Abstract
Number and operations form the backbone of the school mathematics curriculum. A high school graduate should comfortably and capably meet an expression like, “Let f(x) be a function of a real variable x,” implying that the student has a robust sense of the real number continuum. This understanding is a central objective of the school mathematics curriculum, taken as a whole. Yet there are reasons to doubt whether typical US high school graduates fully achieve this understanding. Why? And what can be done about this? I argue that there are obstacles already at the very foundations of number in the first grade. The construction narrative of the number line, characteristic of the prevailing curriculum, starts with cardinal counting and whole numbers and then builds the real number line through successive enlargements of the number systems studied. An alternative proposed by V. Davydov, the occupation narrative, begins with pre-numerical ideas of quantity and measurement, from which the geometric (number) line, as the environment of linear measure, can be made present from the beginning and wherein new numbers progressively take up residence. I will compare these two approaches, including their cognitive premises, and suggest some advantages of the occupation narrative.
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References
Bass, H. (1998). Algebra with integrity and reality. Keynote address at the NRC Symposium on the Nature and Role of Algebra in the K-14 Curriculum. Proceedings published by the NRC.
Brousseau, G. (1997). Foundations and methods of didactique. In N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield (Eds. and Trans., pp. 21–75), Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990. Dordrecht, The Netherlands: Kluwer.
Butterworth, B. (2015, June). Low numeracy: From brain to education, Plenary address at ICMI Study 23. Macau, China.
Clements, D. H., & Stephan, M. (2001). Measurement in PreK-2 mathematics. In D. H. Clements, J. Sarama, & A.-M. DiBiase (Eds.), Engaging young children in mathematics. Reston, VA: NCTM.
Davydov, V. V. (1975). Logical and psychological problems of elementary mathematics as an academic subject. In L. P. Steffe (Ed.), Children’s capacity for learning mathematics (pp. 55–107). Chicago: University of Chicago.
Davydov, V. V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Reston, VA: National Council of Teachers of Mathematics. (Original published in 1972).
Dougherty, B. J., & Slovin, H. (2004). Generalized diagrams as a tool for young children’s problem solving. In Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 295–302). Bergen, Norway: Bergen University College.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel.
Harel, G. (2003). The DNR system as a conceptual framework for curriculum development and instruction. In R. Lesh, J. Kaput, E. Hamilton, & J. Zawojewski (Eds.), Foundations for the future. Mahwah, NJ: Lawrence Erlbaum Associates.
I, J.Y., & Dougherty, B. (2015). Linking multiplication models to conceptual understanding in meassurement approach. 11th Hawaii International Conference of Education, Honolulu, HI. https://www.researchgate.net/publication/270279409_Linking_Multiplication_Models_to _Conceptual_Understanding_in_Measurement_Approach
Moxhay, P. (2008). Assessing the scientific concept of number in primary school children. Paper presented at ISCAR 2008, San Diego. lchc.ucsd.edu/MCA/Mail/xmcamail.2009_05.dir/pdfxxr6FcSxae.pdf
National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. In C. T. Cross, T. A. Woods, & H. Schweingruber (Eds.), Common early childhood mathematics, Center for Education, division of behavioral and social sciences and education. Washington, DC: The National Academies Press.
Schmittau, J. (2005). The development of algebraic thinking: A Vygotskian perspective. ZDB, Vol. 37. lchc.ucsd.edu/mca/Mail/xmcamail.2014-11.dir/pdfv7AHOJXcb2.pdf
Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary school mathematics, For the Learning of Mathematics 34. http://flm-journal.org/Articles/739D3FD8C95A0A3770B35494FA3327.pdf
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Bass, H. (2018). Is the Real Number Line Something to Be Built or Occupied?. In: Li, Y., Lewis, W., Madden, J. (eds) Mathematics Matters in Education. Advances in STEM Education. Springer, Cham. https://doi.org/10.1007/978-3-319-61434-2_10
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DOI: https://doi.org/10.1007/978-3-319-61434-2_10
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