Inquiry in Mathematics Learning and Teaching Barbara Jaworski Loughborough University, UK.

Inquiry in Mathematics Inquiry in Mathematics Learning and TeachingLearning and Teaching

Barbara JaworskiBarbara JaworskiLoughborough University, Loughborough University,

UKUK

Summer School -- Alexandropoulis -- 2009Summer School -- Alexandropoulis -- 2009 22

Better mathematics?Better mathematics?

How can How can students (pupils)students (pupils) learn mathematics learn mathematics better?better?

How can How can teachersteachers provide better opportunities for provide better opportunities for students to learn mathematics?students to learn mathematics?

What kinds of activity in classrooms contribute to What kinds of activity in classrooms contribute to deeper mathematical understandings?deeper mathematical understandings?

How can How can didacticiansdidacticians (mathematics educators) (mathematics educators) contribute to improving mathematics learning and contribute to improving mathematics learning and teaching?teaching?

What roles should/can What roles should/can students, teachers and students, teachers and didacticiansdidacticians play in the developmental process play in the developmental process


This sessionThis session

10 minutes – introduction10 minutes – introduction

20 minutes – working as a group20 minutes – working as a group

20 minutes – feedback from 20 minutes – feedback from groupsgroups

30 minutes – input from BJ30 minutes – input from BJ

10 minutes – questions/discussion 10 minutes – questions/discussion


Group TaskGroup Task

OrangesOranges

The mystery of The mystery of the missing orangethe missing orange

FractionsFractions

22 11

3 3 2 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?


Learning communitiesLearning communitiesWorking

Thinking

Exploring

TOGETHER

Asking questions

Seeking new possibilities

Seeking answers

Tackling problems

In learning mathematics

In teaching mathematicsIn researchingmathematics learning and teaching

Discussing outcomes

Looking critically


InquiryInquiryAsk questionsAsk questions

Seek answersSeek answers Recognise problemsRecognise problems

Seek solutionsSeek solutionsInvent …Invent … Wonder …Wonder …

Imagine …Imagine …Look criticallyLook critically

Inquiry as a tool

Inquiry as a way of being


Inquiry in mathematics Inquiry in mathematics learning and teachinglearning and teaching

Taking a rich mathematical task Taking a rich mathematical task (one in which people with experience know (one in which people with experience know there is rich potential for doing there is rich potential for doing mathematics)mathematics)

Working on the task in inquiry mode with a Working on the task in inquiry mode with a small group and reflecting with others on small group and reflecting with others on the group workthe group work

Relating the task to other areas of Relating the task to other areas of mathematics or mathematical activitymathematics or mathematical activity

Designing further tasks to motivate and Designing further tasks to motivate and challenge learnerschallenge learners


Challenge from a teacherChallenge from a teacher

xx + 4 + 4 = 4 = 4 xx

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

What can we do about it?


xx + 4 + 4 = = xx

xx + 4 + 4 = 4 = 4 xx

WHY?

xx + 4 + 4 = 4 = 4 xx

xx + 4 + 4 = 4 = 4 xx

What does this mean? What does this mean? Is it true? Is it true? For what values of x?For what values of x?

1 + 41 + 4, , 3 + 43 + 4 , , 9 + 4 9 + 4 , … , … 11 3 9 3 9

If x ≠ 0If x ≠ 0

xx + 4 = 4 + 4 = 4xx

4 = 34 = 3xx

4/3 = 4/3 = xx

≠ ≠ 444/3 + 44/3 + 4

4/34/3

= = 16/316/3

4/34/3

= = 16 16 .. 3 3

3 43 4

= = 1616

44

= 4= 4


What can inquiry bring to such a What can inquiry bring to such a situationsituation

1.1. Seeking ways to address a problemSeeking ways to address a problem2.2. Thinking deeply about the problem, Thinking deeply about the problem,

what is involved and what is neededwhat is involved and what is needed3.3. Taking some action to solve the Taking some action to solve the

problemproblem4.4. Looking critically at what we do and Looking critically at what we do and

what it achieveswhat it achieves5.5. Undertaking further systematic Undertaking further systematic

inquiry directed at specific learninginquiry directed at specific learning


Three layers of inquiryThree layers of inquiry Inquiry in learning mathematics:Inquiry in learning mathematics:

• Teachers and didacticians exploring mathematics Teachers and didacticians exploring mathematics together in tasks and problems in workshops;together in tasks and problems in workshops;

• Pupils in schools learning mathematics through Pupils in schools learning mathematics through exploration in tasks and problems in classrooms.exploration in tasks and problems in classrooms.

Inquiry in teaching mathematics:Inquiry in teaching mathematics:• Teachers using inquiry in the design and Teachers using inquiry in the design and

implementation of tasks, problems and mathematical implementation of tasks, problems and mathematical activity in classrooms in association with didacticians.activity in classrooms in association with didacticians.

Inquiry in developing the teaching of Inquiry in developing the teaching of mathematics:mathematics:

• Teachers and didacticians researching the processes Teachers and didacticians researching the processes of using inquiry in mathematics and in the teaching of using inquiry in mathematics and in the teaching and learning of mathematics.and learning of mathematics.


Inquiry transitionInquiry transition

FromFrom

Inquiry as a mediational tool in Inquiry as a mediational tool in practicepractice

ToTo

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


Inquiry as paradigmInquiry as paradigm

The idea of inquiry as ‘a way of The idea of inquiry as ‘a way of being’ can be seen as paradigmatic.being’ can be seen as paradigmatic.

Paradigms (world views)Paradigms (world views) PositivismPositivism InterpretivismInterpretivism Critical TheoryCritical Theory Post modernismPost modernism

Inquiry


PositivismPositivism

Seeking objectivity and truth through Seeking objectivity and truth through defining social situations in scientific terms defining social situations in scientific terms usually involving quantification, measure usually involving quantification, measure and logic: defining measurable variables; and logic: defining measurable variables; designing comparable situations; giving designing comparable situations; giving absolute values; not leaving open to absolute values; not leaving open to interpretation.interpretation.

Justification most often through statistical Justification most often through statistical analysis or study of carefully controlled analysis or study of carefully controlled experimental conditions .experimental conditions .


InterpretivismInterpretivism

Recognising social situations as complex and Recognising social situations as complex and seeking to describe and characterise them seeking to describe and characterise them through interpretation: seeking meaning in through interpretation: seeking meaning in observed actions and interactions; gaining observed actions and interactions; gaining insight to people’s perspectives on who they insight to people’s perspectives on who they are and what they do. are and what they do.

Justification through detailed description and Justification through detailed description and multiple sources of explanation and evidence multiple sources of explanation and evidence to support interpretation and throw light on to support interpretation and throw light on what is studied; being critical about the what is studied; being critical about the perspectives one brings to interpretationperspectives one brings to interpretation


Critical theoryCritical theory Going beyond descriptive interpretation to Going beyond descriptive interpretation to

recognise that social situations embody deeply recognise that social situations embody deeply political human issues and power relationships political human issues and power relationships that research should seek to uncover and address that research should seek to uncover and address such issues: revealing relationships which limit or such issues: revealing relationships which limit or oppress; bringing critical analysis to accepted oppress; bringing critical analysis to accepted traditions to offer opportunities for change.traditions to offer opportunities for change.

Justification through action and interaction that Justification through action and interaction that examine deeply and overtly ways of thinking, examine deeply and overtly ways of thinking, reveal factors and conditions that suppress reveal factors and conditions that suppress individuals or groups and provide individuals or groups and provide emancipatory/empowering opportunity through emancipatory/empowering opportunity through giving voice, enabling and enfranchising.giving voice, enabling and enfranchising.


PostmodernismPostmodernism Going beyond modernism which Going beyond modernism which

rationalises, structures and seeks to rationalises, structures and seeks to explain by categorising and explain by categorising and compartmentalising: bringing and valuing compartmentalising: bringing and valuing multiple perspectives and methods; multiple perspectives and methods; questioning the dominance of any one questioning the dominance of any one view of the world, deconstructing to reveal view of the world, deconstructing to reveal the limiting nature of imposed structures; the limiting nature of imposed structures; revolt against control.revolt against control.

Justification in revellation; coversation and Justification in revellation; coversation and negotiation, opening up; not pretending to negotiation, opening up; not pretending to compartmentalise; revealing complexity compartmentalise; revealing complexity and chaos.and chaos.


Interpretivism

Postmodernism

Critical theory

INQUIRY


tomorrow …tomorrow …

… … inquiry in inquiry in Developmental ResearchDevelopmental Research

Thank YouThank You


Inquiry Inquiry

in Developmental in Developmental Research Research

in Mathematics Educationin Mathematics Education


Better mathematics?Better mathematics? How can How can students (pupils)students (pupils) learn mathematics learn mathematics

better?better? How can How can teachersteachers provide better opportunities provide better opportunities

for students to learn mathematics?for students to learn mathematics? What kinds of activity in classrooms contribute What kinds of activity in classrooms contribute

to deeper mathematical understandings?to deeper mathematical understandings? How can How can didacticiansdidacticians contribute to improving contribute to improving

mathematics learning and teaching?mathematics learning and teaching? What roles should/can What roles should/can students, teachers and students, teachers and

didacticiansdidacticians play in the developmental process play in the developmental process


In a study of disaffection in secondary mathematics In a study of disaffection in secondary mathematics classrooms in the UK, Elena Nardi and Susan Steward found classrooms in the UK, Elena Nardi and Susan Steward found that students on whom the study focused …that students on whom the study focused …

… … apparently engage with mathematical tasks in the apparently engage with mathematical tasks in the classroom mostly out of a sense of professional obligation and classroom mostly out of a sense of professional obligation and under parental pressure. They seem to have a minimal under parental pressure. They seem to have a minimal appreciation and gain little joy out of this engagement. appreciation and gain little joy out of this engagement.

Most students we observed and interviewed view Most students we observed and interviewed view mathematics as a tedious and irrelevant body of isolated, non-mathematics as a tedious and irrelevant body of isolated, non-transferable skills, the learning of which offers little transferable skills, the learning of which offers little opportunity for activity. In addition to this perceived opportunity for activity. In addition to this perceived irrelevance, and in line with previous research that attributes irrelevance, and in line with previous research that attributes student alienation from mathematics to its abstract and student alienation from mathematics to its abstract and symbolic nature, students often found the use of symbolism symbolic nature, students often found the use of symbolism alienating.alienating.

Students resented what they perceived as Students resented what they perceived as rote learningrote learning activity, activity, rule-and-cuerule-and-cue following, and some saw mathematics following, and some saw mathematics as an … as an …

… … elitist subject that exposes the weakness of the intelligence elitist subject that exposes the weakness of the intelligence of any individual who engages with it. of any individual who engages with it. (Nardi & Steward, 2003, p. 361) (Nardi & Steward, 2003, p. 361)


Usable KnowledgeUsable Knowledge

Educational researchers, policymakers, Educational researchers, policymakers, and practitioners agree that educational and practitioners agree that educational research is often divorced from the research is often divorced from the problems and issues of everyday practice problems and issues of everyday practice – a split that creates a need for new – a split that creates a need for new research approaches that speak directly to research approaches that speak directly to the problems of practice…and lead to the problems of practice…and lead to “usable knowledge” (p. 5)“usable knowledge” (p. 5)The Design-Based Research Collective (2003), in the United The Design-Based Research Collective (2003), in the United States: In a special issue of States: In a special issue of Educational Researcher Educational Researcher devoted devoted to papers on design researchto papers on design research


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”?


The revised drawingThe revised drawing


Margaret Brown (1979, p. 362) reports from Margaret Brown (1979, p. 362) reports from research into 11-12 year old children’s research into 11-12 year old children’s solutions to problems involving number solutions to problems involving number operations. A question askedoperations. A question asked

A gardener has 391 daffodils. These are to A gardener has 391 daffodils. These are to be planted in 23 flowerbeds. Each be planted in 23 flowerbeds. Each flowerbed is to have the same number of flowerbed is to have the same number of daffodils. How do you work out how many daffodils. How do you work out how many daffodils will be planted in each flowerbed?daffodils will be planted in each flowerbed?

The following interview took place between a The following interview took place between a student YG and the interviewer MB:student YG and the interviewer MB:

DaffodillsDaffodills


YGYG You er … I know what to do but I can’t say it …You er … I know what to do but I can’t say it …

MBMB Yes, well you do it then. Can you do it?Yes, well you do it then. Can you do it?

YGYG Those are daffodils and these are flowerbeds, Those are daffodils and these are flowerbeds, large you see … Oh! They’re being planted in large you see … Oh! They’re being planted in different flowerbeds, you’d have to put them in different flowerbeds, you’d have to put them in groups …groups …

MBMB Yes, how many would you have in each group? Yes, how many would you have in each group? What would you do with 23 and 391, if you had What would you do with 23 and 391, if you had to find out?to find out?

YGYG See if I had them, I’d count them up … say I had 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 20 of each … I’d put 20 in that one, 20 in that one …one …

MBMB Suppose you had some left over at the end Suppose you had some left over at the end when you’ve got to 23 flowerbeds?when you’ve got to 23 flowerbeds?

YGYG I’d plant them in a pot (!!)I’d plant them in a pot (!!)


New tasks for oldNew tasks for old.. InIn Adapting and Extending Secondary Mathematics Activities: Adapting and Extending Secondary Mathematics Activities:

New tasks for old, New tasks for old, Stephanie Prestage and Pat Perks (2001) Stephanie Prestage and Pat Perks (2001) look at traditional tasks such as one finds in a text booklook at traditional tasks such as one finds in a text book

They suggest an alternative perspective on the task so that it They suggest an alternative perspective on the task so that it offers students something to think about or explore; engaging offers students something to think about or explore; engaging student in mathematical inquiry. An example relating to student in mathematical inquiry. An example relating to Pythagoras Theorem isPythagoras Theorem is

What right angled triangles can you find What right angled triangles can you find with an hypotenuse of 17cm? (Page 25)with an hypotenuse of 17cm? (Page 25)

Such a task is different from traditional exercises which ask Such a task is different from traditional exercises which ask more direct questions with single right or wrong answers. more direct questions with single right or wrong answers.

Solving the problem requires the algorithm to be used many Solving the problem requires the algorithm to be used many times as a pupil makes decisions about the number and types times as a pupil makes decisions about the number and types of solutions. This is better than a worksheet any day, and of solutions. This is better than a worksheet any day, and requires little preparation.requires little preparation. (Prestage and Perks, 2001, p. 25) (Prestage and Perks, 2001, p. 25)


Developmental Research in Developmental Research in Mathematics Education …Mathematics Education …

Research which promotes the Research which promotes the development of mathematics development of mathematics teaching and learning teaching and learning

a)a) while simultaneously studying while simultaneously studying the the practices and processes practices and processes involved; orinvolved; or

b)b) as an integral part of studying as an integral part of studying the the practices and processes involvedpractices and processes involved


ImplicitlyImplicitly

Much research that studies practices Much research that studies practices and processes in mathematics and processes in mathematics learning and/or teaching is learning and/or teaching is implicitlyimplicitly developmental in that it promotes developmental in that it promotes development development withoutwithout this being an this being an intendedintended factor in the research factor in the research design.design.

(Jaworski, 2003)(Jaworski, 2003)


ExplicitlyExplicitly

Research that is Research that is explicitlyexplicitly developmental sets out to promote developmental sets out to promote development as part of the design of development as part of the design of the research.the research.Research and development are often Research and development are often reflexively related to each other, so reflexively related to each other, so that separation of aspects of that separation of aspects of research and development is research and development is difficult. difficult.


Co-learning agreementCo-learning agreement

In a co-learning agreement, researchers In a co-learning agreement, researchers and practitioners are both participants in and practitioners are both participants in processes of education and systems of processes of education and systems of schooling. Both are engaged in action and schooling. Both are engaged in action and reflection. By working together, each might reflection. By working together, each might learn something about the world of the learn something about the world of the other. Of equal importance, however, each other. Of equal importance, however, each may learn something more about his or her may learn something more about his or her own world and its connections to own world and its connections to institutions and schooling (Wagner, 1997, institutions and schooling (Wagner, 1997, p. 16).p. 16).


Examples of Co-Learning InquiryExamples of Co-Learning Inquiry The The Mathematics Teacher Enquiry ProjectMathematics Teacher Enquiry Project – a study of – a study of

teaching development resulting from teachers’ own teaching development resulting from teachers’ own classroom research as insiders Here teachers were classroom research as insiders Here teachers were invited (by outsider researchers) to ask and explore their invited (by outsider researchers) to ask and explore their own questions relating to issues in learning and teaching own questions relating to issues in learning and teaching mathematics. Outsider research showed that teachers’ mathematics. Outsider research showed that teachers’ enquiry, in collaboration with other researchers, led to enquiry, in collaboration with other researchers, led to enhanced thinking and developments in teaching. enhanced thinking and developments in teaching. Outsider researchers themselves learned significantly Outsider researchers themselves learned significantly from their study of teachers’ activity. (Jaworski, 1998). from their study of teachers’ activity. (Jaworski, 1998). See also, See also, Hall, 1997; Edwards, 1998Hall, 1997; Edwards, 1998

Collaboration between teachers and (outsider) Collaboration between teachers and (outsider) researchers to study the use of the researchers to study the use of the teaching triadteaching triad as a as a developmental tool, while using the triad to analyse developmental tool, while using the triad to analyse teaching, led to deeper understandings of the teaching teaching, led to deeper understandings of the teaching triad as a tool for teaching development as well as for triad as a tool for teaching development as well as for analyzing and understanding teaching complexity. (Potari analyzing and understanding teaching complexity. (Potari and Jaworski, 2002; J & P 2009).and Jaworski, 2002; J & P 2009).


Learning Communities in MathematicsLearning Communities in Mathematics

A developmental research project A developmental research project aiming to improve the learning and aiming to improve the learning and teaching of mathematics through a teaching of mathematics through a design involving teachers and design involving teachers and didacticians working together for didacticians working together for mutual learning. mutual learning. (e.g., Jaworski, 2005, 2006, 2008)(e.g., Jaworski, 2005, 2006, 2008)


Co-learning: a learning communityCo-learning: a learning community

Teachers Teachers

Academics/teacher Academics/teacher educators etc.educators etc.

Teacher-Teacher-researchersresearchers

Teacher-educator-Teacher-educator-researchersresearchers

A community of inquiryA community of inquiry

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.


InquiryInquiry I am proposing a I am proposing a

process of critical, process of critical, collaborative co-learning collaborative co-learning - central to this process - central to this process is the theoretical is the theoretical construct of construct of inquiryinquiry..

Inquiry is about asking Inquiry is about asking questions and seeking questions and seeking answers, recognising answers, recognising problems and seeking problems and seeking solutions, exploring and solutions, exploring and investigating to find out investigating to find out more about what we do more about what we do that can help us do it that can help us do it better.better.

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

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


The inquiry cycleThe inquiry cycle

PlanPlan ActAct ObserveObserve ReflectReflect FeedbackFeedback

A basis forA basis forAction researchAction researchDesign researchDesign researchLesson studyLesson studyLearning studyLearning study

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

Developmental ResearchDevelopmental Research


Identity in CommunityIdentity in Community

Wenger (1998) speaks of people Wenger (1998) speaks of people belongingbelonging to a to a community of practice, having community of practice, having identityidentity with regard to with regard to a community of practice, in terms of three a community of practice, in terms of three dimensions: dimensions: engagement, imagination engagement, imagination andand alignment alignment. .

““Identity is a concept that figuratively combines Identity is a concept that figuratively combines the intimate or personal world with the collective the intimate or personal world with the collective space of cultural forms and social relations”. space of cultural forms and social relations”. (Holland, Lachicotte, Skinner and Cain, 1998, p. 5)(Holland, Lachicotte, Skinner and Cain, 1998, p. 5)

Identity refers to Identity refers to ways of beingways of being and we can talk and we can talk about about ways of beingways of being in teaching-learning in teaching-learning situations, which assume situations, which assume alignmentalignment with what is with what is normal and expected in those situations. normal and expected in those situations.

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.

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.


From Alignment to From Alignment to CriticalCritical Alignment Alignment

A community of practice becomes a A community of practice becomes a community of inquiry when participants community of inquiry when participants take on an inquiry identity …take on an inquiry identity …

… … that is, they start overtly to that is, they start overtly to ask questions about their practice, while ask questions about their practice, while still, necessarily, aligning with its norms. still, necessarily, aligning with its norms.

In the beginning, inquiry might be seen as In the beginning, inquiry might be seen as a tool enabling investigation into or a tool enabling investigation into or exploration of aspects of practice – a exploration of aspects of practice – a critical scrutiny of practice. critical scrutiny of practice.


Thus, we see an inquiry identity growing within a Thus, we see an inquiry identity growing within a CoP and the people involved becoming CoP and the people involved becoming inquirersinquirers in their practice; individuals, and the community in their practice; individuals, and the community as a whole, develop as a whole, develop an inquiry way of being in an inquiry way of being in practicepractice,, so that inquiry becomes a norm of so that inquiry becomes a norm of practice with which to align. practice with which to align.

We might see the use of inquiry as a tool to be a We might see the use of inquiry as a tool to be a form of form of critical alignmentcritical alignment; that is engagement in ; that is engagement in and alignment with the practices of the and alignment with the practices of the community, while at the same time asking community, while at the same time asking questions and reflecting critically. questions and reflecting critically.

Critical alignment, through inquiry, is seen to be Critical alignment, through inquiry, is seen to be at the roots of an overt developmental process in at the roots of an overt developmental process in which knowledge grows in practice.which knowledge grows in practice.


Key constructsKey constructs

Co-learning communityCo-learning community Inquiry in theory and in Inquiry in theory and in

practicepractice Community of inquiryCommunity of inquiry Critical alignmentCritical alignment

Developmental researchDevelopmental research Development -- various Development -- various

research projects in the research projects in the literatureliterature

TheoreticalConstructs

DevelopmentalOutcomes

Inquiry in Mathematics Learning and Teaching Barbara Jaworski Loughborough University, UK.

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Transcript of Inquiry in Mathematics Learning and Teaching Barbara Jaworski Loughborough University, UK.