Transcript LCM/ICTML Conference

Inquiry in Mathematics
Learning and Teaching
Barbara Jaworski
Loughborough University, UK
Better mathematics?

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How can students (pupils) learn mathematics better?
How can teachers provide better opportunities for
students to learn mathematics?
What kinds of activity in classrooms contribute to
deeper mathematical understandings?
How can didacticians (mathematics educators)
contribute to improving mathematics learning and
teaching?
What roles should/can students, teachers and
didacticians play in the developmental process
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This session
10 minutes – introduction
20 minutes – working as a group
20 minutes – feedback from groups
30 minutes – input from BJ
10 minutes – questions/discussion
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Group Task
Oranges
Explain!!
Fractions
The mystery of
the missing orange
2
3
÷
1
2
Work on the task yourself.
What did you do? achieve? learn?
Imagine offering the task to pupils.
(How would you offer it?)
What might you expect your pupils to do?
achieve? learn?
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Learning communities
Thinking
TOGETHER
Tackling
problems
Asking
questions
Seeking answers
Seeking new
possibilities
Discussing
outcomes
In learning mathematics
Looking critically
In teaching mathematics
In researching
mathematics learning and teaching
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Inquiry
Ask questions
Seek answers
Recognise problems
Seek solutions
Inquiry
as a way of being
Invent …
Inquiry
as a tool
Wonder …
Imagine …
Look critically
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Inquiry in mathematics
learning and teaching
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Taking a rich mathematical task
(one in which people with experience know
there is rich potential for doing mathematics)
Working on the task in inquiry mode with a
small group and reflecting with others on the
group work
Relating the task to other areas of
mathematics or mathematical activity
Designing further tasks to motivate and
challenge learners
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Challenge from a teacher
x+4 = 4
x
Pupils come to us at upper
secondary level making
mistakes such as this.
What can we
do about it?
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x+4 = 4
x
x+4 =
x
x+4 = 4
x
WHY?
x+4 = 4
x
What does this mean?
Is it true?
For what values of x?
1 + 4,
1
3+4, 9+4 ,…
4/3 + 4 ≠ 4
3
9
4/3
If x ≠ 0
x + 4 = 4x
4 = 3x
4/3 = x
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= 16/3
4/3
= 16 . 3
3 4
= 16
4
= 4
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What can inquiry bring to such a
situation
1.
2.
3.
4.
5.
Seeking ways to address a problem
Thinking deeply about the problem,
what is involved and what is needed
Taking some action to solve the
problem
Looking critically at what we do and
what it achieves
Undertaking further systematic
inquiry directed at specific learning
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Three layers of inquiry
Inquiry in learning mathematics:

•
Teachers and didacticians exploring mathematics together
in tasks and problems in workshops;
•
Pupils in schools learning mathematics through exploration
in tasks and problems in classrooms.
Inquiry in teaching mathematics:
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•
Teachers using inquiry in the design and implementation of
tasks, problems and mathematical activity in classrooms in
association with didacticians.
Inquiry in developing the teaching of mathematics:

•
Teachers and didacticians researching the processes of
using inquiry in mathematics and in the teaching and
learning of mathematics.
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Inquiry transition
From
Inquiry as a mediational tool in practice
To
Inquiry as a way of being – one of the
norms of practice
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Inquiry as paradigm
The idea of inquiry as ‘a way of being’
can be seen as paradigmatic.
Paradigms (world views)

Positivism
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Interpretivism
Inquiry
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Critical Theory
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Post modernism
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Positivism
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Seeking objectivity and truth through
defining social situations in scientific terms
usually involving quantification, measure and
logic: defining measurable variables;
designing comparable situations; giving
absolute values; not leaving open to
interpretation.
Justification most often through statistical
analysis or study of carefully controlled
experimental conditions .
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Interpretivism
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Recognising social situations as complex and
seeking to describe and characterise them
through interpretation: seeking meaning in
observed actions and interactions; gaining
insight to people’s perspectives on who they
are and what they do.
Justification through detailed description and
multiple sources of explanation and evidence
to support interpretation and throw light on
what is studied; being critical about the
perspectives one brings to interpretation
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Critical theory

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Going beyond descriptive interpretation to recognise
that social situations embody deeply political human
issues and power relationships that research should
seek to uncover and address such issues: revealing
relationships which limit or oppress; bringing critical
analysis to accepted traditions to offer opportunities
for change.
Justification through action and interaction that
examine deeply and overtly ways of thinking, reveal
factors and conditions that suppress individuals or
groups and provide emancipatory/empowering
opportunity through giving voice, enabling and
enfranchising.
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Postmodernism
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Going beyond modernism which rationalises,
structures and seeks to explain by
categorising and compartmentalising: bringing
and valuing multiple perspectives and
methods; questioning the dominance of any
one view of the world, deconstructing to
reveal the limiting nature of imposed
structures; revolt against control.
Justification in revellation; coversation and
negotiation, opening up; not pretending to
compartmentalise; revealing complexity and
chaos.
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Critical theory
Interpretivism
INQUIRY
Postmodernism
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tomorrow …
… inquiry in
Developmental Research
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Thank You
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Inquiry
in Developmental Research
in Mathematics Education
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Better mathematics?





How can students (pupils) learn mathematics
better?
How can teachers provide better opportunities
for students to learn mathematics?
What kinds of activity in classrooms contribute
to deeper mathematical understandings?
How can didacticians contribute to improving
mathematics learning and teaching?
What roles should/can students, teachers and
didacticians play in the developmental process
Summer School -- Alexandropoulis -- 2009
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
In a study of disaffection in secondary mathematics classrooms
in the UK, Elena Nardi and Susan Steward found that students on
whom the study focused …
… apparently engage with mathematical tasks in the classroom
mostly out of a sense of professional obligation and under
parental pressure. They seem to have a minimal appreciation and
gain little joy out of this engagement.
Most students we observed and interviewed view mathematics as
a tedious and irrelevant body of isolated, non-transferable skills,
the learning of which offers little opportunity for activity. In
addition to this perceived irrelevance, and in line with previous
research that attributes student alienation from mathematics to
its abstract and symbolic nature, students often found the use of
symbolism alienating.

Students resented what they perceived as rote learning activity,
rule-and-cue following, and some saw mathematics as an …
… elitist subject that exposes the weakness of the intelligence of
any individual who engages with it.
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(Nardi & Steward, 2003,
p. 361)
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Usable Knowledge
Educational researchers, policymakers, and
practitioners agree that educational research
is often divorced from the problems and
issues of everyday practice – a split that
creates a need for new research approaches
that speak directly to the problems of
practice…and lead to “usable knowledge” (p. 5)
The Design-Based Research Collective (2003), in the United
States: In a special issue of Educational Researcher devoted to
papers on design research
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Look at Figure 1 here.
What is it?
What shape is it?
Figure 1: The teacher’s drawing
What would be your reaction to someone
who said “it is a square”?
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The revised drawing
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Daffodills
Margaret Brown (1979, p. 362) reports from
research into 11-12 year old children’s
solutions to problems involving number
operations. A question asked
A gardener has 391 daffodils. These are to
be planted in 23 flowerbeds. Each flowerbed
is to have the same number of daffodils. How
do you work out how many daffodils will be
planted in each flowerbed?
The following interview took place between a
student YG and the interviewer MB:
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YG You er … I know what to do but I can’t say it …
MB Yes, well you do it then. Can you do it?
YG Those are daffodils and these are flowerbeds, large
you see … Oh! They’re being planted in different
flowerbeds, you’d have to put them in groups …
MB Yes, how many would you have in each group? What
would you do with 23 and 391, if you had to find out?
YG See if I had them, I’d count them up … say I had 20
of each … I’d put 20 in that one, 20 in that one …
MB Suppose you had some left over at the end when
you’ve got to 23 flowerbeds?
YG I’d plant them in a pot (!!)
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New tasks for old.
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In Adapting and Extending Secondary Mathematics Activities:
New tasks for old, Stephanie Prestage and Pat Perks (2001) look
at traditional tasks such as one finds in a text book
They suggest an alternative perspective on the task so that it
offers students something to think about or explore; engaging
student in mathematical inquiry. An example relating to
Pythagoras Theorem is
What right angled triangles can you find
with an hypotenuse of 17cm? (Page 25)
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Such a task is different from traditional exercises which ask
more direct questions with single right or wrong answers.
Solving the problem requires the algorithm to be used many
times as a pupil makes decisions about the number and types of
solutions. This is better than a worksheet any day, and requires
little preparation. (Prestage
and Perks, 2001, p. 25)
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Developmental Research in
Mathematics Education …
a)
b)
Research which promotes the
development of mathematics teaching
and learning
while simultaneously studying the
practices and processes involved; or
as an integral part of studying the
practices and processes involved
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Implicitly
Much research that studies practices
and processes in mathematics learning
and/or teaching is implicitly
developmental in that it promotes
development without this being an
intended factor in the research design.
(Jaworski, 2003)
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Explicitly
Research that is explicitly
developmental sets out to promote
development as part of the design of
the research.
Research and development are often
reflexively related to each other, so
that separation of aspects of research
and development is difficult.
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Co-learning agreement
In a co-learning agreement, researchers and
practitioners are both participants in
processes of education and systems of
schooling. Both are engaged in action and
reflection. By working together, each might
learn something about the world of the other.
Of equal importance, however, each may learn
something more about his or her own world
and its connections to institutions and
schooling (Wagner, 1997, p. 16).
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Examples of Co-Learning Inquiry
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The Mathematics Teacher Enquiry Project – a study of
teaching development resulting from teachers’ own classroom
research as insiders Here teachers were invited (by outsider
researchers) to ask and explore their own questions relating
to issues in learning and teaching mathematics. Outsider
research showed that teachers’ enquiry, in collaboration with
other researchers, led to enhanced thinking and developments
in teaching. Outsider researchers themselves learned
significantly from their study of teachers’ activity. (Jaworski,
1998). See also, Hall, 1997; Edwards, 1998
Collaboration between teachers and (outsider) researchers to
study the use of the teaching triad as a developmental tool,
while using the triad to analyse teaching, led to deeper
understandings of the teaching triad as a tool for teaching
development as well as for analyzing and understanding
teaching complexity. (Potari and Jaworski, 2002; J & P 2009).
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Learning Communities in Mathematics
A developmental research project
aiming to improve the learning and
teaching of mathematics through a
design involving teachers and
didacticians working together for
mutual learning.
(e.g., Jaworski, 2005, 2006, 2008)
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Co-learning: a learning community
Teacher-
common goal – to improve
 Teachers
researchers
opportunity
for students to
engage with mathematics in the
Because I talk here about
best possible ways to support and
complex practices, it seems
build their mathematical

Academics/teacher
clear to me that the best
 Teacher-educatorconcepts and fluency
educators
etc.
possible ways are what we
researchers
are all striving to know.

A community of inquiry
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Inquiry


I am proposing a
process of critical,
collaborative co-learning
- central to this process
is the theoretical
construct of inquiry.

Inquiry is about asking
questions and seeking
answers, recognising
problems and seeking
solutions, exploring and
investigating to find out
more about what we do
that can help us do it
better.
The overt use of inquiry
 Such use of inquiry
in practice has the aim
starts off as a mediating
- of disturbing practice
tool in the practice, and
on the inside,
shifts over time to
- of challenging the
become an inquiry stance
status quo,
or an inquiry way of
- of questioning
being
in practice
accepted ways of Summer School -- Alexandropoulis
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-- 2009
being and doing.
The inquiry cycle
We implement a cycle of

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Plan
Act
Observe
Reflect
Feedback
planning, action, observation,
reflection, feedback.
A basis for
Action research
Design research
Lesson study
Learning study
Developmental Research
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Identity in Community



Wenger (1998)
ofthe
people
belonging teachers
to a
For speaks
example,
mathematics
community within
of practice,
havingschool
identity
with
regard to
a particular
have
identity
a community
ofalignment
practice, related
in termstooftheir
three
and
school as a
dimensions:social
engagement,
imagination
alignment.
system and
group ofand
people.
individual
teacher
or teacher
“Identity isAny
aFor
concept
that
figuratively
combines the
example, in practices of
has
identity
related
to their
intimate oreducator
personal
world
with
the
collective
space
mathematics learning and teaching,
involvement
in day to day practice,
of cultural direct
forms
and social
relations”.
participants
engage
in their practice
but constituted
through
the
many other
(Holland, Lachicotte,
Skinner and
Cain,
1998, p.use
5)
alongside
their
peers,
communities with which the individual
imagination
in
interpreting
their
own
Identity refers
to
ways
of
being
and
we
can
talk
aligns to some degree.
roles
in in
the
practice and align
about ways of
being
teaching-learning
situations,
with what
established
norms
which assumethemselves
alignment with
is normal
and
and values
of teaching within school
expected in those
situations.
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and educational system.
From Alignment to Critical Alignment

A community of practice becomes a
community of inquiry when participants take
on an inquiry identity …
… that is, they start overtly to ask
questions about their practice, while still,
necessarily, aligning with its norms.

In the beginning, inquiry might be seen as a
tool enabling investigation into or exploration
of aspects of practice – a critical scrutiny of
practice.
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


Thus, we see an inquiry identity growing within a CoP
and the people involved becoming inquirers in their
practice; individuals, and the community as a whole,
develop an inquiry way of being in practice, so that
inquiry becomes a norm of practice with which to
align.
We might see the use of inquiry as a tool to be a form
of critical alignment; that is engagement in and
alignment with the practices of the community, while
at the same time asking questions and reflecting
critically.
Critical alignment, through inquiry, is seen to be at
the roots of an overt developmental process in which
knowledge grows in practice.
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Key constructs

Theoretical
Constructs

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Developmental
Outcomes
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Co-learning community
Inquiry in theory and in practice
Community of inquiry
Critical alignment
Developmental research
Development -- various research
projects in the literature
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