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Fragments and coherence
Anne WatsonATM/MA/NANAMIC/AMET
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 • where you want to differentiate, manage this by providing work at no more than three or four levels of difficulty across the class • target a small number of pairs, groups or individuals for particular questioning and support, rather than monitoring them all • make sure that pupils working independently know where to find resources, what to do before asking for help and what to do if they finish
early • brief any supporting adults about their role, making sure that they have plenty to do with the pupils they are assisting
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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 children’s 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 whatthey have achieved;_ return to the lesson objective(s) and reinforce key words, facts, ideas and notation;_ re-emphasise teaching points and vocabulary;_ identify key points and methods for children to remember, and to resolve any mistakes andmisunderstandings;
_ give the children a clear idea of what they are moving onto next, and sometimes to sethomework;_ relate the mathematics children have learned to other subjects in order to help themaccess the whole curriculum;
_ continue to teach – not just have children reporting back
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?? Mystery document• 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
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Knowing multiplication when I see it
x 2 = 24x 3 = 24
e x = 24
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Knowing multiplication when I see it
24
2
6
3
2
212
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Knowing multiplication when I see it
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Knowing multiplication when I see it
xy = 24
24yx =
24xy =
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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 - don’t 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 don’t think we have to do that for this module.’
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Joining up mathematics:it’s 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 mathematics:it’s how you see it and what you do• Doing and undoing
• Mathematical repertoire
• Relating properties
• Discrete / continuous
• Mathematical reasoning
• Exemplification / generalisation
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A lesson without:
is not a maths lesson
• Doing and undoing
• Mathematical repertoire
• Relating properties
• Discrete / continuous
• Mathematical reasoning
• Exemplification / generalisation
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8th Annual Institute of Mathematics PedagogyJuly 28th to 31st
Cuddesdon near [email protected]
John Mason, Malcolm Swan, Anne Watson