Fragments and coherence Anne Watson ATM/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 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 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

x 2 = 24 x 3 = 24 e x = 24

Knowing multiplication when I see it

Knowing multiplication when I see it

xy = y x = 24 x y =

Knowing multiplication when I see it What two numbers multiply to give 24? …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

8 th Annual Institute of Mathematics Pedagogy July 28 th to 31 st Cuddesdon near Oxford John Mason, Malcolm Swan, Anne Watson