Teaching mathematics mathematically: a reflection on

research and current thinking

Anne Watson Professor of Mathematics Education

University of OxfordFebruary 16th 2009

My claim

• Personal engagement in mathematics, and reflection on the nature of that engagement, is at the heart of good mathematics teaching, and may make much of what is written about pedagogy redundant.

Long Train Running

• http://uk.youtube.com/watch?v=kEwENEppsAM

Linear functions: constructing and representing

• multiply by something and add something• knowing the number of trucks, and the length of

trucks and engine, gives me the length of the train• Different kinds of trucks, each one a variable, or

assume all the same length• the number of trucks is the independent variable;

the length of the train is the dependent variable• y = mx + c

Linear functions: general class and special cases

• All functions of the form y = mx + c• Polynomial whose coefficients are all zero

apart from the coefficient of x (perhaps) and a constant term (doesn’t feel like the train)

• e.g. 0x4 + 0x 3 + 0x 2 + mx + c• Special case m = 0: e.g. y = 6• To know x = 6 isn’t, and why it isn’t

Linear function: is it a line?

• iconic; isomorphic; metaphoric; semantic; syntactic; encapsulation; interpretation; unpacking

• What about x = 6?• Situation looks like a line? The seductions

of visual representations• Skateboarder

• http://uk.youtube.com/watch?v=KCsz2k7mOsU&feature=related

Designing the learning experience

• Identifying key mathematical understandings

• Understanding how they might arise in response to stimuli

• Understanding how prior knowledge might or might not be helpful

• Understanding what else can be seen and assumed – seeing as others might see

How to be a ‘good’ teacher: according to Ofsted

• 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 – to do what?

• 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 ….

• So how does this get translated into action

Whole class interactive teaching is where the emphasis has been.

What the Strategy says.• Directing and telling• Demonstrating and modelling• Explaining and illustrating• Questioning and discussing• Exploring and investigating• Consolidating and embedding• Reflecting and evaluating• Summarising and reminding – about what?• Does this collection of urgings say all we need to

know about conceptual development? Let’s see what else about ‘well-structured’

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

CMTP: How are teachers structuring lessons so students learn new mathematical concepts?

What happens in the “main part”

• How the lesson is structured• How students are helped to ‘do’ mathematics• How discussion is managed • How ideas are shared • How right/wrong answers are dealt with • What public and individual writing is done/displayed• What kinds of questions and prompts are used• How independent learning is encouraged and supported• What task types are used • What habits of working appear to have been established • What ideas are emphasised • What questions are answered quickly/slowly• What seems to be important in mathematics.

I define this mish mash of whole class interaction and emphasis on language and talk and inclusion as:English

interactive mathematics teaching

Typified by:

individual and collective sense-makingstructure this towards conventional understandings talk between students, in small groupstalk between students and teacher in the whole classexploratory taskscollaboration choice of groupingpractice exercisestelling the students things eliciting their understandingsmaking conjectures

How did teachers who made a difference teach?: From research

• Deep Progress (IAMP)• belief that all students can learn mathematics• commitment to developing mathematical thinking

• CMTP• belief that all students can learn mathematics• commitment to developing mathematical thinking• English interactive mathematics teaching• team discussion about what it means to do

mathematics, and how students learn

Critical shift of teachers’ attention in CMTP

• From how maths is taught but what maths is taught

• Too little too late – but did happen spontaneously in all schools eventually.

My doubts about English interactive style

• why would motivational and inclusive social modes of working lead to better mathematics learning rather than just more? (language; discussion; recycling given ways of thinking)

• more teaching, harder curriculum, rewards, social coercion, performance-focused teaching too (Kumon; Escalante; JUMP – one to one enthusiasts, hierarchical completeness - but more about that)

• the basic belief of ‘transformability’, equity and common learning power lead to more effort and organising to bring about better learning

• TIMSS: mathematical coherence, sustained complexity, teaching harder mathematics

• failure of NAGTY (??) – coercion, teaching more, teaching faster, special treats and trips.

• Growth of student understanding through discussion does not match with the curriculum coverage model of observation (learning objective) and assessment.

• not whether they are learning but what they are learning, and is it sustainable?

Stoyanova’s study• ‘Working Mathematically’ rolled-out statewide in Western Australia, n=1600

(exploratory style – not necessarily the same as English interactive style but some similar features)

• Higher achievement was associated with: • problem-posing• checking by alternative methods• asking ‘what if..?’ questions• giving explanations• testing conjectures• checking answers for reasonableness• splitting problems into subproblems• teachers’ beliefs about learners’ ability to learn• higher levels of teachers’ own knowledge

• Not associated with: • explicit teaching of problem-solving strategies• making conjectures• sharing strategies

• Negatively associated with use of real life contexts• Achievement falling since initial rise – professional development support?

Enthusiasm?

Why? My analysis is that:

• real life contexts, making own conjectures, sharing own strategies, and teachers who do not know enough maths cannot shift ways of thinking or objects of attention (discussion and interaction not enough – the question is how thinking is changed by interaction)

• checking answers, suggesting new variations, testing conjectures, recognising subproblems, engaging with structure to pose problems, require new ways of thinking

Learning shifts: my epistemological analysis of what can be done with

mathematical ideas in maths lessons• Remembering something familiar• Seeing something new• Public orientation towards concept, method and properties• Personal orientation towards concept, method or properties• Analysis, focus on outcomes and relationships, generalising• Indicate synthesis, connection, and associated language• Rigorous restatement (note reflection takes place over time, not in one

lesson, several experiences over time)• Being familiar with a new object• Becoming fluent with procedures and repertoire (meanings, examples,

objects..)

Lesson analysis: the basics are the focus of attention

Repertoire: terms; facts; definitions; techniques; procedures

Representations and how they relate Examples to illustrate one or many features Collections of examples Comparison of objects Characteristics & properties of classes of objects Classification of objects Variables; variation; covariation

What the teacher does

• Make or elicit declarative/nominal/factual/technical statements

• Tell learners to do things• Invite perception/direct attention• Ask for learners for particular responses: objects,

structures, examples, behaviours• Discuss implications • Integrate and connect • Affirm: meaning later

What learners can do naturally

Imitate Visualise Seek pattern Follow pattern Compare, classify Draw on prior experience and repertoire Describe Explore variation Explore covariation Informal induction Informal deduction Create objects with one or more features Exemplify Express in ‘own words’

Unnaturally - practise and apply technique

Mathematical unnatural things to do next. What happens next?

• Discuss implications: Vary the variables, adapt procedures, identify relationships, explain and justify, induction and prediction, deduction• Learner shift: Analysis, focus on outcomes and

relationships• Integrate and connect: Associate ideas, generalise,

abstract, objectify, formalise, define• Learner shift: Synthesis, connection

• Affirm: Adapt/ transform ideas, apply to more complex maths and to other contexts, prove, evaluate process• Learner shift: Rigour, objectification

CMTP: we found differences in teaching: what kinds of teacher?

Attention to objects/names/recognition

Attention to structures/relations

Reflection on methods in terms of whether they worked

Reflection on methods and answers in terms of mathematical implications

Affirmation in terms of finishing & correct answers

Affirmation in terms of consistency and coherence of mathematics

Implications in terms of curriculum and usefulness

Implications in terms of contribution to and relationship with other mathematical ideas

Mathematical knowledge in teaching

Why would there be that clear difference? Departments working and planning together and discussing maths together.

• what comes to mind from a structured personal space of possible productive actions

• actions made available through recognising words, diagrams, settings: explicit or tacit

• potential actions dynamically gather in the moment – become available

• habitual or deliberate action

What comes to mind

• Phenomenological primitives (di Sessa) • Embodied awareness (Maturana) often not associated with

formal mathematical language and situations – edited out of selfhood in maths classroom

• Intuition (Fischbein) and quasi-intuitions• Naïve concept (Pratt & Noss; Noss & Hoyles)• Enculturation (e.g. Threlfall 1998)• Habitual

• expectations of relationships between properties• ways of seeing and responding• actions, expectations, assumptions

• Acculturation (e.g. Kitcher)

Cuoco: Mathematical habits of Mind

• Pattern seeking• Experimenting: thought experiments; controlling variables• Describing: steps of a process; notations; arguments for

and against; notes• Tinkering: range of application; domain of truth; what if?• Inventing: rule-structures; isomorphisms• Visualising: relationships, processes, change, calculations,

covariation (models, diagrams, imagination)• Conjecturing: from data, relations, properties• Guessing

Algebraic habits

• See calculation as structure of operations• Represent classes of mathematical objects

and their relations• Algorithms• Extend meaning over new domains• Abstraction

Geometric habits

• Like shapes• Worry about change and invariance• Proportional reasoning• Explore systems and distinctions• Reasoning about properties

Shifts (mentioned by Cuoco et al. but not explicitly – my analysis)

• Between generalities and examples• From looking at change to looking at

change mechanisms (functions)• Between various points of view• Between deduction and induction• Between domains of meaning and extreme

values as sources of structural knowledge

Shifts (Watson: work in progress)

• from proximal, ad hoc, and sensory methods of solution to abstract concepts

• from inductive learning of structure to understanding abstract relations

• to focusing on properties instead of visible characteristics; verbal and kinaesthetic socialised responses to sensory stimuli are often inadequate for abstract tasks

• from ideas that can be modelled iconically to those that can only be represented symbolically: length of train if we consider each type of carriage as different length, hence different variable.

• … take one of these first

What has gone wrong?

• Social norms• Mathematical norms of that classroom• The teacher knows what an equilateral triangle is, and so

do the students, but do they focus on properties enough to know what is NOT equilateral?

• Focus on visual – focus on properties• Discussion, interaction, participation, ICT use, groups,

main part, etc. etc.• That’s just one of the shifts that learners have to make ….

QTS standards

• 27 statements about professional responsibilities, social aspects of school and classroom, and diversity

• 12 about teaching and learning the subject of which• 4 are about assessment• 4 general statements about teaching• 2 general statements about planning • 1 about curriculum structure• and …

Q 14

• “Have a secure knowledge and understanding of their subjects/curriculum areas and related pedagogy to enable them to teach effectively across the age and ability range for which they are trained.”

Nothing about

• Understanding how people learn• Understanding conceptual development in a

subject• Anticipating and understanding learners’

responses• How to use subject knowledge to design

teaching

Q14 for mathematics

“Have a secure knowledge and understanding of … • concepts beyond those being taught & how these unify earlier

ideas• how mathematical ideas are linked & how to identify links• central concepts and techniques & how the way these are taught

shapes future learning and potential progress• development of topics in typical UK textbooks & mathematical

development• how different topics are taught and learnt in different ways• how understandings & misunderstandings arise• the powers and limitations of: models, analogies, examples,

metaphors, comparisons, representations• how to extend their own repertoire of the above”

• Personal engagement in the necessary shifts inherent in mathematics, and reflection on the nature of that engagement (how I made those shifts), is at the heart of good mathematics teaching, and may make much of what is written about pedagogy redundant.