research on numeracy skills

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Early numeracy skills in preschool-aged children: A review of neurocognitive findings and implications for assessment and intervention

Kimberly p. raghubar.

1 Department of Pediatrics, Baylor College of Medicine and Psychology Service, Texas Children’s Hospital

Marcia A. Barnes

2 Department of Special Education, and the Meadows Center for Preventing Educational Risk, The University of Texas at Austin

Objectives:

The goals are to 1) provide a review of the typical and atypical development of early numeracy; 2) present what is known about the neurocognitive underpinnings of early numeracy; and 3) discuss the implications for early assessment and intervention.

Studies on the development of typical and atypical early numeracy are reviewed with a particular focus on longitudinal findings including those from our work on spina bifida myelomeningocele. Implications of this research for assessment are presented. The paper ends with a discussion of early math interventions.

Learning to count, identify numbers, and compare and manipulate quantities are key early numeracy skills. These are powerful predictors of school-age mathematical learning and performance. General neurocognitive abilities such as working memory and language, are also important for the development of early numeracy. It is recommended that early assessment for risk of mathematical learning difficulties include tests of both early number knowledge and key neurocognitive abilities. Math-specific interventions are most effective for improving early numeracy. There is currently little evidence that training of general cognitive functions transfers to mathematical learning.

Conclusion:

Understanding the development of early numeracy skills and their neurocognitive predictors offer important insights into early assessment and intervention for children at risk for or with mathematical learning difficulties.

Although numeracy is far less studied than literacy, it is as important as literacy for predicting productivity, wages, and employment outcomes ( Rivera-Batiz, 1992 ), and the cost of innumeracy to individuals and society is high ( Hudson, Price, & Gross, 2009 ). In recent years, analyses of data from large national longitudinal databases such as the Early Childhood Longitudinal Study, have shown that children’s mathematical knowledge at school entry is the strongest predictor of both later math success as well as success in other academic domains ( Duncan et al., 2007 ). Similar findings have been obtained with respect to risk for math learning disability or MLD. Morgan, Farkas and Wu (2009) found that 70 percent of children who started and ended kindergarten below the 10 th percentile in mathematics were also below the 10 th percentile in 5 th grade, underlining the importance of early mathematical knowledge to later mathematical achievement. Together, these findings suggest that early numeracy is critically important for later mathematical development ( Aunola, Leskinen, Lerkkanen, & Numri, 2004 ; Duncan et al., 2007 ; Toll, van der Ven, Kroesbergen, & van Luit, 2011 ). Given the essential role that early numeracy plays in the development of later mathematical ability and disability, it is important to understand: 1) what early numeracy skills are important for later mathematical development; 2) what early number-specific and general cognitive abilities are involved in the acquisition of these key early numeracy skills; and 3) the implications of such knowledge for early assessment and intervention.

Although studies on the typical and atypical development of literacy skills and interventions for reading have historically outnumbered those in the area of numeracy ( Siegler, 2007 ), there has been a strong upsurge in research in mathematics over the past two decades ( Geary, 2013 ). Given the importance of early numeracy skills to later mathematical development along with rapid changes in knowledge in this area, the goal of this paper is to review the emerging literature on the acquisition of early numeracy skills in typically developing children and in children at risk for math difficulties. Specifically, we provide a description of commonly studied early numeracy skills shown to be important for later mathematical achievement and present findings on number-specific and general neurocognitive abilities that are important for the development of these early numeracy skills and later mathematical achievement. Because there are more studies of early math in young typically developing children and in neurologically intact children at risk for math difficulties (e.g., children with socio-economic disadvantage) than there are of preschool children with neurological disorders, most of the review focuses on the former groups. However, we also present findings on select groups of preschool children with neurological disorders. These include spina bifida myelomeningocele (SBM), a neurodevelopmental disorder associated with high rates of math learning disability (about 50%), but low rates of reading disability ( Fletcher et al., 2005 ). Because the rate of MLD at school-age is known and SBM is diagnosed before or at birth, we have been able to study these children’s mathematical development over a much longer developmental time window than is typical of most other longitudinal studies. The high rate of specific MLD in this population, coupled with the fact that they have been followed from toddlerhood into the middle elementary school years makes the findings of relevance to understanding math disabilities more generally ( Barnes & Raghubar, in press ). We also briefly discuss math difficulties in two high incidence clinical populations; children with low birth weight and children with Fragile X syndrome. Importantly, we discuss the implications of the findings across populations for early assessment of risk, and discuss evidence for and against the effectiveness of particular approaches to early numeracy interventions.

Early Numeracy Skills and Their Relation to Mathematical Achievement

Early numeracy is an umbrella term that encompasses several skills such as verbal counting, knowing the number symbols, recognizing quantities, discerning number patterns, comparing numerical magnitudes, and manipulating quantities (i.e., adding and subtracting objects from a set). Such informal math skills are acquired prior to or outside of the school setting. In comparison, formal math knowledge is acquired through explicit teaching within the school setting such as instruction in the concepts and steps involved in regrouping in multi-digit addition and subtraction. For most children, acquisition and mastery of early numeracy skills occurs spontaneously through activities in the home and other experiences in the child’s everyday environment ( LeFevre et al., 2009 ), though this is not true for all children. For example, children from low socioeconomic status households often demonstrate less well developed numerical knowledge during the preschool years and kindergarten than their middle-income peers ( Griffin, Case, & Siegler, 1995 ; Jordan, Huttenlocher, & Levine, 1994 ; Starkey, Klein, & Wakeley, 2004 ), which has been associated with exposure to significantly fewer and less complex everyday number activities and experiences (e.g., Blevins-Knabe & Musin-Miller, 1996 ). Children with learning difficulties related to specific neurodevelopmental disorders or to other developmental factors also often struggle with the acquisition of informal mathematical knowledge. Consequently, children with low number skills at kindergarten entry, regardless of the source of this lack of mathematical knowledge, are at high risk for low math achievement over the early elementary school grades ( Jordan, Kaplan, Ramineni & Lokuniak, 2009 ).

Early numeracy skills involve the understanding and manipulation of both symbolic and non-symbolic number. Early symbolic number skills include learning the count sequence and understanding the numerical meaning of number words (e.g., “three”) and Arabic numerals (e.g., “3”). Children are considered to know the meaning of symbols once they have acquired the cardinality principle, or the understanding that the last number word used when counting a set indicates the number of objects in the set. Symbolic number knowledge in the preschool years has been reliably associated with later math achievement ( Gobel, Watson, Lervag, & Hulme, 2014 ; also see Merkley & Ansari, 2016 for review).

N on-symbolic number skills and representations refer to ways of representing numbers without using symbols and typically involve numerical manipulations or transformations on objects as well as comparisons of the magnitude of sets of objects. For example, young children can perform simple addition and subtraction with non-symbolic numerical representations (e.g., with actual objects or pictures of objects; Bisanz, Sherman, Rasmussen, & Ho, 2005 ). These non-symbolic tasks that involve quantity manipulations (adding and subtracting from sets of objects) are related to later achievement on symbolic arithmetic tasks (adding and subtracting Arabic numerals). In contrast, the evidence for a relationship between performance on magnitude comparison tasks using non-symbolic representations of number (e.g., presenting two arrays and determining which one has more) and later math achievement is not as strong ( Leibovich & Ansari, 2016 ). These symbolic and nonsymbolic number skills are discussed in more detail below drawing on studies of typical and atypical development to demonstrate the relationship of these early numeracy abilities to later math achievement.

Symbolic Number Skills

The development of counting skills and its impact on arithmetic skill development has been well studied. Sequential counting refers to the ability to recite the number word sequence (e.g., 1, 2, 3, 4, 5…10) and acknowledge the position of a number word in this sequence (e.g., 1, 2, 3…what comes next? 4; or 4 comes after 3 and before 5) without explicitly understanding cardinal meaning (how many are there?). Gradually, children apply their knowledge of the counting sequence to enumerate sets of objects. This serial quantification process is referred to as cardinal counting and involves mapping each number word onto each item in a set (one-to-one correspondence) to acknowledge the exact number of items in a collection ( Fuson, 1988 ; Gelman & Gallistel, 1978 ). Ultimately, children demonstrate an understanding of the numerical meaning of number words with acquisition of the cardinality principle ( Gelman & Gallistel, 1978 ). In the research literature, counting skills in preschool and kindergarten children have commonly been assessed by asking young children to watch a hand puppet point to and count objects or dots on a page, and to tell the puppet whether or not he counted correctly. Incorrect counts typically violate one of three counting principles: one-to-one correspondence (one counting tag is applied to each object); stable order/ordinality (number tags must be must be applied in an invariant order); and cardinality (the last number counted refers to the total quantity). Early studies demonstrated that typically developing preschool-aged children are sensitive to violations of the one-to-one and cardinal principles, correcting the puppet when he double counted, skipped an item, or repeated an incorrect cardinal value ( Gelman, Meck, & Merkin, 1986 ).

Young children employ counting procedures when solving arithmetic problems and rely on fingers or other external referents (e.g., blocks, drawings, stickers, etc.) ( Siegler & Shrager, 1984 ). The most common counting procedures regardless of the use fingers are counting-all (counting both addends starting at 1, for example, for a 2 and 3 problem, counting 1,2, then 3,4,5), counting-on max (stating the smaller valued addend and then counting a number of times equal to the value of the larger addend, for example, 2, then 3,4,5), and counting-on min (stating the larger addend and then counting the smaller addend, for example, 3, then 4,5). Despite the mix of strategies employed, young children tend to shift to use of more efficient strategies, doing away with the more laborious counting all procedure in favor of the counting-on min procedure, which involves the least amount of counting. This shift in procedures is related, in part, to improvements in children’s conceptual knowledge of counting ( Siegler, 1987 ). The use of counting also results in the development of memory representations of basic math facts, facilitating direct retrieval of answers from memory (e.g., stating “five” without having to count when asked to solve 2 +3).

Early knowledge about number symbols predicts formal math skills and later math achievement. Number identification (i.e., children are told a number and they have to select the correct number from 4 or 5 options) assessed at the beginning of the first year of school was a powerful predictor of longitudinal growth in arithmetic skills over the next 11 months ( Gobel et al., 2014 ). Similarly, longitudinal studies examining early number skills including number identification and counting assessed at the end of kindergarten were strong predictors of math outcomes at the end of first grade ( Chard et al., 2005 ; Clarke & Shinn, 2004 ). Longitudinal work conducted by Jordan and colleagues found that growth in early numeracy skills including symbolic skills from the start of kindergarten through the middle of first grade was highly predictive of first grade math achievement ( Jordan, Kaplan, Locuniak, & Ramineni, 2007 ) and predicted both the rate of growth and levels of math achievement between 1 st and 3 rd grades ( Jordan et al., 2009 ).

Longitudinal studies of young children at risk for MLD have largely focused on school-age children in the early grades. Findings from these studies suggest that children with low math achievement scores understand basic counting principles, such as cardinality and ordinality but have difficulty understanding unessential features of counting such as the idea that counting of objects can proceed in any order ( Geary, Hamson, & Hoard, 2000 ; Geary, Hoard, & Hamson, 1999 ). Our longitudinal work with children with SBM and their typically developing peers ( Barnes et al., 2011 ) shows that difficulties in symbolic aspects of arithmetic are discernable early in development as early as 36 months of age, children with SBM were significantly less able than their typically developing peers to answer the entry-level questions on the Test of Early Mathematics Ability (TEMA-2; Ginsburg & Baroody, 1990 ), which involve counting small sets of objects, showing the number of fingers corresponding to spoken number words, and understanding cardinality ( Barnes et al., 2011 ). At 60 months of age; these same children with SBM were unable to count as high as their typically developing peers and they were also less skilled than their peers at detecting incorrect counts by the puppet in the counting task described above. In all, these longitudinal studies of children at risk for MLD show that 1) early difficulties in conceptual counting knowledge and in counting procedures are characteristic of children who go on to have difficulties in mathematics at school-age; and 2) difficulties in counting knowledge and procedures can be discerned quite early in development, before formal schooling.

Non-Symbolic Arithmetic

Preschool-aged children demonstrate success on non-symbolic addition and subtraction problems using relatively small numbers ( Huttenlocher, Jorden, & Levine, 1994 ; Levine, Jordan, & Huttenlocher, 1992 ). These problems typically consist of having the examiner place an array of objects on a mat and then a screen is placed in front of the array so as to occlude the array from the child. The child watches while the examiner either adds chips to the array (addition) or removes chips from the array (subtraction) in a single trial. Although children cannot see the quantity behind the screen, they see the quantity that was added or removed. Finally, children are asked to use their chips to match the quantity that remains hidden after the transformation ( Jordan, Huttenlocher, & Levine, 1992 ).

Typically developing preschoolers aged three years and older can add and subtract on these arithmetic tasks using non-symbolic quantities, though accuracy tends to be related to age and problem size, for example, accuracy is higher for smaller problems such as 2 + 2 than it is for larger problems such as 4 + 3 ( Huttenlocher et al., 1994 ). When non-symbolic arithmetic problems (e.g., 2 chips are placed on a mat, a screen occludes the child’s view and three more chips are added. “How many do I have under here now? Show me on your mat?”) and symbolic arithmetic problems (i.e., simple word problems of the type, “How much is 2 and 3”?; or “John had 2 balls. He got 3 more. How many does he have altogether?”) were examined in children ages 4–6 years old, non-symbolic problems were found to be easier for the younger children ( Levine et al., 1992 ; Huttenlocher et al., 1994 ), but this difference was diminished for the older children ( Levine et al., 1992 ; Rasmussen & Bisanz, 2005 ).

Non-symbolic arithmetic tasks are considered to tap early conceptual knowledge about arithmetic such as the concepts of addition and subtraction as demonstrated above, well before children are able to articulate this conceptual knowledge or solve similar sized problems using symbols such as Arabic numerals or number words. In addition to early conceptual knowledge about addition and subtraction, typically developing preschoolers have also been shown to understand mathematical concepts such as inversion (the principle that a + b – b must equal a). Klein and Bisanz (2000) demonstrated that 4-year-old children solve these three-term inversion problems more quickly than standard problems (e.g., a + b – c), suggesting the use of a selective strategy for inversion problems, which is confirmed by spontaneous reporting by children that counting was not needed because the second and third numbers were the same.

In terms of individual differences, non-symbolic arithmetic is correlated with math achievement throughout kindergarten to the middle of first grade ( Jordan et al., 2007 ) and is a commonly used preschool math task in longitudinal studies of the prediction of later math achievement ( Jordan et al., 2007 ; 2009 ). In our longitudinal studies of children with SBM, performance on this non-symbolic arithmetic task at 60 months of age strongly predicted performance on a standardized measure of multi-digit symbolic arithmetic 4 to 5 years later (Barnes & Raghubar, in press), suggesting developmental continuity in the ability to manipulate non-symbolic and symbolic quantities.

Non-symbolic Representations and Approximate Number System Acuity

Several recent studies on typical and atypical math development have focused on the concept of “number sense”, which has been defined as the ability to rapidly indicate which of two sets of non-symbolic quantities is larger without counting ( Butterworth, Varma, & Laurillard, 2011 ). This ability is referred to as approximate number system (ANS) acuity (for a review of this literature see Geary, Berch & Mann Koepke, 2015 ). In a typical ANS task, two large arrays of quantities are presented briefly and their magnitudes compared. For an example, see www.panamath.org ( Halberda & Feigenson, 2008 ; Halberda, Mazzocco, & Feigenson, 2008 ). As the ratio between the quantities to be compared decreases, accuracy on ANS acuity tasks also decreases. The ability of human infants and nonhuman animals to compare large approximate quantities is presumed to be evolutionarily important and, in the case of humans, to provide the “infant’s starter kit” for the development of number skills ( Butterworth, 1999 ). Although this ability is presumed to be innate, there are both developmental changes and individual differences in ANS acuity, and the latter have been proposed to underlie individual differences in the acquisition of both early numeracy skills as well as math achievement and MLD at school-age ( Geary et al., 2015 ).

Despite the recent intense interest in whether ANS acuity is importantly related to growth in mathematical abilities, there is currently a lack of evidence for a unidirectional causal relationship between ANS abilities and the acquisition of symbolic math skills. Instead, evidence suggests that children who have acquired symbolic skills such as the cardinality principle are more likely to succeed on non-symbolic tasks ( Batchelor, Keeble, & Gilmore, 2015 ; Slusser, Ditta, & Samecka, 2013 ), leading some to conclude that acquiring symbolic knowledge likely influences non-symbolic skills, rather than the other way around ( Merkley & Ansari, 2016 ). Similarly, relations between ANS acuity and math achievement are mediated by symbolic number knowledge such as Arabic numeral naming and cardinality ( Chu, van Marle, & Geary, 2015 ; see also Merkley & Ansari for a review). Indeed, preschoolers’ cardinality knowledge emerged as an important mediator of the relationship between ANS acuity and early math achievement ( van Marle, Chu, Li, & Geary, 2014 ; Chu et al., 2015 ). Based on these recent findings, it has been suggested that explicit knowledge of the cardinal value of number symbols (i.e., that the last item counted represents the cardinal value of the set) may act as a critical early link between ANS acuity and math achievement given that the core implicit knowledge represented by the ANS is cardinal value, albeit an approximate value, of collections of objects. Consequently, Chu and colleagues (2015) proposed that young children who easily discriminate relative quantities of collections of objects also easily achieve insight that quantities represented by different number symbols differ in quantity.

In terms of individual differences, although performance on the ANS task correlates with mathematical achievement ( Bonny & Lourenco, 2013 ; Halberda, et al., 2008 ; Mazzocco, Feigenson, & Halberda, 2011b ), three meta-analyses ( Chen & Li, 2014 ; Fazio, Bailey, Thompson, & Siegler, 2014 ; Schneider et al., 2016 ) have found that the relation between ANS acuity and mathematical performance, while significant and consistent, is small (r = ~ .20). Studies of children with MLD and of young children at risk for MLD have begun to address whether a deficit in ANS acuity is a defining feature of difficulties in mathematical learning. Some studies ( Mazzocco, Feigenson, & Halberda, 2011a ; Piazza et al., 2010 ) have reported poorer ANS acuity in children with MLD, but other studies have not ( Iuculano, Tang, Hall, & Butterworth, 2008 ; Rousselle & Noel, 2007 ). In preschool children, ANS acuity at the beginning of prekindergarten was found to be predictive of math achievement at the end of the prekindergarten year, but only for children in the lowest quantile of the mathematics performance continuum ( Purpura & Logan, 2015 ). In a micro-analytic approach to studying these relations, Bugden and Ansari (2015) found that children with MLD did not differ from typically developing peers on items in which number and area are correlated (i.e., where 10 dots take up more area than 5 dots), but did differ on items in which number and area are uncorrelated (i.e., where 10 dots take up the same area as 5 dots). Group differences on these uncorrelated trials were mediated by visual-spatial working memory, calling into question the idea that it is deficits in ANS acuity per se that discriminate children with and without MLD.

In sum, these studies of non-symbolic number skills suggest that ANS acuity (comparison of non-symbolic numerical magnitudes) may not have a strong and direct relationship with later math ability and disability. In contrast, performance on tasks in which non-symbolic quantities are manipulated arithmetically (the nonverbal addition and subtraction tasks described above) do seem to be related to later math achievement in the domain of calculation. In keeping with the predictive value of such nonverbal arithmetic tasks, items involving non-symbolic arithmetic are part of the most recent version of the Test of Early Mathematics Ability, the TEMA-3 ( Ginsburg & Baroody, 2003 ), an assessment of number and operations that covers the age range from 36 months to 8 years.

Neurocognitive Skills and Their Relation to Math Achievement

A variety of neurocognitive factors have been related to mathematical performance in school-age children, including language abilities, processing speed, working memory, attention, and other executive skills such as set switching ( Peng, Namkung, Barnes, & Sun, 2015 ; Peterson et al., 2016 ; Willcutt, et al., 2013 ). Fewer of these neurocognitive correlates have been examined in relation to early numeracy skill development. We provide a review of the math-related neurocognitive correlates that have been related to the acquisition of early numeracy, namely, working memory, language, and finger skills.

Working Memory

Working memory refers to an individual’s ability to hold information in memory while simultaneously processing other information ( Baddeley, 1992 ; Engle, Tuholski, Laughlin, & Conway, 1999 ). Working memory capacity increases from preschool through the elementary school years, and it has been shown to be a significant predictor of children’s academic achievement in general, and math achievement specifically. Moreover, children with MLD typically do not perform as well as their typically developing peers on a variety of working memory measures ( Bull, Johnston, & Roy 1999 ; Geary, Hoard, Byrd-Craven, & DeSoto, 2004 ; see Raghubar, Barnes, & Hecht, 2010 for review).

Most research connecting working memory and math achievement has focused on school-aged children and formal mathematics but there is some evidence that working memory is just as important to the development of informal math concepts and early numeracy skills in preschoolers and kindergarteners. Purpura and Ganley (2014) examined the relation of verbal working memory to several early math skills in children in preschool and kindergarten. After accounting for many other math-related variables (age, grade, sex, broad calculation ability, and single word expressive vocabulary), verbal working memory emerged as a significant predictor of cardinality, guessing the number of objects without counting (with set sizes ranging from 1 to 7), set comparison (determining which set of four has the most dots), and number order. These findings were in line with the authors’ hypothesis that working memory would be related to that subset of early math skills that are more complex and require multiple steps; in contrast, working memory was not related to more basic skills such as rote verbal counting, one-to-one counting (counting sets of objects), number identification, and simple story problems without extraneous information (e.g., “Johnny had one cookie and his mother gave him one more cookie. How many cookies does he have now?”).

In one longitudinal study that compared the neurocognitive preschool predictors of later academic achievement in math and reading, visual working memory assessed during preschool was found to predict growth specific to math achievement in the early school grades ( Bull, Espy, & Wiebe, 2008 ). Visual-spatial working memory appears to support the ability of young children to solve non-symbolic arithmetic problems such as the type described above in which items are added to or subtracted from a hidden set of objects ( Levine et al., 1992 ; Rasmussen & Bisanz, 2005 ). To solve these problems, it has been suggested that preschoolers use mental models or nonverbal representations of the mathematical situation involving transformation on number when adding to or subtracting from sets of objects ( Bisanz et al., 2005 ). By first grade, however, visual-spatial working memory was no longer a significant predictor of these same nonverbal math problems, perhaps signaling a switch to the use of verbal rather than visual-spatial codes during problem solving ( Bisanz et al., 2005 ).

A recent meta-analysis by Peng and colleagues (2015) has helped to specify the relation between math and working memory as well as potential moderators of this relation. This meta-analysis revealed that the relation between math and working memory was not impacted by domain of working memory; that is, verbal, visual-spatial, or numerical working memory. However, different types of math skills were found to draw on working memory resources to different degrees: Whole number calculations and word problem solving showed the strongest relation with working memory whereas geometry showed the weakest relation with working memory. Importantly, the relation between working memory and mathematics was stronger among individuals with MLD and a co-occurring disorder, such as attention-deficit/hyperactivity disorder, Turner syndrome, or velo-cardio-facial syndrome, compared to typically developing individuals or individuals with specific MLD alone. It may be that individuals with a co-occurring disorder demonstrate significant impairments in neurocognitive processes (such as working memory), which in turn, further impacts mathematical development.

In summary, working memory is an important predictor of several early numeracy skills including cardinality, set comparison, number order, and non-symbolic arithmetic ( Bisanz et al., 2005 ; Purpura & Ganley, 2014 ). The continued importance of working memory for later mathematics achievement at school-age is demonstrated for a number of math skills at school age most notably calculation and word problem solving skills ( Peng et al., 2015 ). Furthermore, relationships between working memory and school-aged math skills appear to be particularly strong for children with math difficulties in the context of comorbid neurodevelopmental disorders ( Peng et al., 2015 ).

Language is a strong predictor of several early numeracy skills including number identification, cardinality, number comparison, number order, and story problems ( LeFevre et al., 2010 ; Purpura & Ganely, 2014 ), but not verbal counting, one-to-one counting, subitizing ( Purpura & Ganley, 2014 ) and non-symbolic arithmetic ( LeFevre et al., 2010 ). That children’s language skills are related to many though not all aspects of early numeracy is not surprising considering that most early numeracy skills require children to know number names; connect number names with specific quantities and written numerals; connect written numerals with quantities; connect number names, quantities, and written numerals; and demonstrate an understanding of comparative terms such as “more”, “less”, and “equal to”. Consequently, language may play a key role in the acquisition of new knowledge and the integration of that knowledge with prior knowledge. Moreover, language may play a critical role in the integration of early numeracy skills and formal mathematical learning.

Phonological awareness, a specific type of language-based ability that is strongly related to the acquisition of reading, refers to the ability to encode, access, and manipulate speech sounds within words. Phonological awareness, more so than measures of vocabulary or word meaning, tap the quality of lexical representations (perhaps because they require fine distinctions between phonological representations such as cat vs. car or cat vs. rat) and is related to young children’s sequential counting skills ( Barnes et al., 2011 ; Koponen, Salmi, Eklund, & Aro, 2013 ; Krajewski & Schneider, 2009 ; Soto-Calvo, Simmons, Willis, & Adams 2015 ). A longitudinal study conducted by Krajewski and Schnieder (2009) followed children from preschool to third grade and examined the relation of phonological awareness to the development of early numeracy skills. Findings indicated that phonological awareness assessed in preschool was differentially related to the development of early numeracy skills. Phonological awareness predicted individual differences in learning the number sequence; but not individual differences on tasks where the number sequence is mapped onto quantity (e.g., deciding which of two number words represented “more” or “less”; matching quantities to the corresponding Arabic numerals and vice versa). Along these lines, Soto-Calvo and colleagues (2015) found that phonological awareness assessed in the first year of pre-kindergarten (at 4 years, 8 months on average) no longer predicted sequential counting skills 14-months later after including the autoregressor (sequential counting skills from the previous year). If a cognitive predictor remains significant when autoregressor effects are controlled, it can be concluded that the cognitive predictor predicts growth in the outcome measure. In this case, phonological awareness influenced sequential counting at school entry but not growth in sequential counting during the first year of schooling, suggesting that while phonological awareness is related to early counting, it is no longer a unique predictor of counting later in development.

Taken together, the evidence collected to date suggests that the quality of language-based representations, tapped by measures of phonological awareness, is important for acquisition of the counting sequence and may be less relevant for the formation of higher-order mathematical competencies. Moreover, the role of phonological awareness in learning the counting sequence may be rather circumscribed: Phonological awareness may influence the rate at which children acquire the first few number words, but later extension of the number sequence may depend to a greater extent on other conceptual factors, such as knowledge of the base 10 ( Soto-Calvo et al., 2015 ). For example, it is likely that conceptual number knowledge, rather than phonological awareness is important for understanding that 20 follows 19 and comes before 21.

In addition to the quality of language representations and their role in early counting, there has been a more recent focus on the language of mathematics itself and how mathematical language is related to math achievement in both very young and older children. There is recent evidence to suggest that mathematical language is an important predictor of early mathematical performance among preschool-aged children even when accounting for other cognitive processes, such as executive functioning, general vocabulary knowledge, and rapid automatized naming ( Purpura & Logan, 2015 ; Purpura & Reid, 2016 ). Examples of math-specific language for preschoolers include comparative (e.g., “take away”, “more”, “less”) and spatial (e.g., “nearest”, “far away”, “under”) terms ( Purpura & Logan, 2015 ). The importance of general language and mathematical language to early math development brings up strong consideration of children who are English language learners and/or from families of lower socioeconomic status (SES), as these children have both less word knowledge including mathematically-relevant vocabulary and are at greater risk for early math difficulties. For example, among 6–9 year-old native English-speaking and English language learners from low SES families, language ability was found to predict gains in some areas of math such as data analysis/probability and geometry but not in other areas such as algebra or arithmetic, suggesting that language ability is not directly involved in learning how to manipulate quantities and execute algorithms but is involved in how children learn to make meaning of mathematical content ( Vukovic & Lesaux, 2013 ). Moreover, the relation between language ability and math concepts was similar for both English language learners and their native English speaking peers, though this relation was somewhat more pronounced for English language learners.

In sum, recent evidence suggests that mathematical language, may be particularly important even for preschool children’s mathematical competence. The link between math language and early numeracy skills is likely due to the fact that understanding math-specific terms (e.g., more, fewer) is inherently necessary for the completion of basic mathematical tasks ( Purpura, Hume, Sims, & Lonigan 2011 ). Despite its importance for early numeracy skill development, math specific vocabulary assessments are currently not available, though they are being developed.

Finger Skills

A variety of finger skills have been linked to early numeracy and math achievement in the preschool and early school grades. Fingers act as a support to a number of early numeracy skills such as reciting the numerical chain ( Sato & Lalain, 2008 ), understanding the cardinal meaning of number words ( Butterworth, 1999 ), establishing one-to-one correspondence by pointing to each object when counting ( Gallistel & Gelman, 1992 ), and keeping track of counted items in mental calculations ( Geary, 2005 ). Such findings have been used to argue for common neural representations of fingers and numbers because of their functional developmental connections through use of fingers to count and calculate (e.g., Butterworth, 1999 ). Fine motor skills involved in finger counting and pointing may also help children compensate for limited working memory capacity by avoiding having to internally store a mental representation of each counted object ( Alibali & DiRusso, 1999 ). Finger gnosis assessed at 5 years of age was found to be a powerful predictor of numerical ability up to three years later ( Fayol, Barrouillet, & Marinthe, 1998 ). Among school-aged children, finger localization/gnosis has been shown to predict performance on standard mathematical tests, including number system knowledge and calculation ( Noel, 2005 ; Penner-Wilger et al., 2007 ) as well as tasks tapping numerical representations, such as number-line estimation and magnitude comparison ( Penner-Wilger et al., 2008 ). Moreover, training in finger identification in first grade was associated with improvements in mathematical tasks including representation of numerosities with fingers, processing Arabic digits, and quantification ( Gracia-Bafalluy & Noel, 2008 ).

Although we have mainly considered the predictive value of early number-specific and general neurocognitive abilities for early numeracy and later math achievement, a few more recent studies have attempted to address which number-specific and general neurocognitive abilities uniquely predict which early and later math skills by including a range of potential predictors in their studies (e.g. Bailey, Watts, Littlefield, & Geary, 2014 ; Purpura & Logan, 2015 ). In our longitudinal studies of SBM ( Barnes et al., 2011 ), for example, the ability to detect counting errors, to count orally, and to perform nonverbal arithmetic problems at 60 months of age was best predicted by a combination of 36-month early number knowledge (as measured by the first few items on the TEMA) and neurocognitive factors (e.g., visual spatial, fine motor, and language abilities). Although the combination of predictors varied somewhat for counting and nonverbal arithmetic, the factors that predicted early numeracy in children at high risk for MLD (the children with SBM) were the same as those that predicted early numeracy in the typically developing control group. We also asked whether later math achievement at 9–10 years of age (multi-digit calculation, math fluency, math problem solving) is best predicted by early numeracy at 60 months (oral counting, detection of counting errors, nonverbal or nonsymbolic arithmetic), early neurocognitive abilities assessed at 36 and 60 months of age (visual working memory, phonological abilities, finger skills), or a combination of the two ( Barnes et al., 2014 ; Barnes & Raghubar, 2016 ; in press). For complex calculation at 9–10 years of age, phonological awareness and working memory were significant longitudinal predictors along with non-symbolic arithmetic. These findings suggest that preschool working memory and language abilities are important developmental precursors of later math calculation ability, but also that there is developmental continuity in the ability to manipulate number because the ability to add and subtract non-symbolic representations of number in the preschool years predicted the ability to add and subtract multi-digit numbers at school-age. For mixed math problem solving, working memory was the only significant longitudinal predictor, again underlining the importance of working memory abilities for solving novel quantitative problems ( Fuchs et al., 2016 ; Geary, 2013 ). For single-digit arithmetic fluency, only counting knowledge (detection of counting errors) was a significant predictor suggesting that a number-specific developmental precursor - counting knowledge - is an important early determinant of later math fact acquisition.

Taken together, both early general neurocognitive skills and number-specific skills appear to be important for the acquisition of preschool numeracy skills as well as for later school-age math achievement. Given the importance of early numeracy skills and early neurocognitive abilities to later math learning and performance, these findings have implications for assessment and intervention during the preschool years, which we discuss following a brief description of early math skills in neurodevelopmental disorders.

Mathematical Development in Children with Neurodevelopmental Disorders.

Neurodevelopmental disorders stemming from congenital, genetic, traumatic, or acquired origins are often associated with difficulties in academic achievement. Although these disorders may be associated with difficulties in both math and reading, specific or more severe difficulties tend to be associated with math more so than reading (e.g., Fragile X, SBM, traumatic brain injury, very low birth weight). We have argued that this is the case because math is not a unitary skill and several neurocognitive systems are likely involved in math learning and performance ( Barnes & Raghubar, 2014 ). As such, neurological insults or disorders that affect brain development can be expected to affect one or more of the neurocognitive systems that has been implicated in one or more mathematical skills/domains. Although a review of mathematical learning and performance in children with neurodevelopmental disorders is beyond the scope of this review, in this section, we provide a brief description of early numeracy skills in two high incidence clinical populations associated with math difficulties; namely, children with low birth weight and Fragile X syndrome. Although there is increased incidence of math disability in these populations, mathematical development and its neurocognitive correlates are considerably less well studied than in children with SBM. For these populations, we limit our description to what is known about early numeracy and/or how their difficulties in early numeracy are related to later MLD.

Similar to SBM, children with very low birth weight or very preterm birth have higher rates of MLD than normal birth weight term-born children (reviewed in Taylor, Espy, & Anderson, 2009 ). As a group, preterm children demonstrate difficulties in formal mathematical skills at school-age, assessed using standardized tests or measures. However, there is limited information on early numeracy skill development and basic numerical processing in this population ( Simms, Cragg, Gilmore, Marlow, & Johnson, 2013 ). Available information indicates that young children with low birth weight performed less well than their peers or national normative standards on measures of early numerical skills (e.g., classifying, sorting, comparing, and counting of objects; Aarnoudse-Moens, Oosterlaan, Duivenvoorden, van Goudoever, & Weisglas-Kuperus 2011 ) and number identification and sequencing ( Pritchard et al., 2009 ). Similarly, very preterm 6-year-olds performed less well than their term-born peers on measures of symbolic magnitude comparison (i.e., determining which of two Arabic numerals is larger – 7 vs 9) and explicit number knowledge (i.e., counting, seriation, matching dots to numerals, Arabic numeral reading and writing), though 8-year-old very preterm children performed as well as their peers on these basic skills (Guarini et al., 2013). As such, findings suggest that very preterm children acquire these foundational skills but at a slower rate than their peers. However, this does not exclude that preterm children might remain behind their full-term peers at 8 years of age in more complex or formal mathematical skills.

Fragile X syndrome is a genetic condition that is associated with poor math performance and the prevalence of MLD among girls with Fragile X exceeds that in the general population during and beyond the primary school years ( Mazzocco, 1998 ; 2001 ). Given that Fragile X syndrome affects females less severely than males, with only approximately 50% of females having intellectual disability compared to nearly all males, research on MLD in this population has typically involved only females without MLD. Kindergarten-aged girls with Fragile X syndrome demonstrate age-appropriate mastery of rote counting skills (e.g. counting by ones) and number recognition, but have more difficulty than their same-age peers with aspects of applied counting such as using one-to-one correspondence when counting to identify the n th item in a set (e.g. identifying the 5 th item in a set) ( Murphy, Mazzocco, Gerner, & Henry, 2006 ). Difficulties with applied counting have been observed to extend into the third grade among girls with Fragile X syndrome ( Mazzocco, Bhatia, & Lesniak-Karpiak, 2006 ). Fifth grade girls with Fragile X syndrome continued to demonstrate difficulties with counting principles at a higher rate than their typically developing peers, and demonstrated decreased accuracy on a problem verification task assessing number facts (e.g., 4 + 5 = 9 or 2 × 2 = 10; Right or Wrong?) and number knowledge (e.g., 200 – 150 = 300?), and complex calculation ( Murphy & Mazzocco, 2008 ).

Children with neurodevelopmental disorders often demonstrate neurocognitive difficulties that are predictive of early numeracy acquisition and later math achievement. Children with very preterm birth (Litt et al., 2010; Taylor et al., 2000 ) and Fragile X syndrome ( Mazzocco, Quintero, Murphy, & McCloskey, 2016 ) demonstrate difficulties with visual-spatial skills and working memory for example which have been shown to be related to mathematical achievement in these populations. Of importance is the suggestion that there may be particularly strong relationships between math achievement and neurocognitive skills such as working memory ( Peng et al., 2015 ) in individuals with MLD and a co-occurring neurodevelopmental disorder.

Overall, relatively little research has been conducted specifically with young children with neurodevelopmental disorders to explore early numeracy development and relations to later math achievement. Similar to children at risk for math difficulties, children with neurodevelopmental disorders including SBM, Fragile X syndrome, and low birth weight exhibit difficulties with early numeracy skills, which are discernable at young age. These early difficulties have important implications for later math achievement. Given that children with neurodevelopmental disorders demonstrate deficits in core neurocognitive skills and early numeracy difficulties they are at particular risk for later math difficulties. Because math risk can be discerned at an early age in these neurodevelopmental disorders as well as in children without neurodevelopmental disorders, the implications for assessment and early intervention are clear; first, risk for later MLD can be assessed in the preschool years, and this also opens the possibility for early intervention. We turn now to a fuller discussion of assessment and intervention strategies to promote early numeracy development.

Implications for Early Assessment and Intervention

Early assessment.

Direct assessment of early numeracy skills is important in young children given their strong association with later mathematical achievement. Assessment of early math skills should include measures of numerical symbol knowledge such as number identification based on numerals and arrays, and counting emphasizing cardinality and ordinality ( Merkley & Ansari, 2016 ). Given findings of developmental continuity in preschool non-symbolic arithmetic and later symbolic arithmetic, manipulation of non-symbolic number should also be part of early math assessment. These types of early numeracy skills are assessed separately in research-based studies, but are not currently available as separate normative based tests. However, there are some standardized measures of math that are suitable for preschool children and which contain some of these types of items. One advantage of one of these, the Test of Early Mathematics Ability – Third Edition (TEMA-3; Ginsburg & Baroody, 2003 ), is that it was created based on theory and empirical studies of mathematical development so that it contains items that tap early numeracy skills that correspond to the sequence in which such skills are typically acquired; it is suitable for assessing numeracy very early - from 36 months of age and contains an adequate sampling of early numeracy skills at this early entry point; and it explicitly measures the early numeracy skills shown to be important for later mathematics, such as counting (ordinal and cardinal knowledge), number identification, non-symbolic arithmetic, and understanding of rudimentary mathematical words such as “more”.

Brief screening instruments for assessing risk for reading difficulties in the preschool years are now available (e.g., Test of Preschool Early Literacy or TOPEL; Lonigan, Wagner, Torgesen, & Rashotte, 2007 ), and similar brief, reliable and easy to score screening instruments for preschool math are in the process of being standardized (e.g., Preschool Early Numeracy Skills Screener-Brief or PENS-B; Purpura, Reid, Eiland, & Baroody, 2015 ). Although mathematically relevant vocabulary (e.g., more, less, take away, nearest, under, equals) is important in the preschool and early school years such math-specific vocabulary measures do not currently exist, although they are being developed. Other aspects of mathematics such as early spatial sense and geometry, early informal measurement skills, and so forth, are not yet well-represented in standardized math assessments suitable for preschool children.

Including an assessment of neurocognitive processes has not been shown to improve the validity or reliability of diagnosis of learning disabilities ( Miciak, Fletcher, Stuebing, Vaughn, & Tolar, 2014 ). In recent diagnostic schemes such as DSM-5, testing of neurocognitive processes is considered to be largely unnecessary for diagnosing a learning disability and to increase assessment burden without improving diagnosis (reviewed in Barnes, Raghubar, & Martinez-Lincoln, in press; Tannock, 2013 ). However, we suggest that there is a role for the assessment of a small set of neurocognitive abilities among preschool-aged children at risk for MLD. Assessment of neurocognitive difficulties in preschoolers may be of particular importance for assessing risk for MLD because: 1) As the review above shows, difficulties in working memory and language early in development are powerful risk factors for poor development of preschool numeracy skills, and, in longitudinal research, for later MLD; and 2) Assessment of early math skills alone may be misleading for some young children who have had limited exposure to early numeracy concepts in the home or their preschool care settings making assessment of early math for these children possibly less reliable and valid than it is for older children. Thus for preschool children, we recommend a dual approach where both early numeracy skills as well as key neurocognitive abilities are assessed to determine risk for MLD.

Early Intervention

Mathematical concepts and procedures can be instructed in young children, and this is true for both typically developing preschool and kindergarten children and preschoolers and kindergarteners identified as at risk for mathematical difficulties ( Barnes et al., 2016 ; Toll & Van Luit, 2012 ; van de Rijt, & van Luit, 1998 ), though the extent of transfer to later, formal math skills remains a matter for debate, particularly for children at significant risk for MLD. It is beyond the scope of this paper to review early numeracy interventions. Therefore, we present a sampling of various early interventions that may hold promise for very young children who are at risk for later math difficulties.

The Building Blocks Intervention ( Clements & Samara, 2007 ) has been shown to increase the mathematics knowledge of preschoolers from low-income communities more than “business as usual curricula”. This intervention stems from developing math around children’s everyday activities. Educational goals included developing competence in two foundational domains: 1) number concepts and arithmetic operations; and 2) spatial and geometric concepts and processes. Additionally, some studies report promising results using numerical games to improve early numeracy skills. For example, Siegler and Ramani (2008) conducted a series of studies examining the utility of an intervention based on a linear board game with squares labeled 1 through 10 for four, 15-minute sessions across a two-week time period among 4-year-olds from low-income families. Large effects on children’s early numeracy skills were observed, including improvements in number line estimation, counting, numerical magnitude comparison, and numeral identification ( Ramani & Siegler, 2008 ; Siegler & Ramani, 2008 ), which are aspects of early numeracy that are related to later mathematical achievement. One thing to keep in mind is that these early interventions have been conducted with children who may be considered to be at heightened risk due to low SES and who enter preschool with low math knowledge. However, they are not specifically designed to address the needs of children who continue to have difficulties with math learning once a research-based curriculum is in place.

More intensive interventions for preschool children at high risk for later learning disabilities are just beginning to be tested for effectiveness (for reading see Lonigan & Phillips, 2016 ). In a randomized controlled trial, we have tested the effectiveness of a tutorial-based mathematics intervention for preschool children who scored at a very low level on a preschool math screening measure compared to their other low-income peers ( Barnes et al., 2016 ). This intensive tutorial-based mathematics intervention focused largely on numbers and operations and early spatial sense and geometry. The intervention began with direct instruction in very rudimentary symbolic (e.g., counting to 3) and non-symbolic (e.g., non-symbolic arithmetic with small set sizes such as 2 + 1) number skills and proceeded in a sequential fashion from these foundational mathematics activities to more complex preschool math activities. There were significant medium sized effects on a broad assessment of early mathematics ability and small, but significant effects on a standardized measure of early math ability (TEMA-3). Because one of the sites had a more supportive Tier 1 early numeracy program than the other, we could also see the effect of combining a highly supportive mathematics instruction at the classroom level with a more intensive higher tier of early math instruction. The children who received intensive math instruction through their Tier 1 classroom curriculum and through the tutorial program had the best math outcomes ( Barnes et al., 2016 ). These results suggest that early interventions for children at highest risk for later MLD are most effective when they include a strong classroom-based program in combination with additional higher tiers of math instruction. We also suggest that in the absence of evidence to the contrary, early math interventions with evidence of efficacy for young children without neurodevelopmental disorders may also be useful for young children at risk for MLD with neurodevelopmental disorders. This has been shown to be the case for adolescents with SBM who demonstrated gains in mathematical problem solving after instruction with a math word problem solving intervention originally designed for children with MLD and no neurodevelopmental disorder ( Coughlin & Montague, 2011 ).

Given the relationship between neurocognitive skills and mathematical learning and achievement, it begs the question of whether children with strengths versus weaknesses in specific neurocognitive domains may derive varying levels of benefit from different types of math intervention. One such study ( Toll & van Luit, 2013 ) examined the effects of an early math intervention in children with poor early numeracy skills accompanied by either very low or adequately developed working memory. Both groups were found to benefit from the math intervention, however, children with higher verbal working memory at the beginning of the intervention benefitted more from the intervention, suggesting that underlying neurocognitive factors may moderate training or intervention effectiveness. Similarly, recent intervention studies with school-aged children have found that working memory level moderates intervention effects ( Fuchs et al., 2014 ; Swanson, 2014 ). Swanson (2014) found that third grade children at risk for math difficulties with relatively higher working memory were more likely to benefit from strategy training in a word problem solving intervention whereas children with lower working memory may have had their already low cognitive resources overtaxed by strategy training. Fuchs and colleagues (2014) found that fourth graders with very weak working memory learned fractions better with conceptual activities whereas students with more adequate, though still low, working memory learned fractions with fluency activities, meant to build automaticity.

These studies have focused on math-specific treatments and measured whether general neurocognitive processes (i.e., working memory) moderate treatment effects. Other studies have examined the effect of training math-related neurocognitive processes such as working memory and examining transfer of cognitive training to math performance. The premise for such interventions is that given the prominence of general neurocognitive skills such as working memory in predicting math performance, improvements in neurocognitive functioning (i.e. training working memory) may also lead to improved math performance. For the most part, interventions targeting working memory demonstrate improvements in working memory (though maintenance is an issue as is transfer to other related cognitive domains), with limited transfer to math performance (reviewed in Melby-Lervag & Hulme, 2012; Melby-Lervag, Redick, & Hulme, 2016 ).

Another recent class of studies have examined the effect of interventions that combine training of math-specific and general neurocognitive processes. Studies that compare the effectiveness of number-specific training alone (e.g., counting training) or a combination of the two (e.g., with simultaneous working memory training) on early numeracy skills in typically developing children, show that number-specific training appears to produce the greatest improvements ( Kroesbergen et al., 2012 ; Kyttala et al., 2015 ). This was also found to be the case in the tutorial-based math intervention study discussed above for very low performing preschool children ( Barnes et al., 2016 ). In this study, attention training combined with the math intervention did not produce any added benefit for mathematics learning than the same math intervention alone; however, similar to many other studies that just employ cognitive training with preschool children, the attention intervention was remarkably non-intensive in comparison to the typical number-specific math interventions that have been designed and tested for children with or at risk for MLD.

At the present time, findings are consistent with the idea that specific training of early numeracy skills is not only effective, but also more effective in improving early numerical performance than training of general neurocognitive processes in typically developing preschool children and in preschool children at high risk for MLD. Nor is there convincing evidence thus far that combined math-specific and neurocognitive intervention are more beneficial than math-specific interventions alone. It is interesting to note that we know much more about issues around scope and sequence of content and dosage (how much treatment is needed at what level of intensity and for how long) for specific math interventions than we do for training of neurocognitive processes such as attention and working memory. We suggest that further basic research is needed to test whether there are any benefits of combining neurocognitive training with math interventions. For example, such studies would ideally test whether effects of such combined interventions vary as a function of variables such as: 1) when the cognitive training is provided, for example, prior to math instruction or concurrent with it, at very young versus older ages etc; 2) the intensity and scheduling of training, such as the dosage of cognitive training and whether that training involves massed versus distributed practice; and 3) the nature of neurocognitive training, for example, whether cognitive training uses materials that are devoid of mathematical content or whether cognitive training is more strongly integrated with the mathematical intervention.

In sum, given the current state of evidence, early number-specific interventions are recommended for young children at risk for MLD. There is not enough evidence to recommend either general neurocognitive interventions or combined number-specific and general neurocognitive interventions for preschool children at risk for MLD.

Conclusions

In this review we have defined early numeracy skills, emphasized their association with later math achievement, and identified neurocognitive correlates of early numeracy and math achievement. We have done this, highlighting our longitudinal work with children with SBM, a congenital disorder diagnosed in utero and associated with increased risk of MLD at school-age. Our work along with other longitudinal studies indicates that children who are at risk for MLD demonstrate difficulties with numeracy very early in development, as evidenced by early difficulties in oral counting and counting knowledge, and imprecise representations and transformations on quantity. We argued that the association between early numeracy skills and formal mathematics learning warrants assessment of early numeracy skills, particularly those assessing symbolic number knowledge. A number of neurocognitive processes are related to early numeracy development, namely working memory and language. In school-aged children, assessment of neurocognitive processes offers limited utility for diagnosing MLD. However, among preschool-aged children poor working memory and/or low language abilities place children at increased risk for difficulties in acquiring early numeracy skills, with significant negative implications for later formal math learning and achievement. Knowledge of weaknesses in specific neurocognitive processes may lead to instructional modifications of research-based math interventions that reduce working memory burden or provide direct instruction in basic mathematical vocabulary. However, given the lack of evidence for the efficacy of training of specific neurocognitive processes for early math learning, such neurocognitive training is not recommended for improving early numeracy in young children at risk for math learning difficulties.

Acknowledgments

This research was supported by grants from the National Institute of Child Health and Development, P01 HD35946, Spina Bifida: Cognitive and Neurobiological Variability and R01HD046609, Longitudinal Effects of Spina Bifida on Learning, and the Canadian Institutes of Health Research, and by grant R342A110270 from the Institute of Education Sciences (NCSER), U.S. Department of Education. The opinions expressed are those of the authors and do not necessarily represent views of the Institute or the U.S. Department of Education.

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An IERI – International Educational Research Institute Journal

• Open access
• Published: 24 May 2023

Early numeracy and literacy skills and their influences on fourth-grade mathematics achievement: a moderated mediation model

• Isabelle Chang   ORCID: orcid.org/0000-0002-8061-121X 1  

Large-scale Assessments in Education volume  11 , Article number:  18 ( 2023 ) Cite this article

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This study explored the influence of early literacy and numeracy skills on fourth-grade math achievement using the Trends in International Mathematics and Science Study (TIMSS). The study utilized valuable information collected by TIMSS about context related questionnaires such as home resources for learning, early literacy and numeracy development, readiness for school, and students’ home and school lives in a cross cultural and linguistic framework. The main purpose of this study was aligned with those of TIMSS to improve math learning and performance and strengthen future employees’ skills in the global workplace. Participants were comprised of mostly Asian and European students. Results show that (1) early literacy skills have a stronger effect on G4 math scores than early numeracy skills; (2) Home resources for learning impact more on children’s early literacy skills than early numeracy skills; (3) both early literacy and numeracy activities have progressed to early literacy skills but demonstrated limited advancement to early numeracy skills, a missing link; (4) students’ confidence in math emerged as the strongest predictor of G4 math scores; (5) students with stronger early literacy skills and early numeracy skills are more confident in math; and (6) The moderated mediation analysis revealed that (a) early literacy skills have stronger direct effects on G4 math achievement than early numeracy skills; (b) the effects of early numeracy skills on G4 math scores become more pronounced for children with weaker early literacy skills (i.e., conditional effects); and (c) the effect of early numeracy skills on G4 math achievement is transmitted through students’ confidence (i.e., mediator) and the effect is more prominent for those who had more proficient early literacy skills (i.e., conditional indirect effects). Findings from the conditional direct and indirect effects of early numeracy skills on G4 math achievement suggest that children who had more proficient early literacy skills utilize strategies beyond just early numeracy skills to solve G4 math problems and that children’s strategies to solve math problems may be enhanced by the proficiency of their literacy skills.

Introduction

A prolific body of literature has identified the family as the key dynamic influencing the early development of children’s academic skills including literacy and numeracy (e.g., Liu et al., 2019 ; Manolitsis et al., 2013 ; McCormick et al., 2020 ; Munro et al., 2021 ; Van Voorhis et al., 2013 ). To further understanding of the influence of early literacy skills (ELS) and early numeracy skills (ENS) stemming from the home learning environment on children’s academic achievement, many researchers have focused on identifying the specific skills which may predominantly impact later math performance. Considerable research has confirmed ENS are the strongest predictor for later math achievement (e.g.,Duncan et al., 2007 ; Kiss et al., 2019 ; Nelson & McMaster, 2019 ; Nguyen et al., 2016 ; Scalise & Ramani, 2021 ). In contrast, other studies have found that ELS were the foremost predictor in later math achievement (e.g., Aragón et al., 2016 ; Birgisdottir et al., 2020 ; Purpura et al., 2011 ; Yang et al., 2021 ). In addition, topics related to children’s family resources and confidence in math have also received research attention seeking to identify factors that might affect the gaps in math achievement observed among various ethnic groups in the United States and among countries in the international comparisons.

Additionally, multiple studies have utilized mediation models to illustrate the indirect effects of ENS on later math achievement. For example, the effect of ENS on later math achievement has been found to transmit through a third variable which is math language skills (King & Purpura, 2021 ). However, the extant studies have rarely, if ever, examined whether the effect of ENS on later math achievement becomes more (or less) pronounced for children with different levels of ELS (i.e., moderation effect). Likewise, few if any studies have been initiated utilizing a moderated mediation model to explore whether the effects of ENS on later math achievement transmitted through a mediator differs depending upon the levels of a moderator (e.g., ELS). The present study will address these gaps by constructing a moderated mediation model utilizing the Trends in International Mathematics and Science Study (TIMSS, 2019 ) data to explore the conditional direct and indirect effects of ENS on G4 math achievement.

Early literacy and numeracy activities

There is substantial literature demonstrating that home learning environments with parent–child early literacy activities (ELA) and early numeracy activities (ENA) encouraged children’s learning and enhanced their school achievement. Respective studies found differences in children’s outcomes resulted from distinct approaches in which parents interact with their children and have consistently confirmed home numeracy environment predicted ENS (e.g., Clerkin & Gilligan, 2018 ; Fazio et al., 2014 ; Segers et al., 2015 ; Siegler, 2016 ; Susperreguy et al., 2020 ). Parent–child activities in basic addition and subtraction lead to improved academic ability (Munro et al., 2021 ) and the home numeracy environment was predictive of numeracy (Napoli & Purpura, 2018 ). Children engaged in ELA and ENA became more confident and curious about math at age 10 (Clerkin & Gilligan, 2018 ) and later school academic achievement benefitted (Lehrl et al., 2020 ; Niklas & Schneider, 2017 ). Mothers who shared picture books with their children enhanced their math and literacy skills (Ribner et al., 2020 ) whereas fathers in Hong Kong reported higher frequencies of engagement in more real-life number game activities than mothers (Liu et al., 2019 ). Parents posed more complex questions about numbers to their sons than daughters (Uscianowski et al., 2020 ) while maternal use of relevant math facts during games were associated with their daughters’ later addition accuracy (Casey et al., 2020 ). Parent–child letter–sound interactions predicted growth in counting skills (Soto-Calvo et al., 2020 ) and formal home ENA (e.g., comparing numerals) predicted children’s symbolic number system knowledge (Skwarchuk et al., 2014 ). Moreover, symbolic knowledge was found to relate to broader mathematical competence (Chen & Li, 2014 ; Scalise & Ramani, 2021 ; Schneider et al., 2017 , 2018 ). Parents reported engaging in more literacy than numeracy practices as they considered literacy development more important than numeracy development (Napoli et al., 2021 ). Other parents reported prioritizing early numeracy and providing more math activities at home (Zippert & Rittle-Johnson, 2020 ).

McCormick et al. ( 2020 ) affirmed that unconstrained language and math activities predicted gains in children’s language and math skills. Examples of unconstrained skills are vocabulary and problem solving, while constrained skills would include competencies like letter knowledge and counting. Constrained skills are directly teachable and have a ceiling, while unconstrained skills are limitless and are acquired gradually through experience (Snow & Matthews, 2016 ). Everyday contexts such as shopping in grocery stores can promote math related conversations and potentially provide learning opportunities for children (Hanner et al., 2019 ). It appears that negligible research has been conducted exploring attainable ways for ELA to evolve to ELS or for ENA to advance to ENS much less how to develop curricula and remedial programs that help parents and teachers facilitate ELA and ENA to extend to ELS and ENS.

Early literacy and later math achievement

Studies on the influence of early literacy to later math achievement have largely focused on phonological awareness. Researchers have demonstrated the associations between phonological awareness and math achievement (De Smedt et al., 2010 ; Fuchs et al., 2006 ; Krajewski & Schneider, 2009 ) and fifth-grade computation skills (Hecht et al., 2001 ). Studies show that ENS and phonological processing influenced growth in math from kindergarten through third grade (Vukovic, 2012 ) and elementary school children with mild intellectual disabilities were found to profoundly rely on phonological awareness when solving math problems (Foster et al., 2015 ). However, other studies found early written language skills, but not phonological awareness, are unique predictors of Icelandic fourth graders’ (Birgisdottir et al., 2020 ) and Finnish third and fourth graders’ math achievement (). Verbal skills and visuomotor skills uniquely predicted fourth-grade math achievement (Kurdek & Sinclair, 2001 ) and reading skills significantly predicted Canadian children’s third-grade reading and math performance (Romano et al., 2010 ). Manfra et al. ( 2017 ) reported that writing and counting skills were consistently strong predictors of third-grade reading and math achievement among low-income, ethnically diverse children while Fuchs et al. ( 2016 ) found that second-grade language comprehension is more critical for fourth-grade word-problem solving than pre-algebraic knowledge. Furthermore, Kiss et al. ( 2019 ) affirmed that reading skills explained significant variations in third graders’ performance on number operations, algebra, data analysis, and geometry measurement. Third grade reading comprehension was related to a conceptual understanding and the application of math knowledge at eighth grade (Grimm, 2008 ). In addition, specific early literacy skills (e.g., vocabulary and print knowledge) were found to be influential in math development (e.g., Purpura & Napoli, 2015 ; Purpura et al., 2011 ). Purpura and Reid ( 2016 ) observed that math language (e.g., “more,” “many,” and “fewer”) is more proximal to numeracy skills than general language. Vanbinst et al. ( 2020 ) suggest that early reading and early arithmetic have a shared underlying cognitive basis for Flemish children. Furthermore, English literacy (second strongest to working memory) predicted Singaporean 11-year-olds’ algebraic word problem solving skills (Lee et al., 2009 ) and bilingualism is beneficial for children's math reasoning and problem-solving skills (Hartanto et al., 2018 ). Peng and colleagues’ (2020) meta-analysis results affirmed that more complicated language and math skills were associated with stronger relations between language and math.

Early numeracy and later math achievement

Early numeracy is a common term that is comprised of skills such as verbal counting, recognizing number patterns, comparing numerical magnitudes, manipulating quantities, and adding and subtracting objects. The associations between children’s ENS and their later math achievement have been well documented (e.g., Clerkin & Gilligan, 2018 ; Duncan et al., 2007 ; Raghubar & Barnes, 2017 ). While preliteracy skills were more strongly related to word reading, sensitivity to relative quantity was more strongly related to first-grade math (Chu et al., 2016 ). Basic numerical cognition was predictive of later procedural calculation skills and word problem development (Fuchs et al., 2010 ) and most strongly related to the ability to solve applied math problems (Jordan et al., 2010 ). Students who received early numeracy intervention improved their second-grade math performance (Bryant et al., 2021 ). Similarly, advanced counting competencies (Nguyen et al., 2016 ) and fluency in solving addition problems (Geary, 2010 ) were found to be strong predictors of fifth-grade math achievement. A meta-analysis of longitudinal studies revealed that number acuity may prospectively predict later math performance and is also retrospectively associated with early math performance (Chen & Li, 2014 ). Nevertheless, studies demonstrated that early math and reading skills positively influence each other in later school years (e.g., Cirino et al., 2018 ; Hooper et al., 2010 ) but when deficient also displayed comorbidities with each other.

Mediation effects

Some studies have employed more complex statistical techniques to identify and explain the mechanism or procedure that underlies an observed relation between ENS and math achievement via a third hypothetical variable (i.e., a mediator). For example, Purpura et al. ( 2017 ) detected that the relation between early math and literacy skills is mediated by children’s mathematical language skills and partially mediated by print knowledge (Purpura & Napoli, 2015 ). Math language mediates the relation between the direct home numeracy environment and numeracy skills (King & Purpura, 2021 ) and language abilities impact formal math skills partially through informal math skills (). In addition, verbal analogies were indirectly related to arithmetic knowledge through symbolic number skill (Vukovic & Lesaux, 2013 ) and reading comprehension skills may be partially mediating the relation between problem solving ability and growth in math (Vista, 2013 ).

It appears that the extant studies barely, if ever, explored how ELA and ENA directly or indirectly enhanced later math performance nor inquired into how ELA and ENA progress into ELS and ENS which in turn advance later math achievement. Furthermore, insufficient research has been undertaken examining whether a third variable influences the strength or direction of the relations between ENS and math achievement. For instance, if found to be significant, ELS can cause an amplifying or weakening effect between ENS and math achievement as the child’s ELS level increases (i.e., moderation). Hence, the question as to whether children with weaker ELS might rely more on ENS to solve math problems remains unexplored. This study will explore the moderation effect of ELS for ENS on G4M. For instance, whether the effect of ENS on G4M becomes less pronounced for children possessing stronger ELS.

HRL and math achievement

Many studies have confirmed the relation between home resources and children’s school achievement. Increases in school resources are consistently associated with improvements in math achievement for all groups, regardless of their individual resources (McConney & Perry, 2010 ). In China, increased levels of family and school resources were favorably related to student math achievement, and the association became more pronounced for levels of school resources in rural areas (Xue et al., 2020 ). In Japan, the student level of resources (i.e., number of books, computer access, paternal, and maternal educational levels) were positively related to student math achievement, whereas the school level of resources (i.e., rural and economically disadvantaged) were inversely related to student math achievement (Takashiro, 2017 ). In contrast, in Australia correlations between math performance and resources are far weaker in the nonmetropolitan schools than for those in the metropolitan areas (Murphy, 2019 ). In Tennessee, the higher the percentage of disadvantaged (determined by the percentage of students receiving federally subsidized free and reduced-priced lunches) the lower the achievement was. Hence, it appears there is a locale difference. Students with few resources in rural areas outperformed their economically disadvantaged nonrural peers. The author surmised that it is possible that support in the economically disadvantaged rural locales provides a sense of community not found in other economically disadvantaged areas which enables rural students to achieve higher in math than their nonrural peers (Hopkins, 2005 ). Furthermore, the effect of cognitive activation on math achievement was transmitted through self-efficacy which was moderated by resources at both the student level and the teacher level (Li et al., 2020 , 2021 ).

Researchers tested the Information Distortion Model (IDM) and hypothesized that in Australia children with a few resources would have higher academic interest compared to those with many resources and who were academically equally prepared. Results support the model and indicate that children with a few resources had higher numeracy interest than those who were equally prepared with many resources (Parker et al., 2021 ). In contrast, in China few family resources were shown to hamper middle school student math achievement, however high levels of subjective social mobility can buffer the adverse effects of low family resources on math achievement. Subjective social mobility reflects students’ personal beliefs in their ability to attain a higher social and economic status in the future and is regarded as a motivational resource (Zhang et al., 2020 ). Math and reading ability heightened future adult SES attainment (Ritchie & Bates, 2013 ). Further studies examining the underlying mechanism of family resources and its impact on academic achievement, as well as the varying influences on student achievement in different locales (e.g., rural vs. urban) in a cross-culture paradigm (e.g., European descents vs. Asian countries) are necessary. It appears that children with few resources in rural areas of mostly European descent countries (e.g., Australia and U.S.) outperformed their urban counterparts, whereas the reverse association was observed in Asian countries (e.g., China and Japan).

Confidence in math

Studies examining confidence and math achievement often focused on the reciprocal relations between the two. Typically, math self-efficacy improved math achievement and earlier math achievement was a consistent predictor of later confidence and interest, suggesting a reciprocal relation between confidence and math performance (e.g., Arens et al., 2017 ; Ganley & Lubienski, 2016 ; Grigg et al., 2018 ; Schöber et al., 2018 ; Sewasew et al., 2018 ). Nonetheless, Vogt et al. ( 2020 ) analyzed the NICHD SECCYD data and reported that children who scored higher on applied problems at 54 months had lower non-ability-based confidence (or overconfidence) at age 15 than those with a lower achievement. The associations between non-ability-based confidence and earlier development were similar for boys and girls. NICHD ( 2006 ) reported that the participants in SECCYD were comprised of “76.4% white, 12.7% African-American, 6.1% Hispanic…” (p. 30). Hence, caution is needed in generalization of the findings as the Hispanic group was underrepresented with whites overrepresented. In a cross-cultural study, self-rating in math and absence of disappointment with poor performance were associated with better performance in the English group, whereas no significant relationships between attitudes and performance were detected among Chinese first graders (Dowker et al., 2019 ). Fourth through sixth graders who perceived their classroom environments as more caring, challenging, and mastery oriented had significantly higher levels of math self-efficacy, which favorably impacted math performance. Note that the participants consisted of “Latino/a (62%) and Caucasian (31%) students” (Fast et al., 2010 , p. 731). Students from a sample of “5th grade to early college students (41% female, 80% white)” (Rice et al., 2013 , p. 1028) with greater social support for math and science from parents, teachers, and friends had more constructive attitudes and a higher sense of their own competence. Marked overconfidence was observed within the world regions that had lower scores on measures of cognitive ability whereas less inflated levels of overconfidence were noted among the high-achieving world regions based on samples from 33 nations (Stankov & Lee, 2014 ). Erickson and Heit ( 2015 ) cautioned that both overconfidence and anxiety can adversely affect metacognitive ability and can lead to math avoidance. Another cross-cultural study reported that teachers with greater self-efficacy in teaching math had higher job satisfaction and class levels of math achievement and interaction quality. At the student level, a higher individual self-concept advanced math achievement, and individual perceptions of interaction quality enhanced self-concept (Perera & John, 2020 ). It appears that the extant studies have given minimal attention as to whether the effect of ENS on later math achievement is transmitted through confidence (i.e., mediation). Furthermore, whether the effect of ENS on later math achievement becomes more pronounced for children with stronger ELS (i.e., moderation effect) is a topic deserving more comprehensive research.

The present study

The review of the existing studies on related topics has shown that complex relations among multiple variables influence students’ math achievement. However, few studies have explored moderators of ENS on later math performance and whether the effects might be inordinately impacted by the levels of the moderator (e.g., mean and ± 1SD). For example, the effect of ENS on math might become less pronounced as children’s ELS increased. In addition, whether the effect of ENS is transmitted through other variables (e.g., confidence) to enhance later math performance and if this effect may be stronger for children with stronger ELS (i.e., moderated mediation) deserves comprehensive study. This study uses the TIMSS 2019 data in a cross-linguistic and cross-cultural framework to examine the variables comprising (1) home environment support and (2) student engagement and attitudes that lead to the differing outcomes in students’ fourth-grade math achievement. Specifically, this study targeted the top eight performing countries on variables in home environment support including ELA, ENA, ELS, ENS, and HRL, as well as student engagement and attitudes (e.g., SCM), and fourth-grade math plausible values as the outcome variable. The hypotheses explored in this study are as follows:

Home environment support, student engagement, and attitudes are predictors of G4M. In particular, ELS are more effective than ENS in predicting G4M.

Home environment support and student engagement and attitudes are intercorrelated.

There are conditional direct effects of ENS on G4M.

ELS have a direct effect on G4M.

ENS have a direct effect on G4M.

ELS moderate the effect of ENS on G4M, and the effect of ENS on G4M is more prominent for children with weaker ELS

HRL (a covariate) have a direct effect on G4M.

There are conditional indirect effects of ENS on G4M (i.e., ENS → SCM → G4M).

ENS have a direct effect on SCM.

ELS have a direct effect on SCM.

ELS moderate the effect of ENS on SCM, and the effect of ENS on SCM is more pronounced for children with stronger ELS.

SCM have a direct effect on G4M.

SCM mediate the relation between ENS and G4M, and the effect is more evident for children with stronger ELS.

HRL (a covariate) have a direct effect on SCM.

Note that HRL serve as a covariate to control for ENS differences on SCM and G4M. The conceptual and statistical moderated mediation model diagrams are exhibited in Fig.  1 .

Moderated Mediation Diagrams. Predictor: ENS Early numeracy tasks (skills) beginning school/SCL, Moderator: ELS Early literacy tasks (skills) beginning school/SCL, Mediator: SCM Students confident in mathematics/SCL, Covariate: HRL Home Resources for Learning/SCL, Outcome variable: G4M Fourth-grade mathematics

Participants

UNESCO’s International Standard Classification of Education (ISCED) 2011 (UNESCO, 2012 ) provides an internationally accepted classification scheme for describing levels of schooling across countries. All students enrolled in the grade that represents 4 years of schooling counting from the first year of ISCED Level 1 (providing the mean age at the time of testing is at least 9.5 years) are eligible to participate in TIMSS. Approximately 600,000 students from 64 countries and 8 benchmarking systems (e.g., Hong Kong Special Administrative Region, SAR) participated in TIMSS, 2019 . Approximately 32,000 participants from the top eight performing countries were included in the present study and their mean PVs are shown in Fig.  2 .

TIMSS 2019 Top Eight Fourth Grade Average Math Scale Scores. SGP Singapore, HKG Hong Kong, KOR Korea, TWN Taiwan, JPN Japan, RUS Russia, IRL Ireland, LAT Latvia

Assessments

The TIMSS, 2019 assessment migrated to eTIMSS, a digital version of the assessment administered to students on computers and tablets. eTIMSS also included a novel section consisting of problem solving and inquiry tasks (PSIs) designed to exploit the digital environment to its fullest. About half the participating countries chose to transition to eTIMSS. The eTIMSS participating countries also administered the paper version of their trend items to a sample of schools, providing a bridge that helped link the two test-taking modes. For TIMSS, 2019 , comparing the item statistics for eTIMSS and paperTIMSS was integral in identifying items that were psychometrically equivalent under the IRT scaling. This process enabled the eTIMSS and paperTIMSS achievement results to be reported on the same achievement scale (i.e., fourth-grade math). The TIMSS assessment was categorized into content and cognitive domains. The content domains were comprised of (1) numbers, (2) geometric shapes and measures, and (3) data display, while the cognitive domains consisted of (1) knowing, (2) applying, and (3) reasoning. Constructed response items were scored independently by two trained scorers and TIMSS, 2019 trend scoring reliability for human scored items was 97%. The agreement across items was 98% or higher across countries, whereas within-country scoring reliability was 98% (Fishbein et al., 2020 ).

TIMSS utilized Item Response Theory (IRT) models to compute proficiency scores to provide valid comparisons across student populations based on broad coverage of the achievement domain. These test items were arranged in blocks that were assembled into student booklets that contained different (but systematically overlapping) sets of item blocks. Because each student received only a fraction of the achievement items for 72 min test time, statistical and psychometric methods were required to link these different booklets together so that student proficiency could be reported on a comparable numerical scale even though no student answered all the tasks. IRT was well suited to handle such data collection designs in which not all students were tested with all items. In addition, TIMSS used the three-parameter logistic (3PL) for multiple-choice items, the 2PL IRT model for constructed response items worth 1 score point, and the generalized partial credit model for constructed response items worth up to 2 score points (von Davier, 2020 ).

Home Environment Support

Participating countries (except U.S. and England) administered an early learning survey to the parents/guardians and a questionnaire to students, both of which were linked to the achievement booklet. The scales were constructed using the partial credit model IRT scaling methods. Each context scale separated students into regions corresponding to high, middle, and low values on the construct using the combinations of responses. Partial credit IRT scaling was based on a statistical model that related the probability that a student would respond to an item to that student’s math ability on the underlying construct. The average Cronbach’s alpha reliability coefficients for countries with data is approximately 0.88 for the early literacy and numeracy activities and the early literacy and numeracy tasks scales, and 0.70 for the home resources for learning scale (Yin & Fishbein, 2020 ).

Early literacy activities scale (9 items).

This survey asked parents, “ Before your child began elementary school, how often did you or someone else in your home do the following activities with him/her? ” Such an item included in this scale is “ Play with alphabet toys (e.g., blocks with letters of the alphabet) .” Students whose parents “ often ” engaged in five activities with them while “ sometimes ” doing the other four activities scored 10.8 or higher. Those whose parents “ never or almost never ” did five activities with them while “ sometimes ” doing the other four activities scored 6.2 or lower. All others sometimes engaged them in early literacy activities.

Early numeracy activities scale (9 items)

This survey asked parents, “ Before your child began elementary school, how often did you or someone else in your home do the following activities with him/her? ” Such an item included in this scale is “ Play games involving shapes (e.g., shape sorting toys, puzzles) .” Students whose parents “ often ” engaged in five activities with them while “ sometimes ” doing the other four activities scored 11.0 or higher. Those whose parents “ never or almost never ” did five activities with them while “ sometimes ” doing the other four activities scored 6.6 or lower. All others sometimes engaged them in early numeracy activities.

Early Literacy tasks scale (7 items)

This survey asked parents, “ How well could your child do the following when s/he began the first grade of elementary school? ” Such an item included in this scale is “ Write words other than his/her name .” Students whose parents rated “ very well ” to four tasks and “ moderately well ” to the other three tasks scored 11.2 or higher. Those whose parents rated “ not very well ” to four tasks while “ moderately well ” to three tasks scored 8.8 or lower.

Early numeracy tasks scale (5 items)

This survey asked parents, “ Could your child do the following when s/he began the first grade of elementary school? ” Such an item included in this scale is “ Recognize written numbers .” Students whose parents’ responses were “ very well ” to two tasks and “ moderately well ” to one task and “ yes ” to two tasks scored 11.7 or higher. Those whose parents’ responses were “ not very well ” to two tasks and “ moderately well ” to one task and “ no ” to two tasks scored 8.1 or lower (P. Foy, Personal Communication, July 26, 2021).

Home resources for learning scale (5 resources, 22 items)

One of the five resource questions comprised of this scale asked parents, “ Highest level of occupation of either parent (parents) .” Such an item included in this resource subscale was “ Professional (corporate manager or senior official, professional, or technician or associate professional). ” Students with many resources scored 11.8 or higher, having more than 100 books at home and both home study supports, more than 25 children’s books at home, at least one parent having finished university, and at least one parent having a professional occupation. Students with few resources scored 7.4 or lower, having 25 or fewer books at home and neither of the home study supports, 10 or fewer children’s books at home, neither parent having achieved beyond upper-secondary education, and neither parent owning a small business or holding a clerical or professional occupation. All others had some resources.

Student engagement and attitudes

Confidence in mathematics scale (9 items).

This scale asked students, “ How much do you agree with these statements about mathematics? ” Such an item included in this scale is “ I am good at working out difficult mathematics problems .” Those very confident in mathematics scored 10.7 or higher reporting “ agree a lot ” with five statements and “ agree a little ” with the other four. Students not confident in mathematics scored 8.5 or lower reporting “ disagree a little ” with five statements and “ agree a little ” with the other four. All others were somewhat confident in mathematics (Yin & Fishbein, 2020 ). The average Cronbach’s alpha reliability coefficients for the students confident in math scale for these eight countries is 0.87.

Data analysis

The present study employed the IEA IDB Analyzer (Version 4.0.42, IEA, 2021 ) and PROCESS Macro (v4.0, 2022) (Hayes, 2022 ) in conjunction with IBM SPSS 28 ( 2022 ) to test on the public-use version of the TIMSS, 2019 data. The proportions of the sample are approximately 8% from Hong Kong, 18% from Singapore, and 12% from each of the remaining six countries with relatively similar contributions. IDB Analyzer was utilized to compute descriptive statistics and bivariate correlations. A moderated mediation model (Hayes, 2017 , model 8) was constructed to estimate the conditional effect of ENS on SCM (mediator) varied as a function of ELS (moderator). In addition, the conditional direct effect of ENS on G4M is also dependent upon the levels of ELS. An index of moderated mediation was used to test the significance of the moderated mediation. Confidence intervals did not contain zero between the lower and upper limits of the 95% confidence intervals indicating that the direct and indirect effects were conditional on the level of the moderator (ELS). The number of bootstrap samples for percentile bootstrap confidence intervals was 5,000.

Descriptive statistics and correlations

The average bivariate correlations comprised of mostly Asian and European students showed that ELS ( r  = 0.36) were effective than ENS ( r  = 0.21) in predicting G4M and ELA and ENA were inadequate indicators of G4M ( r s = 0.16 and 0.15). SCM dominated with the highest impact in G4M ( r  = 0.44) with HRL also prominent ( r  = 0.39). Hypothesis 1 has been supported.

Bivariate correlations between home learning environment and student attitudes showed that children who were proficient in ELS were also likely to be proficient in ENS ( r  = 0.30). ELS benefitted from both ELA ( r  = 0.34) and ENA ( r  = 0.32) while ENS were facilitated ineffectively from ELA and ENA ( r s = 0.09 and 0.11), and ENA evolved more to ELS ( r  = . 32) than ENS ( r  = 0.11). Parents with many HRL engaged in more early literacy ( r  = 0.30) and numeracy ( r  = 0.25) activities with their children than their counterparts who possessed fewer HRL, and parents engaged both activities with their children in proportional amounts ( r  = 0.77). HRL enriched children’s ELS ( r  = 0.34) more than ENS ( r  = 0.16), and inadequately boosted SCM ( r  = 0.19). Children who engaged in more ELA and ENA with their parents were not much more confident in math ( r s = 0.14 and 0.15). In a similar vein, students with more HRL, and who entered school with higher levels of ELS and ENS, were not more confident in math with r s = 0.17, 0.11, and 0.19 respectively. A correlation matrix with home environment support, student attitudes, and plausible values (PVs) is exhibited in Table 1 . Hypothesis 2 has been supported. Note that ELA and ENA were not included in further analysis mostly due to the weak correlation coefficients with G4M.

• Moderated mediation

A moderated mediation model was tested using the PROCESS macro (Hayes, 2022 ) in SPSS to estimate the conditional direct and indirect effects of ENS on G4M through SCM (i.e., mediator) as moderated by ELS. All variables were standardized prior to entering the analysis to facilitate interpretation of any effects resulting in the estimation of standardized coefficients (β). Betas are comparable in magnitude within a model as well as between studies with higher absolute β coefficients presenting stronger effects. Results from the present study exhibited that ENS and ELS both had direct effects on SCM (βs = 0.05 and 0.10 ps < 0.001, respectively) signifying that ELS had a stronger influence on SCM than ENS. Moreover, ELS moderated the relations between ENS and SCM (β = 0.02, p  < 0.001) suggesting that the effect of ENS on SCM became more pronounced for students with stronger ELS (β = 0.07, p  < 0.001) than those with lower ELS (i.e., mean and -1SD) (βs = 0.05 and 0.03, respectively, p s < 0.001). The results suggested that there were conditional effects of the focal predictor (ENS) at different levels of the moderator (i.e., ± 1SD and mean ELS) on SCM. Furthermore, HRL was significantly associated with SCM (β = 0.13, p  < 0.001).

ENS → SCM → G4M

ENS, SCM, and ELS had direct effects on G4M (βs = 0.09, 0.28, and 0.16, respectively, p s < 0.001) indicating that ELS was a much more prominent predictor of G4M than ENS despite SCM having the strongest effect on G4M. ELS moderated the relations between ENS and G4M (β = -0.04, p  < 0.001) suggesting that the effect of ENS on G4M was more evident for those with weaker ELS (βs = 0.13 and 0.09, respectively, p s < 0.001) than those with stronger ELS (β = 0.06, p  < 0.001). The results suggested that there were conditional direct effects of the focal predictor (ENS) varying with different levels of the moderator (ELS) on G4M implying that students who possessed stronger ELS utilized strategies more than ENS to solve fourth grade math problems. Further, HRL had a strong direct effect on G4M (β = 0.20, p  < 0.001). Sub-hypotheses ( a ) through ( d ) of H3 have been supported, suggesting that there are conditional direct effects of ENS at different levels of ELS on G4M.

The estimation of the moderated mediation models suggested that SCM was a mediator of the relation between ENS and G4M and had significant direct effects on G4M (ENS → SCM → G4M: Index of moderated mediation = 0.0056, BootSE  = 0.0013, BootLLCI  = 0.0030, BootULCI  = 0.0081). That is, the effect of ENS on G4M was transmitted through SCM. This denotes that (1) children who have stronger ENS had higher math achievement at fourth grade, (2) this relationship was explained by stronger SCM, and (3) this effect was stronger for children with stronger ELS. Sub-hypotheses ( a ) through ( f ) of H4 have been supported suggesting that there are conditional indirect effects of ENS on G4M transmitted through SCM at different levels of ELS.

In summary, the results indicate that (1) the mechanism linking ENS to G4M through a mediator SCM is related to ELS, and (2) the conditional direct effects of ENS on G4M vary at different levels of ELS suggesting that literacy skills might enhance students’ approaches in analyzing, interpreting, and solving math problems. The findings suggest that students with lower ELS benefitted more from ENS whereas those with higher ELS relied less on ENS to solve fourth-grade math problems. The full moderated mediation model with estimations of conditional direct and indirect effects are presented in Table 2 . The hypotheses that have been explored and supported are summarized in Table 3 .

IEA TIMSS has employed advanced techniques in methodology to derive the scale scores of home environment support and student engagement and attitudes and sub-categories in cognitive and content domains. This has provided data for researchers to explore multifaceted relations among variables using advanced statistical techniques to test models with moderated and mediated relations with the intent of improving math teaching and learning around the world. This study adds greater understanding of math learning and how early literacy and numeracy skills influence G4 math performance. The participants were comprised of mostly Asian and European students with results showing that ELS were stronger than ENS in predicting G4M and lend support to extant studies including Aragón et al. ( 2016 ), Birgisdottir et al. ( 2020 ), Purpura et al. ( 2011 ), and Yang et al. ( 2021 ). In contrast, it appears that the results from this study did not confirm children’s early numeracy skills as the strongest predictor of their later math achievement that have been reported by others (e.g., Bryant et al., 2021 ; Duncan et al., 2007 ; Raghubar & Barnes, 2017 ). Parents reported engaging in higher frequencies of ELA than ENA before their children started school, which is coherent with recent findings by Napoli et al. ( 2021 ). Both parent–child ELA and ENA developed into ELS, while bafflingly neither ELA nor ENA advanced into ENS which did not confirm that home numeracy environment predicted ENS (e.g., Clerkin & Gilligan, 2018 ; Fazio et al., 2014 ; Segers et al., 2015 ; Siegler, 2016 ; Susperreguy et al., 2020 ). Parents with many HRL engaged in moderately more ELA than ENA with their children than those with fewer HRL. In addition, HRL contributed more to ELS than ENS although children’s proficiencies in ELS and ENS were correlated. The findings that HRL fostered G4 math achievement are consistent with previous studies (e.g., McConney & Perry, 2010 ; Xue et al., 2020 ). Moreover, higher individual SCM advanced math achievement, in agreement with existent studies (e.g., Perera & John, 2020 ).

The positive moderation effect of the ELS on the relations between ENS and SCM indicate that the effect of ENS on SCM becomes more pronounced for children possessing stronger ELS. The negative moderation effect of the ELS on the relations between ENS and G4M appears to be in line with results from Fuchs et al. ( 2016 ) and Peng et al. ( 2020 ). The results suggest that children with weaker ELS utilized more ENS to assist them when approaching math problems whereas those with stronger ELS employed strategies more than ENS to answer math problems. Findings from this study suggest that children’s strategies to solve math problems might be enhanced by their literacy skills. Future studies that explore the dynamics between ELS and ENS and their combined influences and strengths on math achievement are urgently needed. In addition, studies to uncover how ELS help children answer math problems deserve comprehensive examination. It is necessary to separate ELS into different levels (e.g., ± 1SD and mean) when examining relations between ENS and math attainment. Moderated mediation analysis is an informative technique for assessing whether direct and indirect effects are conditional on values of a moderating variable such as ELS.

Limitations

The limitations of this study include characteristics of design or methodology that impacted or influenced the interpretation of the findings. For instance, parents reported low percentages of early literacy and numeracy activities with their children before school in the top five performing East Asian countries. Intriguingly, how Asian children have mastered their early literacy and numeracy skills and excelled to the top in international assessments remain undetermined. Likewise, low percentages of East Asian students reported that they were confident in math, nevertheless their performance was exceptional. In addition, participating countries included in this study were comprised of mostly Asian and European students and as such the findings should not be generalized to black and Hispanic children’s math learning. Future studies should explore whether the conditional direct and indirect effects are equal or more evident in black and/or Hispanic children’s math learning. If so, parents and educators should try to strengthen children’s literacy skills in tandem with numeracy skills to enhance their later math achievement. It would also be beneficial if the results of this moderated mediation model can be replicated using large-scale U.S. data which often include participants from more diverse ethnic backgrounds.

Implications

Both ELA and ENA advanced to children’s ELS and ELS are stronger than ENS in predicting children’s math achievement. Moreover, HRL have a stronger association with ELS than ENS. This study seeks to prompt awareness among parents and educators and should foster children’s literacy skills in concert with numeracy skills thanks to the critical role of ELS in their later math achievement. Most critically, parents are urged to engage in comparable levels of early literacy and numeracy practices utilizing instantaneous everyday situations with their children before entering school to enhance their children’s literacy and numeracy skills regardless of the availability of learning resources they can provide. However, whether strengthening literacy skills for students from disadvantaged groups with few resources can narrow the achievement gap deserves comprehensive studies.

Results from this study suggest that children’s early literacy skills have a critical role in their later math achievement in a cross-linguistic and cross-cultural framework. The effect of ENS on SCM becomes more pronounced when the level of ELS increases whereas the effect of ENS on SCM diminishes for those with weaker ELS. In addition, the effect of ENS is transmitted through SCM, and confidence has a strong effect on G4M. In contrast, the effect of ENS on G4M diminishes when the level of ELS increases. It appears that children with weaker ELS employ ENS whereas those with stronger ELS utilize strategies more than ENS to tackle math problems. This suggests that children’s math problem solving strategies might be advanced by their early literacy skills. Furthermore, ELS are more effective than ENS in predicting children’s G4 math achievement. The main takeaways from this study are: (1) ELS have a stronger effect on G4M than ENS; (2) HRL impact more on children’s ELS than ENS; (3) both early literacy and numeracy activities have progressed to ELS but demonstrated limited advancement to ENS, a missing link; (4) SCM emerged as the strongest predictor of G4M; (5) students with stronger ELS and ENS are more confident in math; and (6) The moderated mediation analysis revealed that (a) ELS have stronger direct effects on G4 math achievement than ENS; (b) the effects of ENS on G4M become more pronounced for children with weaker ELS (i.e., conditional effects); and (c) the effect of ENS on G4 math achievement is transmitted through SCM (ENS → SCM → G4M) and the effect is more prominent for those who had more proficient ELS (i.e., conditional indirect effects). Findings from the conditional direct and indirect effects of ENS on G4M suggest that children who had more proficient ELS utilize strategies beyond just ENS to solve G4 math problems and that children’s strategies to solve math problems may be enhanced by the proficiency of their literacy skills. Parents and educators should facilitate children’s literacy skills and numeracy skills to enhance their later math achievement.

Availability data and materials

The datasets generated and/or analyzed during the current study are available in the TIMSS 2019 International Database repository, https://timss2019.org/international-database/.

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Chang, I. Early numeracy and literacy skills and their influences on fourth-grade mathematics achievement: a moderated mediation model. Large-scale Assess Educ 11 , 18 (2023). https://doi.org/10.1186/s40536-023-00168-6

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Developing numeracy skills using interactive technology in a play-based learning environment

• Tess Miller   ORCID: orcid.org/0000-0001-6984-9326 1  

International Journal of STEM Education volume  5 , Article number:  39 ( 2018 ) Cite this article

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The purpose of this study was to measure the impact of interactive technology in the form of mathematical applications (apps) delivered using iPads on kindergarten children’s learning of number sense in a play-based learning environment. Secondly, factors influencing the use of interactive technology in a play-based environment were examined. This technology was introduced to a small ( n  = 13) rural kindergarten classroom using an experimental design embedded in a mixed methods approach.

The teacher was keen to introduce technology to her class but was self-described as a beginner in using iPads for personal or teaching tasks. Small gains were noted between the control and intervention groups but they were not significant. Further, children were observed collaborating which supported prior research. Another observation was related to attention span, when an app became too challenging children would abandon the app or use a trial and error method to move to the next level. Lastly, when given choice, children were drawn to creative and entertaining apps rather than apps that were more pedagogically accurate but less creative. Although there was not a large gain in achievement, using interactive technology promoted student collaboration and engagement in a play-based learning environment.

Conclusions

Small gains in mathematics achievement and high levels of engagement suggest that using interactive technology in the kindergarten classroom enhances learning of mathematics. Factors influencing the use of interactive technology included the quality of the app such that creative and fun apps promoted children’s engagement in learning mathematics. The level of difficulty of an app was a second factor influencing children’s use of interactive technology. If the difficulty level was too challenging, children became disengaged with the app.

In the next era of technological advancements, we can only envision futuristic developments such as no-touch interfaces and sensors that read and predict our movements (Loganathan 2013 ). However, for the current generation, we are enjoying the developments in technology that have focused on interactive tablets among a host of other interactive gadgets currently on the market. The intuitive design of iPads positions them for use in educational settings, including early years classrooms (Warmoth 2013 ). Their design was built on mental models of how we perceive an experience (Weinschenk 2011 ). For example, when reading a book, we turn pages by using the index finger to flip to the next page. This method of turning pages has been modeled on the iPad where the user also uses his/her index finger to flip to the next page. Interacting with an iPad in this manner facilitates data flow through the interface (i.e., what the user sees on the screen) linking the user and technology and is referred to as interactive technology (Large 2016 ). Page flipping and other interactive features (e.g., voice recognition) on an iPad make using this form of technology simple or intuitive even for the youngest of learners.

The introduction of iPads into early learning classrooms has revealed gains in student achievement (Bebell et al. 2012 ); however, measuring gains in achievement should not be the sole factor determining the merit of interactive technology. Leveraging student engagement in a discipline like mathematics also makes iPads a worthy investment for the classroom given that they can be used to gather information, read a book, take photos, record physical activity, make artistic drawings, and learn about literacy or numeracy through the use of stimulating and creative applications (apps). Given the abundance of apps currently available, examples of how iPads can be used to engage children and promote learning is much larger than what is posited here. In the next era of educational technology, we need to think of the iPad as a manipulative that children can choose from a host of other manipulatives to discover new concepts.

The purpose of this study is to measure the impact of interactive technology in the form of mathematical applications delivered using iPads on kindergarten children’s learning of numeracy in a play-based learning environment (Fesseha and Pyle 2016 ). In particular, numeracy concepts were focused on the number-sense strand (Department of Education, Early Learning and Culture of PE 2008 ). Secondly, factors influencing the use of interactive technology in a play-based environment were examined. The research questions posed in this study are as follows: To what extent does the use of mathematical apps using iPads enhance children’s learning of numeracy in kindergarten? What factors influence children’s use of interactive technology in a play-based learning environment?

Considering the relatively short evolution of iPads that were introduced in 2010 with Apple’s launch of iPad 1 (Ritchie 2014 ), a significant body of literature has been published on the use of iPads in education, building on literature reporting on the use of desktop computers in education. In a systematic review of what the authors described as mobile learning, Crompton et al. ( 2017 ) reviewed 113 studies of which four were at the pre- or kindergarten levels and these four studies were conducted in 2014 and 2015 suggesting interactive technology was slowly making its way into the early grades. Unfortunately, these authors did not cross list their review to identify the subject domains the four studies were conducted in.

Of the research focusing on technology in early learning, studies focused on children and teachers’ perceptions of technology (Knezek and Christensen 2002 ; Tsitouridou and Vryzas 2003 ) as well as the use of technology in literacy development (e.g., Chiong and Shuler 2010 ; Flewitt et al. 2015 ; Primavera et al. 2001 ). Fewer studies have been in the area of numeracy (Clements and Sarama 2007 ). This literature review begins with teachers’ perceptions towards using technology given that teachers’ comfort with technology, including the teacher in this study, has been shown to impact the use of technology in the classroom (Simon et al. 2013 ). The second part of this literature review focuses on literature exploring the effective use of iPads in the area of literacy and numeracy. This review concludes with a look at studies that advocated against the use of technology in early learning.

Teachers’ perceptions towards using technology

For many preschool teachers, their use of technology is less than teachers in higher grades and was limited to downloading images for instructional purposes and digital cameras (Public Broadcasting Service and Grunwald Associates 2009 , 2011 ). This absence of technological integration into early years classrooms is most likely due to limited opportunities for professional development on interactive technology as well as the lack of technology itself. Hence, to advance interactive technology in early years classrooms, it is important to recognize that teachers need professional development on the appropriate use of technology in the classroom (Simon et al. 2013 ), as well as opportunities to acquire technology. In a case study of four kindergarten teachers by Lu et al. ( 2017 ), teachers’ experiences using iPads in a literacy context was also found to be beneficial as it allowed teachers to meet the demands of creating individual assessments or work on lesson preparation as the iPads “functioned like an extra teaching assistant, providing feedback to students” (Lu et al. 2017 , p.19).

Studies employing interactive technology in the form of tablets reported improved motivation, supported learning in small groups, and independent work as well as gains in vocabulary and phonological awareness (Dobler 2012 ; Hutchison et al. 2012 ; Flewitt et al. 2015 ; Hutchison and Reinking 2011 ; Simon et al. 2013 ; Takac et al. 2015 ). For example, Simon et al. ( 2013 ) concluded that tablets in addition to desktop computers supported learning in small groups or individually. These researchers also noted a tendency for longer periods of use especially when children had choice in their activity. Similarly, Flewitt et al. ( 2015 ) introduced iPads in early learning for the purpose of literacy development, which included writing and video recording of stories for sharing with the class. One iPad was given to a classroom with 3- and 4-year-olds and another iPad to a class of 4- and 5-year-olds for 2 months. Each iPad contained a story-creation app as well as a number of other learning apps. Training was provided to both instructors and data collection consisted of surveys, observations, and conversations. These researchers reported increased motivation and use of iPads. This was especially noted among children who were not easily engaged in traditional writing tasks. Further, the touch screen interface was reported to be easier than a keyboard. Similar to findings reported by Simon et al. ( 2013 ), the iPads fostered small group and independent learning. At the same time, children were observed helping each other use the iPads which is aligned with the findings from Shifflet et al. ( 2012 ) who reported that pre-school children using tablets developed a collaborative approach to learning which enhanced their social skills.

In terms of academic gains, an experimental study revealed that kindergarten children in both the control and experimental groups (i.e., with iPads) showed gains as measured using the Rigby Benchmark Assessment and the Children’s Progress Academic Assessment but the differences between groups were not statistically significant (Bebell et al. 2012 ). However, on a third measure, the Observation Survey of Early Literacy Achievement assessment, a statistically significant difference in phonemic awareness was reported where children in the iPad group scored higher (Bebell et al. 2012 ). Based on this study, the introduction of iPads in early learning did not appear to hinder learning, and in some areas of literacy, they enhanced children’s understanding of the discipline.

Of the studies that examined early learning of mathematics using tablets, most studies reported gains in achievement or positive experiences (Alade et al. 2016 ; Dejonckheere et al. 2015 ; Hubber et al. 2016 ; Hung et al. 2015 ; Kosko and Ferdig 2016 ; Mattoon et al. 2015 ; Outhwaite et al. 2017 ; Presser et al. ( 2015 ); Reeves et al. 2017 ; Stubbe et al. 2016 ). However, Bebell and Pedulla’s ( 2015 ) longitudinal study examining the impact of mathematics apps on achievement in grades kindergarten (K) to 2 did not reveal any consistent gains. Despite different outcomes, each of these studies dispelled the argument that iPads and other digital manipulatives were not edutainment but rather effective learning aids (Baird and Henninger 2011 ).

A few studies narrowed their focus to a small number of apps or a specific mathematical skill, which provided insight on the connection between the app and student gains for young children. For example, Reeves et al. ( 2017 ) selected apps that focused on skills related to counting, sequencing, and early addition. In each area of skill development, gains were reported. Dejonckheere et al. ( 2015 ) also focused on a specific skill and reported a gain in achievement. These researchers used tablets to allow 4- to 6-year-olds to play on a digital number line exploring concepts related to estimation. This focus on one numeracy concept applying different strategies for estimation revealed a significant gain in accuracy of estimation for the two groups that were given strategies but not for the control group. Likewise, Presser et al. ( 2015 ) focused on skills related to subitizing (recognizing how many in a set without counting) and equi-partitioning (splitting an area or set into equal groups) using an app known as Next Generation Preschool . Gains in numeracy skills were also reported in their study.

Along the same focus, Kosko and Ferdig ( 2016 ) examined gains in achievement but approached their study from the perspective of selecting apps that were pedagogically accurate and aligned with curriculum. These researchers reported that well-designed mathematics apps improved achievement and concluded that well-designed mathematics apps can support student learning but more research was needed to explore the extent to which these apps improved learning. Given the number of mathematics apps available on the iPad, it is particularly important to consider characteristics of mathematics apps to better understand the impact, if any, an app has on student learning. Further, neither of these studies centered on a play-based learning environment that allowed for flexible and creative use of time and space or provided choice in selecting an app (Steglin 2005 ). Hence, the characteristics of apps that attracted children are largely unknown.

Advocates against interactive technology

Although Dinehart ( 2015 ) was an advocate against the use of technology in early learning, citing that it diminished children’s fine motor skills, she did not consider apps designed for early childhood education that fostered fine motor skills through writing letters and numbers. Other concerns against the use of technology in early learning focused on the amount of time spent viewing screens. Vanderloo ( 2014 ) reported that children between 4 and 7 years of age spent an average 1.5 to 7.0 h viewing screens each day. Unfortunately, there was no differentiation between viewing screens for different activities such as watching a movie, watching a lesson on the interactive whiteboard, or playing games on an iPad (Vanderloo 2014 ). The sedentary nature of viewing screens was the catalyst for Vanderloo’s work, which is undoubtedly a concern; however, moderation of screen viewing may better guide the use of technology in early learning rather than banning it all together. This position is more aligned with the National Association for Education of Young Children ( 2012 ) who differentiated between screen viewing for interactive activities (e.g., mathematical apps) and non-interactive activities (e.g., movies) to promote active learning using technology.

When drawing on the literature presented above, it is reasonable to conclude that young children experienced gains when using iPads to learn about literacy and numeracy, and in one study, gains were long term (Outhwaite et al. 2017 ). When these gains were compared to a control group, the gains were not always statistically significant. Concerns related to excessive screen viewing were tempered by differentiating between interactive learning versus non-interactive learning. Since most studies narrowed their focus to a few mathematical apps, there is much to learn about the characteristics of apps that children gravitate towards when left to their own accord in a play-based learning environment.

Theoretical framework

The Technological Pedagogical Content Knowledge (TPACK) framework was utilized to explore the implications of using mathematical apps installed on iPads in an early learning context. Integrating technology to enhance learning requires knowledge related to the subject content (i.e., numeracy), pedagogy, and technology (Mishra and Koehler 2006 ). Based on the TPACK framework, technology can be successfully integrated into learning when these three domains are successfully woven together. Hence, when apps were selected for this study on early numeracy concepts, the pedagogy had to be aligned with children’s level of cognitive development, and at the same time, the technology had to be simple and intuitive so that kindergarten children could be successful using it.

This research also builds on Naismith et al.’s ( 2004 ) theories in identifying six theory-based categories of activities for interactive technology: behaviorist, constructivist, situated, collaborative, internal and lifelong, and learning and teaching support. These researchers described behaviorist activities as those that primarily aim to change behavior through reinforcement of a stimulus such as feedback. For example, when a student responds correctly to a mathematics question, a pleasing sound or an animated character may appear in an app designed for kindergarten level students. In contrast, constructivist activities were described as activities that call on the student to apply what they know to new contexts and build new learning. The game, Environmental Detectives , was created for high school students who engage in the game as environmental engineers tasked with solving a problem is an example of a constructist activity/game. For a younger audience, Minecraft is similar in that students create and manipulate objects and engage in a task. When selecting apps for this study, there were no constructivist activities/games for the iPads that were aimed at the early learner.

In returning the focus to the theory-based categories, the remaining four theory-based categories of activities/games for interactive devices involved a higher cognitive engagement and advanced interactive software, which is appropriate for students in the senior grades. Subsequently, interactive activities designed for early learners appears to be founded on the behaviorist theory-based category which is similar to the finding in Bray and Tangney’s ( 2017 ) systematic review of literature focused on using technology in mathematics education. These researchers identified a wide range of technologies being used in different contexts within the middle and senior mathematics classrooms with a predominance of constructivist and social constructivist tasks. Such tasks appear to be aligned with higher grade levels where the curriculum calls for higher levels of inquiry-based, student-centered, and collaborative approaches to learning mathematics leaving the behaviorist types of activities/games for the younger students.

This study was implemented in a small rural Canadian primary school. The kindergarten teacher selected to be involved in the study was a veteran teacher with over 20 years of early years teaching experience. Although the teacher was not fluent with iPad technology, she was keen to learn and introduce the technology to her classroom. Funds to purchase four iPads with protective childproof cases and glass screen protectors, apps, stylus, and child safe headsets (control the volume such that the sound does not increase above 85 dB) were obtained through a small university grant. The agreement between the researcher and the primary school allowed the iPads and supporting technology to remain a property of the kindergarten class. Ethics permission was obtained through the university research ethics board as well as the local school board ethics authority.

A mixed methods design was applied using qualitative and quantitative data to explore the impact of mathematical apps on the learning of numeracy skills and the factors influencing the use of this technology in an early years setting. The qualitative data included field notes documenting conversations during the training session with the kindergarten teacher as well as observation notes of students using iPads. The experimental component provided the quantitative data. In this aspect of the study, the quantitative measures were the pre- and post-test measures for the experimental and control groups. The rationale for choosing a mixed methods design was to provide a wider perspectives of the context of using interactive technology in an early years setting as well as to have greater understanding of the research questions posed in this study (Johnson and Onwuegbuzie 2004 ; Almalki 2016 ). A mixed methods approach allows the researcher to compensate for the fundamental weaknesses that are associated with using only a quantitative or qualitative study (Almalki 2016 ).

Study design

The participating kindergarten teacher was selected for this study because she had demonstrated excellent knowledge about teaching mathematics to kindergarten children and was keen to engage in a research project. Prior to commencing the study, four iPads with several language arts and mathematics apps that were aligned with the curriculum were selected in collaboration with the teacher and researcher. The researcher and teacher met three times prior to commencing the study to select and experiment with the apps.

One week prior to commencing the study, one iPad was placed at each play station for approximately 20 min each day for 1 week. Children had the choice of using the iPads without receiving guided instructions. This pre-exposure to the iPads was intended to remove any novelty effects that might influence a gain in numeracy skills (Gravetter and Forzano 2011 ).

In the experimental phase, 13 children in the kindergarten class, aged four and five, were randomly selected to one of two groups. One group received a 2-week intervention involving the use of iPads to learn numeracy concepts each day and the other group followed the traditional play-based learning activities that focused on numeracy development, in particular, concepts of number sense. In a conversation with the teacher, she believed that students would be able to master the outcomes being learned in the 2-week period.

Children’s mathematical skills from both groups were measured at the beginning of the study (time 1) with 30 items that were aligned with the curriculum being taught and again following the 10-day intervention period (time 2). At the conclusion of the study, the control group was introduced to the iPads in the same manner as the experimental group (i.e., 10 days) for the purpose of ensuring equal opportunity to engage with the iPads.

The intervention consisted of using interactive technology in a play-based mathematics classroom in lieu of the teachers’ originally planned play-based lessons. A teacher-trained, research assistant removed Group 1 children to another classroom during the time designated for learning mathematics, which was approximately 20 min each day. This group of children was introduced to various mathematical apps while the control group, Group 2, children followed the play-based activities that fostered the same skill development that was in the apps. For example, one of the apps fostered the development of writing numerals and a play-based activity required students to trace numerals.

The intervention began using 10 apps for the first week and then a new app was introduced each day thereafter for a total of 15 apps. Children would receive instruction on how to use a particular app at the start of the lesson followed by time to play with the app. In the second part of the lesson, children could choose whatever app they preferred for the remainder of the lesson. The apps were downloaded from the Apple Store for free or for a nominal fee. The app icons used in this study are shown in Fig.  1 .

Apps used in study

Table  1 , below, summarizes the types of skill development in each of the apps.

The items used to measure children’s mathematics ability in the pre- and post-tests were created based on the concepts taught in the classroom, which were aligned with the curriculum outcomes as previously noted. A map of the curriculum outcome and corresponding items is shown in Table 4 in the Appendix . These items were modeled from the exemplars provided in the provincial curriculum document and in consultation with the teacher (Department of Education, Early Learning and Culture of PE 2008 ).

Data collection

Data in the form of children’s numeracy test scores was collected using an application called Explain Everything. This application is an interactive screen-casting whiteboard, which stored the test items and recorded children’s responses to each item. To capture children’s responses, the examiner would orally read instructions that were printed on the bottom of each test page and then the child would respond by speaking, writing, or manipulating objects on the screen; all of which were captured using Explain Everything simultaneous video and audio recording feature known as screen casting. Figure  2 shows an original test item as presented in Explain Everything and how a student manipulated the objects on the screen (on the right) to demonstrate their understanding.

Original test item and test item after answered by student

The researcher and a research assistant independently scored each child’s test by reviewing each screen-cast. When the scores did not match, we discussed our responses and agreed on a score. Each item was scored based on a 4-point rubric: (1) Do not know or responded incorrectly, (2) demonstrated some understanding of the concept but response was not correct, (3) provided a correct response but the strategy was not strategic or efficient, and (4) provided a correct response that was efficient. An example of a level 2 response would be a number written backwards or upside down and an example of a level 3 response is writing the number seven starting from the bottom and moving to the top (i.e., drawing the number from the finish position to the start position). The kindergarten teacher reviewed and agreed with the 4-point rubric. In the sample response shown in Fig.  2 , the child would receive a 4 for the first answer (i.e., 5) but a 3 for the second answer because the child orally indicated the set containing six elements but they had difficulty writing the digit 6.

The pre-test at time 1 and post-test at time 2 contained the same items except that the colors or the shape of objects were changed. Prior to implementing the study, the test was piloted with three children from another school and reviewed by the kindergarten teacher. Small changes to the printed instructions on each slide were made to better align the vocabulary with children’s understanding.

During the experimental phase, the research assistants recorded field notes documenting children’s behaviors and the apps that were most favored. Children’s feedback on what they liked about the apps (i.e., the characteristics of the apps) and other observations were also documented.

Mean scores were calculated for both groups on the pre-test. Due to the small sample size, it was not possible to conduct the stringent inferential analysis of covariance; therefore, group difference scores were analyzed. The observational field notes were conceptually analyzed to determine the presence of common words or phrases to make inferences about the observations. The coding began with predefined categories such as collaboration, level of engagement, and choice of apps but was flexible to allow for the addition of other unanticipated themes.

Cronbach’s alpha was used to measure the internal consistency of the scale. After removing poor performing items due to poor discrimination, 22 items remained. The items that discriminated poorly were due to all students receiving the top score on the item; subsequently, there was no discrimination between ability. Easy items were purposefully included on the assessment to ease students into the testing; hence, it was expected that a number of items would be removed from the test due to poor discrimination. Cronbach’s alpha for the remaining 22-item scale on the pre-test was 0.803 and 0.805 for the post-test. Thus, the coefficient exceeded the absolute minimum threshold of 0.7 but also met the ideal minimum threshold of 0.8 (Tabachnick and Fidell 2012 ), indicating a reliable set of test items.

Descriptive summary of items

Group 1 consisted of four females and three males, and group 2 had two females and four males. All children were 4 or 5 years old. Items ranged in difficulty with the hardest being items 1.7b (write the number 6; M  = 2.46, SD  = 1.27), 1.7c (write the number 5; M  = 2.92, SD  = 1.19), and 1.2h (recognize seven dots on a 10 frame; M  = 2.92, SD  = 0.28). Easier items were 1.4b (create a set of 7; M  = 3.77, SD  = 0.83), 1.7k (count backwards from 5; M  = 3.67, SD  = 0.89), 2.1d (identify repeating and non-repeating patterns; M  = 3.54, SD  = 1.13), 1.2a (identify 3 on a die, M  = 3.54, SD  = 0.52), and 1.2b (identify 4 on a die, M  = 3.54, SD  = 0.52).

Given that four out of 10 apps involved drawing numbers, an increase in this skill was anticipated from the pre- to the post-test. However, of the two items assessing drawing numbers, only one item (i.e., 1.7b drawing the number 6) revealed a significant increase from M  = 2.86 and SD  = 1.46 to M  = 3.29 and SD  = 1.25, following the intervention.

When comparing mean scores, the experimental and control group differed by 0.01 on the pre-test (see Table  2 ). After the intervention, the experimental group increased slightly (+ 0.02) and the control group decreased slightly (− 0.04). On the post-test, the two groups differed by 0.05 with the experimental group having the higher mean score (see Table  2 ). These differences are too small to suggest the intervention had any effect on students’ mathematics ability.

Observational findings

All children were keen to use the iPads over the 10 days of mathematics lessons as they asked the teacher each day when the research team was arriving so that they could use the iPads. During the introduction to a new app phase at the beginning of each guided mathematics lesson, where the children were shown how to use an app (if they needed help), children did not use a headset so that they could hear the instructions (volume was turned down on the ipads). During this guided instruction, children were more apt to collaborate with each other to share what was on their screens and provide help to get to another level or step. When the children wore a headset, there was a greater tendency for children to focus on their own screen as they were not distracted by sounds coming from other iPads or giggles and exclamations coming from their peers.

The four stronger children in the class (as identified by the teacher and confirmed by the pre-test) appeared to have a better understanding of how to maneuver through different levels in an app whereas the weaker children frequently needed guidance on how to proceed to the next level. Another observation related to levels in an app was the difficulty of the level. When an the app level became too challenging, children would either look to abandon the app or randomly select answers until they eliminated all incorrect responses and identified the correct response. An example of this type of question was the equation 2 + 3 = ? which was supplemented by corresponding dots along with responses of 4, 5, and 6. When we debriefed the teacher about this trial and error process of selecting the correct response, she believed that children were learning more than we were giving them credit for which was encouraging; however, we were still concerned that children may be learning by memorizing rather than having a conceptual understanding of the concept. Another finding related to children’s ability was that stronger children exhibited more independence in using apps. These children were able to use a new app by listening to the audio instructions provided in the app or were confident enough to skip over the audio instructions and starting using the app immediately. In comparison, weaker children frequently needed the research assistant to provide oral instruction as well as provide a demonstration (i.e., model using the app) for new apps.

When drawing numbers, three children would opt to use their index finger to trace the number. Given that children were still developing their fine motor skills, we believed that it was important to encourage children to use a stylus; hence, we encouraged children to use a stylus at all times but particularly when they were drawing numbers.

After every 3 days, children were asked, what was their favorite app? The last app that they played was the most common response, likely because it was the most current in their mind. When prompted further to think about the other apps, the most favored apps were not the most pedagogically structured apps but rather apps that had a lot of bling. For example, children were drawn to apps that had exploding stars when they completed a set of tasks or would see a funny character dance across the screen (e.g., Endless 123). In the same vein, children were quick to move from one app to another. For example, all children quickly grew tired of number drawing and when left to their own choices they would not select apps with this skill development. When debriefing this finding with the teacher she noted that an app tended to have a life expectancy of a few days and then children would become tired with the simplicity of the app unless the app captured their attention with bling. Table  3 displays the app and frequency of children’s preference for the app, which was based on the frequency they choose the app during their structured playtime. This is a holistic measure taking into consideration that there were parts of an app children played frequently, while other parts of the app were ignored. The frequency is also influenced by mathematics ability in that more challenging apps were used less frequently by weaker children.

In terms of the teacher’s experience with the iPads, pre-experiment meetings revealed that she was completely new to using iPads and did not own an iPad of her own. She received instruction on how to (a) access the internet, (b) link the iPads so that apps could be downloaded automatically to more than one device, (c) download apps, (d) organize apps into a folder, and (e) delete apps that were no longer being used or accidentally downloaded. Throughout the study, the teacher would periodically sought assistance for these tasks. Following the study, we met twice to solve problems related to purchasing new apps and simultaneously downloading them to all four devices.

This discussion is framed by the two research questions posed in this study. In addition, other insights garnered in the study are discussed.

To what extent does the use of mathematical apps using iPads enhance children’s learning of numeracy in kindergarten?

Considering the small gains in achievement by the experimental group in comparison to a slight decrease in achievement in the control group, there was no significant difference in children’s understanding of numeracy as measured on the pre- and post-tests between the two groups. Although a difference was anticipated based on prior research, these findings were the same as Mattoon et al. ( 2015 ) who also reported small gains but no significant difference between their two groups in a 6-week long study. Despite the absence of significant gains between the two groups, this study provides evidence that using technology in this context did not deter or lessen children’s development of numeracy skills. This study adds to the work of Bebell et al. ( 2012 ) who reported that iPads do not hinder early learning of literacy. We can now conclude that iPads do not hinder early learning of numeracy as well as literacy. This is an important finding that will broaden the utility of iPads in the early years classroom. In summary, the use of mathematical apps on iPads slightly enhanced children’s learning of mathematics as shown by gains from the pre- to the post-test; however, the gains were not significant between groups.

What factors influence children’s use of interactive technology in a play-based learning environment?

Factors that influenced children’s use of interactive technology focused on (a) collaboration, (b) ability, (c) use of a stylus, (d) maturity, and (e) teachers’ skill level. A factor known to influence children’s use of interactive technology was their affinity for collaboration. Given that prior research documented how technology fostered a collaborative learning environment (Shifflet et al. 2012 ), it was not surprising that children naturally collaborated without any guidance to do so from the researcher. During the introduction phase of a new app, children naturally gravitated towards each other to share what was on their iPad as well as to help each other progress to the next level or game. This affinity for collaboration is an asset to learning mathematics that needs to be supported so that when children leave the play-based learning environment they are still drawn to helping each other with tasks in general but specifically, in the learning of mathematics.

Another observation focused on the impact of children’s prior mathematics ability and experience with apps. No research was found that examined the relationship between children’s mathematics ability and interaction with apps. However, Hung et al. ( 2015 ) reported that when children were challenged, they reported higher levels of engagement and satisfaction. Extending Hung et al.’s ( 2015 ) finding, it appears that children with stronger skills in mathematics were more apt to persevere and be engaged with the app. In contrast, weaker children were more apt to abandon the app or use a trial and error process to progress to the next level, in which case, the quality of their learning was questioned, as learning may be memorized rather than conceptual. Children’s ability level can be connected to attention span since apps that required greater concentration would be met with a shorter attention span. This short attention span appeared to be age appropriate considering the attention span for 4- to 5-year-olds is a maximum 6 to 7 min (i.e., chronological age + 1); although attention span for children playing or being socially engaged can exceed, these maximum times are typically reserved for formal instruction (Wesson 2011 ). Hence, challenging and less creative apps (e.g., Montessori Math) might be perceived more like a formal lesson whereas entertaining and creative apps with bling (e.g., Count-up-to-ten) may be perceived as play.

Further endorsing the need for more research in this area was the absence of research focusing on the use of a stylus versus the index finger to interact with iPads. Although children naturally gravitated towards using their index finger, we encouraged children to use the stylus to reinforce printing skills learned with a pencil. More research is needed to corroborate this practice.

The fact that children in this study gravitated towards apps that stimulated laughter through humorous animated characters or bursts of stars or sparkles confirmed our thoughts about their level of maturity and corresponding desire for play. Once children had been exposed to the new app each day, they were able to choose any app to play with for the remaining time. The outcome of children’s decision-making resulted in the selection of apps that were creative and fun in contrast to other apps that were more pedagogically accurate containing appropriate levels of difficulty and sequencing of questions. This finding needs further research to explore the extent to which this affinity for creative and fun apps continues through the elementary and primary grades.

The last factor that influenced children’s use of interactive technology was the teachers’ skill level and interest in implementing apps as curricular learning resources. As noted above, the teacher participating in this study could be described as a beginner in using interactive technology and her skill set was similar to what was previous documented as limited to downloading images for presentations (Public Broadcasting Service and Grunwald Associates 2009 , 2011 ). However, her interest, rather than ability, was the catalyst for implementing technology in her classroom and through opportunities for professional development as called for by other researchers (i.e., Simon et al. 2013 ); this teacher can now be described as experienced and innovative in her use of interactive technology in the classroom. Key to this transformation was providing professional development in a one-to-one session and in an as-needed basis.

With the advancement of interactive technology and more user-friendly touchable interfaces, the use of these devices in early years classrooms is not only suitable but also appropriate in preparing early years children for the technological world they will live in (Gordon and Williams Browne 2016 ). This study revealed that children using interactive technology in the form of mathematics apps as part of a play-based learning environment for mathematics had small gains in achievement as measured using a pre- and post-test. Although the gains in achievement were not significant between the control and intervention groups, learning using interactive technology did not lessen children’s opportunity to learn about numeracy concepts as children were observed collaborating and were highly engaged, particularly with apps that were creative. A longer experiment with a larger sample is needed to validate this finding.

Another important finding related to using interactive technology in a play-based learning environment was the need for guided direction in selecting quality apps that supported learning. When children were left to their own accord, they almost always selected apps or segments of apps that were high in entertainment value and low in educational value. Given the premise of play-based learning environments where play is situated in intentional and specific learning contexts that nurtures learning while providing children with independence to choose activities, it is important to select mathematical apps that are not only aligned with the curriculum but are also creative and fun and provide opportunities to learn. In these learning environments, pleasurable, entertaining play is encouraged in balance with other play-based activities that foster learning in specific domains such as numeracy. This finding was connected to children’s attention span which appeared to be dependent on the creativity of the app as well as the difficulty level of the app such that more creative apps held children’s attention spans longer but also the difficulty level influenced student engagement with apps. This finding corroborates the work of Couse and Chen ( 2010 ) who reported that engagement increased with age and by extension mathematics ability.

In sum, interactive technology in the form of iPads with mathematical apps promoted student collaboration and engagement. However there is still much to learn about the quality of apps and their impact on children’s learning, particularly when children are encouraged to make choices in a play-based learning environment.

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Perspectives on numeracy: reflections from international assessments

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This paper examines perspectives regarding the mathematical skills expected of adults and school graduates, comparing ideas developed as part of two major multinational comparative assessments of skills: the Programme for International Student Assessment and the Programme for International Assessment of Adult Competencies (also known as the OECD Survey of Adult Skills). The paper reflects on the conceptual and assessment frameworks developed for these two programmes, aiming to shed light on the commonalities and differences between the constructs of numeracy and mathematical literacy and to inform current debate about directions for developing mathematical skills in the 21st Century.

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Tout, D., Gal, I. Perspectives on numeracy: reflections from international assessments. ZDM Mathematics Education 47 , 691–706 (2015). https://doi.org/10.1007/s11858-015-0672-9

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DOI : https://doi.org/10.1007/s11858-015-0672-9

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