What is the Discipline of Mathematics Education? Essential Maths

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What is the Discipline of Mathematics
Education?Essential MathsMathematical Essences
John Mason Hobart 2007
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Outline

• Justifying a problem a day keeps the teacher in
  play
• What mathematics is essential?
• What is mathematical essence?

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Grid Movement
((73)x2)3 is a path from 7 to ?. What
expression represents the reverse of this path?
What values can ? have if exactly one - and
one are used? Max value? Min Value?
x2
?
7
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What about other cells?Does any cell have 0?
-7? Does any other cell have 7?
Characterise ALL the possible values that can
appear in a cell
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-3
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Reflections

• What variations are possible?
• What have you gained by working on this task
  (with colleagues)?

• What criteria would you use in choosing whether
  to use this (or any) task?
• What might be gained by working on (a variant of)
  this task with learners?

Tasks gt Activity gt Experience gt Reflection
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More Disciplined Enquiry

• What is the point? (Helen Chick)
• Outer task Inner task
• What is the line? (Steve Thornton)
• Narrative for HoD, Head, parents, self
• What is (the) plain?
• What awarenesses? What outcomes?
• What is the space?
• Domain of related tasks
• Dimensions of possible variation ranges of
  permissible change

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Differences
AnticipatingGeneralising
Rehearsing
Checking
Organising
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Sketchy Graphs

• Sketch the graphs of a pair of straight lines
  whose y-intercepts differ by 2
• Sketch the graphs of a pair of straight lines
  whose x-intercepts differ by 2
• Sketch the graphs of a pair of straight lines
  whose slopes differ by 2
• Sketch the graphs of a pair of straight lines
  meeting all three conditions

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Cubic Construction

• Sketch a cubic
• which has a local maximum
• and which has only one real root
• and which has a positive inflection slope

Note the task structureuse of a constraint to
challengeusual/familiar examples
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Chordal Midpoints

• Where can the midpoint of a chord of your cubic
  get to?(what is the boundary of the region of
  mid-points?)
• What about 1/3 points or 4/3 points?

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Justifying doing maths for oneself and with
others

• Sensitise myself to what learners may be
  experiencing
• Refresh my awareness of the movements of my
  attention
• Remind myself what it is like to be a learner
• Experience the type of task I might use with
  learners

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Awarenesses

• Give a family a fish
• and you feed them for a day
• Show them how to fish,
• and you feed them
• until the stocks run out

Obtaining tasks and lesson plans gets you
through some lessons Becoming aware of
affordances, constraints and attunements, in
terms of mathematical themes, powers
heuristics enables you to promote learning
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More Or Less Altitude Area
Draw a scalene triangle
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More Or Less Rectangles Area
Draw a rectilinear figure which requires at least
4 rectangles in any decomposition
How many can have the same perimeter?
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More Or Less Percent Value
50 of something is 20
Value

60 of 60 is 36
40 of 30 is 12
60 of 30 is 20
50 of 40 is 20
50 of 60 is 30
50 of 30 is 15
40 of 60 is 24
40 of 50 is 20
40 of 40 is 16
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More Or Less Whole Part
? of 35 is 21
Part
Whole
3/4 of 28 is 21
3/5 of 35 is 21
6/7 of 35 is 30
3/5 of 40 is 24
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Magic Square Reasoning
What other configurationslike thisgive one
sumequal to another?
Try to describethem in words
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More Magic Square Reasoning
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Graphical Awareness
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Multiplication as Scaling

• If you stick a pin in Hobart in a map of
  Australia, and scale the map by a factor of 1/2
  towards Hobart
• And if a friend does the same in Darwin, scaling
  by 1/2 towards Darwin
• What will be the difference in the two scaled
  maps?

What if one of you scales by a factor of 2/3
towards Hobart and then by a further 1/2 towards
Darwin, while the other scales by 1/2 towards
Darwin and then by a further 2/3 towards Hobart?
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Raise Your Hand When You See
Something which is 2/5 of something 3/5 of
something 5/2 of something 5/3 of
something 2/5 of 5/3 of something 3/5 of 5/3
of something 5/2 of 2/5 of something 5/3 of
3/5 of something 1 2/5 of something1 3/5
of something
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Essential Conceptual Awarenesses

• Choosing the unit
• Additive actions
• Multiplicative actions
• Scaling multi-ply many-fold, repetition, lots
  of
• Coordinated actions
• Angle actions
• Combining
• Translating
• Measuring actions
• Comparing lengths areas volumes (unit)
• Comparing angles
• Discrete-Continuous
• Randomness

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Essential Mathematical-nesses

• Mathematical Awarenesses underlying topics
• Movement of Attention
• Mathematical Themes
• Mathematical Powers
• Mathematical Strategies
• Mathematical Dispositions

Ways of working on these constitute a (the)
discipline of mathematics education
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Movement of Attention

• Gazing (holding wholes)
• Discerning Details
• Recognising Relationships
• Perceiving Properties
• Reasoning on the Basis of Properties

Compare SOLO van Hiele
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Mathematical Themes

• Doing Undoing
• Invariance in the midst of Change
• Freedom Constraint
• Extending and Restricting Meaning

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Mathematical Powers

• Imagining Expressing
• Specialising Generalising
• Conjecturing Convincing
• Classifying Characterising

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Mathematical Strategies/Heuristics

• Acknowledging ignorance (Mary Boole)
• Changing view point
• Changing (re)presentation
• Working Backwards

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Mathematical Dispositions

• Propensity to see the world mathly
• Propensity to pose problems
• Propensity to seek structure
• Perseverence

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Essential Pedgaogic Awarenesses

• Tasks
• initiate activity
• activity provides immediate experience
• learning depends on connecting experiences, often
  through labelling when standing back from the
  action

• Mathematics develops from engaging in actions on
  objects and those actions becoming objects,
• Actions need to become not just things done under
  instruction or guidance, but choices made by the
  learner

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Choices

• What pedagogic choices are available when
  constructing/selecting mathematical tasks for
  learners?
• What pedagogic choices are available when
  presenting mathematical tasks to learners?
• What criteria are used for making those choices?

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What mathematics is essential?

• Extensions of teaching-maths
• Experience analogously something of what learners
  experience, but enrich own awareness of
  connections and utility
• Extensions of own maths
• Experience what it is like to encounter an
  unfamiliar topic

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It is only after you come to know the surface of
things that you venture to see what is
underneath
but the surface of things is inexhaustible
(Italo Calvino 1983)
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Human Psyche
Only awareness is educable Only behaviour is
trainable Only emotion is harnessable
Mental imagery
Awareness (cognition)
Emotion (affect)
Behaviour (enaction)
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What Can a Teacher Do?

• Directing learner attention
• by being aware of structure of own attention
• (amplifying editing stressing ignoring)
• Invoking learners powers
• Bringing learners in contact with mathematical
  heuristics powers
• Constructing experiences which, when accumulated
  and reflected upon, provide opportunity for
  learners to educate their awareness and train
  their behaviour through harnessing their emotions.

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I am grateful to the organisers for affording me
the opportunity and impetus to contact, develop
and articulate these ideas
For this presentation and othersand other
resources see
http//mcs.open.ac.uk/jhm3