Fragments and coherence

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Fragments and coherence

• Anne Watson
• ATM/MA joint conference
• Keele 2008

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How to be good

• Most learners make good progress because of the
  good teaching they receive
• Behaviour overall is good and learners are well
  motivated
• They work in a safe, secure and friendly
  environment
• Teaching is based on secure subject knowledge
  with a well-structured range of stimulating tasks
  that engage the learners
• The work is well matched to the full range of
  learners needs, so that most are suitably
  challenged.
• Teaching methods are effectively related to the
  lesson objectives and the needs of learners .

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Assessment for learning

• Ensure that every learner succeeds set high
  expectations
• Build on what learners already know structure
  and pace teaching so that they can understand
  what is to be learned, how and why
• Make learning of subjects and the curriculum real
  and vivid
• Make learning enjoyable and challenging
  stimulate learning through matching teaching
  techniques and strategies to a range of learning
  needs
• Develop learning skills, thinking skills and
  personal qualities across the curriculum, inside
  and outside the classroom
• Use assessment for learning to make individuals
  partners in their learning

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Personalisation

• Teaching is focused and structured
• Teaching concentrates on the misconceptions, gaps
  or weaknesses that learners have had with earlier
  work
• Lessons or sessions are designed around a
  structure emphasising what needs to be learnt
• Learners are motivated with pace, dialogue and
  stimulating activities
• Learners progress is assessed regularly (various
  methods)
• Teachers have high expectations
• Teachers create a settled and purposeful
  atmosphere for learning

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Main part of a lesson

• introduce a new topic, consolidate previous work
  or develop it
• develop vocabulary, use correct notation and
  terms and learn new ones
• use and apply concepts and skills
• assess and review pupils' progress
• This part of the lesson is more effective if you
  
• make clear to the class what they will learn
• make links to previous lessons, or to work in
  other subjects
• give pupils deadlines for completing activities,
  tasks or exercises
• maintain pace, making sure that this part of the
  lesson does not over-run and that there is enough
  time for the plenary
• When you are teaching the whole class it helps if
  you
• demonstrate and explain using a board,
  flip-chart, computer or OHP
• highlight the meaning of any new vocabulary,
  notation or terms, and encourage pupils to repeat
  these and use them in their discussions and
  written work
• involve pupils interactively through carefully
  planned and challenging questioning
• ask pupils to offer their methods and solutions
  to the whole class for discussion
• identify and correct any misunderstandings or
  forgotten ideas, using mistakes as positive
  teaching points
• ensure that pupils with particular needs are
  supported effectively.
• When pupils are working on tasks in pairs, groups
  or as individuals it helps if you
• keep the whole class busy working actively on
  problems, exercises or activities related to the
  theme of the lesson
• encourage discussion and cooperation between
  pupils

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Whole class interactive teaching

• Directing and telling
• Demonstrating and modelling
• Explaining and illustrating
• Questioning and discussing
• Exploring and investigating
• Consolidating and embedding
• Reflecting and evaluating
• Summarising and reminding

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Self-evaluation for schools

• Planning and teaching of main part of the
  lesson
• Planning and teaching of plenary part of the
  lesson
• Use of opportunities to assess and diagnose
  childrens learning needs
• Progression from mental to written methods
• Developing questioning skills
• Problem-solving techniques and reasoning skills
• Using a calculator as a teaching tool
• In the best lessons, teachers
• _ give attention to explaining the teaching
  objectives
• _ demonstrate the features of the work to be
  covered
• _ ensure that children are ready to begin work
  with confidence
• _ work with the whole class or organise tasks for
  different groups
• _ use timed tasks and feedback to control the
  pace of the lesson.
• It important to have a plenary at the end of
  every lesson in order to
• _ have a definite conclusion to the lesson, so
  that the children go away positive about what
• they have achieved
• _ return to the lesson objective(s) and reinforce
  key words, facts, ideas and notation

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?? Mystery document

• What I have found to be successful?
• What I have found challenging?
• What changes might I make to practice?
• Firm conceptual basis
• Flexibility
• Encouragement to all
• Exposition by teacher
• Discussion
• Appropriate practical work
• Consolidation and practice of fundamental skills
• Problem solving
• Investigative work
• Resources
• Organisation

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A trip through trig
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What has to be joined up to understand
trigonometry?

• Angle as measure of turn
• Angle as a variable in triangles
• Similarity
• Finding right-angled triangles in various
  orientations
• Conventions about labelling triangles
• Names of sides O and A and H as labels
• Lengths O, A, H as related variables
• Ratio
• Three ways to express the relationship a bc
• Enough about functions to grasp what sin, cos,
  tan mean
• Inverse of sin, cos, tan what inverse means
• and .

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Or

• Is it by doing trig that you come to understand
  all those bits?

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Making a mess of multiplication
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So multiplication appears to be

• .. either times tables or something very advanced

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The missing stuff

• Scaling, stretching, substituting n units for 1
  unit
• Shift from discrete to continuous
• Shifting from binary operator to more elements
  involved distributivity and associativity
• One dimensional two-dimensional n dimensional

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Knowing multiplication when I see it
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Knowing multiplication when I see it
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Knowing multiplication when I see it

• x 2 24
• x 3 24
• e x 24

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Knowing multiplication when I see it

24
2
12
6
2
2
3
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Knowing multiplication when I see it
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Knowing multiplication when I see it

• xy 24
• x 24
• y
• y 24
• x

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Knowing multiplication when I see it

• What two numbers multiply to give 24?
• and another
• and another
• What three numbers multiply to give 24?
• What number squared gives 24?

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Joining up mathematics a dis-content approach

• Year 13 student using graphing software to draw
  graph of sin and cos functions We did trig in
  year 10 for GCSE - dont remember any of it now.
• Me (eventually) How could you change the sine
  curve to get the cosine curve?
• Student (argumentatively) Is that
  transformations? Billy, when did we do
  transformations? I dont think we have to do that
  for this module.

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Joining up mathematicsits how you see it and
what you do

• Additive multiplicative
• Multiplicative exponential
• Discrete continuous
• Intuitive mathematical
• Ad hoc abstract
• Rules and facts tools
• Procedures meaning
• Perceptual conceptual
• Pattern relationship
• Results reflection on results
• Relationship properties

• Operations inverses
• Operations functions
• Functions composition
• Inverses
• Result reflection on procedure/method
• Conjecture proof
• Inductive deductive
• Empiricism reasoning
• Examples generalisations

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Joining up mathematicsits how you see it and
what you do

• Doing and undoing
• Mathematical tools
• Relating properties

• Discrete to continuous
• Mathematical reasoning
• Exemplification/generalisation

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A lesson without

• Doing and undoing
• Mathematical tools
• Relating properties

• Discrete to continuous
• Mathematical reasoning
• Exemplification - generalisation
• is not a maths lesson

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anne.watson_at_education.ox.ac.ukwww.education.ox.ac
.uk8th Annual Institute of Mathematics
PedagogyJuly 28th to 31stCuddesdon near
Oxfords.elliott_at_shu.ac.ukJohn Mason, Malcolm
Swan, Anne Watson