Separable effects of the approximate number system, symbolic number knowledge, and number ordering ability on early arithmetic development

Attout, 2017, Serial order working memory and numerical ordinal processing share common processes and predict arithmetic ability, British Journal of Developmental Psychology, 36, 285, 10.1111/bjdp.12211 Attout, 2014, The relationship between working memory for serial order and numerical development: A longitudinal study, Developmental Psychology, 50, 1667, 10.1037/a0036496 Australian Curriculum, Assessment and Reporting Authority, 2013 Berteletti, 2010, Numerical estimation in preschoolers, Developmental Psychology, 46, 545, 10.1037/a0017887 Booth, 2006, Developmental and individual differences in pure numerical estimation, Developmental Psychology, 42, 189, 10.1037/0012-1649.41.6.189 Brankaer, 2014, Numerical magnitude processing deficits in children with mathematical difficulties are independent of intelligence, Research in Developmental Disabilities, 35, 2603, 10.1016/j.ridd.2014.06.022 Brannon, 2002, The development of ordinal numerical knowledge in infancy, Cognition, 83, 223, 10.1016/S0010-0277(02)00005-7 Brigstocke, 2016 Bull, 2014, Executive functioning and mathematics achievement, Child Development Perspectives, 8, 36, 10.1111/cdep.12059 Burgoyne, 2019, Pattern understanding is a predictor of early reading and arithmetic skills, Early Childhood Research Quarterly, 49, 69, 10.1016/j.ecresq.2019.06.006 Cantlon, 2009, The neural development of an abstract concept of number, Journal of Cognitive Neuroscience, 21, 2217, 10.1162/jocn.2008.21159 Caravolas, 2012, Common patterns of prediction of literacy development in different alphabet orthographies, Psychological Science, 23, 678, 10.1177/0956797611434536 Carey, 2017, Do analog number representations underlie the meanings of young children’s verbal numerals?, Cognition, 168, 243, 10.1016/j.cognition.2017.06.022 Castronovo, 2012, Impact of high mathematics education on the number sense, PLoS One, 7, e33832, 10.1371/journal.pone.0033832 Chen, 2014, Association between individual differences in non-symbolic number acuity and math performance, Acta Psychologica, 148, 163, 10.1016/j.actpsy.2014.01.016 Cohen, 1992, A power primer, Psychological Bulletin, 112, 155, 10.1037/0033-2909.112.1.155 Cragg, 2017, Direct and indirect influences of executive functions on mathematics achievement, Cognition, 162, 12, 10.1016/j.cognition.2017.01.014 Cui, 2017, Examining the relationship between rapid automatized naming and arithmetic fluency in Chinese kindergarten children, Journal of Experimental Child Psychology, 154, 146, 10.1016/j.jecp.2016.10.008 De Smedt, 2013, The relationship between symbolic and non-symbolic numerical magnitude processing and the typical and atypical development of mathematics: Evidence from brain and behavior, Trends in Neuroscience and Education, 2, 48, 10.1016/j.tine.2013.06.001 Dehaene, 1997 Dehaene, 1995, Towards and anatomical and functional model of number processing, Mathematical Cognition, 1, 83 Donker, 2016, Alphanumeric and non-alphanumeric rapid automatized naming in children with reading and/or spelling difficulties and mathematical difficulties, Learning and Individual Differences, 47, 80, 10.1016/j.lindif.2015.12.011 Fazio, 2014, Relations of different types of numerical magnitude representations to each other and to mathematics achievement, Journal of Experimental Child Psychology, 123, 53, 10.1016/j.jecp.2014.01.013 Feigenson, 2004, Core systems of number, Trends in Cognitive Science, 8, 307, 10.1016/j.tics.2004.05.002 Feigenson, 2013, Links between the intuitive sense of number and formal mathematics ability, Child Development Perspectives, 7, 74, 10.1111/cdep.12019 Fuhs, 2013, ANS acuity and mathematics ability in preschoolers from low-income homes: Contributions of inhibitory control, Developmental Science, 16, 136, 10.1111/desc.12013 Gallistel, 2000, Non-verbal numerical cognition: From reals to integers, Trends in Cognitive Science, 4, 59, 10.1016/S1364-6613(99)01424-2 Gilmore, 2013, Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement, PLoS One, 8, e67374, 10.1371/journal.pone.0067374 Göbel, 2014, Children’s arithmetic development: It is number knowledge, not the approximate number sense, that counts, Psychological Science, 25, 789, 10.1177/0956797613516471 Goffin, 2016, Beyond magnitude: Judging ordinality of symbolic number is unrelated to magnitude comparison and independently relates to individual differences in arithmetic, Cognition, 150, 68, 10.1016/j.cognition.2016.01.018 Halberda, 2008, 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, 44, 1457, 10.1037/a0012682 Hawes, 2019, Neural underpinnings of numerical and spatial cognition: An fMRI meta-analysis of brain regions associated with symbolic number, arithmetic, and mental rotation, Neuroscience & Biobehavioral Reviews, 103, 316, 10.1016/j.neubiorev.2019.05.007 Hornung, 2017, General and specific contributions of RAN to reading and arithmetic fluency in first graders: A longitudinal latent variable approach, Frontiers in Psychology, 8, 10.3389/fpsyg.2017.01746 Hulme, 2012, The causal role of phoneme awareness and letter–sound knowledge in learning to read: Combining intervention studies with mediation analyses, Psychological Science, 23, 572, 10.1177/0956797611435921 Kaufmann, 2009, Numerical and non-numerical ordinality processing in children with and without developmental dyscalculia: Evidence from fMRI, Cognitive Development, 24, 486, 10.1016/j.cogdev.2009.09.001 Kolkman, 2013, The role of executive functions in numerical skills, Learning and Individual Differences, 24, 145, 10.1016/j.lindif.2013.01.004 Koponen, 2018, Cognitive predictors of counting skills, Journal of Numerical Cognition, 4, 410, 10.5964/jnc.v4i2.116 Koponen, 2017, A meta-analysis of the relation between RAN and mathematics, Journal of Educational Psychology, 109, 977, 10.1037/edu0000182 Koponen, 2013, Counting and RAN: Predictors of arithmetic calculation and reading fluency, Journal of Educational Psychology, 105, 162, 10.1037/a0029285 Koponen, 2016, Counting and rapid naming predict the fluency of arithmetic and reading skills, Contemporary Educational Psychology, 44–45, 83, 10.1016/j.cedpsych.2016.02.004 Krajewski, 2009, 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, 103, 516, 10.1016/j.jecp.2009.03.009 Lervåg, 2009, Rapid automatized naming (RAN) taps a mechanism that places constraints on the development of early reading fluency, Psychological Science, 20, 1040, 10.1111/j.1467-9280.2009.02405.x Libertus, 2016, The precision of mapping between number words and the approximate number system predicts children’s formal math abilities, Journal of Experimental Child Psychology, 150, 207, 10.1016/j.jecp.2016.06.003 Lyons, 2015, Numerical order processing in children: From reversing the distance-effect to predicting arithmetic, Mind, Brain and Education, 9, 207, 10.1111/mbe.12094 Lyons, 2009, Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols, Cognition, 113, 189, 10.1016/j.cognition.2009.08.003 Lyons, 2011, Numerical ordering ability mediates the relation between number-sense and arithmetic competence, Cognition, 121, 256, 10.1016/j.cognition.2011.07.009 Lyons, 2018, Symbolic number skills predict growth in nonsymbolic number skills in kindergarteners, Developmental Psychology, 54, 440, 10.1037/dev0000445 Lyons, 2014, Numerical predictors of arithmetic success in Grades 1–6, Developmental Science, 17, 714, 10.1111/desc.12152 Malone, 2020, Number knowledge and the approximate number system are two critical foundations for early arithmetic development, Journal of Educational Psychology, 112, 1167, 10.1037/edu0000426 Malone, 2019, Learning correspondences between magnitudes, symbols and words: Evidence for a triple code model of arithmetic development, Cognition, 187, 1, 10.1016/j.cognition.2018.11.016 Malone, 2019, The relationship between numerosity discrimination and arithmetic skill reflects the approximate number system and cannot be explained by inhibitory control, Journal of Experimental Child Psychology, 184, 220, 10.1016/j.jecp.2019.02.009 Matejko, 2016, Trajectories of symbolic and nonsymbolic magnitude processing in the first year of formal school, PLoS One, 11, e149863, 10.1371/journal.pone.0149863 Mazzocco, 2011, Preschoolers’ precision of the approximate number system predicts later school mathematics performance, PLoS One, 6, e23749, 10.1371/journal.pone.0023749 Mix, 2008, Children’s numerical equivalence judgments: Crossmapping effects, Cognitive Development, 23, 191, 10.1016/j.cogdev.2007.03.001 Mix, 1996, Do preschool children recognize auditory–visual numerical correspondences, Child Development, 67, 1592, 10.2307/1131720 Mix, 2002, Multiple cues for quantification in infancy: Is number one of them?, Psychology Bulletin, 128, 278, 10.1037/0033-2909.128.2.278 Morsanyi, 2017, Number comparison and number ordering as predictors of arithmetic performance in adults: Exploring the link between the two skills, and investigating the question of domain-specificity, Quarterly Journal of Experimental Psychology, 70, 2497, 10.1080/17470218.2016.1246577 Mundy, 2009, Children’s mapping between symbolic and nonsymbolic representations of number, Journal of Experimental Child Psychology, 103, 490, 10.1016/j.jecp.2009.02.003 Mussolin, 2016, How approximate and exact number skills are related to each other across development: A review, Developmental Review, 39, 1, 10.1016/j.dr.2014.11.001 Muthén, L. K., & Muthén, B. O. (1998–2006). Mplus user’s guide (5th ed.). Los Angeles: Muthén & Muthén. Nieder, 2005, Counting on neurons: The neurobiology of numerical competence, Nature Reviews Neuroscience, 6, 1, 10.1038/nrn1626 Nieder, 2009, Representation of numbers in the brain, Annual Review of Neuroscience, 32, 185, 10.1146/annurev.neuro.051508.135550 Nosworthy, 2013, A two-minute paper-and-pencil test of symbolic and nonsymbolic numerical magnitude processing explains variability in primary school children’s arithmetic competence, PLoS One, 8, e67918, 10.1371/journal.pone.0067918 O’Connor, 2018, Young children’s non-numerical ordering ability at the start of formal education longitudinally predicts their symbolic number skills and academic achievement in maths, Developmental Science, 21, e12645, 10.1111/desc.12645 Pauly, 2011, Domain-specific rapid automatized naming deficits in children at risk for learning disabilities, Journal of Neurolinguistics, 24, 602, 10.1016/j.jneuroling.2011.02.002 Piazza, 2004, Tuning curves for approximate numerosity in the human intraparietal sulcus, Neuron, 44, 547, 10.1016/j.neuron.2004.10.014 Queensland Government (2020). Early childhood education and care: About prep. Retrieved from https://education.qld.gov.au/. Sasanguie, 2017, About why there is a shift from cardinal to ordinal processing in the association with arithmetic between first and second grade, Developmental Science, 21, e12653, 10.1111/desc.12653 Schneider, 2017, Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: A meta-analysis, Developmental Science, 20, 1, 10.1111/desc.12372 Szűcs, 2017, A critical analysis of design, facts, bias and inference in the approximate number system training literature: A systematic review, Trends in Neuroscience and Education, 6, 187, 10.1016/j.tine.2016.11.002 Vanbinst, 2015, Does numerical processing uniquely predict first graders’ future development of single-digit arithmetic?, Learning and Individual Differences, 37, 153, 10.1016/j.lindif.2014.12.004 Verguts, 2004, Representation of number in animals and humans: A neural model, Journal of Cognitive Neuroscience, 16, 1493, 10.1162/0898929042568497 Vogel, 2017, Processing the order of symbolic numbers: A reliable and unique predictor of arithmetic fluency, Journal of Numerical Cognition, 3, 288, 10.5964/jnc.v3i2.55 Vogel, 2015, Differential processing of symbolic numerical magnitude and order in first-grade children, Journal of Experimental Child Psychology, 129, 26, 10.1016/j.jecp.2014.07.010 Wang, 2020, Exploring mechanisms of rapid automatized naming to arithmetic skills in Chinese primary schoolers, Psychology in the Schools, 57, 556, 10.1002/pits.22349 Wise, 2008, Phonological awareness and rapid naming skills of children with reading disabilities and children with reading disabilities who are at risk for mathematics difficulties, Learning Disabilities Research & Practice, 23, 125, 10.1111/j.1540-5826.2008.00270.x Xenidou-Dervou, 2017, Nonsymbolic and symbolic magnitude comparison skills as longitudinal predictors of mathematical achievement, Learning and Instruction, 50, 1, 10.1016/j.learninstruc.2016.11.001 Zhang, 2017, Knowing, applying, and reasoning about arithmetic: Roles of domain-general and numerical skills in multiple domains of arithmetic learning, Developmental Psychology, 53, 2304, 10.1037/dev0000432