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To
develop activities and a framework for classroom
practice that promotes
student self regulation of cognition during
mathematical problem solving |
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School:
Varndean School, Brighton |
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Fellow: Andrew Blair |
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Email: andrew.blair@varndean.brighton-hove.sch.uk |
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A. Project aims: |
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Aim 1: To develop a conceptual framework
for teachers |
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The aim of the project is to develop a lesson
structure that promotes students’ ability
to regulate their own thinking when solving
mathematical problems. The structure I have
developed is based on the following theories:
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| (1)
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Students internalise self-regulatory
statements, prompts and questions through,
first, regulating others’ behaviour
and receiving regulation from others. |
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Activities aimed at just above students’
level of development encourage problem
solving through peer collaboration. |
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Peer collaboration leads to greater
educational advances (often involving
more abstract mathematical thinking)
than does individualised working. |
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Efficient and effective solving processes
are characterised by higher levels of
metacognitive utterances and students’
critical and supportive scrutiny of
peers’ comments (creating ‘transactive
clusters’ of evaluative discussion).
It is the aim of the teacher to make
examples of metacognition (including
monitoring and regulating) an explicit
part of classroom discourse. |
| (5) |
In a classroom of guided inquiry,
the teacher acts as a participant in
the inquiry process, at the same time
as responding critically to and evaluating
students’ contributions. The intensity
of guidance (or ‘cognitive structuring’)
is contingent on the progress of the
class. The teacher is also the arbiter
between contending forms of ‘other-regulation’
and the ultimate authority over the
class’s activity. If necessary,
the teacher instructs students in scientific
concepts relevant to the inquiry, at
which time they are contextualised and
meaningful. |
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(See Appendix 1 for an explanation of my
own Guided Inquiry model.) |
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Aim 2: To develop prompts that promote
‘other-regulation’ |
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The following prompts for inquiry have been
trialled in lessons. |
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24 x 21
= 42 x 12 |
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Any fraction can be turned into a
decimal and any decimal can be turned
into a fraction. |
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The sum of two fractions equals their
product. |
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x – y = 4 |
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4 x 5321 
52 x 431 |
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If an + b is the nth
term of a linear sequence with odd numbers
only, a is always even and b is always
odd. |
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 a
= kb |
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n gets bigger |
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Aim 3: To encourage staff in the
department, school and partner schools to
use the conceptual framework and prompts |
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See below. |
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2. Progress, Achievements and Future
Plans |
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| Success
Criteria and Dissemination |
Changes
to Success Criteria |
Completed |
Future Plans (date for completion) |
Activities
To produce 10 activities (6 at KS3 and
4 at KS4).
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Prompts
To produce lesson and mathematical notes
for 5 prompts.
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Mathematical
and lesson notes written for:
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24 x 21 = 42 x 12 (piloted with mixed-ability
classes in Years 7, 8, Intermediate
11 and individual high-ability student
in Yr. 8)
• The sum of two
fractions equals their product. (Yrs.
10 and 11, Higher and Intermediate GCSE)
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(Yrs 10 and 11, Higher and
Intermediate GCSE and
mixed-ability Yr. 8.) Trials carried
out on:
• x – y = 4 (Yr. 7 mixed-ability
class)
• Any fraction can be turned
into a decimal and any decimal can
be turned into a fraction. (Yr. 10
Intermediate)
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(1)
Further trials to be carried out in
pairs, small groups and whole classes
on:
• x – y = 4 (Yrs.
7 and 8)
• Any fraction can
be turned into a decimal and any decimal
can be turned into a fraction. (Yr.
8 and 10 Higher)
Jan. –
March 2005 (2) Write up mathematical
and lesson notes for prompts above.
April 2005
(3) Whole class inquiries to use
tablet technology to improve interactivity
of lesson.
Jan. – April 2005
(4) Further exploration of methods
for individual students to convert
social regulation into written form.
Jan. – April 2005
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Activities
To trial and evaluate the activities
with small groups and whole classes.
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Prompts
To trial and evaluate the prompts with
small groups and whole classes.
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Conceptual
framework
To produce a conceptual framework (the
teacher’s role, examples of productive
teacher and student responses to metacognitive
statements, the classroom culture and
the form of peer interaction).
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See lesson
notes and Appendix 1 (Guided Inquiry
Lesson Structure)
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Write
up as an introduction to a booklet on
guided inquiry in mathematics classrooms.
June 2005
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Student
performance
To increase the level of student regulatory
dialogue during peer collaboration through
the use of the activities. (Evidence
from pre- and post-analysis of transcripts
of dialogue between two pairs of students.)
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Student
performance
To increase the level of student regulatory
dialogue during peer collaboration through
the use of the inquiry prompts. (Evidence
from transcripts of classroom dialogue
– comparison with National Numeracy
Strategy lessons.) |
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Comparative analysis of ‘teacher
talk’ (project as part of MA module
ED840 ‘Child Development’,
Open University.) Result from small-scale
research: Higher percentage of sustained
cognitive structuring and open questioning
in inquiry structure; greater proportion
of modelling and closed questions in
the National Numeracy Strategy. |
(1) Comparative analysis
of paired student dialogue (project
as part of MA module E835 ‘Educational
Research in Action’, Open University.
Deadline: August 2005)
March
– July 2005 (2) Continued
tracking of the development of Year
8 students withdrawn from lessons.
Jan. – June 2005
(3) Application to research guided
inquiry lesson structure further as
part of a PhD at Brighton University.
Jan. 2005 |
Department
To rewrite the department’s schemes
of work to include the prompts and encourage
their use by members of the department. |
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Two presentations given at departmental
meetings.
• Collaboration
from other teachers who have tried out
the prompts and developed their own. |
Lessons
to be inserted into the schemes of work.
The author is currently looking at the
possibility of covering the whole ‘content’
of the NNS Year 7 programme by using
inquiry prompts. June 2005 |
Dissemination
within the school
As coordinator of Varndean’s cross-curricula
research group on thinking skills and
as a member of the Technology College
group, to report on the conceptual approach
and prompts and to encourage their take-up
by other departments within the school.
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Edited Research Group’s publication
Teaching Thinking at Varndean.
First volume included a section on Guided
Inquiry. Copy of the booklet given to
all teaching staff (September 2004).
• Inset on ‘metacognition
in the classroom’ provided to
staff in 3 twilight sessions. (Summer
2004)
• SMT observation of
Guided Inquiry lesson: enthusiasm for
wider use. |
1)
2004-05: half-termly Research Group
meetings with updates on work. Jan.
– July 2005
(2) To develop link with interested
science teacher who has already tried
out the prompt ‘All metals come
from the ground’. April –
May 2005 |
Dissemination
within the school
To provide INSET to the whole staff
in two ‘twilight’ sessions
(summer term 2005). |
Dissemination
to other schools
To report to / encourage partner schools
(as a Technology College and ‘Leading
Edge’ school) to use activities
through classroom observations across
Key Stages 1 to 4.
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To work with schools
in KS1 and KS2 as part of Technology
College status and one secondary school
as part of support scheme for Specialist
Schools. Jan. – June 2005 |
Dissemination within
LEA
To post project results on LEA intranet
(all schools) and the KS3 strategy team’s
‘good practice’ website,
and to present findings to local maths
coordinators. |
Dissemination
to include PGCE / GTP students at local
university |
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Workshop on ‘cognitive acceleration
and metacognition in the mathematics
classroom’ (including Guided Inquiry)
to PGCE maths students at the University
of Sussex (Oct. 2004) |
(1) Presentation
to GTPs at the University of Sussex
on ‘thinking skills’. April
2005
(2) Post results of project on KS3 numeracy
website. July 2005
(3) To present results to Brighton and
Hove Maths Coordinators meeting. June
2005 |
Article
To write a further journal article for
Teaching Thinking or similar
publication.
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Article in Mathematics Teaching,
journal of the Association of Teachers
of Mathematics (March 2004).
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Article in Teaching Thinking at
Varndean (September 2004).
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Article
for Teaching Thinking and Creativity.
July 2005 |
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3. Changes to the Project (with
Reasons) |
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Evaluation of Student Progress
After a lack of interest from the University
of Sussex, I abandoned the idea of carrying
out pre- and post-tests under the supervision
of a paid researcher. I am know evaluating
the lesson structure as part of my MA course,
carrying out small-scale comparative research
between the NNS model and guided inquiry.
The focus for the first project was the type
and techniques of teacher mediation in classroom
discourse and, in 2005, the second one will
focus on the quality of student interaction. |
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Change to the number of
prompts supported by lesson and mathematical
notes
That the undertaking to research 10 prompts
in depth was too optimistic became apparent
before the year started. I am now concentrating
on five. |
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4. Personal Reflections |
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Below is an edited version of the last report
I sent to the researchers at Sheffield University: |
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1. What is working well? |
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The conceptual framework for the lesson
structure I am developing is complete with
a fuller definition of the teacher’s
role. In fact, my research has led me to the
ideas of the classroom as a ‘community
of inquiry’ or a ‘community of
discourse’ in different branches of
neo-Vygotskian writings – both of which
have similarities with my own structure. Furthermore,
I have an ever-growing list of prompts. The
best ones provide students with the opportunity
to incorporate different areas of maths in
better understanding the prompts. I was confident
of the five I would concentrate on at the
beginning of the year, but it is heartening
to be able to develop others using the pedagogical
approach. |
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2. What are your major frustrations? |
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(1) As Head of Department (in
a department that was ‘underperforming’
when I took up post), there is great pressure
to improve KS3 and GCSE results. Even though
results have improved markedly over the last
three years, I still feel my main focus has
to be internal. Tension has thus
arisen with one of the project’s aims
of disseminating my practice externally
to teachers in other schools. Hopefully this
can be resolved in the summer term when public
exams are over.
(2) Paradoxically, at the same time as demanding
improved results, the Headteacher has cut
the curriculum time given over to maths to
a level below the national average. In such
circumstances (with the content-laden KS3
Framework and GCSE specification), it seems
difficult to fit in my own experimental work.
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3. What is giving you most satisfaction
and pleasure? |
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The lessons that I am able to devote to
creating a ‘community of inquiry’
in the classroom by expecting students to
regulate their own activity gives me most
pleasure. The reaction of students can be
one of bemusement when they are not given
content-based learning objectives, but they
have embraced the class’s joint endeavour
and enjoyed the challenge of setting their
own goals. Additionally, testing and adapting
theory in practice and exploring the mathematical
potential of the prompts I have designed are
other rewarding aspects of the project. |
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Appendix 1: Guided
Inquiry Lesson Structure |
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MATHEMATICS CLASSROOMS OF GUIDED INQUIRY |
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Guided Inquiry is a whole-class
pedagogical model that promotes students’
ability to regulate their thinking when solving
mathematics problems. Inquiry is initiated
with a teacher’s prompt and
generated from students’ queries and
conjectures. |
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Vygotsky’s theory that students learn
to regulate their own thinking during social
interaction underpins the model. Students
develop metacognitive strategies when their
behaviour is regulated by collaborators in
social activity and when they regulate the
thinking of others. The important characteristic
of the Guided Inquiry classroom,
insofar as it is possible, lies in the students’
regulation of their own activity with
the teacher mediating between contending ‘other-regulation’.
A hierarchy of other-regulation
sees each student operating in his or her
individual zone of proximal development with
a more expert peer or the teacher negotiating
a cognitive structure, with the nature of
intervention contingent on a student’s
progress. |
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The lesson follows the four phases of the
problem-solving cycle: orientation, exploration
and planning, solving, and reflecting and
evaluating (see diagram below). It starts
with a prompt (a written, numerical
or algebraic mathematical statement), such
as 24 x 21 = 42 x 12, that suggests different
lines of inquiry at concrete and abstract
mathematical levels. In the sense that the
prompt is open to different levels of inquiry,
it can be revisited in successive years to
develop it further. |
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Pairs of students are expected to make an
observation or ask a question about the prompt,
perhaps with the teacher modelling the self-question:
‘What could I ask myself about this
statement?’ In the example above, questions
and observations that have arisen during lessons
include: |
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Is the
sum correct? |
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An approximation shows the sum is
right because 20 x 20 = 40 x 10 |
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How do you multiply two 2-digit numbers? |
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One number on the right is half one
on the left and the other is double. |
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Does doubling and halving always give
equal amounts in multiplication sums? |
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The digits on the left have been reversed
to make the numbers on the right. |
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Does reversing digits always work
to make an ‘equals sum’? |
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Are there any other sums like this
one? |
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The class as a whole would be invited to
order the questions (and, importantly for
metacognitive development, justify their order)
and then decide how they would analyse the
comments and answer the questions they themselves
have generated. Students are then expected,
by collaborating in small groups, to respond
to the issues raised in the class discussion.
Ground rules for interaction, which emphasise
explaining reasoning, questioning that reasoning,
justifying methods and decisions, and reacting
critically to peers’ comments, aim to
create transactive clusters. |
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Even though there are no conventional ‘content’
learning intentions at the start of the lesson
(as in the National Numeracy Strategy), the
teacher might expect students to set their
own cognitive goals. In the example we are
following, some students might decide to remind
themselves about multiplying two 2-digit numbers
or approximating, although the teacher might
call on students to model methods to the class.
Other students could look for similar examples
that fulfil the conditions identified, or
they could test a general statement or find
a counter-example. More able students might
move on to making algebraic generalisations
about the relationship between the digits.
The final phase sees students decide whether
the approach they took as a class was efficient
and effective. Further questions that have
emerged at this stage involve using 3- and
4-digit numbers, and four 2-digit numbers
(arranged in 2 pairs). |
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The teacher aims to make students’
monitoring and regulation explicit during
classroom discourse. By becoming aware of
peers’ higher-order thinking processes
and others’ regulation of his or her
activity, an individual student internalises
(or ‘appropriates’) the forms
of regulation. The student then adapts them
in ways dependent on their previous ontogenesis
for use in regulating his or her own thinking
during problem solving. Additionally, the
teacher would introduce scientific (conceptual)
knowledge necessary for progress in the inquiry.
When concepts are used as cognitive tools
during activity, they become contextualised
and meaningful. |
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Other advantages of Guided Inquiry
lessons: |
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Improves
motivation because students are answering
their own questions or responding to
peers’ comments. |
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Promotes creative thinking, speculation
and discussion because there is no clear
path at the start of the lesson and
no ‘correct’ procedure to
solve a problem. |
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Promotes divergent thinking to suite
those students whose minds do not instinctively
converge on pre-determined learning
intentions. |
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A Lesson Structure to Promote Self-Regulation
in Classrooms of Guided Inquiry |
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