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# Every lesson a problem solving lesson

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```									Engaging mathematical
minds

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
processes; visualize things (even when the
"things" are not inherently visual); seek to explain
why things are as they seem them; and argue
(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
atmosphere?
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
relationships
• Accepts being stuck as honourable
• Poses questions
• ‘Stickwithitness’
Five ingredients that
contribute to successful
lessons
• Lesson starts: low threshold, high ceiling
activities
• Creative climate and conjecturing atmosphere
• Valuing mathematical thinking
• Purposeful activity and discussion
• Develop expert learners
Beginning                                 End

Pit
uncertainty confusion
cognitive conflict
James Nottingham
Northern Wisdom
Creative Climate

Total                              Energy available
energy of                          for task or success
individual

Energy required for
emotional survival

Ceserani & Greatwood, 1995
Does attending to community
work?

• 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
consolidation
• Thus we need a ‘social pedagogy’
• None of this is possible without trusting
relationships
• 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
alone.
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
individuals.
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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