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Title: Fundamental Constructs Underpinning Pedagogic Actions in Mathematics Classrooms


1
Fundamental ConstructsUnderpinningPedagogic
Actionsin Mathematics Classrooms
John Mason March 2009
2
Outline
  • Raise some pedagogic questions
  • Engage in some mathematical thinking
  • Use this experience to engage with those questions

If you fail to prepare for your surface,prepare
for your surface to fail
3
Learning Doing
  • What do learners need to do in order to learn
    mathematics?
  • What do they think they need to do?
  • What are mathematical tasks for?
  • What do learners think they are for?

Doing ? Construing
Teaching takes place in time Learning takes place
over time
4
Doing Undoing Additively
  • What operation undoes
  • adding 3?
  • subtracting 4?
  • adding 3 then subtracting 4?
  • subtracting from 7?
  • subtracting from 11 then subtracting from 7?

7 - )
11 - )
5
Doing Undoing Multiplicatively
  • What are the analogues for multiplication?
  • What undoes multiplying by 3?
  • What undoes dividing by 2?
  • What undoes dividing by 3/2?
  • What undoes multiplying by 3/2? Now do it
    piecemeal!
  • What undoes dividing into 12?

6
Reflection
  • Doing Undoing (mathematical theme)
  • Dont need particulars as test-bed
  • Recognising relationships but then perceiving
    them as properties
  • Dimensions-of-Possible-VariationRange-of-Permissi
    ble-Change
  • Relationship between adding subtracting
    between multiplying dividing
  • You can work things out for yourself
  • Importance of listening to what is said and
    seeing it in several different ways
  • Worksheet-itis

7
Some Constructs
  • Outer, Inner Meta Task(s)
  • Didactic transposition
  • Expert awareness ? instructions in behaviour
  • Didactic contract
  • Didactic tension
  • The more clearly the teacher specifies the
    behaviour sought,the easier it is for learners
    to display that behaviour without generating it
    from and for themselves

8
Similarly Shapely Cuts
  • What planar shapes have the property that they
    can be cut by a straight line into two pieces
    both similar to the original?

Just ask for similar to each other?
9
Reflection
  • Breaking away from the familiar
  • Switching from edges to angles and back to edges
    (choosing what to attend to)
  • Mathematical similarity angles ratios
  • Asking what are the possibilities?(analysis by
    cases)
  • Reasoning
  • Acknowledging ignorance (Mary Boole)
  • Manipulating familiar diagrams in fresh way
  • ZPD acting for yourself rather than in reaction
    to cue/instruction

10
Magic Square Reasoning
What other configurationslike thisgive one
sumequal to another?
Try to describethem in words
Any colour-symmetric arrangement?
11
More Magic Square Reasoning
12
Reflection
  • What are the inner tasks?
  • Invariance in the midst of change
  • Movements of attention
  • Discerning details
  • Recognising relationships
  • Perceiving these as properties
  • Reasoning with unknown entities based on agreed
    properties
  • Doing Undoing
  • Dealing with unspecified-unknown numbers

13
Leibnizs Triangle
1
14
Reflection
  • Movements of attention
  • Discerning details
  • Recognising relationships
  • Perceiving properties
  • Reasoning on the basis of agreed properties
  • Infinity
  • Connections (Pascals triangle)

15
MGA, DTR Worlds of Experience
DoingTalkingRecording
3 Worlds EnactiveIconicSymbolic
16
Variation
  • Dimensions-of-possible-variationRange-of-permissi
    ble-change
  • Invariance in the midst of change

17
What are tasks for?
  • Tasks generate activity
  • Activity provides experience of engaging in
    (mathematical) actions
  • Inner task is
  • What concepts themes expected to encounter
  • what actions expected to modify or extend
  • What actions to internalise for self
  • In order to learn from experience, it is
    necessary to withdraw from immersion in action
  • Reflection on and reconstruction of highlights

18
Implicit Theories Constructs worthy of
Critique
  • Doing Learning
  • If I get the answers, I must be learning
  • The muscle metaphor
  • Keep exercising and eventually you can do it
  • The Collective Hypothesis
  • Talking produces learning
  • The Jacobs Staircase metaphor
  • Learning progresses steadily and uniformly
  • Worksheets are necessary
  • For managing the classroom
  • For record keeping as evidence of activity
  • For learning

19
Darwininian Metaphor
  • Development when the organism and the environment
    are mutually challenging and when there are
    sufficient mutations to provide variation
  • Excessive challenge leads to loss of species
  • Inadequate challenge leads to loss of flexibility

Birmingham moths
Learners Teachers Institutions
20
Taking Account of the Whole Psyche
  • Enaction Cognition Affect
  • Behaviour Awareness Emotion
  • Doing Noticing Feeling

Being
Being mathematical with and in front of
learners so that they experience what it is like
being mathematical
Change ? doing differently Developing enhancing
and enriching being
Maintaining Complexity
21
For Access to Fundamental Constructs
  • NCETM website (Mathemapedia)
  • Fundamental Constructs in Mathematics Education
    (RoutledgeFalmer)
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