Teaching with Technology: Complexity Theory As a Lens for Reflecting on Practice

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Teaching with Technology Complexity Theory As a
Lens for Reflecting on Practice

• Margaret Sinclair, York University, Toronto,
  Ontario, Canada

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Artigue implementation weak because

• 1) computer technologies, though having strong
  scientific and social legitimacy, have poor
  educational legitimacy
• 2) issues around the computerisation of
  mathematical knowledge have been underestimated
• 3) opposition between technical and conceptual
  dimensions of mathematical activity has been
  affected by the introduction of technologies that
  make the technical aspects easier
• 4) the complexity of instrumentation processes,
  i.e., dealing with the impact (both mathematical
  and technical) of technological tools, has been
  underestimated.

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Complexity vs Complicated

• Complexity theory rests on the idea that order
  emerges through the interactions of organisms or
  agents.
• i.e., it is not the same as complicated which
  is related to mechanistic theories that assume a
  centrally controlled governing structure

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Complexity terms

• Recursion - feedback - self-similarity
• Network - sum greater than parts
• Dynamic interactions links connections
• Emergence bottom up
• Ecological - adaptive

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Advantages of complex systems

• Adaptable
• Evolvable
• Resilient
• Boundless
• Generate novelty

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Disadvantages of complex systems

• Nonoptimal
• Noncontrollable
• Nonpredictable small differences in initial
  conditions wide variation in end results
• Nonunderstandable
• Nonimmediate
• Kelly, K. (1994). Out of control The new
  biology of machines, social systems and the
  economic world. Cambridge, MA Perseus Books.

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Davis and Simmt thesis

• Mathematics classes are adaptive and
  self-organizing complex systems
• That is they are learning systems.

Davis, B., Simmt, E. (2003). Understanding
learning systems Mathematics education and
complexity science. Journal for Research in
Mathematics Education, 34(2), 137-167.
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Necessary (but not sufficient) conditions for a
complex system to be able to LEARN

• Internal diversity
• Redundancy
• Decentralized control
• Organized randomness (Enabling constraints)
• Neighbour interactions

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Internal Diversity

• e.g., in an organization there are people who
  have different skills
• Linked to range of possible innovations

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Redundancy

• e.g., many people know/can do the same thing
• allows others to compensate
• allows for common understanding.

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Decentralized Control

• e.g., bottom up, not top down
• Learning emerges from shared understandings.
• The individual is not the focus.

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Organized Randomness (Enabling Constraints)

• e.g., within boundaries there is freedom to
  choose, innovate
• Proscriptive rather than prescriptive tasks
• .. a shift in thinking about the sorts of
  constraints that are necessary for generative
  activity.
• (Davis Simmt, 2003)

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Neighbour Interactions

• e.g., interactions between agents
• Possibility for ideas to bump up against one
  another
• Interactions
• Peer - peer
• Teacher student
• Also student task

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Observations - NI Independent Study
Partner
Other Pairs
strong
weak
Teacher
strong
Student
strong
Computer Program
Problem/ Instructions
Task
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Micro-level Interplay between task questions and
technology (here a pre-constructed sketch)

• Question
  JavaSketch
• Focuses attention - Draws attention
• Prompts action - Provides affordances
• Invites exploration - Provides alternate paths
• Introduces uncertainty - Supports
• experimentation

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Response

• Complexity theory challenges us to see the whole
  system in a new way as if it were a living
  thing.
• Many papers of this group already use the
  language of complexity - focus on analysis of the
  whole and on the interrelatedness of elements in
  the environment.
• And we see evidence that small differences in
  initial conditions are often associated with wide
  variation in end results.

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Connections

• Valsiners ZFM and ZPA
• Jill TRTLEs
• Merrilyn Sociocultural theories view learning
  as the product of interactions with other people
  and with material and representational tools
  offered by the learning environment.
• Affordances/constraints
• Jill -- charting manifestations of affordances,
  affordance bearers, and circumstances in which
  they occur
• Anne ICT tools represent affordances and
  possibilities for the user and constraints

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Connections

• Personal link to ecological
• Anne emphasis on seeing ICT as a personal
  technology, developing into an instrument for the
  learner and with considerations of affordance and
  constraints to analyse the activities.
• Luly work of Borba people with media
• Margaret grad students natural use of
  Spherical Easel
• Interactions
• Many!
• Lulu between dimensions (epistemological,
  technological, cognitive, pedagogical)

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Connections

• Distributed control
• Lulu not top down or bottom up but Filling
  Inwards and Filling outwards
• FO ..interventions are intended to guide
  personal understandings towards institutional
  knowledge..understandings arise out of students
  constructive efforts
• FI ..interventions aimed at supporting learners
  in internalising institutionalized knowledge
  mathematically significant issues appropriated
  during the learners constructive efforts

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Other ideas

• Mohan Chinnapan schemas knowledge structures
  or networks.
• According to Anderson (2000), two variables
  determine the quality of a schema the spread of
  the network and the strength of the links between
  the various components of informtaion located
  within the network.
• Jaag reference to Simons HLTs for development
  of activities. relates to redundancy issue

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Other ideas

• Cultural considerations
• John
• The interplay between different values in
  traditional and technology classes introduces
  many complexities.
• The work of Lins (2002) and Kendal (2001)
  highlights respectively how individual teachers
  within similar institutions in a single country
  appropriate software differently and privilege
  different aspects of the same software with their
  students.

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Thank you