1. 1 Phenomenal Knowledge and Knowing Phenomenologically Research Day on subject knowledge in teaching John Mason Feb 16 2009 The Open University Maths Dept University of Oxford Dept of Education

2. 2 Thesis: What is needed in order to … … do mathematics effectively is … …awareness of mathematical actions … teach mathematics effectively is … …awareness of those awarenesses … teach people to teach mathematics effectively is … …awareness of those awarenesses of those awarenesses

3. 3 Visiting  People set off at the same time to walk between two towns, some in each direction. They all meet at noon. They carry on as before; some reach their destination at 4pm, and the others at 6:15 pm.  At what time did they all set out? Let the speeds be v and w, and the time before noon be h The am distances are vh and wh The pm distances are 4v and 25w/4 So wh = 4v and vh = 25w/4 Whence h 2 = 25 and h = 5 So they set out at 7:00 am h h 4:00 6:1 5 wh vh 12:00 distance time

4. 4 Reflections on Mathematical Actions  Addressing questions such as: must everyone go the same speed, at least in the same direction? At what speeds do they go after they meet (assume they maintain their same speed)?  Articulating the relationship between speed, distance and time  Denoting what is not known: acknowledging ignorance (Mary Boole)  Deciding what is known  Presenting the information to yourself in symbols or as a graph  Calculating relevant quantities or invoking similar triangles or  Choosing which relationships to equate so as to locate fruitful ones  Moving to ratio of walking speeds  Possibly, solving a quadratic equation  Wondering about relevance of 3 2 + 4 2 = 5 2  Generalising (changing the 4 and the 6:15); adding a third town in some way, …)

5. 5 Further Reflections  Crossed Ladders –Crossing height is Harmonic sum of ladder heights Geometric mean of ladder excess heights 4:00 6:1 5 12:00 distance time

6. 6 Interlude on Tasks  tasks are for initiating activity;  through engaging in activity people have the opportunity to experience things, particularly mathematical actions and their effects.  Teachers have intentions when they choose tasks: –it is not that the task itself will promote learning, but that the inner task will involve the directing or re-directing of attention –and perhaps the internalising or integrating of actions previously dependent on being triggered by some outside agency (ZPD).  It is valuable, even necessary to draw back from the action and to become aware of the actions and their effects (utility)

7. 7 Reflections on Pedagogical Actions  How to direct attention to mathematical choices?  Draw a diagram; calculate distances;  How to direct attention to a fruitful choice of approach?  What to label?  Seeking invariants  Avoiding didactic transposition and the worksheet syndrome  Developing habits of mind, body and affect

8. 8 Reflections on Teacher Education Actions  How to draw attention to the choices in ways of working so as to draw attention to fruitful choices  What features of a task afford opportunities for directing attention usefully?  Drawing attention to ways to avoid didactic transposition

9. 9 Awarenesses  1: awarenesses which enable mathematical actions to take place  2: awarenesses of these awarenesses which makes it possible to prepare pedagogical strategies for use with others (colleagues, students) (awareness in discipline)  3: awarenesses of second order awarenesses making it possible to work sensitively with teachers on the choice of pedagogic strategies and didactic tactics pertinent to different mathematical topics and to teaching investigatively (awareness-in-counsel).

10. 10 Learning through Lived Experience  Phenomenological knowledge is obtained experientially  Reflective withdrawal from action in order to become aware of the action (and its effect) –One thing we don’t often learn from experience … … is that we don’t often learn from experience alone  Educating Awareness Training Behaviour Harnessing Emotion  Developing habits of mind, body and affect

11. 11 Structure of the Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

12. 12 Structure of a Topic Language Patterns & prior Skills Imagery/Sense- of/Awareness; Connections Different Contexts in which likely to arise; dispositions Techniques & Incantations Root Questions predispositions Standard Confusions & Obstacles Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable BehaviourBehaviour EmotionEmotion AwarenessAwareness

13. 13 Follow-Up  Access to these slides and to a written version: –//mcs.open.ac.uk/jhm3 –Choose option 8