Effective Mathematics Teaching

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Effective Mathematics Teaching Learning

• Educating Awareness Training Behaviour
• Through Harnessing Emotions

Exeter Sept 03
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• Learning is maximally effective when
• all aspects of psyche are involved
• learners are active (doing, construing)
• learners are using their natural powers
• Teaching is maximally effective when
• ethos is mathematical
• being mathl with in front of learners
• evident caring for learners for maths

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Write down two numbers which sum to ten
Write down another two numbers which sum to ten
and another two numbers which sum to ten
What did you notice?
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Write down two numbers which sum to ten
What can change, and still some feature is
preserved?
Change the TEN Change SUM Change TWO Change
NUMBERS
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Dimensions-of-Possible-Variation label for a
collection of questions every learner can ask for
themselves What can change and still is
preserved? What is the Range-Of-Permissible-Change
in each case? This applies to every task to
every concept, even to how tasks are presented
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I have written down two numbers which sum to
ONE. I square the larger and add the smaller
(A) I square the smaller and add the larger
(B) Which of my two answers A B will be the
biggest? Make a conjecture !!
What did you notice?
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Depicting

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Depicting

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Depicting

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Depicting

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Depicting

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Depicting

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Depicting

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Depicting

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Generalising
The product of the largest of each pair the
smallest of one pair is the same as
The product of the smallest of each pair the
other largest
What if they sum to something else?
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Relation to Curriculum
A lesson without the opportunity for learners to
generalise is NOT a mathematics lesson!
Generalisation is NOT something to be taught,
but a power to be developed and invoked in
EVERY lesson
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Imagining Expressing (communicating) Specialisin
g Generalising Conjecturing Convincing
(reasoning) Organising Characterising
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Invariance in the Midst of Change Freedom
Constraint Doing Undoing Extending
Contracting Meaning
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• Learning is maximally effective when
• all aspects of psyche are involved
• Awareness, behaviour, emotions
• learners are active (doing, construing)
• doing ? construing
• making choices experiencing creative
  energy flow
• learners are using their natural powers

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Learners Theory

• If I complete the tasks I am set then learning
  will (presumably) take place

Learners Development

• Assenting gt Asserting, Anticipating
• Following gt Formulating
• Taking initiative Making choices

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What is the point of teaching if there is little
learning? Major pressure obligation to cover
everything difficult to get through in time
available ask myself who is covering the
syllabus? Is it the learners, or just
me? Sometimes not much is written down when
learners become creative, misconceptions and
confusions surface and life gets messy building
sites are messy places From Malcolm Swan
(Nottingham)
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• Teaching is maximally effective when
• ethos is mathematical
• conjecturing atmosphere
• exposure to themes heuristics
• being mathl with in front of learners
• displaying use of powers
• struggling sometimes exploring together
• evident caring
• for learners
• for mathematics (mathematical thinking)

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Teaching Traps

• Doing for learners what they can already do for
  themselves
• doing ? construingworking through ? working on
• The more clearly I specify behaviour sought from
  learners, the easier it is for them to display it
  without generating it for themselves

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Perhaps the greatest of all pedagogical
fallaciesis the notion that a person learns only
the particular thing being studied at the
time.Collateral learning may be and often is
much more important the the (actual) lesson.
John Dewey
We are wise to create systems for spin-offs
rather than for pay-offs. Bill Brookes