The effect of first school years on mathematical skill profiles
Main Article Content
Abstract
This study investigated the effect of children’s first formal school years on mathematical skill profiles, measured by a variety of arithmetical skills and Spontaneous Focusing On Numerosity (SFON) tasks. By using person-centered approach the aim was to investigate whether the amount of formal schooling is associated with mathematical skills in the same way for all children, or, whether the associations differ according to the children’s mathematical skill profiles. Data was analyzed from 652 4–7-year-old children from four European countries with different school entrance ages. A person-centered approach with latent profile regression analyses on four-factor score variables identified six mathematical skill profiles with both qualitative and quantitative differences. The results revealed significant, but small effects of the amount of schooling on mathematical profiles when chronological age and country-specific school entrance age were controlled for. Educational implications of the findings emphasize regarding the heterogeneity in children’s mathematical skill profiles and the potentially different effects of starting formal schooling across different profiles.
Article Details
How to Cite
Nanu, C., Laakkonen, E., & Hannula-Sormunen, M. (2020). The effect of first school years on mathematical skill profiles. Frontline Learning Research, 8(1), 56–75. https://doi.org/10.14786/flr.v8i1.485
Section
Articles
FLR adopts the Attribution-NonCommercial-NoDerivs Creative Common License (BY-NC-ND). That is, Copyright for articles published in this journal is retained by the authors with, however, first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
References
Abenavoli, R. M., Greenberg, M. T., & Bierman, K. L. (2017). Identification and validation of school readiness profiles among high-risk kindergartners. Early Childhood Research Quarterly, 38, 33–43. https://doi.org/10.1016/j.ecresq.2016.09.001
Altwicker-Hámori, S., & Köllő, J. (2012). Whose children gain from starting school later?—Evidence from Hungary. Educational Research and Evaluation, 18(October), 459–488. https://doi.org/10.1080/13803611.2012.695142
Aunio, P., Korhonen, J., Bashash, L., & Khoshbakht, F. (2014). Children’s early numeracy in Finland and Iran. International Journal of Early Years Education, 22(4), 423–440.
Aunola, K., Leskinen, E., Lerkkanen, M.-K., & Nurmi, J.-E. (2004). Developmental dynamics of math performance from preschool to grade 2. Journal of Educational Psychology, 96(4), 699–713. https://doi.org/10.1037/0022-0663.96.4.699
Authors (submitted). The effect of school starting age on children`s spontaneous focusing on numerosity and mathematical skills.
Batchelor, S., Inglis, M., & Gilmore, C. (2015). Spontaneous focusing on numerosity and the arithmetic advantage. Learning and Instruction, 40, 79–88. https://doi.org/10.1016/j.learninstruc.2015.09.005
Bisanz, J., Morrison, F. J., & Dunn, M. (1995). Effects of age and schooling on the acquisition of elementary quantitative skills. Developmental Psychology, 31(2), 221-236. http://dx.doi.org/10.1037/0012-1649.31.2.221
Black, S. E., Devereux, P. J., & Salvanes, K. G. (2011). Too young to leave the nest? The effects of school starting age. Review of Economics and Statistics, 93(2), 455–467. https://doi.org/10.1162/REST_a_00081
Authors (2016). Development of SFON in Ecuadorian kindergartners. European Journal of Psychology of Education, 1–14. https://doi.org/10.1007/s10212-016-0306-9
Caro, D. H., & Cortés, D. (2012). Measuring family socioeconomic status: An illustration using data from PIRLS 2006. IERI Monograph Series: Issues and Methodologies in Large-Scale Assessments, 5, 9–33. https://doi.org/10.1787/9789264091504-en
Chew, C. S., Forte, J. D., & Reeve, R. A. (2016). Cognitive factors affecting children`s nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis. Journal of Experimental Child Psychology, 152, 173-191. https://doi.org/10.1016/j.jecp.2016.07.001
Clements, D. H., & Sarama, J. (2007). Early childhood mathematics learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 461–488). Charlotte, NC: Information Age Publishing.
Clements, D. H., & Sarama, J. (2014). The importance of early years. In R. E. Slavin (Ed.), Science, technology & mathematics (STEM) (pp. 5-9). Thousand Oaks, CA: Corwin.
Cliffordson, C., & Gustafsson, J.-E. (2010). Effects of schooling and age on performance in mathematics and science: A between-grade regression discontinuity design with instrumental variables applied to Swedish TIMSS 1995 data. Fourth IEA International Research Conference (IRC-2010), Gothenburg, Sweden, 1–13.
Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition analysis. New Jersey, NJ: John Wiley & Sons.
Crosnoe, R., Morrison, F., Burchinal, M., Pianta, R., Keating, D., Friedman, S. L., … The Eunice Kennedy Shriver National Institute of Child Health and Human Development Early Child Care Research Network. (2010). Instruction, teacher-student relations, and math achievement trajectories in elementary school. Journal of Educational Psychology, 102(2), 407–417. http://doi.org/10.1037/a0017762
Datar, A. (2006). Does delaying kindergarten entrance give children a head start? Economics of Education Review, 25, 43-62.
Dowker, A. (2005). Individual differences in arithmetic. Implications for psychology, neuroscience and education. New York, NY: Psychology Press.
Dowker, A. (2008). Individual differences in numerical abilities in preschoolers. Developmental Science, 11(5), 650–654. https://doi.org/10.1111/j.1467-7687.2008.00713.x
Edens, K. M., & Potter, E. F. (2013). An exploratory look at the relationships among math skills, motivational factors and activity choice. Early Childhood Education Journal, 41(3), 235–243. https://doi.org/10.1007/s10643-012-0540-y
FitzSimons, G. (2013). Doing mathematics in the workplace. A brief review of selected literature. Adult Learning Mathematics: An International Journal, 8(1), 7-19.
Fuson, K. (1988). Children’s counting and concepts of number. New York, NY: Springer Verlag.
Ginsburg, H. P., & Seo, K.-H. (1999). Mathematics in children’s thinking. Mathematical Thinking and Learning, 1, 113-129.
Gray, S. A., & Reeve, R. A. (2016). Number-specific and general cognitive markers of preschoolers’ math ability profiles. Journal of Experimental Child Psychology, 147, 1–21. https://doi.org/10.1016/j.jecp.2016.02.004
Hamre, B. K., & Pianta, R. C. (2005). Can instructional and emotional support in the first grade classroom make a difference for children at risk of school failure? Child Development, 76(5), 949–967. https://doi.org/10.1111/j.1467-8624.2005.00889.x
Authors(2015). 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, 17(2–3), 155–177. https://doi.org/10.1080/10986065.2015.1016814
Authors(2009). Neural correlates of Spontaneous Focusing on Numerosity (SFON) in a 9-year-longitudinal study of children’s mathematical skills, paper presented at the biennial meeting of the European Association for Research in Learning and Instruction, Amsterdam, The Netherlands.
Authors(2009). Neural correlates of spontaneous focusing on numerosity (SFON). NeuroImage, 47, S39–S41.
Authors(2005). Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction, 15(3), 237–256. https://doi.org/10.1016/j.learninstruc.2005.04.005
Authors(2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107(4), 394–406. https://doi.org/10.1016/j.jecp.2010.06.004
Authors(2005). Does social interaction influence 3-year-old children’s tendency to focus on numerosity? A quasi-experimental study in day care. In L. Verschaffel, E. De Corte, G. Kanselaar, & M. Valcke (Eds.), Powerful environments for promoting deep conceptual and strategic learning (Studia Paedagogica 41) (pp. 63–80). Leuven, The Netherlands: Leuven University Press.
Authors(2017). Early mathematical skill profiles of prematurely and full-term born children. Learning and Individual Differences, 55, 108-119. https://doi.org/10.1016/j.lindif.2017.03.004
Authors(2007). Development of counting skills: Role of spontaneous focusing on numerosity and subitizing-based enumeration. Mathematical Thinking and Learning, 9(1), 51–57. https://doi.org/10.1207/s15327833mtl0901_4
Herbst, M., & Strawiński, P. (2015). Early effects of an early start: Evidence from lowering the school starting age in Poland. Journal of Policy Modeling, 38, 256–271. https://doi.org/10.1016/j.jpolmod.2016.01.004
Hickendorff, M., Edelsbrunner, P. A., McMullen, J., Schneider, M., & Trezise, K. (2017). Informative tools for characterizing individual differences in learning: Latent class, latent profile, and latent transition analysis. Learning and Individual Differences, (November). https://doi.org/10.1016/j.lindif.2017.11.001
Hu, L. & Bentler, P. (1999). Cutoff criteria for fit indices in covariance structure analysis: conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1-55.
Huttenlocher, J., Jordan, N. C., & Levine, S. C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123(3), 284–296. https://doi.org/10.1037//0096-3445.123.3.284
Kavkler, M., Tancig, S., Magajna, L., Aubrey, C., Göbel, S. M., Watson, S. E., … Oeckert. (2000). Is early learning really more productive? The effect of school starting age on school and labor market performance. European Early Childhood Education Research Journal, 8(1659), 45. https://doi.org/10.1177/0956797613516471
Locuniak, M. N., & Jordan, N. C. (2008). Using kindergarten number sense to predict calculation fluency in second grade. Journal of Learning Disabilities, 41(5), 451–459.
Luyten, H. (2006). An empirical assessment of the absolute effect of schooling: regression‐discontinuity applied to TIMSS‐95. Oxford Review of Education, 32(3), 397–429. https://doi.org/10.1080/03054980600776589
Magidson, J., & Vermunt, J. K. (2002). Latent class models for clustering: A comparison with K-means. Canadian Journal of Marketing Research, 20, 37–44.
Magnusson, D. (2003). The person approach: Concepts, measurement models, and research strategy. New Directions for Child and Adolescent Development, 2003(101), 3–23. https://doi.org/10.1002/cd.79
Marsh, H. W., Lüdtke, O., Trautwein, U., & Morin, A. J. S. (2009). Classical latent profile analysis of academic self-concept dimensions: Synergy of person- and variable-centered approaches to theoretical models of self-concept. Structural Equation Modeling: A Multidisciplinary Journal, 16. http://dx.doi.org/10.1080/10705510902751010
Martin, R. B., Cirino, P. T., Sharp, C., & Barnes, M. (2014). Number and counting skills in kindergarten as predictors of grade 1 mathematical skills. Learning and Individual Differences, 34, 12–23. https://doi.org/10.1016/j.lindif.2014.05.006
Masyn, K. E. (2013). Latent class analysis with finite mixture modeling. In T. D. Little (Ed.), The Oxford handbook of quantitative methods. New York, NY: Oxford University Press.
Melhuish, E., Sylva, K., Sammons, P., Siraj-Blatchford, I., Taggart, B., Phan, M., & Malin, A.
(2008). Preschool influences on mathematics achievement. Science, 321, 1161-1162.
Muthén, L. K., & Muthén, B. O. (n.d.). Mplus user’s guide (7th ed.). Los Angeles, CA: Muthén & Muthén.
Authors (2018). Spontaneous focusing on numerosity in preschool as a predictor of mathematical skills and knowledge in the fifth grade. Journal of Experimental Child Psychology, 169, 42-58. https://doi.org/10.1016/j.jecp.2017.12.011
Nylund-Gibson, K., & Masyn, K. E. (2016). Covariates and mixture modeling: Results of a simulation study exploring the impact of misspecified effects on class enumeration. Structural Equation Modeling: A Multidisciplinary Journal, 23(6), 782–797. https://doi.org/10.1080/10705511.2016.1221313
Nylund, K. L., Asparouhov, T., & Muthén, B. O. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling, 14(4), 535–569. https://doi.org/10.1080/10705510701575396
Purpura, D. J., Baroody, A. J., & Lonigan, C. J. (2013). The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge. Journal of Educational Psychology, 105(2), 453–464. https://doi.org/10.1037/a0031753
Authors (2016). Kindergartners’ spontaneous focusing on numerosity in relation to their number-related utterances during numerical picture book reading. Mathematical Thinking and Learning, 18(2), 125-141. doi: 10.1080/10986065.2016.1148531
Authors (2017). Verbal and action-based measures of kindergartners’ SFON and their associations with number-related utterances during picture book reading. British Journal of Educational Psychology.. https://doi.org/10.1111/bjep.12201
Sprietsma, M. (2010). Effect of relative age in the first grade of primary school on long‐term scholastic results: International comparative evidence using PISA 2003. Education Economics, 18(1), 1–32. https://doi.org/10.1080/09645290802201961
Sylva, K., Melhuish, E., Sammons, P., Siraj-Blatchford, I. & Taggart, B. (2008). Effective pre-school and primary education 3-11 Project (EPPE 3-11)—Final report from the primary phase: Pre-school, school and family influences on children’s development during key stage, 2(7-11). London, United Kingdom: Department for Children Schools and Families.
Van de Rijt, B., Godfrey, R., Aubrey, R., van Luit, J. E. H., Ghesquière, P., Torbeyns, J., … Tzouriadou, M. (2003). The development of early numeracy in Europe. Journal of Early Childhood Research, 1(2), 155–180. https://doi.org/10.1177/1476718X030012002
Watts, T. W., Clements, D. H., Sarama, J., Wolfe, C. B., Spitler, M. E., & Bailey, D. H. (2017). Does early mathematics intervention change the processes underlying children’s learning? Journal of Research on Educational Effectiveness, 10(1), 96–115. https://doi.org/10.1080/19345747.2016.1204640
Wolke, D., Strauss, V. Y. C., Johnson, S., Gilmore, C., Marlow, N., & Jaekel, J. (2015). Universal gestational age effects on cognitive and basic mathematic processing: 2 cohorts in 2 countries. Journal of Pediatrics, 166, 1410–1416. http://dx.doi.org/10.1016/j.jpeds.2015.02.065
Altwicker-Hámori, S., & Köllő, J. (2012). Whose children gain from starting school later?—Evidence from Hungary. Educational Research and Evaluation, 18(October), 459–488. https://doi.org/10.1080/13803611.2012.695142
Aunio, P., Korhonen, J., Bashash, L., & Khoshbakht, F. (2014). Children’s early numeracy in Finland and Iran. International Journal of Early Years Education, 22(4), 423–440.
Aunola, K., Leskinen, E., Lerkkanen, M.-K., & Nurmi, J.-E. (2004). Developmental dynamics of math performance from preschool to grade 2. Journal of Educational Psychology, 96(4), 699–713. https://doi.org/10.1037/0022-0663.96.4.699
Authors (submitted). The effect of school starting age on children`s spontaneous focusing on numerosity and mathematical skills.
Batchelor, S., Inglis, M., & Gilmore, C. (2015). Spontaneous focusing on numerosity and the arithmetic advantage. Learning and Instruction, 40, 79–88. https://doi.org/10.1016/j.learninstruc.2015.09.005
Bisanz, J., Morrison, F. J., & Dunn, M. (1995). Effects of age and schooling on the acquisition of elementary quantitative skills. Developmental Psychology, 31(2), 221-236. http://dx.doi.org/10.1037/0012-1649.31.2.221
Black, S. E., Devereux, P. J., & Salvanes, K. G. (2011). Too young to leave the nest? The effects of school starting age. Review of Economics and Statistics, 93(2), 455–467. https://doi.org/10.1162/REST_a_00081
Authors (2016). Development of SFON in Ecuadorian kindergartners. European Journal of Psychology of Education, 1–14. https://doi.org/10.1007/s10212-016-0306-9
Caro, D. H., & Cortés, D. (2012). Measuring family socioeconomic status: An illustration using data from PIRLS 2006. IERI Monograph Series: Issues and Methodologies in Large-Scale Assessments, 5, 9–33. https://doi.org/10.1787/9789264091504-en
Chew, C. S., Forte, J. D., & Reeve, R. A. (2016). Cognitive factors affecting children`s nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis. Journal of Experimental Child Psychology, 152, 173-191. https://doi.org/10.1016/j.jecp.2016.07.001
Clements, D. H., & Sarama, J. (2007). Early childhood mathematics learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 461–488). Charlotte, NC: Information Age Publishing.
Clements, D. H., & Sarama, J. (2014). The importance of early years. In R. E. Slavin (Ed.), Science, technology & mathematics (STEM) (pp. 5-9). Thousand Oaks, CA: Corwin.
Cliffordson, C., & Gustafsson, J.-E. (2010). Effects of schooling and age on performance in mathematics and science: A between-grade regression discontinuity design with instrumental variables applied to Swedish TIMSS 1995 data. Fourth IEA International Research Conference (IRC-2010), Gothenburg, Sweden, 1–13.
Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition analysis. New Jersey, NJ: John Wiley & Sons.
Crosnoe, R., Morrison, F., Burchinal, M., Pianta, R., Keating, D., Friedman, S. L., … The Eunice Kennedy Shriver National Institute of Child Health and Human Development Early Child Care Research Network. (2010). Instruction, teacher-student relations, and math achievement trajectories in elementary school. Journal of Educational Psychology, 102(2), 407–417. http://doi.org/10.1037/a0017762
Datar, A. (2006). Does delaying kindergarten entrance give children a head start? Economics of Education Review, 25, 43-62.
Dowker, A. (2005). Individual differences in arithmetic. Implications for psychology, neuroscience and education. New York, NY: Psychology Press.
Dowker, A. (2008). Individual differences in numerical abilities in preschoolers. Developmental Science, 11(5), 650–654. https://doi.org/10.1111/j.1467-7687.2008.00713.x
Edens, K. M., & Potter, E. F. (2013). An exploratory look at the relationships among math skills, motivational factors and activity choice. Early Childhood Education Journal, 41(3), 235–243. https://doi.org/10.1007/s10643-012-0540-y
FitzSimons, G. (2013). Doing mathematics in the workplace. A brief review of selected literature. Adult Learning Mathematics: An International Journal, 8(1), 7-19.
Fuson, K. (1988). Children’s counting and concepts of number. New York, NY: Springer Verlag.
Ginsburg, H. P., & Seo, K.-H. (1999). Mathematics in children’s thinking. Mathematical Thinking and Learning, 1, 113-129.
Gray, S. A., & Reeve, R. A. (2016). Number-specific and general cognitive markers of preschoolers’ math ability profiles. Journal of Experimental Child Psychology, 147, 1–21. https://doi.org/10.1016/j.jecp.2016.02.004
Hamre, B. K., & Pianta, R. C. (2005). Can instructional and emotional support in the first grade classroom make a difference for children at risk of school failure? Child Development, 76(5), 949–967. https://doi.org/10.1111/j.1467-8624.2005.00889.x
Authors(2015). 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, 17(2–3), 155–177. https://doi.org/10.1080/10986065.2015.1016814
Authors(2009). Neural correlates of Spontaneous Focusing on Numerosity (SFON) in a 9-year-longitudinal study of children’s mathematical skills, paper presented at the biennial meeting of the European Association for Research in Learning and Instruction, Amsterdam, The Netherlands.
Authors(2009). Neural correlates of spontaneous focusing on numerosity (SFON). NeuroImage, 47, S39–S41.
Authors(2005). Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction, 15(3), 237–256. https://doi.org/10.1016/j.learninstruc.2005.04.005
Authors(2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107(4), 394–406. https://doi.org/10.1016/j.jecp.2010.06.004
Authors(2005). Does social interaction influence 3-year-old children’s tendency to focus on numerosity? A quasi-experimental study in day care. In L. Verschaffel, E. De Corte, G. Kanselaar, & M. Valcke (Eds.), Powerful environments for promoting deep conceptual and strategic learning (Studia Paedagogica 41) (pp. 63–80). Leuven, The Netherlands: Leuven University Press.
Authors(2017). Early mathematical skill profiles of prematurely and full-term born children. Learning and Individual Differences, 55, 108-119. https://doi.org/10.1016/j.lindif.2017.03.004
Authors(2007). Development of counting skills: Role of spontaneous focusing on numerosity and subitizing-based enumeration. Mathematical Thinking and Learning, 9(1), 51–57. https://doi.org/10.1207/s15327833mtl0901_4
Herbst, M., & Strawiński, P. (2015). Early effects of an early start: Evidence from lowering the school starting age in Poland. Journal of Policy Modeling, 38, 256–271. https://doi.org/10.1016/j.jpolmod.2016.01.004
Hickendorff, M., Edelsbrunner, P. A., McMullen, J., Schneider, M., & Trezise, K. (2017). Informative tools for characterizing individual differences in learning: Latent class, latent profile, and latent transition analysis. Learning and Individual Differences, (November). https://doi.org/10.1016/j.lindif.2017.11.001
Hu, L. & Bentler, P. (1999). Cutoff criteria for fit indices in covariance structure analysis: conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1-55.
Huttenlocher, J., Jordan, N. C., & Levine, S. C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123(3), 284–296. https://doi.org/10.1037//0096-3445.123.3.284
Kavkler, M., Tancig, S., Magajna, L., Aubrey, C., Göbel, S. M., Watson, S. E., … Oeckert. (2000). Is early learning really more productive? The effect of school starting age on school and labor market performance. European Early Childhood Education Research Journal, 8(1659), 45. https://doi.org/10.1177/0956797613516471
Locuniak, M. N., & Jordan, N. C. (2008). Using kindergarten number sense to predict calculation fluency in second grade. Journal of Learning Disabilities, 41(5), 451–459.
Luyten, H. (2006). An empirical assessment of the absolute effect of schooling: regression‐discontinuity applied to TIMSS‐95. Oxford Review of Education, 32(3), 397–429. https://doi.org/10.1080/03054980600776589
Magidson, J., & Vermunt, J. K. (2002). Latent class models for clustering: A comparison with K-means. Canadian Journal of Marketing Research, 20, 37–44.
Magnusson, D. (2003). The person approach: Concepts, measurement models, and research strategy. New Directions for Child and Adolescent Development, 2003(101), 3–23. https://doi.org/10.1002/cd.79
Marsh, H. W., Lüdtke, O., Trautwein, U., & Morin, A. J. S. (2009). Classical latent profile analysis of academic self-concept dimensions: Synergy of person- and variable-centered approaches to theoretical models of self-concept. Structural Equation Modeling: A Multidisciplinary Journal, 16. http://dx.doi.org/10.1080/10705510902751010
Martin, R. B., Cirino, P. T., Sharp, C., & Barnes, M. (2014). Number and counting skills in kindergarten as predictors of grade 1 mathematical skills. Learning and Individual Differences, 34, 12–23. https://doi.org/10.1016/j.lindif.2014.05.006
Masyn, K. E. (2013). Latent class analysis with finite mixture modeling. In T. D. Little (Ed.), The Oxford handbook of quantitative methods. New York, NY: Oxford University Press.
Melhuish, E., Sylva, K., Sammons, P., Siraj-Blatchford, I., Taggart, B., Phan, M., & Malin, A.
(2008). Preschool influences on mathematics achievement. Science, 321, 1161-1162.
Muthén, L. K., & Muthén, B. O. (n.d.). Mplus user’s guide (7th ed.). Los Angeles, CA: Muthén & Muthén.
Authors (2018). Spontaneous focusing on numerosity in preschool as a predictor of mathematical skills and knowledge in the fifth grade. Journal of Experimental Child Psychology, 169, 42-58. https://doi.org/10.1016/j.jecp.2017.12.011
Nylund-Gibson, K., & Masyn, K. E. (2016). Covariates and mixture modeling: Results of a simulation study exploring the impact of misspecified effects on class enumeration. Structural Equation Modeling: A Multidisciplinary Journal, 23(6), 782–797. https://doi.org/10.1080/10705511.2016.1221313
Nylund, K. L., Asparouhov, T., & Muthén, B. O. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling, 14(4), 535–569. https://doi.org/10.1080/10705510701575396
Purpura, D. J., Baroody, A. J., & Lonigan, C. J. (2013). The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge. Journal of Educational Psychology, 105(2), 453–464. https://doi.org/10.1037/a0031753
Authors (2016). Kindergartners’ spontaneous focusing on numerosity in relation to their number-related utterances during numerical picture book reading. Mathematical Thinking and Learning, 18(2), 125-141. doi: 10.1080/10986065.2016.1148531
Authors (2017). Verbal and action-based measures of kindergartners’ SFON and their associations with number-related utterances during picture book reading. British Journal of Educational Psychology.. https://doi.org/10.1111/bjep.12201
Sprietsma, M. (2010). Effect of relative age in the first grade of primary school on long‐term scholastic results: International comparative evidence using PISA 2003. Education Economics, 18(1), 1–32. https://doi.org/10.1080/09645290802201961
Sylva, K., Melhuish, E., Sammons, P., Siraj-Blatchford, I. & Taggart, B. (2008). Effective pre-school and primary education 3-11 Project (EPPE 3-11)—Final report from the primary phase: Pre-school, school and family influences on children’s development during key stage, 2(7-11). London, United Kingdom: Department for Children Schools and Families.
Van de Rijt, B., Godfrey, R., Aubrey, R., van Luit, J. E. H., Ghesquière, P., Torbeyns, J., … Tzouriadou, M. (2003). The development of early numeracy in Europe. Journal of Early Childhood Research, 1(2), 155–180. https://doi.org/10.1177/1476718X030012002
Watts, T. W., Clements, D. H., Sarama, J., Wolfe, C. B., Spitler, M. E., & Bailey, D. H. (2017). Does early mathematics intervention change the processes underlying children’s learning? Journal of Research on Educational Effectiveness, 10(1), 96–115. https://doi.org/10.1080/19345747.2016.1204640
Wolke, D., Strauss, V. Y. C., Johnson, S., Gilmore, C., Marlow, N., & Jaekel, J. (2015). Universal gestational age effects on cognitive and basic mathematic processing: 2 cohorts in 2 countries. Journal of Pediatrics, 166, 1410–1416. http://dx.doi.org/10.1016/j.jpeds.2015.02.065