Inquiry in Mathematics Learning and Teaching

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Inquiry in Mathematics Learning and Teaching

• Barbara Jaworski
• Loughborough University, UK

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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 (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
• The mystery of the missing orange

• Fractions
• 2 1
• 3 2

Explain!!

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
Working
Asking questions
Thinking
TOGETHER
Tackling problems
Seeking answers
Exploring
Seeking new possibilities
Discussing outcomes
Looking critically
In learning mathematics
In teaching mathematics
In researching mathematics learning and teaching
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Inquiry

• Ask questions
• Seek answers
• Recognise problems
• Seek solutions
• Invent
• Wonder
• Imagine
• Look critically

Inquiry as a way of being
Inquiry as a tool
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Inquiry in mathematics learning and teaching

• 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 What does this mean? Is it
true? For what values of x?
x 4 x
x 4 4 x
1 4, 3 4 , 9 4 , 1 3 9
? 4
4/3 4 4/3 16/3 4/3 16 . 3 3
4 16 4 4
WHY?
If x ? 0 x 4 4x 4 3x 4/3 x
x 4 4 x
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What can inquiry bring to such a situation

• 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
• 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
• Interpretivism
• Critical Theory
• Post modernism

Inquiry
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Positivism

• 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

• Recognising social situations as complex and
  seeking to describe and characterise them through
  interpretation seeking meaning in observed
  actions and interactions gaining insight to
  peoples 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

• 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

• 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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(No Transcript)
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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

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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. (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 practiceand 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 teachers 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 childrens 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 cant 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! Theyre being planted in
  different flowerbeds, youd 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, Id count them up say I
  had 20 of each Id put 20 in that one, 20 in
  that one
• MB Suppose you had some left over at the end when
  youve got to 23 flowerbeds?
• YG Id plant them in a pot (!!)

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New tasks for old.

• 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)
• 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

• 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

• 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-researchers
• Teacher-educator-researchers

common goal to improve opportunity for students
to engage with mathematics in the best possible
ways to support and build their mathematical
concepts and fluency

• Teachers
• Academics/teacher educators etc.

Because I talk here about complex practices, it
seems clear to me that the best possible ways are
what we are all striving to know.
A community of inquiry
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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.

• I am proposing a process of critical,
  collaborative co-learning - central to this
  process is the theoretical construct of inquiry.

• The overt use of inquiry in practice has the aim
  - of disturbing practice on the inside, - of
  challenging the status quo, - of questioning
  accepted ways of being and doing.

• Such use of inquiry starts off as a mediating
  tool in the practice, and shifts over time to
  become an inquiry stance or an inquiry way of
  being in practice

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The inquiry cycle

• We implement a cycle of planning, action,
  observation, reflection, feedback.

• Plan
• Act
• Observe
• Reflect
• Feedback

A basis for Action research Design
research Lesson study Learning study
Developmental Research
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Identity in Community

• For example, the mathematics teachers within a
  particular school have identity and alignment
  related to their school as a social system and
  group of people.
• Any individual teacher or teacher educator has
  identity related to their direct involvement in
  day to day practice, but constituted through the
  many other communities with which the individual
  aligns to some degree.

• Wenger (1998) speaks of people belonging to a
  community of practice, having identity with
  regard to a community of practice, in terms of
  three dimensions engagement, imagination and
  alignment.

• For example, in practices of mathematics learning
  and teaching, participants engage in their
  practice alongside their peers, use imagination
  in interpreting their own roles in the practice
  and align themselves with established norms and
  values of teaching within school and educational
  system.

• Identity is a concept that figuratively combines
  the intimate or personal world with the
  collective space of cultural forms and social
  relations. (Holland, Lachicotte, Skinner and
  Cain, 1998, p. 5)
• Identity refers to ways of being and we can talk
  about ways of being in teaching-learning
  situations, which assume alignment with what is
  normal and expected in those situations.

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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

• Co-learning community
• Inquiry in theory and in practice
• Community of inquiry
• Critical alignment
• Developmental research
• Development -- various research projects in the
  literature

Theoretical Constructs
Developmental Outcomes