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Arithmetic Concepts in the Early School Years

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Mathematical Learning and Cognition in Early Childhood

Abstract

Children’s knowledge of arithmetic concepts is considered critical in the development of current and later mathematic skills. If children understand numbers, operations, and the relations among them, they are more likely to make sense of mathematics and be more successful problem solvers. This chapter discusses the importance of arithmetic concepts and how they fit with children’s knowledge of arithmetic facts and arithmetic problem-solving procedures. How concepts are assessed by researchers is presented as well as current research on the development of six specific arithmetic concepts: identity, negation , commutativity, inversion , associativity, and equivalence. Several ways in which teachers and parents can increase children’s knowledge and understanding of arithmetic and can promote the use of that knowledge to improve their mathematical skills are proposed.

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References

  • Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9, 221–247. https://doi.org/10.1080/10986060701360902.

    Article  Google Scholar 

  • Alibali, M. W., Phillips, K. M. O., & Fischer, A. D. (2009). Learning new problem-solving strategies leads to changes in problem representation. Cognitive Development, 24, 89–101. https://doi.org/10.1016/j.cogdev.2008.12.005.

    Article  Google Scholar 

  • Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9, 249–272. https://doi.org/10.1080/10986060701360910.

    Article  Google Scholar 

  • Baroody, A. J., & Gannon, K. E. (1984). The development of commutativity principle and economical addition strategies. Cognition and Instruction, 1, 321–329.

    Article  Google Scholar 

  • Baroody, A. J., Lai, M., Li, X., & Baroody, A. E. (2009). Preschoolers’ understanding of subtraction-related principles. Mathematical Thinking and Learning, 11, 41–60. https://doi.org/10.1080/1098606082583956.

    Article  Google Scholar 

  • Bisanz, J., & LeFevre, J. (1990). Strategic and nonstrategic processing in the development of mathematical cognition. In D. F. Bjorklund (Ed.), Children’s strategies: Contemporary views of cognitive development (pp. 213–244). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Canobi, K. H., Reeve, R. A., & Pattison, P. E. (1998). The role of conceptual understanding in children’s addition problem solving. Developmental Psychology, 34, 882–891. https://doi.org/10.1037/0012-1649.34.5.882.

    Article  PubMed  Google Scholar 

  • Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2002). Young children’s understanding of addition concepts. Educational Psychology, 22, 513–532. https://doi.org/10.1080/0144341022000023608.

    Article  Google Scholar 

  • Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39, 521–534. https://doi.org/10.1037/0012-1649.39.3.521.

    Article  PubMed  Google Scholar 

  • Carpenter, T. P., Fennema, E., Loef Franke, M., Levi, L., & Empson, S. B. (2014). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann.

    Google Scholar 

  • Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34, 344–377. https://doi.org/10.1016/j.dr.2014.10.001.

    Article  Google Scholar 

  • Dubé, A. K, & Robinson, K. M. (2010). The relationship between adults’ conceptual understanding of inversion and associativity. Canadian Journal of Experimental Psychology, 64, 60–66. https://doi.org/10.1037/a0017756.

    Article  Google Scholar 

  • Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232–236.

    Google Scholar 

  • Fyfe, E. R., Matthews, P. G., Amsel, E., McEldoon, K. L., & McNeil, N. M. (2017). Assessing formal knowledge of math equivalence among algebra and pre-algebra students. Journal of Educational Psychology. https://doi.org/10.1037/edu0000208.

    Article  Google Scholar 

  • Gelman, S., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Gilmore, C. K., & Bryant, P. (2006). Individual differences in children’s understanding of inversion and arithmetical skill. British Journal of Educational Psychology, 76, 309–331. https://doi.org/10.1348/000709905X39125.

    Article  PubMed  Google Scholar 

  • Gilmore, C., & Bryant, P. (2008). Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skills. British Journal of Developmental Psychology, 26, 301–316. https://doi.org/10.1348/026151007X236007.

    Article  Google Scholar 

  • Ginsburg, H. P., Cannon, J., Eisenband, J., & Pappas, S. (2006). Mathematical thinking and learning. In K. McCartney & D. Philips (Eds.), The handbook of early child development (pp. 208–229). Malden, MA: Blackwell.

    Google Scholar 

  • Groen, G. J., & Resnick, L. B. (1977). Can preschool children invent addition algorithms? Journal of Educational Psychology, 69, 645–652.

    Article  Google Scholar 

  • Hiebert, J., & Lefevre, P. (1986). Procedural and conceptual knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence.

    Google Scholar 

  • Hudson, C., Price, D., & Gross, J. (2009). The long term costs of numeracy difficulties. London, UK: Every Child a Chance Trust.

    Google Scholar 

  • Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

    Google Scholar 

  • Klein, J. S., & Bisanz, J. (2000). Preschoolers doing arithmetic: The concepts are willing but the working memory is weak. Canadian Journal of Experimental Psychology, 54, 105–116. https://doi.org/10.1037/h0087333.

    Article  PubMed  Google Scholar 

  • LeFevre, J., Sadesky, G. S., & Bisanz, J. (1996). Selection of procedures in mental addition: Reassessing the problem size effect in adults. Journal of Experimental Psychology. Learning, Memory, and Cognition, 22, 216–230.

    Article  Google Scholar 

  • McNeil, N. M. (2014). A change-resistance account of children’s difficulties understanding mathematical equivalence. Child Development Perspectives, 8, 42–47. https://doi.org/10.1111/cdep.12062.

    Article  Google Scholar 

  • McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind: Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 883–889. https://doi.org/10.1111/j/1467-8624.2005.00884.x.

    Article  PubMed  Google Scholar 

  • McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., et al. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24, 367–385. https://doi.org/10.1207/s1532690xci2403_3.

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

    Google Scholar 

  • National Council of Teachers of Mathematics, & National Association for the Education of Young Children. (1999). Mathematics in the early years. Reston, VA: NCTM.

    Google Scholar 

  • National Governors’ Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common core state standards for mathematics. www.corestandards.org/wp-content/uploads/Math_Standards1.pdf.

  • National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, D.C.: U.S. Department of Education.

    Google Scholar 

  • Newton, K. J., Star, J. R., & Lynch, K. (2010). Understanding the flexibility of struggling algebra students. Mathematical Thinking and Learning, 12, 282–305. https://doi.org/10.1080/10986065.2010.482150.

    Article  Google Scholar 

  • Nunes, T., & Bryant, P. (1998). Children doing mathematics. Cambridge, MA: Blackwell.

    Google Scholar 

  • Nunes, T., Bryant, P., Burman, D., Bell, D., Evans, D., Hallett, D., et al. (2008). Deaf children’s understanding of inverse relationships. In M. Marschark & P. C. Hauser (Eds.), Deaf cognition: Foundations and outcomes (pp. 201–225). Oxford: New York, NY.

    Chapter  Google Scholar 

  • OECD. (2016). PISA 2015 results (Volume 1): Excellence and equity in education. Paris: PISA, OECD Publishing. https://doi.org/10.1787/9789264266490-en.

    Book  Google Scholar 

  • Perry, M., Church, R. B., & Goldin-Meadow, S. (1988). Transitional knowledge in the acquisition of concepts. Cognitive Development, 3, 359–400.

    Google Scholar 

  • Piaget, J. (1952). The child’s conception of number. London, UK: Routledge & Kegan Paul.

    Google Scholar 

  • Prather, R. W., & Alibali, M. W. (2009). The development of arithmetic principle knowledge: How do we know what learners know? Developmental Review, 29, 221–248. https://doi.org/10.1016/j.dr.2009.09.001.

    Article  Google Scholar 

  • Rasmussen, C., Ho, E., & Bisanz, J. (2003). Use of the mathematical principle of inversion in young children. Journal of Experimental Child Psychology, 85, 89–102. https://doi.org/10.1016/S0022-0965(03)0031-6.

    Article  PubMed  Google Scholar 

  • Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175–189. https://doi.org/10.1037/0022-0663.91.1.175.

    Article  Google Scholar 

  • Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge in mathematics. Educational Psychology Review, 27, 587–597. https://doi.org/10.1007/s10648-015-9302-x.

    Article  Google Scholar 

  • Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills: Studies in developmental psychology (pp. 75–101). Hove, UK: Psychology Press.

    Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362. https://doi.org/10.1037/0022-0663.93.2.346.

    Article  Google Scholar 

  • Robinson, K. M. (2001). The validity of verbal reports in children’s subtraction. Journal of Educational Psychology, 93, 211–222. https://doi.org/10.1037/0022-0663.93.1.211.

    Article  Google Scholar 

  • Robinson, K. M., & Dubé, A. K. (2009). Children’s understanding of addition and subtraction concepts. Journal of Experimental Child Psychology, 103, 532–545. https://doi.org/10.1016/j.jecp.2008.12.002.

    Article  PubMed  Google Scholar 

  • Robinson, K. M., & Dubé, A. K. (2012). Children’s use of arithmetic shortcuts: The role of attitudes in strategy choice. Child Development Research, 10. https://doi.org/10.1155/2012/459385.

    Article  Google Scholar 

  • Robinson, K. M., & Dubé, A. K. (2013). Children’s additive concepts: Promoting understanding and the role of inhibition. Learning and Individual Differences, 23, 101–107. https://doi.org/10.1016/j.lindif.2012.07.016.

    Article  Google Scholar 

  • Robinson, K. M., Dubé, A. K., & Beatch, J.-A. (2017). Children’s understanding of additive concepts. Journal of Experimental Child Psychology, 156, 16–28. https://doi.org/10.1016/j.jecp.2016.11.009.

    Article  PubMed  Google Scholar 

  • Robinson, K. M., & LeFevre, J. (2012). The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework. Educational Studies in Mathematics, 79, 409–428. https://doi.org/10.1007/s10649-011-9330-5.

    Article  Google Scholar 

  • Robinson, K. M., Ninowski, J. E., & Gray, M. L. (2006). Children’s understanding of the arithmetic concepts of inversion and associativity. Journal of Experimental Child Psychology, 94, 349–362. https://doi.org/10.1016/j.jecp.2006.03.004.

    Article  PubMed  Google Scholar 

  • Robinson, K. M., Price, J. A. B., & Demyen, B. (2018). Understanding of arithmetic concepts: Does operation matter? Manuscript under review.

    Google Scholar 

  • Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525–1538. https://doi.org/10.1037/a0024997.

    Article  PubMed  Google Scholar 

  • Schneider, M., & Stern, E. (2009). The inverse relation of addition and subtraction: A knowledge integration perspective. Mathematical Thinking and Learning, 11(1), 92–101. https://doi.org/10.1080/10986060802584012.

    Article  Google Scholar 

  • Sherman, J. (2007, April). From failure to success on equivalence problems and how teachers perceive the process. Paper presented at the biennial conference for the Society for Research in Child Development, Boston, MA.

    Google Scholar 

  • Sherman, J., & Bisanz, J. (2007). Evidence for use of mathematical inversion by three-year-old children. Journal of Cognition and Development, 8, 334–344. https://doi.org/10.1080/15248370701446798.

    Article  Google Scholar 

  • Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology, 101, 88–100. https://doi.org/10.1037/a0013156.

    Article  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, 250–264. https://doi.org/10.1037/0096-3445.116.3.250.

    Article  Google Scholar 

  • Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press.

    Google Scholar 

  • Siegler, R. S., & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. In R. V. Kail (Ed.), Advances in child development and behaviour (Vol. 33, pp. 1–42). New York, NY: Elsevier.

    Google Scholar 

  • Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Siegler, R. S., & Shipley, C. (1995). Variation, selection, and cognitive change. In T. Simon & G. Halford (Eds.), Advances in child development and behaviour (Vol. 16). New York: Academic Press.

    Google Scholar 

  • Siegler, R. S., & Shrager, J. (1984). Strategy choice in addition: How do children know what to do? In C. Sophian (Ed.), Origins of cognitive skills (pp. 27–46). Hove, UK: Psychology Press.

    Google Scholar 

  • Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology: General, 127, 377–397. https://doi.org/10.1037/0096-3445.127.4.377.

    Article  Google Scholar 

  • Starkey, P., & Gelman, R. (1982). The development of addition and subtraction abilities prior to formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 99–116). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Stevenson, H. W., Lee, S., Chen, C., Stigler, J. W., Hsu, C.-C, Kitamura, S., & Hatano, G. (1990). Contexts of achievement: A study of American, Chinese, and Japanese children. Monographs of the Society for Research in Child Development, 55(1–2, 221), 123. https://doi.org/10.2307/1166090.

    Article  Google Scholar 

  • Vilette, B. (2002). Do young children grasp the inversion relationship between addition and subtraction? Evidence against early arithmetic. Cognitive Development, 17, 1365–1383. https://doi.org/10.1016/S0885-2014%2802%2900125-9.

    Article  Google Scholar 

  • Watchorn, R. P. D., Bisanz, J., Fast, L., LeFevre, J., Skwarchuk, S.-L., & Smith-Chant, B. L. (2014). Development of mathematical knowledge in young children: Attentional skill and the use of inversion. Journal of Cognition and Development, 15, 161–180. https://doi.org/10.1080/15248372.2012.742899.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Psychology, Campion College at the University of Regina, 3737 Wascana Parkway, Regina, SK, S4S 0A2, Canada

    Katherine M. Robinson

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  1. Katherine M. Robinson
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Correspondence to Katherine M. Robinson .

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Editors and Affiliations

  1. Department of Psychology, University of Regina, Regina, SK, Canada

    Katherine M. Robinson

  2. Department of Education, Concordia University, Montreal, QC, Canada

    Helena P. Osana

  3. Department of Mathematics, Wilfred Laurier University, Waterloo, ON, Canada

    Donna Kotsopoulos

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Robinson, K.M. (2019). Arithmetic Concepts in the Early School Years. In: Robinson, K., Osana, H., Kotsopoulos, D. (eds) Mathematical Learning and Cognition in Early Childhood. Springer, Cham. https://doi.org/10.1007/978-3-030-12895-1_10

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