Fragments and coherence

Anne WatsonATM/MA/NANAMIC/AMET

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 • 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 ….

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

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

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

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

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

?? 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

A trip through trig

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 …….

Or

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

Making a mess of multiplication

So multiplication appears to be…

….. either times tables or something very advanced

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

Knowing multiplication when I see it

Knowing multiplication when I see it

Knowing multiplication when I see it

Knowing multiplication when I see it

x 2 = 24x 3 = 24

e x = 24

Knowing multiplication when I see it

24

2

6

3

2

212

Knowing multiplication when I see it

Knowing multiplication when I see it

xy = 24

24yx =

24xy =

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?

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.’

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

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

A lesson without:

is not a maths lesson

• Doing and undoing

• Mathematical repertoire

• Relating properties

• Discrete / continuous

• Mathematical reasoning

• Exemplification / generalisation

anne.watson@education.ox.ac.ukwww.education.ox.ac.uk

8th Annual Institute of Mathematics PedagogyJuly 28th to 31st

Cuddesdon near Oxfords.elliott@shu.ac.uk

John Mason, Malcolm Swan, Anne Watson