The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge

Journal of Educational Psychology

Volumn 105, Issue 2, 2013, Pages 453-464

  Purpura, David J ^a   Baroody, Arthur J ^b   Lonigan, Christopher J ^c  

^a PURDUE UNIVERSITY   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^c FLORIDA STATE UNIVERSITY   (United States)

Author keywords

Formal; Informal; Learning trajectory; Mathematics; Preschool

Indexed keywords

EID: 84881094820     PISSN: 00220663     EISSN: None     Source Type: Journal    
DOI: 10.1037/a0031753     Document Type: Article

References (94)

1. Arnold, D. H., Fisher, P. H., Doctoroff, G. L., & Dobbs, J. (2002). Accelerating math development in Head Start classrooms. Journal of Educational Psychology, 94, 762-770. doi:10.1037/0022-0663.94.4.762
2. Aunio, P.,& Niemivirta, M. (2010). Predicting children's mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20, 427-435. doi:10.1016/j.lindif.2010.06.003
3. Aunola, K., Leskinen, E., Lerkkanen, M.,& Nurmi, J. (2004). Developmental dynamics of math performances from preschool to Grade 2. Journal of Educational Psychology, 96, 699-713. doi:10.1037/0022-0663.96.4.699
4. Baron, R. M.,& Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182. doi:10.1037/0022-3514.51.6.1173
5. Baroody, A. J. (1987). Children's mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York, NY: Teachers College Press.
6. Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody& A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1-34). Mahwah, NJ: Erlbaum.
7. Baroody, A. J., Eiland, M. D., Purpura, D. J.,& Reid, E. E. (2012). Fostering at-risk kindergarten children's number sense. Cognition and Instruction, 30, 435-470. doi:10.1080/07370008.2012.720152
8. Baroody, A. J., Eiland, M. D., Purpura, D. J.,& Reid, E. E. (in press). Can discovery learning foster first graders' fluency with the most basic addition combinations. American Educational Research Journal.
9. Baroody, A. J., Eiland, M.,& Thompson, B. (2009). Fostering at-risk preschoolers' number sense. Early Education and Development, 20, 80-128. doi:10.1080/10409280802206619
10. Baroody, A. J., Feil, Y.,& Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115-131.
11. Baroody, A. J.,& Ginsburg, H. P. (1990). Children's mathematical learning: A cognitive view. Journal for Research in Mathematics Education, 21(Monograph No. 4), 79-90.
12. Baroody, A. J., Lai, M.,& Mix, K. S. (2006). Development of young children's early number and operation sense and its implications for early childhood education. In B. Spodek& O. N. Saracho (Eds.), Handbook of research on the education of young children (2nd ed., pp. 187-221). Mahwah, NJ: Erlbaum.
13. Baroody, A. J.,& Wilkins, J. L. M. (1999). The development of informal counting, number, and arithmetic skills and concepts. In J. V. Copley (Ed.), Mathematics in the early years (pp. 48-65). Washington, DC: National Association for the Education of Young Children.
14. Basista, B.,& Matthews, S. (2002). Integrated science and mathematics professional development programs. School Science and Mathematics, 102, 359-370. doi:10.1111/j.1949-8594.2002.tb18219.x
15. Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York, NY: Guilford Press.
16. Bryant, D. P., Bryant, B. R., Kim, S. A.,& Gersten, R. (2006, February). Three-tier mathematics intervention: Emerging model and preliminary findings. Poster presented at the 14th annual meeting of the Pacific Coast Research Conference, San Diego, CA.
17. Butterworth, B.,& Reigosa, V. (2007). Information processing deficits in dyscalculia. In D. B. Berch& M. M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 65-81). Baltimore, MD: Brookes.
18. Chard, D. J., Clarke, B., Baker, S., Otterstedt, J., Braun, D.,& Katz, R. (2005). Using measures of number sense to screen for difficulties in mathematics: Preliminary findings. Assessment for Effective Intervention, 30, 3-14. doi:10.1177/073724770503000202
19. Cirino, P. T. (2011). The interrelationships of mathematical precursors in kindergarten. Journal of Experimental Child Psychology, 108, 713-733. doi:10.1016/j.jecp.2010.11.004
20. Clarke, B.,& Shinn, M. R. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33, 234-248.
21. Clarke, B., Smolkowski, K., Baker, S. K., Fien, H., Doebler, C. T.,& Chard, D. J. (2011). The impact of a comprehensive Tier 1 core kindergarten program on the achievement of students at risk in mathematics. The Elementary School Journal, 111, 561-584. doi:10.1086/659033
22. Clements, D. H. (2007). Curriculum research: Toward a framework for "research-based curricula". Journal for Research in Mathematics Education, 38, 35-70.
23. Clements, D. H.,& Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81-89. doi:10.1207/s15327833mtl0602_1
24. Clements, D. H.,& Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38, 136-163.
25. Clements, D. H.,& Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.
26. Common Core State Standards. (2011). Common Core State Standards: Preparing America's students for college and career. Retrieved from http://www.corestandards.org/
27. Cross, C. T., Woods, T. A.,& Schweingruber, H. (Eds.). (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: Committee on Early Childhood Mathematics, National Research Council.
28. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1-42. doi:10.1016/0010-0277(92)90049-N
29. Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P.,... Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428-1446. doi:10.1037/0012-1649.43.6.1428
30. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C.,& McDowell, J. (2013). Teaching math to young children: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.
31. Fuchs, L. S., Compton, D. L., Fuchs, D., Hollenbeck, K. N., Craddock, C. F.,& Hamlett, C. L. (2008). Dynamic assessment of algebraic learning in predicting third graders' development of mathematical problem solving. Journal of Educational Psychology, 100, 829-850. doi: 10.1037/a0012657
32. Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D.,& Hamlett, C. L. (2005). The prevention, identification, and cognitive determinants of math difficulty. Journal of Educational Psychology, 97, 493-513. doi:10.1037/0022-0663.97.3.493
33. Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., Fletcher, J. M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98, 29-43. doi: 10.1037/0022-0663.98.1.29
34. Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., Zumeta, R. O. (2009). Remediating number combination and word problem deficits among students with mathematical difficulties: A randomized control trial. Journal of Educational Psychology, 101, 561-576. doi:10.1037/a0014701
35. Fuson, K. C. (1988). Children's counting and concepts of number. New York, NY: Springer-Verlag. doi:10.1007/978-1-4612-3754-9
36. Geary, D. C. (1994). Children's mathematical development: Research and practical applications. Washington, DC: American Psychological Association. doi:10.1037/10163-000
37. Gelman, R.,& Gallistel, C. R. (1978). The child's understanding of number. Oxford, England: Harvard University Press.
38. Gersten, R.,& Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematic difficulties. The Journal of Special Education, 33, 18-28. doi:10.1177/002246699903300102
39. Gilmore, C. K., McCarthy, S. E.,& Spelke, E. S. (2007). Symbolic arithmetic knowledge without instruction. Nature, 447, 589-591. doi: 10.1038/nature05850
40. Ginsburg, H. P. (1975). Young children's informal knowledge of mathematics. Journal of Children's Mathematical Behavior, 1, 63-156.
41. Ginsburg, H. P. (1977). Children's arithmetic: The learning process. Oxford, England: Van Nostrand.
42. Ginsburg, H. P.,& Baroody, A. J. (2003). Test of Early Mathematics Ability (3rd ed.). Austin, TX: Pro-Ed.
43. Ginsburg, H. P., Klein, A.,& Starkey, P. (1998). The development of children's mathematical thinking: Connecting research with practice. In D. Williams, I. E. Sigel,& K. Renninger (Eds.), Child psychology in practice (pp. 401-476). Hoboken, NJ: Wiley.
44. Gravemeijer, K. (2002, April). Learning trajectories and local instruction theories as a means of support for teachers in reform mathematics education. Paper presented at the annual meeting of the American Educational Research Association, Las Vegas, NV.
45. Greenes, C., Ginsburg, H. P.,& Balfanz, R. (2004). Big math for little kids. Early Childhood Research Quarterly, 19, 159-166. doi:10.1016/j.ecresq.2004.01.010
46. Griffin, S.,& Case, R. (1997). Re-thinking the primary school math curriculum: An approach based on cognitive science. Issues in Education, 2, 1-49.
47. Griffin, S. A., Case, R.,& Siegler, R. S. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 25-49). Cambridge, MA: MIT Press.
48. Hatano, G. (2003). Forward. In A. J. Baroody& A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. xi-xiii). Mahwah, NJ: Erlbaum.
49. Hu, L.,& Bentler, P. M. (1999). Cutoffcriteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1-55. doi:10.1080/10705519909540118
50. James, W. (1958). Talks to teachers on psychology and to students on some of life's ideals. New York, NY: Norton.
51. Jordan, N. C., Hanich, L. B.,& Uberti, H. Z. (2003). Mathematical thinking and learning difficulties. In A. J. Baroody& A. Dowker (Eds.), The development of arithmetic concepts and skills: Recent research and theory (pp. 359-383). Mahwah, NJ: Erlbaum.
52. Jordan, N. C., Kaplan, D., Locuniak, M. N.,& Ramineni, C. (2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research& Practice, 22, 36-46. doi:10.1111/j.1540-5826.2007.00229.x
53. Jordan, N. C., Kaplan, D., Ramineni, C.,& Locuniak, M. N. (2009). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45, 850-867. doi:10.1037/a0014939
54. Jordan, N. C.,& Levine, S. C. (2009). Socioeconomic variation, number competence, and mathematics learning difficulties in young children. Developmental Disabilities Research Reviews, 15, 60-68. doi:10.1002/ddrr.46
55. Krajewski, K. (2008). Prä vention der Rechenschwä che [The early prevention of math problems]. In W. Schneider& M. Hasselhorn (Eds.), Handbuch der Pä dagogischen Psychologie (pp. 360-370). Goettingen, Germany: Hogrefe.
56. Krajewski, K.,& Schneider, W. (2009). 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, 19, 513-526. doi:10.1016/j.learninstruc.2008.10.002
57. Kroesbergen, E. H., Van Luit, J. E. H., Van Lieshout, E. C. D. M., Van Loosbroek, E.,& Van de Rijt, B. A. M. (2009). Individual differences in early numeracy: The role of executive functions and subitizing. Journal of Psychoeducational Assessment, 27, 226-236. doi:10.1177/0734282908330586
58. LeFevre, J., Fast, L., Skwarchuk, S., Smith-Chant, B. L., Bisanz, J., Kamawar, D.,& Penner-Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81, 1753-1767. doi:10.1111/j.1467-8624.2010.01508.x
59. Lembke, E.,& Foegen, A. (2006, February). Monitoring student progress in early math. Paper presented at the 14th annual meeting of the Pacific Coast Research Conference, San Diego, CA.
60. Lembke, E.,& Foegen, A. (2009). Identifying early numeracy indicators for kindergarten and first-grade students. Learning Disabilities Research& Practice, 24, 12-20. doi:10.1111/j.1540-5826.2008.01273.x
61. Leong, C. K.,& Jerred, W. D. (2001). Effects of consistency and adequacy of language information on understanding elementary mathematics word problems. Annals of Dyslexia, 51, 275-298. doi:10.1007/s11881-001-0014-1
62. Libertus, M. E., Feigenson, L. H.,& Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14, 1292-1300. doi:10.1111/j.1467-7687.2011.01080.x
63. Lonigan, C. J., Schatschneider, C.,& Westberg, L. (2008). Identification of children's skills and abilities linked to later outcomes in reading, writing, and spelling. In Developing early literacy: Report of the National Early Literacy Panel (pp. 55-106). Washington, DC: National Institute for Literacy.
64. Martinez, R. S., Missall, K. N., Graney, S. B., Aricak, O. T.,& Clarke, B. (2008). Technical adequacy of early numeracy curriculum-based measurement in kindergarten. Assessment for Effective Intervention, 34, 116-125. doi:10.1177/1534508408326204
65. Mazzocco, M.,& Thompson, R. (2005). Kindergarten predictors of math learning disability. Learning Disabilities Research& Practice, 20, 142-155. doi:10.1111/j.1540-5826.2005.00129.x
66. Mix, K. S. (2009). How Spencer made number: First uses of the number words. Journal of Experimental Child Psychology, 102, 427-444. doi: 10.1016/j.jecp.2008.11.003
67. Morris, R. D., Stuebing, K. K., Fletcher, J. M., Shaywitz, S. E., Lyon, G. R., Shankweiler, D. P., Shaywitz, B. A. (1998). Subtypes of reading disability: Variability around a phonological core. Journal of Educational Psychology, 90, 347-373. doi:10.1037/0022-0663.90.3.347
68. Mueller, R. O.,& Hancock, G. R. (2010). Structural equation modeling. In G. R. Hancock& R. O. Mueller (Eds.), The reviewer's guide to quantitative methods in the social sciences (pp. 371-383) New York, NY: Routledge.
69. Muldoon, K., Lewis, C.,& Freeman, N. (2009). Why set-comparison is vital in early number learning. Trends in Cognitive Sciences, 13, 203-208. doi:10.1016/j.tics.2009.01.010
70. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
71. National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through Grade 8 mathematics. Reston, VA: Author.
72. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U. S. Department of Education.
73. Palmer, A.,& Baroody, A. J. (2011). Blake's development of the number words "one," "two," and "three". Cognition and Instruction, 29, 265-296. doi:10.1080/07370008.2011.583370
74. Piaget, J. (1964). Development and learning. In R. E. Ripple& V. N. Rockcastle (Eds.), Piaget rediscovered (pp. 7-20). Ithaca, NY: Cornell University.
75. Preacher, K. J.,& Hayes, A. F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. Behavior Research Methods, Instruments& Computers, 36, 717-731. doi:10.3758/BF03206553
76. Purpura, D. J. (2010). Informal number-related mathematics skills: An examination of the structure of and relations between these skills in preschool. (Doctoral dissertation). Retrieved from ProQuest Dissertations and Theses. (Accession Order No. AAT 3462344)
77. Purpura, D. J., Hume, L., Sims, D.,& Lonigan, C. J. (2011). Emergent literacy and mathematics: The value of including emergent literacy skills in the prediction of mathematics development. Journal of Experimental Child Psychology, 110, 647-658. doi:10.1016/j.jecp.2011.07.004
78. Purpura, D. J.,& Lonigan, C. J. (2013). Informal numeracy skills: The structure and relations among numbering, relations, and arithmetic operations in preschool. American Educational Research Journal, 50, 178-209. doi:10.3102/0002831212465332
79. Rousselle, L.,& Noel, M. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs. nonsymbolic number magnitude. Cognition, 102, 361-395. doi:10.1016/j.cognition.2006.01.005
80. Sarama, J.,& Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.
81. Sarnecka, B. W.,& Carey, S. (2008). How counting presents number: What children must learn and when they learn it. Cognition, 108, 662-674. doi:10.1016/j.cognition.2008.05.007
82. Siegler, R. S.,& Ramani, G. B. (2008). Playing linear numerical board games promotes low-income children's numerical development. Developmental Science, 11, 655-661. doi:10.1111/j.1467-7687.2008.00714.x
83. Siegler, R. S.,& Ramani, G. B. (2009). Playing linear number board games-but not circular ones-improves low-income preschoolers' numerical understanding. Journal of Educational Psychology, 101, 545-560. doi:10.1037/a0014239
84. Sinclair, A., Sigrist, F.,& Sinclair, H. (1983). Young children's ideas about the written number system. In D. Rogers& J. A. Sloboda (Eds.), The acquisition of symbolic skills (pp. 535-541). New York, NY: Plenum Press.
85. Song, M. J.,& Ginsburg, H. P. (1987). The development of informal and formal mathematical thinking in Korean and U.S. children. Child Development, 58, 1286-1296.
86. Stanovich, K. E., Siegal, L. S.,& Gottardo, A. (1997). Converging evidence for phonological and surface subtypes of reading disability. Journal of Educational Psychology, 89, 114-127. doi:10.1037/0022-0663.89.1.114
87. Starkey, P., Klein, A.,& Wakeley, A. (2004). Enhancing young children's mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19, 99-120. doi:10.1016/j.ecresq.2004.01.002
88. Stock, P., Desoete, A.,& Roeyers, H. (2007). Early markers of arithmetic difficulties. Educational and Child Psychology, 24, 28-39.
89. Swanson, H. L. (2004). Working memory and phonological processing as predictors of children's mathematical problem solving at different ages. Memory& Cognition, 32, 648-661. doi:10.3758/BF03195856
90. Swanson, H. L.,& Beebe-Frankenberger, M. (2004). The relationship between working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471-491. doi:10.1037/0022-0663.96.3.471
91. Swanson, H. L.,& Kim, K. (2007). Working memory, short-term memory, and naming speed as predictors of children's mathematical performance. Intelligence, 35, 151-168. doi:10.1016/j.intell.2006.07.001
92. VanDerHeyden, A. M., Broussard, C.,& Cooley, A. (2006). Further development of measures of early math performance for preschoolers. Journal of School Psychology, 44, 533-553. doi:10.1016/j.jsp.2006.07.003
93. Woodcock, R., McGrew, K. S.,& Mather, N. (2001). Woodcock-Johnson Tests of Achievement (3rd ed.). Itasca, IL: Riverside.
94. Zhao, X., Lynch, J. G.,& Chen, Q. (2010). Reconsidering Baron and Kenny: Myths and truths about mediation analysis. Journal of Consumer Research, 37, 197-206. doi:10.1086/651257