Mathematics departments making autonomous change

www.cmtp.co.ukAnne WatsonUniversity of OxfordWarwick 24 Nov 2009

The school aims Improving achievement of PLAS (not

borderline D/C) Altruism & social justice Political pressure

Role of all-attainment groupings

Research-informed Equity Timetable constraints

All schools in year 7, one in year 8, none in year 9 - BUT

... this study is not about ‘mixed-ability’ teaching

How do departments work when making change?

Data: observations and videos of lessons interviews with teachers fieldnotes of meetings audiotapes of meetings between three

heads of department interviews with sample of PLAs internal and external test scripts and results

Complex qualitative data

Activity theory – systems with shared object (intended outcome) of activity, identifiable community and common tools

How the community operates : division of labour and rules

Interacting activity systems

Tools Tools

Rules Rules

Object

Community

Subject

CommunityLabour Labour

Subject

Maths departmentClassroom

“The triangle” affordances

descriptive: helps organise data at a collective level analytical: encapsulates a range of perceptions and

interpretations synthetic: constructs an overall picture of activity and

suggests other connections and potential systemic disruptions

what it does not do explain expose potential disruptions due to individual

differences show how objects and tool-use change

Tools of maths departments

Normal activity internal and external documents resource banks technological resources communication mechanisms

Change activity formal and informal meetings grounded PD opportunities reading meeting structure (affordances) each other’s knowledge and experience

Relation between tools and object

Tools used directly to teach students Normal department tools Tools used to make change - BUT ... those who do not use the ‘make

change’ tools have a different object

Rules and expectations

External and normal rules Expectations which develop as unwritten

community rules Contradictions among rules Expectations of division of labour versus

actual division of labour Transformation of division of labour Labour for the collective, or not

TOOLS

OBJECT SUBJECT

DIVISION OF LABOUR

RULESCOMMUNITY

INTERPRETATIONCOMMUNICATION

INDIVIDUALISM

CLASSROOMTEACHING

CONTROL & AUTONOMY

JOB POLICY

PROFESSIONALISM

ASSESSMENT REGIME

ACCOUNTABILITY

Marginalisation institutional ideological epistemological self- generated

Task-talk as a change tool Task-talk was inclusive, gave everyone a voice,

focused on object, not on each other or on hierarchy Task-talk enabled teachers to discuss own maths

without being too vulnerable Task-talk shifted from what students will do (not do) to

teachers’ practices, expectations and pedagogic habits Task-talk took place post-teaching as well as pre-

teaching Task-talk eventually became talk about how students

learn, given the affordances of task ‘Proficiency’ and ‘deficiency’ views of students were

exposed

Structuring task-talk focus of meetings each other’s knowledge other communication opportunities team planning: parallel groups changed nature of activity of maths

departments TLCs; task-based learning communities –

the tasks of teacher education

Features of the successful departments a team approach to teaching particular topics, discussing what might be done better a stable team learning together take the trouble to be well-informed detailed discussions about learning mathematics research-based ideas to organise, teach and plan teaching parallel groups enables common commitment shared focus use of non-specialist teachers marginalisation

Different lessons Tasks in action in classrooms

ways in which teachers structure work on concepts in lessons

microdifferences in teaching specific topics

Structuring work on concepts: a sequence of microtasks Visualise spatial movement Students create objects with two given features T names the general class T draws objects with imagined features T says what the lesson is about and how this fits with previous

and future lessons T shows multiple objects with same feature Students describe a procedure, in own words T asks for clarification Students think about how a procedure will give them the desired

outcome Students then practise procedures T demonstrates new object with multiple features Students make shapes by varying variables T indicates application to more complex maths which will come

next T shows one object which is nearly finished & students predict

how to complete it by identifying missing features Students deduce further facts.

Another sequence of microtasks T says what this lesson will be about and how it relates to

last lesson Interactive recap of definitions, facts, and other observations. T introduces new aspect & asks what it might mean T offers example, gets them to identify its properties T gives more examples with multiple features; students

identify properties of them Students have to produce examples of objects with several

features Three concurrent tasks for individual and small group work T varies variables deliberately They then do a classification task & identify relationships T circulates asking questions about concepts and properties

Topic-specific contrasts parallel classes team planning shared purpose: to understand and learn how to

construct some loci task A: making loci by following instructions in open

space (e.g. ‘find a place to stand so that you are the same distance from these two points’);

task B: compass and straight-edge constructions teachers chose: order of tasks, language, how links are

made, whether all or some involved in physical task, whether rulers are allowed …

Similarities asking, prompting, telling, showing, giving

reasons referring students to other students’ work explaining choices and actions working out how to do the constructions, variation offered was similar within each locus choice of loci was shared teachers’ stated intentions all teachers praised accuracy written work similar: range of rough sketches

and neat constructions

Florence In the previous lesson they constructed loci with

compasses and straight-edge, i.e. lines. Compasses are ‘an extraordinary tool’ for getting equal lengths.

She says that locus is a set of points obeying a rule. What you get when you ‘model’ with people-points IS a locus in the sense that every point that obeys the rule is on that line and the line joining the points indicates all the points that obey the rule.

Her overall lesson aims had been: reasoning the connections and relationships between people-points and constructed loci.

Alice Alice wrote on the board ‘to be able to visualise

and construct a set of points that satisfy a given set of instructions’. She uses the phrase ‘same distance’ over and over again in the physical activity and the later constructions, so that the aural memory of the lesson is ‘same distance’.

She offers a mixture of physical, visual, aural, verbal experiences. Her view is that they need this physical lesson to give them a vivid experience before understanding what the compasses are really for

Her overall plan had been that they should have multiple memories of how to get ‘same distance’

Differences order of tasks different sub-tasks different things said at different points in activity what was said to whole group or small

group/individuals different order of loci different emphases different tools at different times different conceptualisations afforded