Abstract
This paper makes a case for placing knowledge at the centre of the school mathematics curriculum, and for knowledge building and knowledge differentiation as critical for both equity and excellence, emphasising that knowledge is much more than a set of descriptions of content as might typically be found in a curriculum document or textbook. The paper commences by discussing the implications of the traditional epistemological view of knowledge as justified true belief for mathematics education and uses this to build a preliminary description of knowledge building. Ideas from critical realism are then used to show that it is not so much the content of knowledge that matters but the production of knowledge and to build an enhanced conception of knowledge building in school mathematics. A distinction is made between knowledge and knowing that provides a non-relativist yet fallible view of knowledge, recognising its emergent but directed nature through its production and legitimation within established fields. The importance of knowledge building as a democratic right is then discussed, highlighting the importance of specialised knowledge and arguing that knowledge differentiation provides a basis for a conception of school mathematics curriculum that is dynamic and empowering. The paper concludes by discussing a range of potential theoretical and empirical research projects arising from a focus on knowledge and knowledge building in school mathematics.
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Notes
The term “knowledge building” is used in this paper as a generic phrase to describe the collective production and legitimation of knowledge, to be distinguished from specific realisations or technologies described, for example, by Scardamalia and Bereiter (2006).
This paper does not take a position on fallibilist or absolutist philosophies, whether mathematics is “discovered” or “invented”, nor particular philosophies of mathematics such as Platonism, formalism, constructivism (in the mathematical sense) or social constructivism as described by Ernest (1998). Each contributes important perspectives to how we understand the nature of mathematical knowledge and its development and learning; however, debates favouring one position over another have arguably made little impact on the knowledge practices in school mathematics. The argument made in this paper is that critical realism cuts through these positionings and provides a productive way forward that honours both the distinctive truth warrants of mathematics and the epistemic relativism inherent in its development.
Keno is a gambling game commonly played in hotels or clubs across Australia. Twenty numbers are drawn from a possible 80, with the numbers appearing on screens. Players mark up to 10 numbers that they predict might appear, and win varying amounts depending on how many numbers they choose and how many appear. The most and least commonly drawn numbers on any given day are frequently displayed on the screen.
Moore (2013) distinguishes between social realism and its philosophical ancestor, critical realism. For simplicity, we do not make that distinction, as the fundamental tenets are the same. Social realism emphasises the existence, not only of physical objects but also of social structures and phenomena, independent of our conception or perception of them.
References
Adam, S., Alangui, W., & Barton, B. (2003). A comment on: Rowlands & Carson “Where would formal, academic mathematics stand in a curriculum informed by Ethnomathematics? A critical review”. Educational Studies in Mathematics, 52, 327–335.
Anderson, D. J. (2012). Knowledge and conviction. Synthese, 187(2), 377–392. https://doi.org/10.1007/s11229-010-9831-2.
Atweh, B., Goos, M., Jorgensen, R., & Siemon, D. (Eds.). (2012a). Engaging the Australian National Curriculum: mathematics – perspectives from the field. Online Publication: Mathematics Education Research Group of Australasia.
Atweh, B., Miller, D. & Thornton, S. (2012b). The Australian curriculum: Mathematics – world class or Déjà Vu? In B. Atweh, M. Goos, R. Jorgensen, & D. Siemon. (Eds.), Engaging the Australian national curriculum: Mathematics – perspectives from the field (pp. 1–18) . Online publication: MERGA.
Australian Curriculum, Assessment and Reporting Authority (ACARA) (n.d.). Australian Curriculum: Mathematics. Retreived 18 November 2019 from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/.
Bereiter, C. (2002). Education and mind in the knowledge age. New York: Routledge.
Bernstein, B. B. (1999). Vertical and horizontal discourse: an essay. British Journal of Sociology of Education, 20(2), 157–173.
Bhaskar, R. (2008). Dialectic: the pulse of freedom. New York: Routledge.
Brown, M. W. (2011). The teacher–tool relationship: theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work (pp. 37–56). New York: Routledge.
Carmichael, R. D. (1923). Fermat numbers Fn = 2 2n + 1. The American Mathematical Monthly, 30(7), 137–146.
Chapman, O. (2011). Elementary school teachers’ growth in inquiry-based teaching of mathematics. ZDM - International Journal on Mathematics Education, 43(6), 951–963. https://doi.org/10.1007/s11858-011-0360-3.
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1/2), 113–163.
Corbel, C. (2014). The missing ‘voice’of knowledge in knowledge and skills. In B. Barrett & E. Rata (Eds.), Knowledge and the future of the curriculum (pp. 104–119). London: Palgrave Macmillan.
D’Ambrosio, U. (1997). Where does ethnomathematics stand nowadays? For the Learning of Mathematics, 17(2), 13–18.
Dowling, P. (1998). The sociology of mathematics education: mathematical myths/pedagogic texts. London: Routledge Falmer.
Education Council. (2015). National STEM School Education Strategy. National STEM school education strategy. A comprehensive plan for science, technology, engineering and mathematics education in Australia. Retrieved from http://www.educationcouncil.edu.au/site/DefaultSite/filesystem/documents/National STEM School Education Strategy.Pdf.
Galbraith, P. (2014). Custodians of Quality : Mathematics Education in Australasia Where from? Where at? Where to? In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), J. Anderson, M. Cavanagh, & A. Curriculum in focus: Research guided practice, (Proceedings of the 37th annual conference of the Mathematics Education Research Group of Australasia) (pp. 38–53). Sydney: MERGA.
Gettier, E. L. (1963). Is justified true belief knowledge? Analysis, 23(6), 121–123.
Gonski, D., Arcus, T., Boston, K., Gould, V., Johnson, W., O’Brien, L., … Roberts, M. (2018). Through growth to achievement: report of the review to achieve educational excellence in Australian schools. Canberra. Retrieved from https://docs.education.gov.au/, Australian schools.%0ADocuments/through-growth-achievement-reportreview-achieve-educational-excellence-australian-0.
Groff, R. (2004). Critical realism, post-positivism and the possibility of knowledge. New York: Routledge.
Gutiérrez, R. (2017). Political conocimiento for teaching mathematics: why teachers need it and how to develop it. In S. E. Kastberg, A. M. Tyminski, A. E. Lischka, & W. B. Sanchez (Eds.), Building support for scholarly practices in mathematics methods (pp. 11–38). Charlotte: Information Age Publishing.
Hogan, J. (2012). Mathematics and numeracy: Has anything changed? Are we any clearer? Are we on track? Australian Mathematics Teacher, 68(4), 4–7.
Joseph, G. G. (2000). The crest of the peacock: non-European roots of mathematics. Princeton: Princeton University Press.
Kinnear, V., & Wittmann, E. C. (2018). Early mathematics education: a plea for mathematically founded conceptions. In V. Kinnear, M. Y. Lai, & T. Muir (Eds.), Forging connections in early mathematics teaching and learning (pp. 17–35). Singapore: Springer.
Lakatos, I. (1976). In J. Worrall & E. Zahar (Eds.), Proofs and refutations: the logic of mathematical discovery. Cambridge: Cambridge University.
Lamb, E. (2014). Extrapolation gone wrong: the case of the Fermat primes - Scientific American Blog Network. Retrieved July 20, 2018, from https://blogs.scientificamerican.com/roots-of-unity/extrapolation-gone-wrong-the-case-of-the-fermat-primes/.
Loewenberg Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554.
Lundin, S. (2012). Hating school, loving mathematics: on the ideological function of critique and reform in mathematics education. Educational Studies in Mathematics, 80(1–2), 73–85.
Maton, K. (2000). Languages of legitimation: the structuring significance for intellectual fields of strategic knowledge claims. British Journal of Sociology of Education, 21(2), 147–167.
Maton, K. (2014). Building powerful knowledge: the significance of semantic waves. In B. Barrett & E. Rata (Eds.), Knoweldge and the future of the curriculum (pp. 181–197). Basingstoke: Palgrave Macmillan. https://doi.org/10.1093/em/cau127.
Maton, K., Hood, S., & Shay, S. (2015). Knowledge-building: educational studies in legitimation code theory. New York: Routledge.
McConaghy, C. (2000). Rethinking indigenous education: culturalism, colonialism and the politics of knowing. Brisbane: Post Pressed.
Meyer, J. H. F., & Land, R. (2005). Threshold concepts and troublesome knowledge (2): epistemological considerations and a conceptual framework for teaching and learning. Higher Education, 49(3), 373–388.
Moore, R. (2007). Going critical: the problem of problematizing knowledge in education studies. Critical Studies in Education, 48(1), 25–41. https://doi.org/10.1080/17508480601120970.
Moore, R. (2013). Social realism and the problem of the problem of knowledge in the sociology of education. British Journal of Sociology of Education, 34(3), 333–353. https://doi.org/10.1080/01425692.2012.714251.
Morgan, C. (2006). What does social semiotics have to offer mathematics education research? Educational Studies in Mathematics, 61(1–2), 219–245.
Moses, R. P. (1994). Remarks on the struggle for citizenship and math/science literacy. The Journal of Mathematical Behavior, 13(1), 107–111.
Noorloos, R., Taylor, S. D., Bakker, A., & Derry, J. (2017). Inferentialism as an alternative to socioconstructivism in mathematics education. Mathematics Education Research Journal, 29(4), 437–453. https://doi.org/10.1007/s13394-017-0189-3.
Popper, K. (1978). Three worlds. The Tanner Lecture on Human Values: Delivered at the University of Michigan. Retrieved from http://tannerlectures.utah.edu/_documents/a-to-z/p/popper80.pdf.
Rowlands, S., & Carson, R. (2002). Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review of ethnomathematics. Educational Studies in Mathematics, 50(1), 79–102.
Sarra, C. (2014). Strong and smart – towards a pedagogy for emancipation: education for first peoples. New York: Routledge.
Scardamalia, M., & Bereiter, C. (2006). Knowledge Building : Theory , Pedagogy , and Technology. In K. Sawyer (Ed.), Cambridge handbook of the learning sciences (pp. 97–118). New York: Cambridge University Press.
Siemon, D., Bleckly, J., & Neal, D. (2012). Working with the big ideas in number and the Australian Curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen, & D. Siemon (Eds.), Engaging the australian national curriculum: mathematics – perspectives from the field (pp. 19–45). Online publication: Mathematics Education Research Group of Australasia.
Swan, M. (2014). Improving the alignment between values, principles and classroom realities. In Y. Li & G. Lappan (Eds.), Mathematics curriculum in school education (pp. 621–636). Dordrecht: Springer.
Swidler, A., & Arditi, J. (1994). The new sociology of knowledge. Annual Review of Sociology, 20, 305–329.
Thornton, S. (2008). Speaking with different voices: knowledge legitimation codes of mathematicians and mathematics educators. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting directions (Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia) (pp. 523–530). Brisbane: MERGA.
Turner, F., & Rowland, T. (2011). The knowledge quartet as an organising framework for developing and deepening teachers’ mathematics knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp. 195–212). Dordrecht: Springer.
Watson, A. (2008). School mathematics as a special kind of mathematics. For the Learning of Mathematics, 28(3), 3–7.
Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1–14.
Young, M. F. D. (2009). Education, globalisation and the “voice of knowledge”. Journal of Education and Work, 22(3), 193–204. https://doi.org/10.1080/13639080902957848.
Young, M. F. D. (2010b). The future of education in a knowledge society: the radical case for a subject-based curriculum. Journal of the Pacific Circle Consortium for Education, 22(1), 21–32.
Young, M. F. D., & Muller, J. (2010). Three educational scenarios for the future: lessons from the sociology of knowledge. European Journal of Education, 45(1), 11–27.
Young, M. F. D., & Muller, J. (2013). On the powers of powerful knowledge. The Review of Education, 1(3), 229–250. https://doi.org/10.1002/rev3.3018.
Acknowledgements
The author acknowledges the support of colleagues Kristen Tripet, Ruqiyah Patel, Valerie Barker, Virginia Kinnear and Peter Galbraith as well as the anonymous reviewers who provided feedback on the first draft of this paper.
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Thornton, S. (Re)asserting a knowledge-building agenda in school mathematics. Math Ed Res J 34, 69–85 (2022). https://doi.org/10.1007/s13394-020-00322-1
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DOI: https://doi.org/10.1007/s13394-020-00322-1