Mathematics
departments making
autonomous change
www.cmtp.co.uk
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:
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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
Object
Subject
Rules
Community
Classroom
Labour
Subject
Labour
Rules
Community
Maths department
“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
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internal and external documents
resource banks
technological resources
communication mechanisms
 Change activity
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formal and informal meetings
grounded PD opportunities
reading
meeting structure (affordances)
each other’s knowledge and experience
Relation between tools
and object
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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
COMMUNICATION
INTERPRETATION
INDIVIDUALISM
OBJECT
SUBJECT
CONTROL &
AUTONOMY
ASSESSMENT
REGIME
CLASSROOM
TEACHING
PROFESSIONALISM
ACCOUNTABILITY
DIVISION
OF
LABOUR
JOB
POLICY
COMMUNITY
RULES
Marginalisation
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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 preteaching
 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
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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
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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
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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 peoplepoints 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