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

2. 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

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

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

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

6. 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

7. 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.

8. Reading Graphs

9. 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?

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

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

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

13. 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

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

15. 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

16. 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

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

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