Preschoolers exercising mathematical competencies

Authors

Anna Ida Säfström

Abstract

The mathematical ideas that emerge in children’s free and guided play can be both complex and sophisticated, and if they are linked to formal mathematics, they can be a powerful basis for mathematical development. To form such links, one needs knowledge of how children use and express these ideas. This is especially true in the intersection of arithmetic and geometry, where the intermingling of numerical and spatial concepts and skills is not yet fully understood. This study aims to gain understanding of children’s mathematical practices by describing the interplay of key mathematical ideas, and more specifically how young children exercise mathematical competencies in the intersection of early arithmetic and geometry. The results show that children can use spatial representations when reasoning about numbers, and that they are able to connect spatial and numerical structures. Furthermore, it is shown that children not only use and invent effective procedures, but also are able to explain, justify and evaluate such procedures.

References

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.

Battista, M. T., Clements, D. H., Arnoff, J., Battista, K. & Borrow, C. V. A. (1998). Students' spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29, 503-532. https://doi.org/10.2307/749731

Björklund, C. (2008). Toddlers' opportunities to learn mathematics. International Journal of Early Childhood, 40 (1), 81-95. https://doi.org/10.1007/BF03168365

Clements, D. & Sarama, J. (2007). Early childhood mathematics learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 461-555). Charlotte: Information Age Publishing.

Clements, D. & Sarama, J. (2011). Early childhood teacher education: the case of geometry. Journal of Mathematics Teacher Education, 14, 133-148. https://doi.org/10.1007/s10857-011-9173-0

Ellis, A. B. (2007). A taxonomy for categorizing generalizations: generalizing actions and reflection generalizations. The Journal of the Learning Sciences, 16, 221-262. https://doi.org/10.1080/10508400701193705

Fisher, K. R., Hirsh-Pasek, K., Newcombe, N. & Golinkoff, R. M. (2013). Taking shape: supporting preschoolers' acquisition of geometric knowledge through guided play. Child Development, 84, 1872-1878. https://doi.org/10.1111/cdev.12091

Gersten, R., Jordan, N. C. & Flojo, J. R. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304. https://doi.org/10.1177/00222194050380040301

Ginsburg, H. P. & Seo, K.-H. (1999). Mathematics in children's thinking. Mathematical Thinking and Learning, 1, 113-129. https://doi.org/10.1207/s15327833mtl0102_2

Goldin, G. A. (2003). Representations in school mathematics: a unifying research perspective. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275-285). Reston: NCTM.

Hachey, A. C. (2013). The early childhood mathematics education revolution. Early Education and Development, 24, 419-430. https://doi.org/10.1080/10409289.2012.756223

Handa, Y. (2012). Teasing out repetition from rote: an essay on two versions of will. Educational Studies in Mathematics, 79, 263-272. https://doi.org/10.1007/s10649-011-9343-0

Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-98). New York: Macmillan. https://doi.org/10.2307/749496

Jacobs, V. R., Lamb, L. L. C. & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41, 169-202. https://doi.org/10.5951/jresematheduc.41.2.0169

Johanning, D. I. (2008). Learning to use fractions: examining middle school students' emerging fraction literacy. Journal for Research in Mathematics Education, 39, 281-310.

Jordan, N. C., Glutting, J. & Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20, 82-88. https://doi.org/10.1016/j.lindif.2009.07.004

Kilhamn, C. (2011). Making sense of negative numbers (Ph. D. Thesis). Gothenburg: Acta Universitatis Gothoburgensis. Availble from http://hdl. handle.net/2077/24151

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

Kirsch, D. (2010). Thinking with external representations. AI & Society, 25, 441-454. https://doi.org/10.1007/s00146-010-0272-8

Lee, J. S. & Ginsburg, H. P. (2009). Early childhood teachers' misconceptions about mathematics education for young children in the United States. Australasian Journal of Early Childhood, 34 (4), 37-45. https://doi.org/10.1177/183693910903400406

Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67, 255-276. https://doi.org/10.1007/s10649-007-9104-2

Lithner, J., Bergqvist, E., Bergqvist, T., Boesen, J., Palm, T. & Palmberg, B. (2010). Mathematical competencies: a research framework. In C. Bergsten, E. Jablonka & T. Wedege (Eds.), Mathematics and mathematics education: cultural and social dimensions. Proceedings of MADIF 7, the seventh Mathematics Education Research Seminar, Stockholm, January 26-27 (pp. 157-167). Linköping: SMDF.

Mason, J., Stephens, M. & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21, 10-32. https://doi.org/10.1007/BF03217543

Moss, J., Bruce, C. D. & Bobis, J. (2016). Young children's access to powerful mathematical ideas. In L. D. English & D. Kirschner (Eds.), Handbook of international research in mathematics education (3rd Ed.) (pp. 153-190). New York: Taylor & Francis.

Moss, J., Hawes, Z., Naqvi, S. & Caswell, B. (2015). Adapting Japanese lesson study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study. ZDM, 47, 377-390. https://doi.org/10.1007/s11858-015-0679-2

Mueller, M. F. (2009). The co-construction of arguments by middle-school students. The Journal of Mathematical Behavior, 28, 138-149. https://doi.org/10.1016/j.jmathb.2009.06.003

Mulligan, J. T. & Mitchelmore, M. C. (2015). Early awareness of mathematical pattern and structure. In L. D. English & J. T. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 29-45). Dordrecht: Springer. https://doi.org/10.1007/978-94-007-6440-8_3

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

Nes, F. van & Lange, J. de (2007). Mathematics education and neurosciences: relating spatial structures to the development of spatial sense and number sense. The Mathematics Enthusiast, 4, 210-229. https://doi.org/10.54870/1551-3440.1072

Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM Project. In A. Gagatsis & S. Papastavridis (Eds.), Proceedings of the third Mediterranean conference on mathematics education (pp. 115-124). Athens: Greek Mathematical Society.

Niss, M. & Jensen, T. H. (Eds.). (2002). Kompetencer og matematiklæring: ideer og inspiration til udvikling af matematikundervisning i Danmark [Competencies and mathematics learning: ideas and inspiration for the development of mathematics teaching in Denmark]. København: Undervisningsministeriets forlag.

Oers, B. van (2010). Emergent mathematical thinking in the context of play. Educational Studies in Mathematics, 74, 23-37. https://doi.org/10.1007/s10649-009-9225-x

Perry, B. & Dockett, S. (2015). Reflecting on young children's mathematics learning. In L. D. English & J. T. Mulligan (Eds.), Reconceptualizing early mathematics learning (p. 149-163). Dordrecht: Springer. https://doi.org/10.1007/978-94-007-6440-8_8

Samuelsson, I. P. & Carlsson, M. A. (2008). The playing learning child: towards a pedagogy of early childhood. Scandinavian Journal of Educational Research, 52 (6), 623-641. https://doi.org/10.1080/00313830802497265

Sfard, A. (2009). What's all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70, 191-200. https://doi.org/10.1007/s10649-008-9161-1

Shein, P. P. (2012). Seeing with two eyes: a teacher's use of gestures in questioning and revoicing to engage English language learners in the repair of mathematical errors. Journal for Research in Mathematics Education, 43, 182-222. https://doi.org/10.5951/jresematheduc.43.2.0182

Sherin, M. G., Jacobs, V. R. & Philipp, R. A. (2011). Mathematics teacher noticing: seeing through teachers' eyes. New York: Taylor & Francis. https://doi.org/10.4324/9780203832714

Skolverket. (2016). Läroplan för förskolan Lpfö 98. Reviderad 2016 [Curriculum for preschool Lpfö98. Revised 2016]. Stockholm: Skolverket

Speiser, B., Walter, C. & Sullivan, C. (2007). From test cases to special cases: four undergraduates unpack a formula for combination. The Journal of Mathematical Behavior, 26, 11-26. https://doi.org/10.1016/j.jmathb.2007.03.003

Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404-411.

Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38, 132-135.

Stylianou, D. A. (2011). An examination of middle school students' representational practices in mathematical problem solving through the lens of expert work: towards an organizing scheme. Educational Studies in Mathematics, 76, 265-280. https://doi.org/10.1007/s10649-010-9273-2

Sumpter, L. & Hedefalk, M. (2015). Preschool children's collective mathematical reasoning during free outdoor play. The Journal of Mathematical Behavior, 39, 1-10. https://doi.org/10.1016/j.jmathb.2015.03.006

Säfström, A. I. (2013a). Exercising mathematical competence: practising representation theory and representing mathematical practice (Ph. D. Thesis). Gothenburg: Chalmers University of Technology and University of Gothenburg. Availble from http://hdl.handle.net/2077/32484

Säfström, A. I. (2013b). A competency framework for analysis of mathematical practice. In A. M. Lindemeier & A. Heinze (Ed.), Mathematics learning across the life span: proceedings of PME 37 (Vol.4, p. 129-136). Kiel: IPN, Leibniz Institute for Science and Mathematics Education.

Truxaw, M. P. & DeFranco, T. (2008). Mapping mathematics classroom discourse and its implications for models of teaching. Journal for Research in Mathematics Education, 39, 489-525.

Verdine, B. N., Golinkoff, R. M., Hirsh-Pasek, K., Newcombe, N. S., Filipowicz, A. T. & Chang, A. (2014). Deconstructing building blocks: preschoolers' spatial assembly performance relates to early mathematical skills. Child Development, 85, 1062-1076. https://doi.org/10.1111/cdev.12165

Verschaffel, L., Greer, B. & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 557-628). Charlotte: Information Age Publishing.

Wager, A. A. (2015). Practices that support mathematics learning in a play-based classroom. In L. D. English & J. T. Mulligan (Eds.), Reconceptualizing early mathematics learning (pp. 164-181). Dordrecht: Springer.

Wager, A. A. & Parks, A. N. (2016). Assessing early number learning in play. ZDM, 48, 991-1002. https://doi.org/10.1007/s11858-016-0806-8

Downloads

Published

2024-11-19

How to Cite

Säfström, A. I. (2024). Preschoolers exercising mathematical competencies. NOMAD Nordic Studies in Mathematics Education, 23(1), 5–27. Retrieved from https://tidsskrift.dk/NOMAD/article/view/148927