1 Fundamental Constructs Underpinning Pedagogic Actions in Mathematics Classrooms John Mason March 2009 The Open University Maths Dept University of Oxford Dept of Education

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 Teaching takes place in time Learning takes place over time 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

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’? (11 - ) 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)  Don’t need particulars as test-bed  Recognising relationships but then perceiving them as properties  Dimensions-of-Possible-Variation Range-of-Permissible-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 –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2 Any colour-symmetric arrangement?

11 More Magic Square Reasoning –= 0Sum( )Sum( )

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 Leibniz’s Triangle 1

14 Reflection  Movements of attention –Discerning details –Recognising relationships –Perceiving properties –Reasoning on the basis of agreed properties  Infinity  Connections (Pascal’s triangle)

15 MGA, DTR & Worlds of Experience Doing–Talking– Recording 3 Worlds: Enactive–Iconic– Symbolic

16 Variation  Dimensions-of-possible-variation Range-of-permissible-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 Birmingha m moths Learners Teachers Institutions

20 Maintaining Complexity Taking Account of the Whole Psyche  Enaction – Cognition – Affect  Behaviour – Awareness – Emotion  Doing – Noticing – Feeling Change ≠ doing differently Developing = enhancing and enriching being Being mathematical with and in front of learners so that they experience what it is like being mathematical Being

21 For Access to Fundamental Constructs  NCETM website (Mathemapedia)  Fundamental Constructs in Mathematics Education (RoutledgeFalmer)