Early Numeracy: The Transition from Infancy to Early Childhood

Introduction

By preschool age, most children exhibit a range of numeracy skills, including verbal skills, such as counting, and nonverbal skills, such as recognizing equivalence of object sets.^1,2,3,4,5 Although researchers agree that these abilities are present in early childhood, they continue to debate when, and by what mechanisms, these abilities emerge. In other words, what are the developmental origins of verbal and nonverbal numerical competencies?

Subject

Research on numeracy traditionally focused on verbal counting. However, the notion that numeracy might emerge in infancy and toddlerhood shifted the focus toward concepts that can be measured nonverbally. This shift expanded the range of behaviours included in early numeracy—a change that has direct implications for early childhood education and assessment. This shift also raised questions about the developmental origins of mathematics learning difficulties and gaps in mathematical achievement.

Problems

Most children acquire basic symbolic number skills by 5 years of age, such as reciting the count list to 20 or more,^1,2,4 using the count list to enumerate various sets,^1,2,4,5 understanding that the last word in a count stands for the numerosity of the set (i.e., Cardinal Word Principle or CWP),^1,2,5,6 identifying written numerals,⁵ and judging ordinality of single-digit numerals.⁷ There is also emerging evidence that children can interpret multidigit numbers starting at 3 years of age—correctly judging, for example, which of two multidigit numerals is larger.^8,9

Prior to mastery of symbolic skills, preschool children also exhibit understanding of quantitative relations on nonverbal measures, such as matching equivalent sets of objects,¹⁰ performing simple calculations with objects,¹¹ or indicating which of two dot clouds has more.¹² Children perform object-based number tasks earlier than they demonstrate similar understandings in verbal tasks. For example, preschoolers solve simple object-based addition and subtraction problems (e.g., 2 + 2) years before they can solve analogous verbal problems.^11,13 Similarly, children judge ordinality and equivalence in forced choice tasks earlier than they can compare the same sets verbally, via counting, with nonverbal competence emerging between 2-1/2 and 3 years of age.^11,14,15

A major research focus has been understanding the developmental origins of these nonverbal number concepts. Researchers have shown, using habituation and preferential looking methods, that infants are sensitive to quantity as well,^16,17 with some studies demonstrating this sensitivity in newborns.^18,19 Various proposals have linked individual variation in this early sensitivity to later numeracy and mathematics outcomes. However, open questions remain about the representations underlying this early sensitivity, how the representations themselves develop, and what role these representations may play in subsequent development.

Research Context

One candidate for early nonverbal number representation is the Approximate Number System, or ANS—a representation proposed to underlie the discrimination of different set sizes—particularly large set sizes (e.g., 16 vs. 32).^20,21 Although the ANS is thought to operate over discrete number, it is also inexact and ratio-dependent, similar to non-numerical dimensions such as surface area, meaning that quantities are easier to tell apart when their ratio is higher (e.g., 16 is easier to discriminate from 32 than from 24).²² The ANS is considered innately available because even newborns respond to variations in set sizes as long as the ratios are large enough.¹⁹ However, research also shows that with age and schooling, the ANS becomes more precise.^23,24,25,26

Another proposed nonverbal number representation is based on object individuation, also described as object tracking, mental models, or subitizing—the immediate perception of number in small quantities (e.g., 1 to 4 objects).^11,21,27,28 On these accounts, children incidentally represent number when they differentiate objects in a scene and keep track of the objects’ movements and spatial positions. Set size limits on object individuation have been explained by constraints on working memory²⁹ or attention.²⁷ Some have argued that like the ANS, object-based representations are an innate endowment, with ongoing debate about whether the two systems are distinct^11,30 or simply different instantiations of the same evolutionarily primitive representational system.³¹ Still others have suggested object-based representations could emerge from experiences observing and manipulating objects without necessarily arising from an innate quantification system.³²

A third contributor to early numeracy is exposure to number words and the verbal counting system. Prior to the advent of research on infant quantification, seminal research by Piaget suggested that children lacked a conceptual understanding of quantitative relations until well after they had mastered conventional counting³³ and studies showed that children did not understand numeracy principles until after they had mastered counting procedures.^34,35 Although subsequent research has shown that precounting children understand much more about quantities than Piaget claimed, symbolic number understanding remains a strong predictor of later mathematics achievement,^{36,37,38,39,40} and indeed, stronger than nonverbal quantification skills.^41,42,43,44 Research has also suggested children can extract information about numbers and their meanings from numeric symbols themselves, showing for example, that preschool children can match written multidigit numerals to multidigit number words and compare magnitudes of written multidigit numerals independent of performance using nonverbal measures.⁹

Key Research Questions

Most researchers agree that children respond to changes in number early in life via nonverbal processes. Furthermore, there is general agreement about the stages of verbal number acquisition. Current research is now focused on the underlying nature of nonverbal quantification and whether variation in nonverbal processes is related to later mathematics achievement. In this research, investigators also consider whether children bootstrap between verbal and nonverbal quantification as they learn.⁴⁵ Finally, there is growing interest in the verbal numeracy environment at home and in preschool, and its connection to later child outcomes.

Recent Research Results

Early number discrimination and non-numerical quantitative dimensions

There is ongoing debate about whether infants’ responses to quantitative changes are based on an awareness of discrete number per se, or one of many perceptual variables that correlate with discrete number, such as surface area, convex hull, brightness, duration, temporal density, and spatial frequency.^46,47,48 Researchers have attempted to control these perceptual variables to obtain a clean test of numerical sensitivity,^24,49,50 but it is difficult to control all of these perceptual variables simultaneously, as others have pointed out,^46,47 leading some to suggest that future research should focus on ways to account for non-numerical responses rather than attempting to control them.^46,50,51 Thus, it remains unclear whether infants’ quantitative sensitivity is based on discrete number, as some have claimed, or a combination of other perceptual information that is correlated with discrete number. Similar issues arise in research testing whether infants respond to changes in quantity across dimensions—for example, learning to associate certain visual patterns with larger and smaller numerical sets and transferring this association to objects differing in size,⁵² or expecting that if quantitative pairs (e.g., number and spatial extent) both increase or decrease, they will both change in the same direction¹⁸  —research which has led to the proposal that quantification arises from a generalized magnitude representation. Such a representation is one way to characterize an undifferentiated sense of quantity based on multiple input streams, but the claim that children can switch from one quantitative cue to another would require controls that can isolate each cue effectively.

Making connections

Research has documented how children acquire several distinct verbal enumeration skills (e.g., counting, cardinality, ordinality), as well as how they represent quantities nonverbally. However, to achieve a coherent number concept, children must eventually make connections among these skills and representations (e.g., verbal number words, physical quantities, mental models).^{43,53,54,55,56,57} Small number words may play a critical role in children’s first mappings because the quantities one, two, and three can be immediately perceived and represented nonverbally with less error than representations of larger quantities. Thus, small sets may offer clear perceptual referents that can be labeled with a number word.^28,58,59,60

Once the labels for small sets have been learned, children are positioned to notice that the same words are used as labels and in counting, thereby discovering the Cardinal Word Principle (CWP)—the idea that the last word in a count stands for its cardinal number. In the absence of targeted instruction, most children naturally attain the CWP by age 4 years, but studies have shown the CWP can be induced from practice labeling small sets as well as instruction that juxtaposes counting and labeling.^28,61 In the n-knower framework, CWP has been measured using the Give-n task (e.g., “Give me 5 counters.”), and early research findings suggested children learn number-to-quantity mappings one by one and in order, prior to making the connection between counting and cardinality, which itself is followed by a rapid logical generalization of the CWP to all the numbers within a child’s counting range.^45,62,5 However, diary studies have reported correct use of small number words in certain contexts even earlier, as well as evidence that children may acquire these number meanings in a different order.^63,64 Moreover, recent studies have raised questions about the validity of Give-n performance and the meaning of n-knower classifications based on it.^65,66,67,68 Thus, although much has been learned about these important connections, key questions remain unresolved.

Early predictors of mathematical achievement

Evidence of quantitative sensitivity in infancy has inspired researchers to examine how this sensitivity relates to acquisition of verbal numeracy in early childhood, as well as eventual mathematical achievement in school. Some have argued that the ANS provides a representational foundation for acquisition of later symbolic numeracy and mathematics skills^21,45 and longitudinal studies linking ANS acuity in infancy and preschool to later mathematical achievement in childhood and adolescence seem to bolster this claim.^69,70,71,72 However, other studies examining longitudinal and concurrent associations have failed to find evidence connecting ANS acuity to mathematics achievement,^{73,74,75,76,77} and indeed, accumulating neurological and behavioural evidence points to separate mechanisms.^12,26,78 Finally, when children acquire symbolic numeracy skills, ANS acuity improves concurrently, perhaps as a result.^25,79 Thus, if ANS and symbolic mathematics skills are causally related, the relation could be from symbolic number to ANS rather than the reverse, or perhaps, bidirectional.

Similar patterns have been reported for spontaneous focusing on number (SFON)—the tendency of children to notice exact number in their daily experiences.⁸⁰ Tests of SFON carefully avoid verbal number cues in order to tap children’s self-directed attention to numerosity, but because children in these studies are generally preschool aged or older,^80,81 it is unclear whether the mechanism driving SFON is nonverbal quantification (e.g., object individuation), verbal counting, or both. Concurrent correlational studies indicate strong associations between SFON tendencies and verbal numeracy,^80,82 and longitudinal studies demonstrate that performance on SFON measures in early childhood is correlated with symbolic number knowledge in elementary school;^83,84 however, whereas both SFON and symbolic numeracy predict subsequent mathematics achievement, performance on symbolic number tasks is the stronger predictor.⁸⁵ Also, attempts to improve SFON have been successful when interventions included symbolic number activities,^86,87 suggesting that SFON itself may be driven by symbolic numeracy acquisition, rather than the reverse. Additional research using interventions based on nonverbal activities is needed to draw firm conclusions, but early symbolic numeracy remains the clearest and most potent predictor of later mathematics achievement.

Home Numeracy Environment

Acquisition of children’s first numeracy skills takes place largely in the family home, so the number-related activities of children and their caregivers have received increasing attention.^88,89 Most research on this topic has used either parent report of numeracy activities^{90,91,92,93,94} or coding parent speech from direct observations.^95,96,97 Studies have demonstrated an association between the frequency of home numeracy activities based on parent report and children’s numeracy outcomes,^90,91 though this association is not always obtained.^89,92,93 Existing studies also indicate that although parents talk about number infrequently,⁹⁵ even when activities are designed to elicit such talk,^60,97 there are significant associations between the frequency of incidental number talk and children’s numeracy outcomes.^{90,95,96,97,98} Child outcomes have also been linked to variation in qualitative differences, such as conversation length,⁹⁹ and focusing on large set sizes (e.g., 4-10) or advanced concepts such as cardinality.^97,100,101 A few observational studies have targeted infancy in particular, demonstrating that parental number talk is present at the youngest ages observed to date (i.e., 12 to 14 months).^95,96 Thus, although infants are themselves nonverbal, their emerging understandings of numeracy may be shaped by exposure to verbal numeracy early on.³²

Research Gaps

Although research has generated extensive information about developmental changes in various quantitative skills, such as the CWP, SFON, and nonverbal set size discriminations, less is known about the mechanisms that drive these changes, and particularly, the mechanisms by which children make connections among various concepts and representations to achieve a coherent sense of number. Related to this issue, more research is needed to test proposed mechanisms experimentally, by providing inputs that are consistent with hypotheses about developmental mechanism. For example, though it has been argued that small set sizes offer an opportunity to unite verbal and nonverbal quantification, the next step is to demonstrate that this is the case experimentally. Intervention studies that test the effects of specific input types may also be helpful in this regard. Similarly, more research examining the relations between verbal number and nonverbal number are needed to determine what directions of influence are at play, at what ages, and under what conditions.

Another persistent issue that remains unresolved is whether nonverbal quantification is based on discrete number or attention to non-numerical variables, such as surface area. Although researchers have focused on attempts to control these non-numerical variables, a promising alternative may be to design measures that account for non-numerical responses rather than attempting to control for them.^46,50,51

Finally, intriguing new research about the home numeracy environment as well as the origins of multidigit number concepts have raised a host of new questions that bear investigation. For example, most studies of children’s home numeracy environment have focused on preschool age, but much could be gained by tracing these experiences back into infancy, particularly given the longstanding evidence of nonverbal quantification in this age range. Are infants directed to attend to number much earlier in life than we have documented to date? If so, how might this change our understanding of SFON, for example. Similarly, the unexpectedly early acquisition of multidigit number meanings raises new questions about the presence of multidigit numeracy in parents’ number talk, as well as whether variation in these informal insights is related to subsequent mathematics outcomes. Interventions targeting either the home numeracy environment, early multidigit numeracy, or both, would be exciting new directions for future research.

Conclusions

Evidence of numerical competence in infants has raised intriguing questions about the origins of numeracy and the conceptual resources young children use to acquire verbal counting. However, further research is needed to reveal precisely how this infant competence connects to subsequent nonverbal and verbal development and whether these mechanisms can be leveraged to help all children enter school with a strong foundation of numeracy concepts.

References

1. Gelman R, Gallistel CR. The child’s understanding of number. Cambridge, MA: Harvard University Press; 1978.
2. Fuson KC. Children’s counting and concepts of number. New York, NY: Springer-Verlag; 1988.
3. Mix KS, Huttenlocher J, Levine SC. Quantitative development in infancy and early childhood. New York, NY: Oxford University Press; 2002.
4. Clements DH, Sarama J. Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge; 2009.
5. Litkowski EC, Duncan RL, Logan J AR, Purpura DL. When do preschoolers learn specific mathematics skills? Mapping the development of early numeracy knowledge. Journal of Experimental Child Psychology. 2020;195:104846.
6. Wynn K. Children’s understanding of counting. Cognition 1990;36(2):155-193.
7. Ramani GB, Siegler RS Playing linear numerical board games promotes low-income children’s numerical development. Cognition 2008;11(5):655-661.
8. Mix KS, Prather RW, Smith LB, Stockton JD. Young children’s interpretation of multidigit number names: From emerging competence to mastery. Child Development 2014;85(3):1306-1319.
9. Yuan L, Prather RW, Mix KS, Smith LB. Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves. Cognition 2019;189:89-104.
10. Mix KS. Children’s equivalence judgments: Crossmapping effects. Cognitive Development 2008;23(1):191-203.
11. Huttenlocher J, Jordan NC, Levine SC. A mental model for early arithmetic. Journal of Experimental Psychology: General. 1994;123(3):284-296.
12. Yuan L, Prather RW, Mix KS, Smith LB. Number representations drive number-line estimates. Child Development. 2020;91(4):e952-e967.
13. Rasmussen C, Bisanz J. Representation and working memory in early arithmetic. Journal of Experimental Child Psychology 2005;91:137-157.
14. Cantlon J, Fink R, Safford K, Brannon EM. Heterogeneity impairs numerical matching but not numerical ordering in preschool children. Developmental Science 2007;10:431-440.
15. Mix KS. Similarity and numerical equivalence: Appearances count. Cognitive Development 1999;14:269-297.
16. Lipton JS, Spelke ES. Origins of number sense: Large-number discrimination in human infants. Psychological Science, 2003;14(5):396-401.
17. Xu F. Numerosity discrimination in infants: Evidence for two systems of representations. Cognition, 2003;89(1):B15-B25.
18. de Hevia MD, Izard V, Coubart A, Spelke ES, Streri A. Representations of space, time, and number in neonates. Proceedings of the National Academy of Sciences. 
    2014;111(13):4809-4813.
19. Izard V, Sann C, Spelke ES, Streri A. Newborn infants perceive abstract numbers. Proceedings of the National Academy of Sciences of the United States of America. 2009;106(25):10382-10385.
20. Dehaene S. The number sense: How the mind creates mathematics. New York, NY: Oxford University Press; 1997.
21. Feigenson L, Dehaene S, Spelke ES. Core systems of number. Trends in Cognitive Science 2004;8(7):307-314.
22. Xu F, Spelke ES, Goodard S. Number sense in human infants. Developmental Science 2005;8(1):88-101.
23. Halberda J, Feigenson L. Developmental change in the acuity of the “number sense”: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology 2008;44(5):1457-1465.
24. Piazza M, DeFeo V, Panzeri S, Dehaene S. Learning to focus on number. Cognition 2018;181:35-45.
25. Shusterman A, Slusser E, Halberda J, Odic D. Acquisition of the cardinal principle coincides with improvement in approximate number system acuity in preschoolers. PLoS ONE;11(4):e0153072.
26. Wilkey ED, Ansari D. Challenging the neurobiological link between number sense and symbolic numerical abilities. Annals of the New York Academy of Sciences 2020;1464(1):76-98.
27. Burr DC, Turi M, Anobile G. Subitizing but not estimation of numerosity requires attentional resources. Journal of Vision 2010;10(6):20.
28. Paliwal V, Baroody AJ. Cardinality principal understanding: The role of focusing on the subitizing ability. ZDM 2020;52(4):649-661.
29. Trick LM, Pylyshyn ZW. Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision. Psychological Review 1994;101(1):80.
30. Revkin SK, Piazza M, Izard V, Cohen L, Dehaene S. Does subitizing reflect numerical estimation? Psychological Science 2008 Jun;19(6):607-614.
31. Cheyette SJ, Piantadosi ST. A unified account of numerosity perception. Nature Human Behaviour 2020;4(12):1265-1272.
32. Mix KS, Levine SC, Newcombe N. Development of quantitative thinking across correlated dimensions. In: Henik A, ed. Continuous issues in numerical cognition: How many or how much. London, UK: Academic Press: 2016;1-33.
33. Piaget J. The child’s conception of number. New York NY: Norton; 1941.
34. Briars DJ, Siegler RS. A featural analysis of preschoolers’ counting knowledge. Developmental Psychology 1984;20:607-618.
35. Frye D, Braisby N, Lowe J, Maroudas C, Nicholls J. Young children’s understanding of counting and cardinality. Child Development 1989;60:1158-1171.
36. Conoyer SJ, Foegen A, Lembke ES. Early numeracy indicators. Remedial Special Education 2016;37(3):159-171.
37. Jordan NC, Kaplan D, Ramineni C, Locuniak MN. Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology 2009;45(3):850-867.
38. Krajewski K, Schneider W. Exploring the impact of phonological awareness, visual–spatial working memory, and preschool quantity–number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study. Journal of Experimental Child Psychology 2009;103(4):516-31.
39. Missall KN, Mercer SH, Martínez RS, Casebeer D. Concurrent and longitudinal patterns and trends in performance on early numeracy curriculum-based measures in kindergarten through third grade. Assessment for Effective Intervention 2012;37(2):95-106.
40. Schneider RM, Brockbank E, Feiman R, Barner D. Counting and the ontogenetic origins of exact equality. Cognition 2022;218:104952.
41. Caviola S, Colling LJ, Mammarella IC, Szucs D. Predictors of mathematics in primary school: Magnitude comparison, verbal and spatial working memory measures. Developmental Science 2020;23:e12957.
42. Gobel SM, Watson SE, Lervag A, Hulme C. Children’s arithmetic development: It is number knowledge, not the approximate number system, that counts. Psychological Science 2014;25(3): 789-798.
43. Kolkman ME, Kroesbergen EH, Leseman PP. Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction. 2013;25:95-103.
44. Toll SW, Kroesbergen EH, Van Luit JE. Visual working memory and number sense: Testing the double deficit hypothesis in mathematics. British Journal of Educational Psychology 2016;86(3):429-445.
45. Carey S. Where our number concepts come from. The Journal of Philosophy 2009;106(4):220-254.
46. Leibovich T, Katzen N, Harel M, Henik A. (2017) From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition. Behavioral and Brain Sciences 2017;40:e164.
47. Mix KS, Huttenlocher J, Levine SC. Multiple cues for quantification in infancy: Is number one of them? Psychological Bulletin 2002;128:278-294.
48. Smets K, Sasanguie D, Szücs D, Reynvoet B. The effect of different methods to construct non-symbolic stimuli in numerosity estimation and comparison. Journal of Cognitive 
    Psychology 2015;27(3):310-325.
49. Cordes S, Brannon EM. Crossing the divide: infants discriminate small from large numerosities. Developmental Psychology 2009;45(6):1583-1594.
50. Cantrell L, Smith LB. Open questions and a proposal: A critical review of the evidence on infant numerical abilities. Cognition 2013;128(3):331-352.
51. Sella F, Slusser E, Odic D, Krajcsi A. The emergence of children’s natural number concepts: Current theoretical challenges. Child Development Perspectives 2021;15(4):265-273.
52. Lourenco SF, Longo MR. General magnitude representation in human infants. Psychological Science 2010;21(6):873-881.
53. Baroody AJ, Dowker A. The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Erlbaum; 2003.
54. Dehaene S. Varieties of numerical abilities. Cognition 1992;44(1-2):1-42.
55. Krajewski K, Schneider W. Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: Findings from a four-year longitudinal study. Learning and instruction 2009;19(6):513-526.
56. Malone SA, Heron-Delaney M, Burgoyne K, Hulme C. Learning correspondences between magnitudes, symbols and words: Evidence for a triple code model of arithmetic development. Cognition 2019;187:1-9.
57. Mix KS, Sandhofer CM, Baroody A. Number words and number concepts: The interplay of verbal and nonverbal processes in early quantitative development. In: Kail RV, ed. 
    Advances in child development and behavior. New York, NY: Academic Press; 2005:305-345.
58. Baroody AJ, Lai M, Mix KS Development of young children’s early number and operation sense and its implications for early childhood education. In: Spodek B, Saracho ON, eds. Handbook of research on the education of young children. Mahwah, NJ: Erlbaum; 2006:187-221.
59. Clements DH, Sarama J, MacDonald BL. Subitizing: The neglected quantifier. In: Anderson A, Alibali MW, eds. Constructing number: Merging perspectives from psychology and mathematics education. Switzerland: Springer Cham; 2019;13-45.
60. Spelke E. What makes us smart? Core knowledge and natural language. In: Gentner D, Goldin-Meadow S, eds. Language in mind. Cambridge, MA: MIT Press; 2003.
61. Mix KS, Sandhofer CM, Moore JA, Russell C. Acquisition of the cardinal word principle: The role of input. Early Childhood Research Quarterly. 2012;27(2):274-283.
62. Sarnecka BW, Lee MD. Levels of number knowledge during early childhood. Journal of Experimental Child Psychology. 2009;103(3):325-337.
63. Mix KS. How Spencer made number: First uses of the number words. Journal of Experimental Child Psychology 2009;102:427-444.
64. Palmer A, Baroody AJ. Blake's development of the number words “one,”“two,” and “three”. Cognition and Instruction 2011;29(3):265-96.
65. Baroody AJ, Mix KS, Kartal G, Lai ML. The development and assessment of early cardinal-number concepts. Journal of Numerical Cognition 2023;9(1):182-95.
66. Krajcsi A. Follow-up questions influence the measured number knowledge in the Give-a-number task. Cognitive Development 2021;57:100968.
67. Marchand E, Lovelett JT, Kendro K, Barner D. Assessing the knower-level framework: How reliable is the Give-a-Number task? Cognition 2022;222:104998.
68. O'Rear CD, McNeil NM, Kirkland PK. Partial knowledge in the development of number word understanding. Developmental Science 2020;23(5):e12944.
69. Libertus ME, Feigenson L, Halberda J. Preschool acuity of the approximate number system correlates with school math ability. Developmental Science 2011;14(6):1292-300.
70. Halberda J, Mazzocco M, Feigenson L Individual differences in non-verbal number acuity predicts maths achievement. Nature 2008;455(7213):665-668.
71. Mazzocco M, Feigenson L, Halberda J. Preschoolers’ precision of the Approximate Number System predicta later school mathematics performance. PLoS ONE 2011;6(9):e23749.
72. Starr A, Libertus M, Brannon EM. Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences 2013;110(45):18116-18120.
73. DeSmedt B, Noel M-P, Gilmore C, Ansari D. How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education 2013;2(2):48-55.
74. Holloway ID, Ansari D. Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology 2009;103(1):17-29.
75. Sasanguie D, Göbel SM, Moll K, Smets K, Reynvoet B. Approximate number sense, symbolic number processing, or number–space mappings: What underlies mathematics achievement? Journal of Experimental Child Psychology 2013;114(3):418-431.
76. Soltész F, Szűcs D, Szűcs L. Relationships between magnitude representation, counting and memory in 4-to 7-year-old children: A developmental study. Behavioral and Brain Functions 2010;6(1):1-4.
77. Iuculano T, Tang J, Hall CW, Butterworth B. Core information processing deficits in developmental dyscalculia and low numeracy. Developmental Science 2008;11(5):669-680.
78. Rousselle L, Noel MP. Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs. non-symbolic magnitude processing. Cognition 2007;102(3):361-395.
79. Elliott L, Feigenson L, Halberda J, Libertus ME. Bidirectional, longitudinal associations between math ability and approximate number system precision in childhood. Journal of Cognition and Development 2019;20(1):56-74.
80. Hannula MM, Lehtinen E. Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction 2005;15(3):237-256.
81. Gray SA, Reeve RA. Number-specific and general cognitive markers of preschoolers’ math ability profiles. Journal of Experimental Child Psychology 2016. 147: 1-21.
82. Edens KM, Potter EF. An exploratory look at the relationships among math skills, motivational factors, and activity choice. Early Childhood Education 2013;41(3):235-243.
83. Hannula MM, Lepola J, Lehtinen E. Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology 2010;107(4):394-406.
84. Hannula-Sormunen MM, Lehtinen E, Räsänen P. Preschool children’s spontaneous focusing on numerosity, subitizing, and counting skills as predictors of their mathematical performance seven years later at school. Mathematical Thinking and Learning 2015;17(2-3):155-177.
85. Gloor N, Leuenberger D, Moser Opitz E. Disentangling the Effects of SFON (Spontaneous Focusing on Numerosity) and Symbolic Number Skills on the Mathematical Achievement of First Graders. A Longitudinal Study. Frontiers in Education 2021;6:629201.
86. Braham EJ, Libertus ME, McCrink K. Children’s spontaneous focus on number before and after guided parent-child interactions in a children’s museum. Developmental Psychology 2018;54(8): 1492-1498.
87. Hannula-Sormunen M, Nanu C, Luomaniemi K, Heinonen M, Sorariutta A, Södervik I, Mattinen A. Promoting spontaneous focusing on numerosity and cardinality-related skills at day care with one, two, how many and count, how many programs. Mathematical Thinking and Learning 2020;22(4):312-331.
88. Hornburg CB, Borriello GA, Kung M, Lin J, Litkowski E, Cosso J, Ellis A, King Y, Zippert E, Cabrera NJ, Davis-Kean P. Next directions in measurement of the home mathematics environment: An international and interdisciplinary perspective. Journal of Numerical Cognition 2021;7(2):195-220.
89. Mutaf-Yıldız B, Sasanguie D, De Smedt B, Reynvoet B. Probing the relationship between home numeracy and children's mathematical skills: A systematic review. Frontiers in Psychology 2020;11:2074.
90. LeFevre JA, Skwarchuk SL, Smith-Chant BL, Fast L, Kamawar D, Bisanz J. Home numeracy experiences and children’s math performance in the early school years. Canadian Journal of Behavioural Science/Revue canadienne des sciences du comportement 2009;41(2):55-66.
91. Anders Y, Rossbach HG, Weinert S, Ebert S, Kuger S, Lehrl S, Von Maurice J. Home and preschool learning environments and their relations to the development of early numeracy skills. Early Childhood Research Quarterly 2012;27(2):231-244.
92. Leyva D, Yeomans-Maldonado G, Weiland C, Shapiro A. Latino kindergarteners' math growth, approaches to learning, and home numeracy practices. Journal of Applied Developmental Psychology 2022;80:101417.
93. Missall K, Hojnoski RL, Caskie GI, Repasky P. Home numeracy environments of preschoolers: Examining relations among mathematical activities, parent mathematical beliefs, and early mathematical skills. Early Education and Development 2015;26(3):356-376.
94. Napoli AR, Purpura DJ. The home literacy and numeracy environment in preschool: Cross-domain relations of parent–child practices and child outcomes. Journal of Experimental Child Psychology 2018;166:581-603.
95. Levine SC, Suriyakham LW, Rowe ML, Huttenlocher J, Gunderson EA. What counts in the development of young children's number knowledge? Developmental Psychology 2010;46(5):1309-1319.
96. Mendelsohn A, Suárez-Rivera C, Suh DD, Tamis-LeMonda CS. Word by word: Everyday math talk in the homes of Hispanic families. Language Learning and Development 2023 Oct 2;19(4):386-403.
97. Ramani GB, Rowe ML, Eason SH, Leech KA. Math talk during informal learning activities in Head Start families. Cognitive Development 2015;35:15-33.
98. Susperreguy MI, Davis-Kean PE. Maternal math talk in the home and math skills in preschool children. Early Education and Development 2016;27(6):841-857.
99. Eason SH, Nelson AE, Dearing E, Levine SC. Facilitating young children’s numeracy talk in play: The role of parent prompts. Journal of Experimental Child Psychology 2021;207:105124.
100. Gibson DJ, Gunderson EA, Levine SC. Causal effects of parent number talk on preschoolers’ number knowledge. Child Development 2020;91(6):e1162-77.
101. Gunderson EA, Levine SC. Some types of parent number talk count more than others: Relations between parents’ input and children’s cardinal‐number knowledge. Developmental Science 2011;14(5):1021-1032.