Common sense and good sense:using complexity theory to visualise

mathematics learning

Diana Coben and Ian StevensonDepartment of Education and Professional Studies

King’s College, London

Common sense and good sense:using complexity theory to visualise mathematics learning

• We shall present dynamic formulations of Gramsci’s 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.

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

Antonio Gramsci1891-1937

Italian political theorist and activist

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).

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)

Gramsci’s 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…

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 philosophysense

common

good sense

Dynamic computer models of putative relationships between common sense and good

sense in mathematics learningusing complexity theory…

…taking movement between ‘common sense’ and ‘good sense’ as a metaphor for the learning process in mathematics learning

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)

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…

Overlap model

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0.2

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0 0.5 1 1.5

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0 0.2 0.4 0.6 0.8 1

Initial condition (-1,0)

Initial condition (-1,1)

good sense

common sense

Overlap model

Enclosed model

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Initial condition (0.5,-0.5)

sense

common

good sense

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0.1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Initial condition (-0.67,0.23)

Enclosed model

Containermodel

good sense

common sense

spontaneous philosophy

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0 1 2 3 4 5 6 7 8

Initial conditions:random position and velocity

Case study: Bryan

• Sketch graph of unknown curve 1/(x2-1)• 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

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

Level 1 Points Level 2 Intervals Level 3 Range

1. Crossing the xaxis,

1. Maxima andMinima

1. Transformations ofcurves

2. Crossing the yaxis.

2. Intervals overwhich the graphincreases/decreases

2. Patterns givingrise to cycles

3. Stationary points 3. Plateau regions(f(x) < c , f(x)=c orf(x) >c),

3. Symmetries

4. Rising/fallingcurves joiningplateaux or extrema

4. Dispersion of thegraph

5. Steady rates ofchange

5. Differences inshape measured byarea

6. Non-steady ratesof change

6. Many curves onthe same axes, eachdisplaying level 2features

7. Extrapolation,interpolation

8. Discontinuities

9. HorizontalAsymptotes

10. VerticalAsymptotes

B: which will become the max of the reciprocal graph and this will curve downwards

His reply when asked to explain is split into t hree parts. First Bryan seems a litt le uncertain, and talks quietly to himself. He is moving quickly between all three levels.

B: because i t’s the… (quietly) I don’t know actually... yeah I know, it’s because... this ermm... hmm.... well as this. lt takes a value between nought and a root where it’s kind of at the bottom of the u under the x axis, as this value here is

Bryan then goes on to draws the reciprocal graph using these features

He deduces that there are a symptotes from the zeros of the original equation at level 1 and 2

Therefore one over that value is going to be larger in magnitude than minus one (marked 6 on the sketch) because... well if it’s… let’s take the y value as nought it’s going to reach infinity….

one over that value is going to be infinity, one over nought, so therefore that’s got to go somewhere between nought and infinity so it’s going to go downwards (marked 7)

R: down there. Why negative?

B: well one over a negative number is just going to give you a

Levels Dialogue

1 2 3

B: all right, we got the graph of x squared minus 1 for a start, OK and then we further reduce that to the graph of x squared

and the graph of x squared is just going to be like that (sketches the graph)

it’s just going to be a u shape, touching the nought OK. Now if you transform it to x squared minus one, (sketches x squared minus one)

it’s going to be just shifted by one

B: well one over a negative number is just going to give you a negative R: a negative OK. And you are look at that bit which is below the x axis? B: yeah He now puts in the behaviour above the x-axis to show how the graph approaches asymptotically B: OK now you’ve got the bit above the x axis and it’s kinda like (sketches above axis)

it’s going to start really high, ‘cos one over just a little bit more than nought

Bryan draws in the asymptote using inverse proportion in relation to the first graph of x-squared, and moves on to approach close to the x-axis B: OK, so we’ve broken off from there (meaning the asymptote marked 8) not continuous, and its going to come all the way along there and its going to go downwards (towards the x-axis on the right) ‘cos as y increases the one over y decreases so you are going to get that kind of thing

He notes the symmetry of the graph to sketch the l eft-hand side of the graph. and here it’s symmetrical and you are going to get that kind of thing R: why is it symmetrical?

B: ermm ‘cos this is symmetrical x squared minus one (referring to sketch 4) so therefore one over x squared minus one has got to be symmetrical, ‘cos the x terms squared therefore it doesn’t matter whether x is positive or negative, so x is squared then the positive values of x are going to map to the same y value as the corresponding negative values

R: right OK B: so you’ve got these curly bits R: and what’s happening here then? B: you got a little undefined value

R: an undefined value? B: where that’s reached nough t, x squared minus one has reached nought OK and we can find out where that is ..

For discussion

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

ReferencesCoben, 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.