Abstract
The overarching theme of this book can be simply stated: Building on a foundation of biologically based abilities, children construct number via sensorimotor and mental activity. In this chapter, we return to this theme, and we connect it to three additional themes that emerge across chapters: comparing competing models for conceptual change; consideration of multiple concepts for natural numbers, fractions, and integers; and understanding interrelations of conceptual and procedural knowledge in the construction of number. We close by suggesting ways that psychologists and mathematics educators might move forward with interdisciplinary research that addresses important questions about the construction of number. Indeed, the chapters in this volume chart many possible paths.
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References
Alibali, M. W., & Knuth, E. J. (2018). Bridging psychology and mathematics education: Reflections on boundary crossing. Journal of Numerical Cognition, 4(1), 09–18. https://doi.org/10.5964/jnc.v4i1.111
Alibali, M. W., & Nathan, M. J. (2018). Embodied cognition in learning and teaching: Action, observation, and imagination. In F. Fischer, S. Goldman, C. Hmelo-Silver, & P. Riemann (Eds.), International handbook of the learning sciences (pp. 75–85). New York, NY: Routledge/Taylor & Francis.
Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills. Mahwah, NJ: Erlbaum.
Beckmann, S., & Izsák, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46(1), 17–38. https://doi.org/10.5951/jresematheduc.46.1.0017
Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–126). New York: Academic.
Carey, S. (1991). Knowledge acquisition: Enrichment or conceptual change? In S. Carey & R. Gelman (Eds.), The epigenesis of mind: Essays on biology and cognition (pp. 257–291). Hillsdale, NJ: Erlbaum.
Cook, V. J. (2003). Introduction: The Changing L1 in the L2 user’s mind. In V. J. Cook (Ed.), Effects of second language on the first (pp. 1–19). Clevedon, Avon: Multilingual Matters.
Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge of mathematics. Developmental Review, 34, 344–377. https://doi.org/10.1016/j.dr.2014.10.001
Ericsson, K. A., Krampe, R. T., & Tesch-Römer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100(3), 363–406. https://doi.org/10.1037/0033-295X.100.3.363
Ganor-Stern, D., Pinhas, M., Kallai, A., & Tzelgov, J. (2010). Holistic representation of negative numbers is formed when needed for the task. Quarterly Journal of Experimental Psychology, 63(10), 1969–1981. https://doi.org/10.1080/17470211003721667
Ganor-Stern, D., & Tzelgov, J. (2008). Negative numbers are generated in the mind. Experimental Psychology, 55(3), 157–163. https://doi.org/10.1027/1618-3169.55.3.157
Gopnik, A., & Bonawitz, E. (2014). Bayesian models of child development. Wiley Interdisciplinary Reviews: Cognitive Science, 6(2), 75–86. https://doi.org/10.1002/wcs.1330
Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: Relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14(5), 1021–1032. https://doi.org/10.1111/j.1467-7687.2011.01050.x
Hohensee, C. (2014). Backward transfer: An investigation of the influence of quadratic functions instruction on students’ prior ways of reasoning about linear functions. Mathematical Thinking and Learning, 16(2), 135–174. https://doi.org/10.1080/10986065.2014.889503
Hohensee, C. (2016). Student noticing in classroom settings: A process underlying influences on prior ways of reasoning. Journal of Mathematical Behavior, 42, 69–91. https://doi.org/10.1016/j.jmathb.2016.03.002
Kieren, T. E. (1980). The rational number construct - Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 125–150). Columbus, OH: ERIC/SMEAC.
Levine, S. C., Ratliff, K. R., Huttenlocher, J., & Cannon, J. (2012). Early puzzle play: A predictor of preschoolers’ spatial transformation skill. Developmental Psychology, 48(2), 530–542. https://doi.org/10.1037/a0025913
Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010). What counts in the development of young children’s number knowledge? Developmental Psychology, 46(5), 1309–1319. https://doi.org/10.1037/a0019671
Macnamara, B. N., Hambrick, D. Z., & Oswald, F. L. (2014). Deliberate practice and performance in music, games, sports, education, and professions: A meta-analysis. Psychological Science, 25(8), 1608–1618. https://doi.org/10.1177/0956797614535810
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. https://doi.org/10.1207/s15326985ep4001_3
Norton, A., & Nurnberger-Haag, J. (2018). Bridging frameworks for understanding numerical cognition. Journal of Numerical Cognition, 4(1), 1–8. https://doi.org/10.5964/jnc.v4i1.160
Obersteiner, A., & Hofreiter, V. (2017). Do we have a sense for irrational numbers? Journal of Numerical Cognition, 2(3), 170–189. https://doi.org/10.5964/jnc.v2i3.43
Prather, R., & Alibali, M. W. (2008). Understanding and using principles of arithmetic: Operations involving negative numbers. Cognitive Science, 32(2), 445–457. https://doi.org/10.1080/03640210701864147
Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27(4), 587–597. https://doi.org/10.1007/s10648-015-9302-x
Sidney, P. G., & Alibali, M. W. (2015). Making connections in math: Activating a prior knowledge analogue matters for learning. Journal of Cognition and Development, 16(1), 160–185. https://doi.org/10.1080/15248372.2013.792091
Sidney, P. G., & Alibali, M. W. (2017). Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division. Journal of Numerical Cognition, 3(1), 31–57. https://doi.org/10.5964/jnc.v3i1.71
Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144-150. https://doi.org/10.1111/cdep.12077
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001
Sobel, D. M., & Kushnir, T. (2013). Knowledge matters: How children evaluate the reliability of testimony as a process of rational inference. Psychological Review, 120(4), 779–797. https://doi.org/10.1037/a0034191
Tsang, J. M., Blair, K. P., Bofferding, L., & Schwartz, D. L. (2015). Learning to “see” less than nothing: Putting perceptual skills to work for learning numerical structure. Cognition and Instruction, 33(2), 154–197. https://dx.doi.org/10.1080/07370008.2015.1038539
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. https://doi.org/10.1016/j.learninstruc.2004.06.013
Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5, 27. https://doi.org/10.1186/s40594-018-0119-2
Xu, F., & Tenenbaum, J. B. (2007). Word learning as Bayesian inference. Psychological Review, 114(2), 245–272. https://doi.org/10.1037/0033-295X.114.2.245
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Alibali, M.W., Norton, A. (2019). Synergizing Research on Constructing Number: Themes and Prospects. In: Norton, A., Alibali, M.W. (eds) Constructing Number. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-00491-0_16
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