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