Confirmatory factor analysis of the indicators of basic early math skills

  1. de León, Sara C.
  2. Hernández-Cabrera, Juan A.
  3. Jiménez, Juan E.
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

Journal:
Current Psychology

ISSN: 1046-1310 1936-4733

Year of publication: 2020

Type: Article

DOI: 10.1007/S12144-019-00596-0 GOOGLE SCHOLAR

More publications in: Current Psychology

Abstract

A study was conducted to analyze the factorial structure and measurement invariance of the Curriculum-Based Measurement (CBM) Indicadores de Progreso de Aprendizaje en Matemáticas (IPAM [Indicators of Basic Early Math Skills]), in 2ndgrade Spanish students. The model proposed is a one-factor model in which the five IPAM tasks (i.e., number comparison, missing number, single-digit computation, multi-digit computation, and place value) serve as observable indicators for a single underlying factor (i.e., number sense). The IPAM, composed of three parallel forms (i.e., A, B, and C), was administered to 252 Spanish second graders, three times throughout the school year (i.e., fall, winter, and spring). Consequently, the goodness of fit of the proposed model was analyzed for each measurement time. Furthermore, longitudinal measurement invariance was explored to analyze whether the measurement model would remain stable throughout the three-time points of measurement. Discriminant, predictive, and concurrent validity were also tested. The results support that the number sense latent factor explains the common variance of the observable indicators throughout the school year. Each latent factor was found highly related to the next. Moreover, the IPAM showed adequate indices for discriminant, concurrent and predictive validity. We conclude that the IPAM is an appropriate measure to assess number sense competence in second grade. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Bibliographic References

  • Andrews, P., & Sayers, J. (2015). Identifying opportunities for grade one children to acquire foundational number sense: Developing a framework for cross cultural classroom analyses. Early Childhood Education Journal, 43(4), 257–267. https://doi.org/10.1007/s10643-014-0653-6.
  • Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333–339. https://doi.org/10.1177/00222194050380040901.
  • Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). New York: The Gildford Press.
  • Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, and Allied Disciplines, 46(1), 3–18. https://doi.org/10.1111/j.1469-7610.2004.00374.x.
  • Chan, W. W. L., Au, T. K., & Tang, J. (2014). Strategic counting: A novel assessment of place-value understanding. Learning and Instruction, 29, 78–94. https://doi.org/10.1016/j.learninstruc.2013.09.001.
  • Cho, S., Ryali, S., Geary, D. C., & Menon, V. (2011). How does a child solve 7+8? Decoding brain activity patterns associated with counting and retrieval strategies. Developmental Science, 14(5), 989–1001. https://doi.org/10.1111/j.1467-7687.2011.01055.x.
  • Clarke, B., Lembke, E. S., Hampton, D. D., & Hendricker, E. (2015). Understanding the R in RTI. What we know and what we need to know about measuring students response in mathematics. In R. Gersten & R. Newman-Gonchar (Eds.), Understanding RTI in Mathematics. Proven methods and applications (5th ed., pp. 35–48). Baltimore: Paul H. Brookes Publishing Co..
  • Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.) New York: Routledge.
  • Cragg, L., & Gilmore, C. (2014). Skills underlying mathematics: The role of executive function in the development of mathematics proficiency. Trends in Neuroscience and Education, 3, 63–68. https://doi.org/10.1016/j.tine.2013.12.001.
  • De Smedt, B., Noël, M. P., Gilmore, C., & Ansari, D. (2013). 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, 2, 48–55. https://doi.org/10.1016/j.tine.2013.06.001.
  • Dehaene, S. (2009). Origins of mathematical intuitions: The case of arithmetic. Annals of the New York Academy of Sciences, 1156, 232–259. https://doi.org/10.1111/j.1749-6632.2009.04469.x.
  • Deno, S. L. (2003). Curriculum-based measures : Development and perspectives. Assessment for Effective Intervention, 28(3–4), 3–12. https://doi.org/10.1177/073724770302800302.
  • Dyson, N. I., Jordan, N. C., Beliakoff, A., & Hassinger-Das, B. (2015). A kindergarten number-sense intervention with contrasting practice conditions for low-achieving children. Journal of Research in Mathematics Education, 46(3), 331–370. https://doi.org/10.5951/jresematheduc.46.3.0331.
  • Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. https://doi.org/10.1016/j.tics.2004.05.002.
  • Friso-van den Bos, I., Van Luit, J. E. H., Kroesbergen, E. H., Xenidou-Dervou, I., Van Lieshout, E. C. D. M., Van der Schoot, M., & Jonkman, L. M. (2015). Pathways of number line development in children. Predictors and risk for adverse mathematical outcome. Zeitschrift für Psychologie, 223(2), 120–128. https://doi.org/10.1027/2151-2604/a000210.
  • Geary, D. C. (2000). From infancy to adulthood: The development of numerical abilities. European Child and Adolescent Psychiatry, 9, 11–16. https://doi.org/10.1007/s007870070004.
  • Geary, D. C. (2013). Early foundations for mathematics learning and their relations to learning disabilities. Current Directions in Psychological Science, 22(1), 23–27. https://doi.org/10.1177/0963721412469398.
  • Geary, D. C., Hoard, M. K., Byrd-Craven, J., & DeSoto, M. C. (2004). Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability. Journal of Experimental Child Psychology, 88(2), 121–151. https://doi.org/10.1016/J.JECP.2004.03.002.
  • Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2012). Mathematical cognition deficits in children with learning disabilities and persistent low achievement: A five-year prospective study. Journal of Educational Psychology, 104(1), 206–223. https://doi.org/10.1037/a0025398.
  • Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge: Harvard University Press.
  • Gunderson, E. A., Ramirez, G., Beilock, S. L., & Levine, S. C. (2012). The relation between spatial skill and early number knowledge: The role of the linear number line. Developmental Psychology, 48(5), 1229–1241. https://doi.org/10.1037/a0027433.
  • Hair Jr., J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2010). Multivariate data analysis (7th ed.). Upper Saddle River: Pearson Prentice-Hall.
  • Hernández-Cabrera, J. A. (n.d.). ULLRToolbox. Retrieved June 14, 2019, from https://sites.google.com/site/ullrtoolbox/home.
  • Jiménez, J. E., & de León, S. C. (2017a). Análisis factorial confirmatorio de Indicadores de Progreso de Aprendizaje en Matemáticas (IPAM) en escolares de primer curso de Primaria [Confirmatory factor analysis of IPAM in first-grade students]. European Journal of Investigation in Health, Psychology and Education, 7(1), 31–45. https://doi.org/10.1989/ejihpe.v7i1.193
  • Jiménez, J. E., & de León, S. C. (2017b). Análisis factorial confirmatorio del IPAM en escolares de tercer curso de primaria [Confirmatory factor analysis of IPAMthird-grade schoolchildren]. Evaluar, 17(2), 81–96. Retrieved from https://revistas.unc.edu.ar/index.php/revaluar
  • Jiménez, J. E., & de León, S. C. (2019).Indicadores de progreso de aprendizaje en matemáticas (IPAM)-2º curso de educación primaria [Indicators of basic early math skills (IPAM)- 2nd grade of primary school]. In J. E. Jimenez (Ed.), Modelo de respuesta a la intervención. Un enfoque preventivo para el abordaje de las dificultades específicas de aprendizaje [Response to intervention model. A preventive approach for learning disabilities]. Madrid: Pirámide.
  • Jordan, N. C., & Levine, S. C. (2009). Socioeconomic variation, number competence, and mathematics learning difficulties in young children. Developmental Disabilities Research Reviews, 15(1), 60–68. https://doi.org/10.1002/ddrr.
  • Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). Arithmetic fact mastery in young children: A longitudinal investigation. Journal of Experimental Child Psychology, 85(2), 103–119. https://doi.org/10.1016/S0022-0965(03)00032-8.
  • Jordan, N. C., Kaplan, D., Oláh, L. N., & Locuniak, M. N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at-risk for mathematics difficulties. Child Development, 77(1), 153–177. https://doi.org/10.1111/j.1467-8624.2006.00862.x.
  • Jordan, N. C., Glutting, J., & Ramineni, C. (2008). A number sense assessment tool for identifying children at risk for mathematical difficulties. In A. Dowker (Ed.), Mathematical Difficulties: Psychology and Intervention (1st ed., pp. 45–58). https://doi.org/10.1016/B978-0-12-373629-1.50005-8.
  • Jordan, N. C., Glutting, J., & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20(2), 82–88. https://doi.org/10.1016/j.lindif.2009.07.004.
  • Kelley, B., Hosp, J. L., & Howell, K. W. (2008). Curriculum-based evaluation and math: An overview. Assessment for Effective Intervention, 33(4), 250–256. https://doi.org/10.1177/1534508407313490.
  • Kline, R. B. (2011). Principles and practice of structural equation modeling (3rd ed.). New York: The Gildford Press.
  • Lee, Y. S., & Lembke, E. (2016). Developing and evaluating a kindergarten to third grade CBM mathematics assessment. ZDM Mathematics Education, 48, 1019–1030. https://doi.org/10.1007/s11858-016-0788-6.
  • LeFevre, J. A., Berrigan, L., Vendetti, C., Kamawar, D., Bisanz, J., Skwarchuk, S. L., & Smith-Chant, B. L. (2013). The role of executive attention in the acquisition of mathematical skills for children in grades 2 through 4. Journal of Experimental Child Psychology, 114, 243–261. https://doi.org/10.1016/j.jecp.2012.10.005.
  • Lemaire, P., & Lecacheur, M. (2011). Age-related changes in children’s executive functions and strategy selection: A study in computational estimation. Cognitive Development, 26, 282–294. https://doi.org/10.1016/j.cogdev.2011.01.002.
  • Lembke, E., Hampton, D., & Beyers, S. J. (2012). Response to intervention in mathematics: Critical elements. Psychology in the Schools, 49(3), 257–272. https://doi.org/10.1002/pits.21596.
  • Lembke, E., Lee, Y.-S., Park, Y. S., & Hampton, D. (2016). Longitudinal growth on curriculum-based measurements mathematics measures for early elementary students. ZDM Mathematics Education, 48, 1049–1063. https://doi.org/10.1007/s11858-016-0804-x.
  • Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13(1), 59–72. https://doi.org/10.1207/s15328007sem1301_3
  • Lyons, I. M., & Ansari, D. (2015). Numerical order processing in children: From reversing the distance-effect to predicting arithmetic. Mind, Brain, and Education, 9(4), 207–221. https://doi.org/10.1111/mbe.12094.
  • Lyons, I. M., Price, G. R., Vaessen, A., Blomert, L., & Ansari, D. (2014). Numerical predictors of arithmetic success in grades 1-6. Developmental Science, 17(5), 714–726. https://doi.org/10.1111/desc.12152.
  • McMullen, J., Brezovszky, B., Hannula-Sormunen, M. M., Veermans, K., Rodríguez-Aflecht, G., Pongsakdi, N., & Lehtinen, E. (2017). Adaptive number knowledge and its relation to arithmetic and pre-algebra knowledge. Learning and Instruction, 49, 178–187. https://doi.org/10.1016/j.learninstruc.2017.02.001.
  • Mix, K. S., Levine, S. C., Cheng, Y. L., Young, C., Hambrick, D. Z., Ping, R., & Konstantopoulos, S. (2016). Separate but correlated: The latent structure of space and mathematics across development. Journal of Experimental Psychology: General, 145(9), 1206–1227. https://doi.org/10.1037/xge0000182.
  • Moeller, K., Pixner, S., Zuber, J., Kaufmann, L., & Nuerk, H. C. (2011). Early place-value understanding as a precursor for later arithmetic performance—A longitudinal study on numerical development. Research in Developmental Disabilities, 32, 1837–1851. https://doi.org/10.1016/J.RIDD.2011.03.012.
  • Mundy, E., & Gilmore, C. (2009). Children’s mapping between symbolic and nonsymbolic representations of number. Journal of Experimental Child Psychology, 103(4), 490–502. https://doi.org/10.1016/j.jecp.2009.02.003.
  • National Center on Response to Intervention. (2012). RTI implementer series: Module 1: Screening - training manual. Retrieved December 31, 2019, from https://rti4success.org/sites/default/files/ImplementerSeries_ScreeningManual.pdf
  • Norris, J. E., McGeown, W. J., Guerrini, C., & Castronovo, J. (2015). Aging and the number sense: Preserved basic non-symbolic numerical processing and enhanced basic symbolic processing. Frontiers in Psychology, 6(999), 1–13. https://doi.org/10.3389/fpsyg.2015.00999.
  • Raghubar, K., Cirino, P., Barnes, M., Ewing-Cobbs, L., Fletcher, J., & Fuchs, L. (2009). Errors in multi-digit arithmetic and behavioral inattention in children with math difficulties. Journal of Learning Disabilities, 42(4), 356–371. https://doi.org/10.1177/0022219409335211.
  • Reigosa-Crespo, V., Valdés-Sosa, M., Butterworth, B., Estévez, N., Rodríguez, M., Santos, E., Torres, P., Suárez, R., & Lage, A. (2012). Basic numerical capacities and prevalence of developmental dyscalculia: The Havana survey. Developmental Psychology, 48(1), 123–135. https://doi.org/10.1037/a0025356.
  • Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. https://doi.org/10.18637/jss.v048.i02.
  • Salaschek, M., Zeuch, N., & Souvignier, E. (2014). Mathematics growth trajectories in first grade: Cumulative vs. compensatory patterns and the role of number sense. Learning and Individual Differences, 35, 103–112. https://doi.org/10.1016/J.LINDIF.2014.06.009.
  • Sasanguie, D., & Vos, H. (2018). About why there is a shift from cardinal to ordinal processing in the association with arithmetic between first and second grade. Developmental Science, 1–13. https://doi.org/10.1111/desc.12653.
  • Siegler, R. S., & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. Advances in Child Development and Behavior, 33, 1–42. https://doi.org/10.1016/S0065-2407(05)80003-5.
  • Siegler, R. S., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444. https://doi.org/10.1111/j.1467-8624.2004.00684.x.
  • Simms, V., Clayton, S., Cragg, L., Gilmore, C., & Johnson, S. (2016). Explaining the relationship between number line estimation and mathematical achievement: The role of visuomotor integration and visuospatial skills. Journal of Experimental Child Psychology, 145, 22–33. https://doi.org/10.1016/J.JECP.2015.12.004.
  • Yuste-Hernanz, C. (2002). BADyG-E1:Batería de aptitudes diferenciales y generales [The battery of differential and general abilities] (2nd ed.). Madrid: Ciencias de la Educación Preescolar y Especial, CEPE.
  • Zhang, J., & Norman, D. A. (1995). The representation of numbers. Cognition, 57, 271–295. Retrieved August 22, 2018, from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.76.2649&rep=rep1&type=pdf