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Phenomenal Knowledge and

Knowing PhenomenologicallyResearch Day

on subject knowledge in teaching

John MasonJohn Mason

Feb 16 2009Feb 16 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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Thesis: What is needed in order to Thesis: What is needed in order to ……

… do mathematics effectively is …do mathematics effectively is …… awareness of mathematical actionsawareness of mathematical actions

… teach mathematics effectively is teach mathematics effectively is ……… awareness of those awarenessesawareness of those awarenesses

… teach people to teach mathematics teach people to teach mathematics effectively is …effectively is …… awareness of those awarenesses of awareness of those awarenesses of those awarenessesthose awarenesses

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VisitingVisiting PPeople set off at the same time to walk eople set off at the same time to walk between two towns, some in each direction. between two towns, some in each direction. They all meet at noon. They carry on as They all meet at noon. They carry on as before; some reach their destination at before; some reach their destination at 4pm, and the others at 6:15 pm.4pm, and the others at 6:15 pm.

At what time did they all set out?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 h2 = 25 and h = 5• So they set out at 7:00 am hh 4:004:00 6:1

56:15

whwh

vhvh

12:0012:00

distancedistance

timetime

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Reflections on Mathematical Reflections on Mathematical ActionsActions Addressing questions such as: must everyone go the same Addressing questions such as: must everyone go the same

speed, at least in the same direction? At what speeds do speed, at least in the same direction? At what speeds do they go after they meet (assume they maintain their same they go after they meet (assume they maintain their same speed)?speed)?

Articulating the relationship between speed, distance and Articulating the relationship between speed, distance and timetime

Denoting what is not known: acknowledging ignorance (Mary Denoting what is not known: acknowledging ignorance (Mary Boole)Boole)

Deciding what is knownDeciding what is known Presenting the information to yourself in symbols or as a Presenting the information to yourself in symbols or as a

graphgraph Calculating relevant quantities or invoking similar Calculating relevant quantities or invoking similar

triangles or triangles or Choosing which relationships to equate so as to locate Choosing which relationships to equate so as to locate

fruitful onesfruitful ones Moving to ratio of walking speedsMoving to ratio of walking speeds Possibly, solving a quadratic equationPossibly, solving a quadratic equation Wondering about relevance of 3Wondering about relevance of 322 + 4 + 422 = 5 = 522

Generalising (changing the 4 and the 6:15); adding a third Generalising (changing the 4 and the 6:15); adding a third town in some way, …)town in some way, …)

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Further ReflectionsFurther Reflections

Crossed LaddersCrossed Ladders– CCrossing height is rossing height is

Harmonic sum of ladder heightsHarmonic sum of ladder heights

GGeometric mean of ladder excess heightseometric mean of ladder excess heights

4:004:00

6:156:15

12:0012:00

distancedistance

timetime

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Interlude on TasksInterlude on Tasks

tasks are for initiating activity; tasks are for initiating activity; through engaging in activity people have the through engaging in activity people have the opportunity to experience things, particularly opportunity to experience things, particularly mathematical actions and their effects. mathematical actions and their effects.

Teachers have intentions when they choose Teachers have intentions when they choose tasks:tasks:– it is not that the task itself will promote it is not that the task itself will promote learning, but that the learning, but that the inner taskinner task will involve the will involve the directing or re-directing of attention directing or re-directing of attention

– and perhaps the internalising or integrating of and perhaps the internalising or integrating of actions previously dependent on being triggered by actions previously dependent on being triggered by some outside agency (ZPD).some outside agency (ZPD).

It is valuable, even necessary to draw back It is valuable, even necessary to draw back from the action and to become aware of the from the action and to become aware of the actions and thactions and theieir effects (utility)r effects (utility)

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Reflections on Pedagogical ActionsReflections on Pedagogical Actions

How to direct attention to How to direct attention to mathematical choices?mathematical choices? Draw a diagram; calculate distances; Draw a diagram; calculate distances;

How to direct attention to a How to direct attention to a fruitful choice of approach?fruitful choice of approach?

What to label?What to label? Seeking invariantsSeeking invariants AAvoiding voiding didactic transposition didactic transposition and the worksheet syndromeand the worksheet syndrome

DDeveloping habits of mind, body eveloping habits of mind, body and affectand affect

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Reflections on Teacher Education Reflections on Teacher Education ActionsActions

How to draw attention to the How to draw attention to the choices in ways of working so as choices in ways of working so as to draw attention to fruitful to draw attention to fruitful choiceschoices

WWhat features of a task afford hat features of a task afford opportunities for directing opportunities for directing attention usefully?attention usefully?

Drawing attention to ways to Drawing attention to ways to avoid avoid didactic transpositiondidactic transposition

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AwarenessesAwarenesses

1: 1: awarenesses which enable mathematical awarenesses which enable mathematical actions to take placeactions to take place

2: 2: awarenesses of these awarenesses which awarenesses of these awarenesses which makes it possible to prepare pedagogical makes it possible to prepare pedagogical strategies for use with others strategies for use with others (colleagues, students) (colleagues, students) ((awareness in discipline)awareness in discipline)

3: 3: awarenesses of second order awarenesses awarenesses of second order awarenesses making it possible to work sensitively making it possible to work sensitively with teachers on the choice of pedagogic with teachers on the choice of pedagogic strategies and didactic tactics pertinent strategies and didactic tactics pertinent to different mathematical topics and to to different mathematical topics and to teaching investigativelyteaching investigatively ( (awareness-in-awareness-in-counselcounsel).).

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Learning through Lived Learning through Lived ExperienceExperience Phenomenological knowledge is obtained Phenomenological knowledge is obtained experientiallyexperientially

Reflective withdrawal from action in Reflective withdrawal from action in order to become aware of the action (and order to become aware of the action (and its effect)its effect)– One thing we don’t often learn from One thing we don’t often learn from experience experience ……

… … is that we don’t often learn from is that we don’t often learn from experience aloneexperience alone

Educating AwarenessEducating AwarenessTraining BehaviourTraining BehaviourHarnessing EmotionHarnessing Emotion

Developing habits of mind, body and Developing habits of mind, body and affectaffect

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Structure of the PsycheStructure of the Psyche

ImageryImageryAwareness (cognition)Awareness (cognition)

WillWill

Body (enaction)Body (enaction)

Emotions Emotions (affect)(affect)

HabitsHabitsPracticesPractices

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Structure of a TopicStructure of a Topic

Language Patterns& prior Skills

Imagery/Sense-of/Awareness; Connections

Different Contexts in which likely to arise;dispositions

Techniques & Incantations

Root Questionspredispositions

Standard Confusions & Obstacles

Only Behaviour is TrainableOnly Behaviour is Trainable

Only Emotion is HarnessableOnly Emotion is Harnessable

Only Awareness is Only Awareness is EducableEducable

Behaviour

Behaviour

Behaviour

Behaviour

EmotionEmotionEmotionEmotion

Awareness

Awareness

Awareness

Awareness

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Follow-UpFollow-Up

Access to these slides and to a Access to these slides and to a written version:written version:– //mcs.open.ac.uk/jhm3//mcs.open.ac.uk/jhm3– CChoose option 8hoose option 8