Title: Fundamental Constructs Underpinning Pedagogic Actions in Mathematics Classrooms
1Fundamental ConstructsUnderpinningPedagogic
Actionsin Mathematics Classrooms
John Mason March 2009
2Outline
- 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
3Learning 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
4Doing 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 - )
5Doing 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?
6Reflection
- 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
7Some 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
8Similarly 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?
9Reflection
- 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
10Magic Square Reasoning
What other configurationslike thisgive one
sumequal to another?
Try to describethem in words
Any colour-symmetric arrangement?
11More Magic Square Reasoning
12Reflection
- 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
13Leibnizs Triangle
1
14Reflection
- Movements of attention
- Discerning details
- Recognising relationships
- Perceiving properties
- Reasoning on the basis of agreed properties
- Infinity
- Connections (Pascals triangle)
15MGA, DTR Worlds of Experience
DoingTalkingRecording
3 Worlds EnactiveIconicSymbolic
16Variation
- Dimensions-of-possible-variationRange-of-permissi
ble-change - Invariance in the midst of change
17What 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
18Implicit 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
19Darwininian 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
20Taking 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
21For Access to Fundamental Constructs
- NCETM website (Mathemapedia)
- Fundamental Constructs in Mathematics Education
(RoutledgeFalmer)