Engaging mathematical minds 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 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 Threatening Adversarial Neutral Cooperative Supportive 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?