Engaging mathematicalminds

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 interactedthe 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

Pit

uncertainty confusion

cognitive conflictJames Nottingham

Northern Wisdom

Beginning End

Creative Climate

Energy available for task or success

Energy required for emotional survival

Total energy of individual

Threatening

Adversarial Neutral Supportive

Cooperative

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 communitylook like?