Is counting hindering learning? An investigation into children’s proficiency with simple addition and their flexibility with mental computation strategies

References

• Andrich, D. (2016). Rasch rating-scale model. In W. J. van der Linden (Ed.), Handbook of item response theory, (Vol. 1). Models. Forida: CRC Press
   Google Scholar
• Ashcraft, M. H. (1995). Cognitive psychology and simple arithmetic: A review and summary of new directions. Mathematical Cognition, 1(1), 3–34.
   Google Scholar
• Australian Curriculum, Assessment, and Reporting Authority. (2010). The Australian curriculum. Retrieved January 1, 2019, from http://www.australiancurriculum.edu.au/
   Google Scholar
• Baroody, A. J., Bajwa, N. P., & Eiland, M. (2009). Why can’t Johnny remember the basic facts? Developmental Disabilities Research Reviews, 15(1), 69–79. https://doi.org/https://doi.org/10.1002/ddrr.45
   PubMedGoogle Scholar
• Bartelet, D., Ansari, D., Vaessen, A., & Blomert, L. (2014). Cognitive subtypes of mathematics learning difficulties in primary education. Research in Developmental Disabilities, 35(3), 657–670. https://doi.org/https://doi.org/10.1016/j.ridd.2013.12.010
   PubMed Web of Science ®Google Scholar
• Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24(4), 294–323. https://doi.org/http://dx.doi.org/10.2307/749464
   Google Scholar
• Blöte, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10(3), 221–247. https://doi.org/https://doi.org/10.1016/S0959-4752(99)00028-6
   Web of Science ®Google Scholar
• Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93(3), 627–638. https://doi.org/https://doi.org/10.1037/0022-0663.93.3.627
   Web of Science ®Google Scholar
• Canobi, K. H. (2009). Concept–procedure interactions in children’s addition and subtraction. Journal of Experimental Child Psychology, 102(2), 131–149. https://doi.org/https://doi.org/10.1016/j.jecp.2008.07.008
   PubMed Web of Science ®Google Scholar
• Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179–202. https://doi.org/https://doi.org/10.2307/748348
   Google Scholar
• Carr, M., & Alexeev, N. (2011). Fluency, accuracy, and gender predict developmental trajectories of arithmetic strategies. Journal of Educational Psychology, 103(3), 617–631. https://doi.org/https://doi.org/10.1037/a0023864
   Web of Science ®Google Scholar
• Carr, M., Jessup, D. L., & Fuller, D. (1999). Gender differences in first-grade mathematics strategy use: Parent and teacher contributions. Journal for Research in Mathematics Education, 30(1), 20–46. https://doi.org/https://doi.org/10.2307/749628
   Web of Science ®Google Scholar
• Cheng, Z.-J. (2012). Teaching young children decomposition strategies to solve addition problems: An experimental study. The Journal of Mathematical Behavior, 31(1), 29–47. https://doi.org/https://doi.org/10.1016/j.jmathb.2011.09.002
   Google Scholar
• Chesney, D. L., McNeil, N. M., Petersen, L. A., & Dunwiddie, A. E. (2018). Arithmetic practice that includes relational words promotes understanding of symbolic equations. Learning and Individual Differences, 64, 104–112. https://doi.org/https://doi.org/10.1016/j.lindif.2018.04.013
   Web of Science ®Google Scholar
• Cowan, R., Donlan, C., Shepherd, D.-L., Cole-Fletcher, R., Saxton, M., & Hurry, J. (2011). Basic calculation proficiency and mathematics achievement in elementary school children. Journal of Educational Psychology, 103(4), 786–803. https://doi.org/https://doi.org/10.1037/a0024556
   Web of Science ®Google Scholar
• De Smedt, B., Torbeyns, J., Stassens, N., Ghesquière, P., & Verschaffel, L. (2010). Frequency, efficiency and flexibility of indirect addition in two learning environments. Learning and Instruction, 20(3), 205–215. https://doi.org/https://doi.org/10.1016/j.learninstruc.2009.02.020
   Web of Science ®Google Scholar
• Department for Education. (2013). The national curriculum (United Kingdom). Retrieved 1 July, 2019, from https://www.gov.uk/government/collections/national-curriculum
   Google Scholar
• Fuson, K. C. (1988). Children’s counting and concepts of number. Springer New York.
   Google Scholar
• Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., … Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162. https://doi.org/https://doi.org/10.2307/749759
   Web of Science ®Google Scholar
• Gaidoschik, M. (2019). Didactics as a source and remedy of mathematical learning difficulties. In A. Fritz, V. G. Haase, & P. Räsänen (Eds.), International handbook of mathematical learning difficulties: From the laboratory to the classroom (pp. 73–89). Springer International Publishing.
   Google Scholar
• Geary, D. C. (2011a). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552. https://doi.org/https://doi.org/10.1037/a0025510
   PubMed Web of Science ®Google Scholar
• Geary, D. C. (2011b). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental and Behavioral Pediatrics: JDBP, 32(3), 250–263. https://doi.org/https://doi.org/10.1097/DBP.0b013e318209edef
   PubMed Web of Science ®Google Scholar
• Geary, D. C., Brown, S. C., & Samaranayake, V. A. (1991). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27(5), 787–797. https://doi.org/https://doi.org/10.1037/0012-1649.27.5.787
   Web of Science ®Google Scholar
• Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and arithmetical cognition: A longitudinal study of process and concept deficits in children with learning disability. Journal of Experimental Child Psychology, 77(3), 236–263. https://doi.org/https://doi.org/10.1006/jecp.2000.2561
   PubMed Web of Science ®Google Scholar
• Geary, D. C., Hoard, M. K., & Bailey, D. H. (2012). Fact retrieval deficits in low achieving children and children with mathematical learning disability. Journal of Learning Disabilities, 45(4), 291–307. https://doi.org/https://doi.org/10.1177/0022219410392046
   PubMed Web of Science ®Google Scholar
• Geary, D. C., Hoard, M. K., Byrd-Craven, J., & Catherine DeSoto, M. (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/https://doi.org/10.1016/j.jecp.2004.03.002
   PubMed Web of Science ®Google Scholar
• Goldman, S. R., Pellegrino, J. W., & Mertz, D. L. (1988). Extended practice of basic addition facts: Strategy changes in learning-disabled students. Cognition and Instruction, 5(3), 223–265. https://doi.org/https://doi.org/10.1207/s1532690xci0503_2
   Web of Science ®Google Scholar
• Gray, E. M. (1991). An analysis of diverging approaches to simple arithmetic: Preference and its consequences. Educational Studies in Mathematics, 22(6), 551–574. https://doi.org/https://doi.org/10.1007/BF00312715
   Google Scholar
• Green, H. J., Lemaire, P., & Dufau, S. (2007). Eye movement correlates of younger and older adults’ strategies for complex addition. Acta Psychologica, 125(3), 257–278. https://doi.org/https://doi.org/10.1016/j.actpsy.2006.08.001
   PubMed Web of Science ®Google Scholar
• Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001). Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology, 93(3), 615–626. https://doi.org/http://dx.doi.org/10.1037//0022–0663.93.3.615
   Google Scholar
• Hecht, A. (2006). Group differences in adult simple arithmetic: Good retrievers, not-so-good retrievers, and perfectionists. Memory & Cognition, 34(1), 207–216. https://doi.org/https://doi.org/10.3758/bf03193399
   Google Scholar
• Heirdsfield, A. M., & Cooper, T. J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies. The Journal of Mathematical Behavior, 21(1), 57–74. https://doi.org/https://doi.org/10.1016/S0732-3123(02)00103-7
   Google Scholar
• Hopkins, S., Russo, J., & Downton, A. (2019). Mental computational fluency: Assessing flexibility, efficiency and accuracy. In M. Graven, H. Venkat, A. Essien, & P. Vale (Eds.). Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education. Pretoria, South Africa: PME.
   Google Scholar
• Hopkins, S., & Bayliss, D. (2017). The prevalence and disadvantage of min-counting in seventh grade: Problems with confidence as well as accuracy? Mathematical Thinking and Learning, 19(1), 19–32. https://doi.org/https://doi.org/10.1080/10986065.2017.1258613
   Web of Science ®Google Scholar
• Imbo, I., Duverne, S., & Lemaire, P. (2007). Working memory, strategy execution, and strategy selection in mental arithmetic. The Quarterly Journal of Experimental Psychology, 60(9), 1246–1264. https://doi.org/https://doi.org/10.1080/17470210600943419
   Web of Science ®Google Scholar
• Imbo, I., & Vandierendonck, A. (2007). The development of strategy use in elementary school children: Working memory and individual differences. Journal of Experimental Child Psychology, 96(4), 284–309. https://doi.org/https://doi.org/10.1016/j.jecp.2006.09.001
   PubMed Web of Science ®Google Scholar
• Jain, A. K. (2010). Data clustering: 50 years beyond K-means. Pattern Recognition Letters, 31(8), 651–666. https://doi.org/https://doi.org/10.1016/j.patrec.2009.09.011
   Web of Science ®Google Scholar
• Jordan, N. C., & Hanich, L. B. (2000). Mathematical thinking in second-grade children with different forms of LD. Journal of Learning Disabilities, 33(6), 567–578. https://doi.org/https://doi.org/10.1177/002221940003300605
   PubMed Web of Science ®Google Scholar
• Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74(3), 834–850. https://doi.org/https://doi.org/10.1111/1467-8624.00571
   PubMed Web of Science ®Google Scholar
• 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), 153–175. https://doi.org/https://doi.org/10.1111/j.1467-8624.2006.00862.x
   PubMed Web of Science ®Google Scholar
• Klein, A. S., Beishuizen, M., & Treffers, A. (1998). The empty number line in Dutch second grades: Realistic versus gradual program design. Journal for Research in Mathematics Education, 29(4), 443–464. https://doi.org/https://doi.org/10.2307/749861
   Google Scholar
• 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(6), 513–526. https://doi.org/https://doi.org/10.1016/j.learninstruc.2008.10.002
   Web of Science ®Google Scholar
• Luwel, K., Onghena, P., Torbeyns, J., Schillemans, V., & Verschaffel, L. (2009). Strengths and weaknesses of the choice/no-choice method in research on strategy use. European Psychologist, 14(4), 351–362. https://doi.org/https://doi.org/10.1027/1016-9040.14.4.351
   Web of Science ®Google Scholar
• McIntosh, A. (2005). Mental computation: A strategies approach (2nd ed.). Department of Education, Tasmania.
   Google Scholar
• NCTM. (2000). Principles and standards for school mathematics. Retrieved 1 July, 2019, from http://www.nctm.org/standards.
   Google Scholar
• NCTM. (2014). Procedural fluency in mathematics: A position of the NCTM. Reston, VA: NCTM Ed. Retrieved 1 July, 2019, from http://www.nctm.org/standards.
   Google Scholar
• Ostad, S. A., & Sorensen, P. M. (2007). Private speech and strategy-use patterns: Bidirectional comparisons of children with and without mathematical difficulties in a developmental perspective. Journal of Learning Disabilities, 40(1), 2–14. https://doi.org/https://doi.org/10.1177/00222194070400010101
   PubMed Web of Science ®Google Scholar
• Reed, H. C., Stevenson, C., Broens-Paffen, M., Kirschner, P. A., & Jolles, J. (2015). Third graders’ verbal reports of multiplication strategy use: How valid are they? Learning and Individual Differences, 37, 107–117. https://doi.org/https://doi.org/10.1016/j.lindif.2014.11.010
   Web of Science ®Google Scholar
• Reeve, R., Reynolds, F., Humberstone, J., & Butterworth, B. (2012). Stability and change in markers of core numerical competencies. Journal of Experimental Psychology: General, 141(4), 649–666. https://doi.org/https://doi.org/10.1037/a0027520
   PubMed Web of Science ®Google Scholar
• Reys, R. E., Reys, B. J., Nohda, N., & Emori, H. (1995). Mental computation performance and strategy use of Japanese students in grades 2, 4, 6, and 8. Journal for Research in Mathematics Education, 26(4), 304–326. https://doi.org/https://doi.org/10.2307/749477
   Web of Science ®Google Scholar
• Rhodes, K. T., Lukowski, S., Branum-Martin, L., Opfer, J., Geary, D. C., & Petrill, S. A. (2019). Individual differences in addition strategy choice: A psychometric evaluation. Journal of Educational Psychology, 111(3), 414–433. https://doi.org/https://doi.org/10.1037/edu0000294
   Web of Science ®Google Scholar
• Shrager, J., & Siegler, R. S. (1998). SCADS: A model of children’s strategy choices and strategy discoveries. Psychological Science, 9(5), 405–410. https://doi.org/https://doi.org/10.1111/1467-9280.00076
   Web of Science ®Google Scholar
• Siegler, R., & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. In R. V. Kail (Ed.), Advances in child development and behavior (Vol. 33, pp. 1–42). Elsevier Academic Press.
   Google Scholar
• Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), Origins of cognitive skills (pp. 229–293). Erlbaum.
   Google Scholar
• Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children’s addition. Journal of Experimental Psychology: General, 116(3), 250–264. https://doi.org/https://doi.org/10.1037/0096-3445.116.3.250
   Web of Science ®Google Scholar
• Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. University Press.
   Google Scholar
• Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Lawrence Erlbaum Associates, Inc.
   Google Scholar
• Siegler, R. S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126(1), 71–92. https://doi.org/https://doi.org/10.1037/0096-3445.126.1.71
   PubMed Web of Science ®Google Scholar
• 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(3), 545–560. https://doi.org/https://doi.org/10.1037/a0014239
   Web of Science ®Google Scholar
• Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565–579. https://doi.org/https://doi.org/10.1016/j.learninstruc.2007.09.018
   Web of Science ®Google Scholar
• Starkey, P., & Cooper, R. G., Jr. (1995). The development of subitizing in young children. British Journal of Developmental Psychology, 13(4), 399–420. https://doi.org/https://doi.org/10.1111/j.2044-835X.1995.tb00688.x
   Web of Science ®Google Scholar
• Sun, X. H., Xin, Y. P., & Huang, R. (2019). A complementary survey on the current state of teaching and learning of whole number arithmetic and connections to later mathematical content. ZDM, 51(1), 1–12. https://doi.org/https://doi.org/10.1007/s11858-019-01041-z
   Google Scholar
• Thevenot, C., Barrouillet, P., Castel, C., & Uittenhove, K. (2016). Ten-year-old children strategies in mental addition: A counting model account. Cognition, 146, 48–57. https://doi.org/https://doi.org/10.1016/j.cognition.2015.09.003
   Google Scholar
• Thompson, I. (1999). Getting your head around mental calculation. In I. Thompson (Ed.), Issues in teaching numeracy in primary schools (pp. 145–156). Open University Press.
   Google Scholar
• Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2004). Strategic aspects of simple addition and subtraction: The influence of mathematical ability. Learning and Instruction, 14(2), 177–195. https://doi.org/https://doi.org/10.1016/j.learninstruc.2004.01.003
   Web of Science ®Google Scholar
• Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2006). The development of children’s adaptive expertise in the number domain 20 to 100. Cognition and Instruction, 24(4), 439–465. https://doi.org/https://doi.org/10.1207/s1532690xci2404_2
   Web of Science ®Google Scholar
• Vanbinst, K., Ceulemans, E., Ghesquière, P., & De Smedt, B. (2015). Profiles of children’s arithmetic fact development: A model-based clustering approach. Journal of Experimental Child Psychology, 133, 29–46. https://doi.org/https://doi.org/10.1016/j.jecp.2015.01.003
   PubMed Web of Science ®Google Scholar
• Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359. https://doi.org/https://doi.org/10.1007/BF03174765
   Web of Science ®Google Scholar