Examining how informal knowledge systems change after formal instruction is imperative to understanding learning processes and conceptual development and to implementing effective educational practices. We used network analyses to determine how the organization of informal knowledge about multidigit numbers in kindergartners (N = 279; mean age = 5.76 years, SD = 0.55; 135 females) supports and is transformed by a year of in-school formal instruction. The results show that in kindergarten, piecemeal knowledge about the surface properties of reading and writing multidigit numbers and the use of base-10 units to determine large quantities are strongly associated with each other and connected in a stringlike manner to other emerging skills. After a year of instruction, each skill becomes connected to the “hub” abilities of reading and writing multidigit numbers, which also become strongly connected to more advanced knowledge of base-10 principles. These findings provide new insights into how partial knowledge provides the backbone on which explicit principles are learned.
BaronchelliA.Ferrer-i-CanchoR.Pastor-SatorrasR.ChaterN.ChristiansenM. H. (2013). Networks in cognitive science. Trends in Cognitive Sciences, 17(7), 348–360. https://doi.org/10.1016/j.tics.2013.04.010
2.
BaroodyA. J. (1990). How and when should place-value concepts and skills be taught?Journal for Research in Mathematics Education, 21(4), 281–286. https://doi.org/10.2307/749526
3.
BoothJ. L.SieglerR. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 42(1), 189–201. https://doi.org/10.1037/0012-1649.41.6.189
4.
BurgerJ.IsvoranuA.-M.LunanskyG.HaslbeckJ. M. B.EpskampS.HoekstraR. H. A.FriedE. I.BorsboomD.BlankenT. F. (2022). Reporting standards for psychological network analyses in cross-sectional data. Psychological Methods. Advance online publication. https://doi.org/10.1037/met0000471
5.
ByrgeL.SmithL. B.MixK. S. (2014). Beginnings of place value: How preschoolers write three-digit numbers. Child Development, 85(2), 437–443. https://doi.org/10.1111/cdev.12162
CohenJ. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Routledge.
9.
ColungaE.SmithL. B. (2005). From the lexicon to expectations about kinds: A role for associative learning. Psychological Review, 112(2), 347–382. https://doi.org/10.1037/0033-295X.112.2.347
10.
CooperL. L.TomaykoM. C. (2011). Understanding place value. Teaching Children Mathematics, 17(9), 558–567.
11.
de LeeuwJ.MairP. (2009). Multidimensional scaling using majorization: SMACOF in R. Journal of Statistical Software, 31(1). https://doi.org/10.18637/jss.v031.i03
12.
DurandM.HulmeC.LarkinR.SnowlingM. (2005). The cognitive foundations of reading and arithmetic skills in 7- to 10-year-olds. Journal of Experimental Child Psychology, 91(2), 113–136. https://doi.org/10.1016/j.jecp.2005.01.003
13.
EpskampS.BorsboomD.FriedE. I. (2018). Estimating psychological networks and their accuracy: A tutorial paper. Behavior Research Methods, 50(1), 195–212. https://doi.org/10.3758/s13428-017-0862-1
14.
EpskampS.CramerA. O. J.WaldorpL. J.SchmittmannV. D.BorsboomD. (2012). qgraph: Network visualizations of relationships in psychometric data. Journal of Statistical Software, 48(4). https://doi.org/10.18637/jss.v048.i04
15.
FaulF.ErdfelderE.BuchnerA.LangA.-G. (2009). Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41, 1149–1160. https://doi.org/10.3758/BRM.41.4.1149
16.
FriedmanJ.HastieT.TibshiraniR. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3), 432–441. https://doi.org/10.1093/biostatistics/kxm045
17.
FusonK. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343–403. https://doi.org/10.1207/s1532690xci0704_4
18.
GrossbergS.RepinD. V. (2003). A neural model of how the brain represents and compares multi-digit numbers: Spatial and categorical processes. Neural Networks, 16(8), 1107–1140. https://doi.org/10.1016/S0893-6080(03)00193-X
19.
HanichL. B.JordanN. C.KaplanD.DickJ. (2001). Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology, 93(3), 615–626. https://doi.org/10.1037/0022-0663.93.3.615
20.
HoC. S.-H.FusonK. C. (1998). Children’s knowledge of teen quantities as tens and ones: Comparisons of Chinese, British, and American kindergartners. Journal of Educational Psychology, 90(3), 536–544. https://doi.org/10.1037/0022-0663.90.3.536
21.
HusseinA. A.ForouzanfarT.BloemenaE.de VisscherJ. G. A. M.BrakenhoffR. H.LeemansC. R.HelderM. N. (2018). A review of the most promising biomarkers for early diagnosis and prognosis prediction of tongue squamous cell carcinoma. British Journal of Cancer, 119(6), 724–736. https://doi.org/10.1038/s41416-018-0233-4
22.
JonesP. J.MairP.McNallyR. J. (2018). Visualizing psychological networks: A tutorial in R. Frontiers in Psychology, 9, Article 1742. https://doi.org/10.3389/fpsyg.2018.01742
23.
MairP.BorgI.RuschT. (2016). Goodness-of-fit assessment in multidimensional scaling and unfolding. Multivariate Behavioral Research, 51(6), 772–789. https://doi.org/10.1080/00273171.2016.1235966
24.
McCloskeyM. (2014). Naive theories of motion. In GentnerD.StevensA. L. (Eds.), Mental models (pp. 299–324). Psychology Press.
25.
McGuireP.KinzieM. B. (2013). Analysis of place value instruction and development in pre-kindergarten mathematics. Early Childhood Education Journal, 41(5), 355–364. https://doi.org/10.1007/s10643-013-0580-y
26.
MixK. S.BowerC. A.HancockG. R.YuanL.SmithL. B. (2022). The development of place value concepts: Approximation before principles. Child Development, 93(3), 778–793. https://doi.org/10.1111/cdev.13724
27.
MixK. S.LevineS. C.ChengY.-L.YoungC.HambrickD. Z.PingR.KonstantopoulosS. (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
28.
MixK. S.PratherR. W.SmithL. B.StocktonJ. D. (2014). Young children’s interpretation of multidigit number names: From emerging competence to mastery. Child Development, 85(3), 1306–1319. https://doi.org/10.1111/cdev.12197
29.
MixK. S.SmithL. B.StocktonJ. D.ChengY.-L.BarterianJ. A. (2017). Grounding the symbols for place value: Evidence from training and long-term exposure to base-10 models. Journal of Cognition and Development, 18(1), 129–151. https://doi.org/10.1080/15248372.2016.1180296
30.
MoellerK.PixnerS.ZuberJ.KaufmannL.NuerkH.-C. (2011). Early place-value understanding as a precursor for later arithmetic performance—A longitudinal study on numerical development. Research in Developmental Disabilities, 32(5), 1837–1851. https://doi.org/10.1016/j.ridd.2011.03.012
31.
National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for mathematical practice. http://www.corestandards.org/Math/Practice/
32.
PearlJ. (2019). The seven tools of causal inference, with reflections on machine learning. Communications of the ACM, 62(3), 54–60. https://doi.org/10.1145/3241036
33.
RaghubarK.CirinoP.BarnesM.Ewing-CobbsL.FletcherJ.FuchsL. (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
34.
SieglerR. S.OpferJ. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237–243. https://doi.org/10.1111/1467-9280.02438
35.
SiewC. S. Q. (2020). Applications of network science to education research: Quantifying knowledge and the development of expertise through network analysis. Education Sciences, 10(4), Article 101. https://doi.org/10.3390/educsci10040101
WellsJ. B.ChristiansenM. H.RaceD. S.AchesonD. J.MacDonaldM. C. (2009). Experience and sentence processing: Statistical learning and relative clause comprehension. Cognitive Psychology, 58(2), 250–271. https://doi.org/10.1016/j.cogpsych.2008.08.002
39.
YangZ.AlgesheimerR.TessoneC. J. (2016). A comparative analysis of community detection algorithms on artificial networks. Scientific Reports, 6, Article 30750. https://doi.org/10.1038/srep30750
40.
YuanL.PratherR. W.MixK. S.SmithL. B. (2019). Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves. Cognition, 189, 89–104. https://doi.org/10.1016/j.cognition.2019.03.013
41.
YuanL.XiangV.CrandallD.SmithL. (2020). Learning the generative principles of a symbol system from limited examples. Cognition, 200, Article 104243. https://doi.org/10.1016/j.cognition.2020.104243
42.
ZuberJ.PixnerS.MoellerK.NuerkH.-C. (2009). On the language specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity. Journal of Experimental Child Psychology, 102(1), 60–77. https://doi.org/10.1016/j.jecp.2008.04.003
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