Learning Mathematics Efficiently at ALevel John Mason Coventry

Learning Mathematics Efficiently at A-Level John Mason Coventry & Warwickshire Feb 2008 1

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 2

Zig. Zag Functions /Sketch the graph of – |x-1| – | |x-1| - 2| –… /Relate number of zigs to number of absolute-value functions 3

Examples /Of what is |x| an example? /Of what is y = x 2 and example? – y = b + (x – a )2 ? 4

Human Psyche /Training Behaviour /Educating Awareness /Harnessing Emotion /Who does these? – Teacher with learners? – Learners! 5

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 6

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

Reading Graphs 8

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? 9

Tangential /How many tangents to the quadratic pass through P? 10

Tangent Power /The tangentpower of a point is the number of tangents through it. /Characterise the regions with fixed tangent-power. 11

Trigonometry /Fundamental awarenesses – Thales theorem – Multiple ways to measure angles – Hence multiple relationships – Ratios as functions 12

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 13 attention, guide energies

Worlds of Experience Inner World of imagery World of Symbol s Material World enactive 14 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 15

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 16

Some Mathematical Powers /Imagining & Expressing /Specialising & Generalising /Conjecturing & Convincing /Stressing & Ignoring /Organising & Characterising 17

Some Mathematical Themes /Doing and Undoing /Invariance in the midst of Change /Freedom & Constraint 18