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Every lesson a problem solving lesson


  • pg 1
									Engaging mathematical

        Mike Askew
   Wiltshire, 2 July 2009
Mathematical power
'Mathematical power is best described by a set of
habits of mind. People with mathematical power
perform thought experiments; tinker with real and
imagined machines; invent things; look for
invariants (patterns); make reasonable
conjectures; describe things both casually and
formally (and play other language games); think
about methods, strategies, algorithms, and
processes; visualize things (even when the
"things" are not inherently visual); seek to explain
why things are as they seem them; and argue
passionately about intellectual phenomena.'
             (Goldenberg, Cuoco and Mark, 1998)
Some maths…

Let’s play a game:
You need a friend to play with.
You need a 0-20 number line:
Creative and conjecturing

• What differences did you notice in:
    the ways you interacted
    the mathematics that emerged?

  Did ‘strike out’ encourage a creative
  climate and a conjecturing
Public conversation

•   Repeat
•   Re-voicing
•   Rephrase
•   Build on
•   Agree/disagree
- centred

• Teacher - centred
• Pupil - centred
• Mathematics - centred
Mathematical Habits of Mind

• Generalising and reasoning
• Creativity
• Curiosity and perseverance
Mathematical Habits of Mind

• Generalising and reasoning
  •   Trying out examples (specialising)
  •   Looking for patterns and connections
  •   Generalising
  •   Explaining and justifying
Mathematical Habits of Mind

• Creativity
  •   Creating representations
  •   Making conjectures
  •   Original approaches
  •   Elegant solutions
Mathematical Habits of Mind

• Curiosity and perseverance
  • Looking for connections and
  • Accepts being stuck as honourable
  • Poses questions
  • ‘Stickwithitness’
Five ingredients that
contribute to successful
• Lesson starts: low threshold, high ceiling
• Creative climate and conjecturing atmosphere
• Valuing mathematical thinking
• Purposeful activity and discussion
• Develop expert learners
Beginning                                 End

            uncertainty confusion
              cognitive conflict
                                    James Nottingham
                                    Northern Wisdom
                     Creative Climate

Total                              Energy available
energy of                          for task or success

             Energy required for
             emotional survival

    Threatening Adversarial   Neutral    Cooperative   Supportive

                                        Ceserani & Greatwood, 1995
Does attending to community

• Necessary for effective group work
• Positive effects of stereotypes
Within groups
• Paired collaborative work good for
  conceptual development
• Small group collaborative work good for
  extension work
• Individual work good for practice and
• Thus we need a ‘social pedagogy’
• None of this is possible without trusting
• Does not happen ‘naturally’ and needs
  continuous attention (Kutnick 2006)
Everyone gains

• Pairs/small groups can produce a solution
  that is more sophisticated than the most
  capable individual in the group can produce
Stereotypes and success
• Success and failure may arise from
  awareness of stereotypical views held about
  groups to which we belong.
• Social identity research is examining how
  we take on (internalise) and live out
  (externalise) identities that are shared with
  peers and how these can change.
• Points to importance of expectations of
  classes, schools, authorities, not just
Community of mathematians

• Bring to mind a time when you had a
  good experience of being part of a
  mathematical community.
• Share your story with 2 others.
• What similarities are there?
• How would you recognise a
  mathematical community (as distinct
  from a polite class)?
Community or class?
What would a mathematical community
look like?

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