ATMMANANAMICAMET Keele 2008 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 ID: 677618
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Slide1
Fragments and coherence
Anne Watson
ATM/MA/NANAMIC/AMET
Keele 2008Slide2
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
….Slide3
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 learningSlide4
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 learningSlide5
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 assistingSlide6
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
Slide7
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 what
they 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 and
misunderstandings;
_ give the children a clear idea of what they are moving onto next, and sometimes to
set
homework
;
_ relate the mathematics children have learned to other subjects in order to help them
access the whole curriculum;
_ continue to teach –
not just have children reporting backSlide8
?? 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
OrganisationSlide9
A trip through trigSlide10Slide11Slide12Slide13Slide14Slide15Slide16Slide17Slide18Slide19Slide20Slide21Slide22Slide23Slide24
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 …….Slide25
Or
Is it by ‘doing trig’ that you come to understand all those bits?Slide26
Making a mess of multiplicationSlide27Slide28Slide29Slide30Slide31Slide32Slide33Slide34Slide35Slide36Slide37Slide38Slide39Slide40
So multiplication appears to be…
….. either times tables or something very advancedSlide41Slide42Slide43Slide44Slide45Slide46Slide47Slide48Slide49Slide50Slide51
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 dimensionalSlide52
Knowing multiplication
when I see itSlide53
Knowing multiplication
when I see itSlide54
Knowing multiplication
when I see itSlide55
Knowing multiplication
when I see it
x
2
= 24
x
3
= 24
e
x
= 24 Slide56
Knowing multiplication
when I see it
24
2
6
3
2
2
12Slide57
Knowing multiplication
when I see itSlide58
Knowing multiplication
when I see it
xy = 24
24
y
x =
24
x
y =Slide59
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?Slide60
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.’Slide61
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 – generalisationsSlide62
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 / generalisationSlide63
A lesson without:
is not a maths lesson
Doing and undoing
Mathematical repertoire
Relating properties
Discrete / continuous
Mathematical reasoning
Exemplification / generalisationSlide64
anne.watson@education.ox.ac.uk
www.education.ox.ac.uk
8
th
Annual Institute of Mathematics Pedagogy
July 28
th
to 31
st
Cuddesdon near Oxford
s.elliott@shu.ac.uk
John Mason, Malcolm Swan, Anne Watson