Fundamental Constructs Underpinning Pedagogic Actions in Mathematics Classrooms

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Fundamental ConstructsUnderpinningPedagogic
Actionsin Mathematics Classrooms
John Mason March 2009
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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
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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
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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 - )
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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?

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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

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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

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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?
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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

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Magic Square Reasoning
What other configurationslike thisgive one
sumequal to another?
Try to describethem in words
Any colour-symmetric arrangement?
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More Magic Square Reasoning
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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

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Leibnizs Triangle
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Reflection

• Movements of attention
• Discerning details
• Recognising relationships
• Perceiving properties
• Reasoning on the basis of agreed properties
• Infinity
• Connections (Pascals triangle)

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MGA, DTR Worlds of Experience
DoingTalkingRecording
3 Worlds EnactiveIconicSymbolic
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Variation

• Dimensions-of-possible-variationRange-of-permissi
  ble-change
• Invariance in the midst of change

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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

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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

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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
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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
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For Access to Fundamental Constructs

• NCETM website (Mathemapedia)
• Fundamental Constructs in Mathematics Education
  (RoutledgeFalmer)