants who interacted with the children only in English. But in the end, the scientific value added by doing this was minimal: The picture of children’s numeracy that we gleaned from testing separately in Spanish and English was not much different from the picture that we would have gotten by testing only in English. The results may well have been different with older students, but we hope this knowledge will be useful to our colleagues as they plan studies with very young bilingual learners in the future.
There are several limitations to the present study. First, these data present the results of concurrent associations among children’s skills, as opposed to longitudinal or cross-sectional analyses. Although the entailment analyses suggest that some skills are mastered by children earlier than others, further research is needed to confirm directionality and potential causal connections among these early numeracy skills. Second, these data do not provide an exhaustive comparison of all early numeracy skills. In particular, future research should test whether these patterns are observed with nonsymbolic comparisons of additional set sizes and symbolic comparisons of pairs of numerals (e.g., which is more, 3 or 4?). Third, our sample includes preschoolers ranging in age (3 - 6 years old) and prior years of preschool (e.g., enrolled in their first or second year of Head Start). It remains an open question as to whether the pattern of results vary based on children’s age or exposure to the language of instruction—for instance, whether children’s comparable performance on measures of number skills is true of the youngest learners with the least exposure to English in the classroom setting.
Overall, our findings with this large, linguistically and socioeconomically diverse sample confirm that early number skills such as counting out loud, constructing sets, and recognizing written numerals form an important base for later mathematics learning. Even the preschool-level tasks included in this study build on each other, with earlier-acquired skills such as counting serving as prerequisites for later skills such as using a number line. Nonsymbolic estimation acuity, on the other hand, seems less important in these early years. Furthermore, unlike some other aspects of language development, early numeracy is hardly affected by a child’s being monolingual or bilingual/multilingual. More important are the preschool experiences that support children’s growing number concepts, whether or not those experiences occur in the child’s home language.
Although this study included a relatively large and diverse sample of participants, the developmental variation among the preschoolers of Orange County does not equal the variation in young children across the U.S. nation and beyond. We hope that this study will be part of a growing, more inclusive literature on development in early childhood. The more we as a field reach beyond convenience samples, the more our scholarship can help educators and policymakers support the growth and well-being of all our children.
This research was supported by NSF grant DRL-0953521 to the first author. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors, and do not necessarily reflect the views of the National Science Foundation.
The authors would like to thank the children, families, preschool teachers and administrators who participated in this research, as well as lab manager Tanya Anaya, Prof. Dr. Lieven Verschaffel, and two anonymous reviewers for their help with this project.
For this article, a data set is freely available (
The Supplementary Materials contain the following items (for access see
Research data
Stimuli and data collection sheets
Participants’ household incomes by preschool type
Descriptive statistics on vocabulary and numeracy measures, by child age
Floor and ceiling effects observed on numeracy tasks
Multiple linear regression models for each numerical skill and the amount of variance explained by each model
Entailment relations among symbolic number skills, with a child’s performance coded as “proficient” on the numeral identification task if they identified 6 or more numerals
The authors have declared that no competing interests exist.