Mathematics departments making autonomous change Anne Watson University of Oxford Warwick 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 Rules Object Community Subject Community Labour Subject Maths department Classroom

“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 OBJECTSUBJECT DIVISION OF LABOUR RULES COMMUNITY INTERPRETATION COMMUNICATION INDIVIDUALISM CLASSROOM TEACHING CONTROL & AUTONOMY JOBPOLICY 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