1 When and How is Mathematics Actually Learned? John Mason Trondheim Oct 2007

2 Outline  Some Phenomena  Some Theorising  Some Implications for Teaching

3 Ways of Working  Experiential  What you ‘get’ will be what you notice happening inside you.

6 Tangent Power  Imagine the graph of a smooth function f  The tangent power of the point P relative to f, is the number of tangents to f through P What are the possible tangent powers, and where are they located?

7 Quintic Encounter: -inflection tangent -sense of curve for large values of |x| -shift from single point to following tangent Invariance in the midst of change

8 Reflecting  Began with phenomena  To intrigue, motivate, initiate  Offered a task, a challenge  first, mental; then physical manipulation of familiar entities  Opportunity to express relationships to self and others  not yet integrated into your functioning

9 Theorising  Didactic Contract (Brousseau)  Students attempt the tasks they are set …  Didactic Tension  The more clearly and precisely the teacher indicates the behaviour being sought, the easier it is for learners to display that behaviour without actually thinking mathematically.  Contributions to Learning  Phenomenon-Surprise ––> task or challenge Activity --> teaching interactions Integrating --> into available actions

10 What then is Learning?  Change in what you are sensitised to notice  Change in how you attend  Holding Wholes (Gazing)  Discerning details  Recognising Relationships  Perceiving Properties  Reasoning on the basis of agreed properties

11 Supporting & Promoting Learning Phenomenon- Surprise leading to a task or challenge Activity Reflecting on Experience Using in fresh actions Sleeping on it Integrating Familiar actions & familiar objects used in fresh ways Initiating, challenging, stimulating

12 What CAN a Teacher Do?  direct learner attention  on the basis of being aware of the movements of their own attention and sensitivities  so that learners use and develop their own powers of sense-making  and encounter significant mathematical themes and heuristics

13 Implications for Teaching Phenomenon- Surprise leading to a task or challenge Activity Integrating Manipulating, Getting-a-sense-of Articulating Doing, Talking, Recording Directing Attention away from action See Experience Master

14 MGA

15 Worlds of Experience Material World World of Symbol s Inner World of Imagery enactiveiconicsymbolic

16 Powers / Specialising & Generalising / Conjecturing & Convincing / Imagining & Expressing / Ordering & Classifying / Distinguishing & Connecting / Assenting & Asserting

17 Themes  Doing & Undoing  Invariance in the midst of change  Freedom & Constraint  Extending & Restricting

18 Teaching Trap  Doing for the learners what they can already do for themselves  Teacher Lust: – desire that the learner learn – allowing personal excitement to drive behaviour

19 Structure of a Topic Only awareness is Educable Only behaviour is Trainable Only emotion is Harnessable

Modes of Interaction Expounding Explaining Exploring Examining Exercising Expressing

Teacher Student Content Expounding Student Content Teacher Exploring Student Content Teacher Examining Student Content Teacher Exercising Student Content Teacher Expressing Teacher Student Explaining Content

22 When & How does Learning Mathematics happen?  A process of maturation not a specific action  Teaching takes place IN time Learning takes place OVER time  Challenge Activity involving actions Integration

23 Reflection  What struck you most about this session?  Imagine yourself in the future making use of something that has arisen for you today  This is really the ONLY way to develop professionally:  imagining yourself acting freshly in some situation in the future

24 Further Resources Designing & Using Mathematical Tasks (Tarquin 2004/2006) MathemaPedia (NCETM) wikipedia for maths education