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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

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

Group Task

Explain!!

Oranges

The mystery of the missing orange

Fractions

21

3 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?

Summer School -- Alexandropoulis -- 2009

Learning communities

Working

Asking

questions

Thinking

TOGETHER

Tackling

problems

Seeking answers

Exploring

Seeking new

possibilities

Discussing

outcomes

Looking critically

In learning mathematics

Inteachingmathematics

In researching

mathematics learning and teaching

Summer School -- Alexandropoulis -- 2009

Inquiry

Ask questions

Seek answers

Recognise problems

Seek solutions

Invent …

Wonder …

Imagine …

Look critically

Inquiry

as a way of being

Inquiry

as a tool

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

x + 4 = 4

x

Pupils come to us at upper secondary level making mistakes such as this.

What can wedo about it?

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

Inquiry transition

From

Inquiry as a mediational tool in practice

To

Inquiry as a way of being – one of the norms of practice

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

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 .

Summer School -- Alexandropoulis -- 2009

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 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

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

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

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)

Summer School -- Alexandropoulis -- 2009

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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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

Summer School -- Alexandropoulis -- 2009

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:

Summer School -- Alexandropoulis -- 2009

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 (!!)

Summer School -- Alexandropoulis -- 2009

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)

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

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)

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

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).

Summer School -- Alexandropoulis -- 2009

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).

Summer School -- Alexandropoulis -- 2009

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)

Summer School -- Alexandropoulis -- 2009

Co-learning: a learning community

- Teachers
- Academics/teacher educators etc.

- 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

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

Summer School -- Alexandropoulis -- 2009

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 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

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

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.

Summer School -- Alexandropoulis -- 2009

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

Summer School -- Alexandropoulis -- 2009

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