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Everyday Context and Mathematical Learning: On the Role of Spontaneous Mathematical Focusing Tendencies in the Development of Numeracy

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Abstract

This chapter summarizes classical studies on numeracy development and highlights the impact of children’s own, self-initiated numerical activities in informal everyday situations on the learning trajectories toward an advanced number sense. Theories and empirical evidence of the role early exact and approximate magnitude processing, relational reasoning, counting, number sequence, and arithmetical skills play in the development of natural and rational number knowledge are described. Furthermore, studies on spontaneous focusing on numerosity (SFON) and spontaneous focusing on quantitative relations (SFOR) are described arguing that individual differences in self-initiated focusing on numerosity and quantitative relations explain some of the individual differences in children’s numerical development during the childhood years and primary school. Based on this literature review, we claim that it is important for formal mathematics education to early on take into account children’s own mathematically relevant activities in both formal and informal situations.

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References

  • 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. 1–33). London: Lawrence Erlbaum.

    Google Scholar 

  • Batchelor, S. (2014). Dispositional factors affecting children’s early numerical development. Loughborough University, Leicestershire, United Kingdom. Retrieved from https://dspace.lboro.ac.uk/2134/17474

  • Batchelor, S., Inglis, M., & Gilmore, C. (2015). Spontaneous focusing on numerosity and the arithmetic advantage. Learning and Instruction, 40(2–3), 116–135.

    Google Scholar 

  • Blevins-Knabe, B., Austin, A. B., Musun, L., Eddy, A., & Jones, R. M. (2000). Family home care providers’ and parents’ beliefs and practices concerning mathematics with young children. Early Child Development and Care, 165(1), 41–58.

    Article  Google Scholar 

  • Bojorque, G., Torbeyns, J., Hannula-Sormunen, M., Van Nijlen, D., & Verschaffel, L. (2016). Development of SFON in Ecuadorian kindergartners. European Journal of Psychology of Education. 32, 449–462.

    Article  Google Scholar 

  • Boyer, T. W., & Levine, S. C. (2012). Child proportional scaling: Is 1/3=2/6=3/9=4/12? Journal of Experimental Child Psychology, 111(3), 516–533.

    Article  Google Scholar 

  • Boyer, T. W., Levine, S. C., & Huttenlocher, J. (2008). Development of proportional reasoning: Where young children go wrong. Developmental Psychology, 44(5), 1478–1490.

    Article  Google Scholar 

  • Braithwaite, D. W., Tian, J., & Siegler, R. S. (2017). Do children understand fraction addition? Developmental Science, e12601. 21, 4.

    Google Scholar 

  • Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607–618.

    Article  Google Scholar 

  • Cahoon, A., Cassidy, T., & Simms, V. (2017). Parents’ views and experiences of the informal and formal home numeracy environment. Learning, Culture and Social Interaction, 15(August), 69–79.

    Article  Google Scholar 

  • Carraher, D., Schliemann, A. D., & Brizuela, B. M. (2001). Can young students operate on unknowns. In Proceedings of the XXV Conference of the International Group for the Psychology of Mathematics Education (pp. 130–140).

    Google Scholar 

  • Chi, M. T. H., & Klahr, D. (1975). Span and rate of apprehension in children and adults. Journal of Experimental Child Psychology, 3, 93–102.

    Google Scholar 

  • Clements, D. H., & Sarama, J. (2014). Learning and teaching early math. The learning trajectories approach (2nd ed.). New York: Routledge.

    Book  Google Scholar 

  • Cowan, R., Dowker, A., Christakis, A., & Bailey, S. (1996). Even more precisely assessing children’s understanding of order-irrelevance principle. Journal of Experimental Child Psychology, 62, 84–101.

    Article  Google Scholar 

  • Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.

    Google Scholar 

  • Dehaene, S., Izard, V., Spelke, E., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320(5880), 1217–1220.

    Article  Google Scholar 

  • Edens, K. M., & Potter, E. F. (2013). An exploratory look at the relationships among Math skills, motivational factors and activity choice. Early Childhood Education Journal, 41(3), 235–243.

    Article  Google Scholar 

  • Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6, 568–584.

    Article  Google Scholar 

  • Frydman, O., & Bryant, P. (1988). Sharing and the understanding of number equivalence by young children. Cognitive Development, 3(4), 323–339.

    Article  Google Scholar 

  • Fuson, K. (1988). Children’s counting and concepts of number. New York: Springer Verlag.

    Book  Google Scholar 

  • Fuson, K., Richards, J., & Briars, D. J. (1982). The acquisition and elaboration of the number word sequence. In C. J. Brainerd (Ed.), Children’s logical and mathematical cognition. Progress in cognitive development research (pp. 33–92). New York: Springer-Verlag.

    Chapter  Google Scholar 

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

    Google Scholar 

  • Ginsburg, H. P., Duch, H., Ertle, B., & Noble, K. G. (2012). How can parents help their children learn math? In B. H. Wasik (Ed.), Handbook of family literacy (pp. 51–65). New York: Routledge/Taylor & Francis Group.

    Google Scholar 

  • Greiffenhagen, C. G., & Sharrock, W. (2008). School mathematics and its everyday other? Revisiting Lave’s ‘Cognition in Practice’. Educational Studies in Mathematic, 69, 1–21 Hannula-Sormunen, M. M. (2014). Spontaneous focusing on numerosity and its relation to counting and arithmetic. In A. Dowker & R. Cohen Kadosh (Eds.), Oxford handbook of mathematical cognition (pp. 275–290). Oxford: Oxford University Press.

    Article  Google Scholar 

  • Hannula, M. M. (2005). Spontaneous focusing on numerosity in the development of early mathematical skills (Annales Un). Turku, Finland: Painosalama.

    Google Scholar 

  • Hannula, M. M., & Lehtinen, E. (2001). Spontaneous tendency to focus on numerosities in the development of cardinality. In M. Panhuizen-Van Heuvel (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 113–120). Amersfoort, The Netherlands: Drukkerij Wilco.

    Google Scholar 

  • Hannula, M. M., & Lehtinen, E. (2005). Spontaneous focusing on numerosity and mathematical skills of young children. Learning and Instruction, 15(3), 237–256.

    Article  Google Scholar 

  • Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107(4), 394–406.

    Article  Google Scholar 

  • Hannula, M. M., Mattinen, A., & Lehtinen, E. (2005). Does social interaction influence 3-year-old children’s tendency to focus on numerosity? A quasi-experimental study in day care. In L. Verschaffel, E. De Corte, G. Kanselaar, & M. Valcke (Eds.), Powerful environments for promoting deep conceptual and strategic learning (pp. 63–80).

    Google Scholar 

  • Hannula, M. M., Räsänen, P., & Lehtinen, E. (2007). Development of counting skills: Role of spontaneous focusing on numerosity and subitizing-based enumeration. Mathematical Thinking and Learning, 9, 51–57.

    Article  Google Scholar 

  • Hannula-Sormunen, M. M. (2014). Spontaneous focusing on numerosity and its relation to counting and arithmetic. In A. Dowker & R. Cohen Kadosh (Eds.), Oxford handbook of mathematical cognition (pp. 275–290). Croydon, UK: Oxford University Press.

    Google Scholar 

  • Hannula-Sormunen, M. M., Alanen, A., McMullen, J., & Lehtinen, E. (2016) Integrating SFON enhancement with computerized arithmetical training – A pilot study. Poster presented at the Workshop “Domain-General and Domain-Specific Foundation of Numerical and Arithmetic Processing”, 28–30 September, Tuebingen.

    Google Scholar 

  • Hannula-Sormunen, M. M., Lehtinen, E., & Räsänen, P. (2015). Preschool children’s spontaneous focusing on numerosity, subitizing, and counting skills as predictors of their mathematical performance seven years later at school. Mathematical Thinking and Learning, 17(2–3), 155–177.

    Article  Google Scholar 

  • Hannula-Sormunen, M. M., Nanu, C., Laakkonen, E., Batchelor, S., Simms, V., De Smedt, B., et al. (in preparation). Spontaneous focusing on numerosity as second-order factor. Manuscript in preparation, 2018.

    Google Scholar 

  • Holgersson, I., Barendregt, W., Emanuelsson, J., Ottosson, T., Rietz, E., & Lindstrom, B. (2016). Fingu-A game to support children’s development of arithmetic competence: Theory, design and empirical research. In International perspectives on teaching and learning mathematics with virtual manipulatives (Vol. 7, pp. 123–145). Cham, Switzerland: Springer.

    Google Scholar 

  • Jevons, S. (1871). The power of numerical discrimination. Nature, 3, 281–282.

    Article  Google Scholar 

  • Kucian, K., Kohn, J., Hannula-Sormunen, M. M., Richtmann, V., Grond, U., Käser, T., et al. (2012). Kinder mit Dyskalkulie fokussieren spontan weniger auf Anzahligkeit [Children with Developmental Dyscalculia Focus Spontaneously Less on Numerosities]. Lernen Und Lernstörungen, 1(4), 241–253.

    Article  Google Scholar 

  • Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. Cambridge, UK: Cambridge University Press.

    Book  Google Scholar 

  • Lefevre, J. A., Clarke, T., & Stringer, A. P. (2002). Influences of language and parental involvement on the development of counting skills: Comparisons of French- and English-speaking Canadian children. Early Child Development and Care, 172(3), 283–300.

    Article  Google Scholar 

  • LeFevre, J.-A., Skwarchuk, S.-L., Smith-Chant, B. L., Fast, L., Kamawar, D., & Bisanz, J. (2009). Home numeracy experiences and children’s math performance in the early school years. Canadian Journal of Behavioural Science/Revue canadienne des sciences du comportement, 41(2), 55–66.

    Article  Google Scholar 

  • Lehtinen, E., Hannula-Sormunen, M., McMullen, J., & Gruber, H. (2017). Cultivating mathematical skills: From drill-and-practice to deliberate practice. ZDM – Mathematics Education, 49(4), 625–636.

    Article  Google Scholar 

  • Lemer, C., Dehaene, S., Spelke, E., & Cohen, L. (2003). Approximate quantities and exact number words: Dissociable systems. Neuropsychologia, 41, 1942–1958.

    Article  Google Scholar 

  • Levine, S. C., Jordan, N. C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72–103.

    Article  Google Scholar 

  • Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Mattinen, A. (2006). Huomio lukumääriin: Tutkimus 3-vuotiaiden lasten matemaattisten taitojen tukemisesta päiväkodissa [Focus on numerosities: A study on supporting 3 year-old children’s mathematical development in day care]. Turku, Finland: Painosalama.

    Google Scholar 

  • McCrink, K., & Wynn, K. (2004). Large-number addition and subtraction by 9-month-old infants. Psychological Science, 15, 776–781.

    Article  Google Scholar 

  • McCrink, K., & Wynn, K. (2007). Ratio abstraction by 6-month- old infants. Psychological Science, 18, 740–745.

    Article  Google Scholar 

  • McMullen, J. (2014). Spontaneous focusing on quantitative relations and the development of rational number conceptual knowledge. Turku, Finland: Painosalama.

    Google Scholar 

  • McMullen, J., Hannula-Sormunen, M., & Lehtinen, E. (2017). Spontaneous focusing on quantitative relations as a predictor of rational number and algebra knowledge. Contemporary Educational Psychology, 51, 356–365.

    Article  Google Scholar 

  • McMullen, J., Hannula-Sormunen, M. M., Kainulainen, M., Kiili, K., & Lehtinen, E. (in press). Moving mathematics out of the classroom: Using mobile technology to enhance spontaneous focusing on quantitative relations. British Journal of Educational Technology.

    Google Scholar 

  • McMullen, J., Hannula-Sormunen, M. M., Laakkonen, E., & Lehtinen, E. (2016). Spontaneous focusing on quantitative relations as a predictor of the development of rational number conceptual knowledge. Journal of Educational Psychology, 108(6), 857–868.

    Article  Google Scholar 

  • McMullen, J., Hannula-Sormunen, M. M., & Lehtinen, E. (2011). Young children’s spontaneous focusing on quantitative aspects and their verbalizations of their quantitative reasoning. In B. Ubuz (Ed.), Proceedings of the 35th conference of the International Group for the Psychology of Mathematics Education (pp. 217–224). Ankara, Turkey: PME.

    Google Scholar 

  • McMullen, J., Hannula-Sormunen, M. M., & Lehtinen, E. (2013). Young children’s recognition of quantitative relations in mathematically unspecified settings. Journal of Mathematical Behavior, 32(3), 450–460.

    Article  Google Scholar 

  • McMullen, J., Hannula-Sormunen, M. M., & Lehtinen, E. (2014). Spontaneous focusing on quantitative relations in the development of children’s fraction knowledge. Cognition and Instruction, 32(2), 198–218.

    Article  Google Scholar 

  • McMullen, J., Hannula-Sormunen, M. M., Lehtinen, E., & Siegler, R. S. (submitted). Adaptive rational number knowledge and spontaneous focusing on multiplicative relations.

    Google Scholar 

  • Merenluoto, K., & Lehtinen, E. (2004). Number concept and conceptual change: Towards a systemic model of the processes of change. Learning and Instruction, 14(5), 519–534.

    Article  Google Scholar 

  • Missall, K., Hojnoski, R. L., Caskie, G. I. L., & Repasky, P. (2015). Home numeracy environments of preschoolers: Examining relations among mathematical activities, parent mathematical beliefs, and early mathematical skills. Early Education and Development, 26(3), 356–376.

    Article  Google Scholar 

  • Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Math without words: Quantitative development in infancy and early childhood. New York: Oxford University Press.

    Book  Google Scholar 

  • Mix, K. S., Levine, S. C., & Huttenlocher, J. (1999). Early fraction calculation ability. Developmental Psychology, 35(5), 164–174.

    Article  Google Scholar 

  • Mullis, I. V. S., Martin, M. O., Goh, S., & Cotter, K. (2016). TIMSS 2015 Encyclopedia: Education policy and curriculum in mathematics and science. Retrieved from http://timssandpirls.bc.edu/timss2015/encyclopedia/

  • Nanu, E. C., McMullen, J., Munck, P., Pipary Study Group, & Hannula-Sormunen, M. M. (2018). Spontaneous focusing on numerosity in preschool as a predictor of mathematical skills and knowledge in the fifth grade. Journal of Experimental Child Psychology, 169, 42–58.

    Article  Google Scholar 

  • Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.

    Article  Google Scholar 

  • Nunes, T., & Bryant, P. (1996). Children doing mathematics. Cornwall, UK: T.J. Press.

    Google Scholar 

  • Nunes, T., & Bryant, P. (2008). Rational numbers and intensive quantities: Challenges and insights to pupils’ implicit knowledge. Anales De Psicologia, 24, 262–270.

    Google Scholar 

  • Piaget, J. (1955). The construction of reality in the child. Psychological Bulletin, 52(1955), 526–528.

    Google Scholar 

  • Pongsakdi, N., Laine, T., Veermans, K., Hannula-Sormunen, M. M., & Lehtinen, E. (2016). Improving word problem performance in elementary school students by enriching word problems used in mathematics teaching. NOMAD, Nordic Studies in Mathematics Education, 21(2). Retrieved from https://www.mendeley.com/profiles/margareta-engvall/

  • Potter, E. F. (2009). Spontaneous focusing on numerosity: Motivational and skill correlates in young children in a public preschool and kindergarten program. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), Proceedings of the 31st annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 152–155). Atlanta, GA: Georgia State University Press.

    Google Scholar 

  • Rathé, S., Torbeyns, J., Hannula-Sormunen, M. M., De Smedt, B., & Verschaffel, L. (2016). Spontaneous focusing on numerosity: A review of recent research. Mediterrean Journal for Research on Mathematics Education, 15, 1–25.

    Google Scholar 

  • Resnick, L. B. (1987). The 1987 presidential address: Learning in school and out. Educational Researcher, 16(9), 13–20; 54.

    Google Scholar 

  • Resnick, L. B. (1992). From protoquantities to operators: Building mathematical competence on a foundation of everyday knowledge. In G. Leinhardt, R. T. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 373–429). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Resnick, L. B., & Greeno, J. G. (1990). Conceptual growth of number and quantity. Unpublished manuscript.

    Google Scholar 

  • Sarnecka, B. W., & Gelman, S. A. (2004). Six does not just mean a lot: Preschoolers see number words as specific. Cognition, 92, 329–352.

    Article  Google Scholar 

  • Sathian, K., Simon, T. J., Peterson, S., Patel, G. A., Hoffman, J. M., & Grafton, S. T. (1999). Neural evidence linking visual object enumeration and attention. Journal of Cognitive Neuroscience, 11, 36–51.

    Article  Google Scholar 

  • Saxe, G. B., Guberman, S. R., & Gearhart, M. (1987). Social processes in early number development. Monographs of the Society for Research in Child Development, 52, Serial No. 216.

    Article  Google Scholar 

  • Schliemann, A. D., & Nunes, T. (1990). A situated schema of proportionality. British Journal of Developmental Psychology, 8(3), 259–268.

    Article  Google Scholar 

  • Siegler, R. S. (2016). Continuity and change in the field of cognitive development and in the perspectives of one cognitive developmentalist. Child Development Perspectives, 10, 128–133.

    Article  Google Scholar 

  • Skwarchuk, S. L., Sowinski, C., & LeFevre, J. A. (2014). Formal and informal home learning activities in relation to children’s early numeracy and literacy skills: The development of a home numeracy model. Journal of Experimental Child Psychology, 121(1), 63–84.

    Article  Google Scholar 

  • Sophian, C. (1998). A developmental perspective on children’s counting. In C. Donlan (Ed.), The development of mathematical skills (pp. 27–41). Hove, UK: Taylor & Francis.

    Google Scholar 

  • Sophian, C. (2007). The origins of mathematical knowledge in childhood. New York: Lawrence Erlbaum.

    Google Scholar 

  • Sophian, C., Harley, H., & Martin, C. S. M. (1995). Relational and representational aspects of early number development. Cognition and Instruction, 13(2), 253–268.

    Article  Google Scholar 

  • Spelke, E. (2003). What makes us smart? Core knowledge and natural language. In D. Gentner & S. Goldin-Meadow (Eds.), Language in mind. Cambridge, MA: MIT Press.

    Google Scholar 

  • Spinillo, A. G., & Bryant, P. E. (1999). Proportional reasoning in young children: Part – Part comparisons about continuous and discontinous quantity. Mathematical Cognition, 5(2), 181–197.

    Article  Google Scholar 

  • Starkey, P., & Cooper, R. G. (1995). The development of subitizing in young children. British Journal of Developmental Psychology, 13, 399–420.

    Article  Google Scholar 

  • Trick, L., Enns, J. T., & Brodeur, D. A. (1996). Life span changes in visual enumeration: The number discrimination task. Developmental Psychology, 32, 925–932.

    Article  Google Scholar 

  • Trick, L. M., & Pylyshyn, Z. W. (1994). Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision. Psychological Review, 101(1), 80–102.

    Article  Google Scholar 

  • Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467.

    Article  Google Scholar 

  • Van Hoof, J., Degrande, T., McMullen, J., Hannula-Sormunen, M. M., Lehtinen, E., Verschaffel, L., et al. (2016). The relation between learners’ spontaneous focusing on quantitative relations and their rational number knowledge. Studia Psychologica, 58(2), 156–170.

    Article  Google Scholar 

  • Wager, A. A. (2012). Incorporating out-of-school mathematics: From cultural context to embedded practice. Journal of Math Teacher Education, 15, 9–23.

    Article  Google Scholar 

  • Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193.

    Article  Google Scholar 

  • Wynn, K. (1992a). Addition and subtraction by human infants. Nature, 358, 749–750.

    Article  Google Scholar 

  • Wynn, K. (1992b). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251.

    Article  Google Scholar 

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Hannula-Sormunen, M.M., McMullen, J., Lehtinen, E. (2019). Everyday Context and Mathematical Learning: On the Role of Spontaneous Mathematical Focusing Tendencies in the Development of Numeracy. In: Fritz, A., Haase, V.G., Räsänen, P. (eds) International Handbook of Mathematical Learning Difficulties. Springer, Cham. https://doi.org/10.1007/978-3-319-97148-3_3

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