Promoting Mathematical Thinking Pro Math 2014 On Being

Promoting Mathematical Thinking Pro. Math 2014 On Being Stuck Problem Solving Makes School Mathematics Enjoyable and Challenging The Open University Maths Dept 1 John Mason Helsinki 2014 University of Oxford Dept of Education

Outline v v v Forms of Being Stuck Getting Unstuck: having-come-to-mind Developing a Repertoire Come to mind: Tulkaa Mieleen Coming to mind: Tullen Mieleen Having come to mind: Tultua Mieleen 2

Conjectures and Assumptions v v v 3 Everything said here is a conjecture … …to be tested in your experience! Enjoyment (in mathematics) arises from acts … … the exercise of internalised skill … the use of natural powers … the development of those powers What you get from this talk is most likely what you notice happening inside you as you seek for examples of your own which parallel mine in some way … …through which you can consider and interrogate my proposed distinctions and connections

Method of Enquiry v v v 4 I prefer to be phenomenologically experiential I shall offer some brief-but-(hopefully)-vivid account of incidents which may resonate your experience, making it available for inspection. Theorising is an epiphenomenon, and will arise as and when it seems apppropriate

Incident v v v 5 When my son was about 8 or 9, he asked where I was going that day, and I told him I was going to Liverpool to give a talk. He asked what it was about so I told him that it was about “What to do when you are stuck”. After a thoughtful silence he announced “Get out and push!”. Being Stuck is an honourable state, yet we are usually in a hurry to ‘get out and push’. ‘get out’ ––> withdraw from the action (or inaction!) ‘push’ ––> invoke some other action

Reflection v v Have you ever been stuck, but only realised it some time later? Have you ever been stuck, with a vague sense of being stuck but without seeming to be able to do anything about it? Accounting-For Multiple selves which harness energies in characteristic ways, invoking characteristic actions, emotions and thoughts Different selves dominate at different times and under different conditions Spectrum of intensities suggests all aspects of psyche are involved: enaction, affect, cognition & will 6

What Keeps Me Stuck? v v Immersed in action; oblivious to possibility Awake to possibility but … … Insufficient energy available to withdraw from action … lack of desire (affect) … lack of sufficient will … lack of action … lack of insight, connection, narrative … Absence of possible action … blocked or obscured by prediliction/habit … unavailable 7

Incident v v Linear functionals; dual of a vector space; dual of the dual … different in finite and infinite cases Struggling over some hours to get to grips with expressions such as LV(f) = f(v) and its meaning. Perisitence; struggle over time; gradual realisation (literally) Things Take Time 8

Incident v I recall in my early years at the Open University being introduced to the function as an example of a function which is differentiable at 0 but which is not differentiable on any interval containing 0. Indeed it has arbitrarily large slope arbitrarily close to 0. v may be 1 in analysis, but not out on the street” Acting As If (William James) Things Take Time (Piet Hein) A solved problem is as useful to the mind as a broken sword on the battlefield (Idries Shah) 9

A Good Expanation Does Not Explain Everything Zen saying (Shigematsu) v v In what sense is How do you know? the same as ? Theorem-in-Action (Vergnaud) Is it sufficient to say “because they give the ame answer? ” 10

Becoming Unstuck: Pushing & Pondering v v v 11 99 heuristics (? !? ) Learn from experience: …One thing we don’t seem to learn from experience, is that we don’t often learn from experience alone. Develop an inner witness-monitor …Two birds, close-yoked companions, Cling to the self-same tree. Of these one eats of the sweet fruit, The other, nothing eating, looks on intent

Reflection Personal: meanings for person; emotions Intellectual: specifics of task Reproductive & Productive Retro-spective On-action Post-paring 12 Maintain habits Russian School Initiated through conflict/dissonance Change of perspective Spective In the moment In-action Paring Actions Emotions Thoughts Initiated through resonance? Pro-spective Pre-paring imagining

Incident v Add 1234, 123. 4, 12. 34, 1. 234, . 1234 123. 4 12. 34 1. 234. 1234 6170 Where should I put the dot? When we don’t know what to do, we enact whatever action is available Bob Davis; Brown & van Lehn; … 13

Incident: Result Spotted in a Journal 9 y = 5 x k=4 1 2 3 B=5 A=9 j=8 1 2 3 4 5 6 7 14 Dolan 2014

Multiple Discourses v v v 15 Piaget: Assimilation & Accommodation; Reflective Abstraction Vygotsky: Internalising higher psychological functions through being in the presence of others Gattegno: integration through subordination Gattegno: Awareness …Only awareness is educable; Only behaviour is trainable; Only emotion is harnessable; Only attention is directable. Reacting & Responding Systems S 1 and S 2

Incident 6 v A baseball bat and ball cost together one dollar and 10 cents. The bat costs one dollar more than the ball. How much does the ball cost? Temptation: $1. 00 and $0. 10 S 1 Automatic, Reactive, Mechanical, Habitual Parking Considered: $1. 05 and $0. 05 16 S 2 Considered; Responsive; Cognised; Justifiable Kahneman, Franklin,

Incident 9 v 17 Stuck on a problem I go down the corridor and ask a colleague if I can explain my problem. The colleague sits there, doing nothing beyond appearing to give me some attention. At the end, I thank them and return to my office with renewed vigour and a fresh view of my problem.

Incident: Stuck in a Rut v 18 How to display a woven copy?

S 1 & S 2: Dichotomy Dual System Theory Reacting Responding S 1 S 2 Body Enaction Emotion Affect Intellect Cognition Will Body Enaction 19 Intellect Cognition Emotion Affect

Reflecting the Complexity of Lived Experience Multiple Selves S 1 Reacting Automaticities Parking Body Enaction Emotion Affect Intellect Cognition Attention So that an alternative self can come to the fore Body Enaction S 2 Responding with a different self Intellect Cognition Chracteristic flows and transformations of energy 20 Emotion Affect Attention & Will

Reflecting the Complexity of Lived Experience Having come-to-mind 21 Body Enaction Coming-toaction Social & Somatic origins Emotion Affect Coming-tofeeling Social & Somatic origins Intellect Cognition Coming-tothought Noticing Opportunity Attention & Will Coming-towill Combining opportunity and possibility Integration Internalising … Repertoire

Reacting & Responding: having come-to-mind Having come-to-mind Dual System Theory Coming-toaction Body Enaction Automatic, Habitual Action Coming-tofeeling Emotion Affect Evoked Emotions Coming-tothought Intellect Cognition Conscious Cognitive Consideration Coming-toattention Will Intentionally Initiated Action 22 S 1 Social & Somatic origins S 2 Repertoire + Noticing Opportunity

Where is the Pleasure? v v v 23 In using your own powers …Not having them usurped by text or teacher By meeting (appropriate challenge) From a sense of freedom, of participating intentionally in choice of action, emotion, thought-pattern, and attention

How Can teachers Foster & Sustain Mathematical Thinking? v v v 24 By using tasks in which many learners spontaneously use some power, some action which the teacher knows is a valuable contribution to learners’ repertoire By allowing gestation time By drawing attention to what was successful and providing a label By Scaffolding & Fading (increasingly indirect prompts) By promoting a re-flective & pro-flective stance … How do you know? … What actions were effective and why? … imagining oneself in the future, enacting a particular action

Follow-Up v v 25 j. h. mason@open. ac. uk mcs. open. ac. uk/jhm 3 … (presentations) Draft paper “On Being Stuck” Thinking Mathematically (2010; Pearson) Fundamental Constructs in Mathematics Education (2004 Chapman)