Learning Mathematics Efficiently at A-Level

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Presentation transcript:

Learning Mathematics Efficiently at A-Level John Mason Coventry & Warwickshire Feb 2008

Conjecturing Atmosphere Everything said is said in order to consider modifications that may be needed Those who ‘know’ support those who are unsure by holding back or by asking informative questions

ZigZag Functions Sketch the graph of |x-1| | |x-1| - 2| … Relate number of zigs to number of absolute-value functions

Examples Of what is |x| an example? Of what is y = x2 and example? y = b + (x – a)2 ?

Human Psyche Training Behaviour Educating Awareness Harnessing Emotion Who does these? Teacher? Teacher with learners? Learners!

Training Behaviour turned into Educating Awareness Practice; Rehearsal against time Sorting Exploring A while using T Construction tasks so as to enrich accessible example spaces Problem construction to explore dimensions-of-possible-variation

Construction Tasks Write down a function which takes the value √7more than 3 times … Exactly 3 times and another Write down an integral which has the value zero.

Reading Graphs

Tangential Sketch the graph of a function which tends to 1 as x goes to infinity How many tangents to a quadratic go through a given point? On a quintic?

Tangential How many tangents to the quadratic pass through P?

Tangent Power The tangent-power of a point is the number of tangents through it. Characterise the regions with fixed tangent-power.

Trigonometry Fundamental awarenesses Thales theorem Multiple ways to measure angles Hence multiple relationships Ratios as functions

What Teachers Can Do aim to be mathematical with and in front of learners aim to do for learners only what they cannot yet do for themselves focus on provoking learners to use and develop their (mathematical) powers encounter (mathematical) themes & heuristics learn about themselves (inner & outer tasks) make mathematically significant choices direct attention, guide energies

Worlds of Experience enactive iconic symbolic Inner World of imagery World of Symbols Material World enactive iconic symbolic

Principal Foci core awarenesses underlying topics familiar actions which need challenging, developing, extending generating reflection through drawing out of immersion in activity getting learners to make significant choices prompting learners to use and develop their natural powers

Task Domains Dimensions-of-possible-variation (what can change without method or approach changing) Ranges-of-permissible-change (over what range can things change) Ways of presenting tasks Ways of interacting during activity Ways of concluding activity

Some Mathematical Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Organising & Characterising

Some Mathematical Themes Doing and Undoing Invariance in the midst of Change Freedom & Constraint