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Common sense and good sense: using complexity theory to visualise mathematics learning Diana Coben and Ian Stevenson Department of Education and Professional Studies Kings College, London

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Common sense and good sense: using complexity theory to visualise mathematics learning We shall present dynamic formulations of Gramscis conceptualisation of common sense and good sense (Gramsci, 1971) and exemplify these formulations using data from secondary students learning mathematics. We shall explore the following questions: –how might various formulations of the relationship between common sense and good sense be expressed visually and dynamically? –could these representations help us to visualise what happens when someone learns (or does not learn) mathematics? These questions are explored through metaphor in the belief that metaphor may illuminate considerations relevant to educators. Gramsci constantly employs spatial metaphors (Jessop,2005) and consequently various spatial metaphors - visual representations of common sense and good sense and the relationship between them - are considered. In earlier work, Coben (2002) concluded that metaphors from the mathematical world of fractals, self-symmetry and dynamical systems may have considerable explanatory power for adult educators. This presentation takes this work forward in collaboration with Stevenson (2008; Stevenson & Noss, 1991), applying it to mathematics learning.

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Common sense and mathematical learning 1.Gramsci on common sense and good sense 2.Common sense and mathematical learning Dynamic models of the relationship between common sense and good sense Explorations using data from students learning mathematics

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Antonio Gramsci 1891-1937 Italian political theorist and activist

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Common sense a conception which, even in the brain of one individual, is fragmentary, incoherent and inconsequential, in conformity with the social and cultural position of those masses whose philosophy it is. (Gramsci, 1971:419) Common sense is the folklore of philosophy and stands midway between real folklore (that is, as it is understood) and the philosophy, the science, the economics of the scholars. Common sense creates the folklore of the future, that is a more or less rigidified phase of a certain time and place. (PN1:173) Nonetheless, it contains a healthy nucleus of good sense [...] which deserves to be made more unitary and coherent. (Gramsci, 1971:328).

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Good sense …an intellectual unity and an ethic in conformity with a conception of reality that has gone beyond common sense and become, if only within narrow limits, a critical conception. (Gramsci, 1971:333)

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Gramscis distinction between good sense and common sense… …is not fully worked out in his prison notebooks. It is: both epistemological and sociological: both a distinction between different forms of knowledge and a distinction between the knowledges characteristic of different social groups. But the distinctions are not mutually exclusive in either case. In epistemological terms, common sense includes elements of good sense. In sociological terms, good sense is not the preserve of an elite, and common sense is common to us all. (Coben, 1998, pp213-214) The relationship between good sense and common sense may be visualised in various ways…

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Visual representations of relationships between common sense and good sense (Coben, 2002) good sense common sense folklore good sense common sense good sense common sense spontaneous philosophy sense common good sense

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Dynamic computer models of putative relationships between common sense and good sense in mathematics learning using complexity theory… …taking movement between common sense and good sense as a metaphor for the learning process in mathematics learning

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In politico-educational terms: After the initial seed - the event which triggers the start of the educative process - the orbit of activities is established and reaches an equilibrium, denoted by the attractors of the system. Common sense and good sense may thus both be considered as classes of equivalence in a dynamic system and the boundary between them may be considered a fractal curve rather than a one-dimensional line. (Coben, 2002, p.282)

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Chaos and dynamic systems Feedback and iteration Sensitivity to initial conditions Self-similar Non-integer dimension Deterministic but not predictable Strange attractors Some dynamic representations of different visualisations of the relationship between common sense and good sense…

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Overlap model Initial condition (-1,0) Initial condition (-1,1) good sense common sense

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Overlap model

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Enclosed model Initial condition (0.5,-0.5) sense common good sense Initial condition (-0.67,0.23)

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Enclosed model

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Container model good sense common sense spontaneous philosophy Initial conditions: random position and velocity

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Case study: Bryan Sketch graph of unknown curve 1/(x 2 -1) Three types of features: points, interval, range Common sense at point level Good sense at range level with interval showing transitions Starts from range and moves to point and back

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Levels of graphing (Stevenson and Noss, 1991, based on Janvier, 1978)

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For discussion How/do these representations help us to visualise what happens when someone learns (or does not learn) mathematics?

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References Coben, D. (1998). Radical Heroes: Gramsci, Freire and the politics of adult education. New York: Garland Publishing Inc./Taylor Francis. Coben, D. (2002). Metaphors for an educative politics: Common sense, good sense and educating adults. In C. Borg, J. A. Buttigieg & P. Mayo (Eds.), Gramsci and Education (pp. 263-290). Boulder, CO: Rowman & Littlefield. Gramsci, A. (1971). Selection from the Prison Notebooks (Q. Hoare & G. Nowell-Smith, Trans.). New York: International Publishers. Janvier, C. (1978) The Interpretation of Complex Cartesian Graphs - Studies and Teaching Experiments. Unpublished PhD. Nottingham: Shell Centre. Jessop, B. (2005). Gramsci as a spatial theorist. Critical Review of International Social and Political Philosophy. Special issue: Images of Gramsci, 8(4), 421-437. Stevenson, I. (2008). Tool, tutor, environment or resource: Exploring metaphors for digital technology and pedagogy using Activity Theory. Computers and Education, 51(2), 836-853. Stevenson, I. J. & Noss, R. (1991) Pupils as expert systems developers. In Furinghetti, F. (Ed.) Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (PME15, Assisi, Italy, June 29-July 4, 1991), 3:294-301.

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