New Directions for Situated Cognition in Mathematics Education

Contents

Contributing Authors

Chapter 1 Introduction

1. INTRODUCTION

2. MATHEMATICS

3. KNOWLEDGE

4. SITUATED PERSPECTIVES: POWER AND LIMITATION

5. WHY SITUATED PERSPECTIVES?

6. THIS BOOK

REFERENCES

Chapter 2 School Mathematics As A Developmental Activity1

1. INTRODUCTION

2. IMPLICIT EPISTEMOLOGY: WHAT DOES IT MEAN ‘ TO BE DOING MATHEMATICS’?

3. LEARNING CLOSE TO PRACTICAL CONTEXTS AND SITUATIONS

4. LEARNING IN THE SCHOOL CONTEXT

4.1 Utilitarian vs. epistemic attitude to the world and to language

4.2 Unreflected, or not consciously developed vs. planned and conscious procedure

4.3 Learning as a relation of operations, tasks and the object of cognitive activity

5. PERFORMANCE IN THE SITUATION VS. DEVELOPMENT

REFERENCES2

Chapter 3 Participating In What? Using Situated Cognition Theory To Illuminate Differences In Classroom Practices

1. INTRODUCTION

2. OUR POSITION ON SITUATED COGNITION

3. CASE STUDIES

3.1 Case 1: Norma

3.2 Case 2: Roisin

3.3 Case 3: Susan

4. ANALYSIS OF CLASSROOM SEQUENCES AS LOCAL COMMUNITIES OF PRACTICE

4.1 How do students seem to be acting in relation to mathematics? What kind of participants do they seem to be within the lesson?

4.2 What developing mathematical competence is publicly recognised, and how?

4.3 Do learners appear to be working purposefully together on mathematics? With what purpose?

4.4 What are the shared values and ways of behaving in relation to mathematics: language, habits, tool- use?

4.5 Does active participation of students and teacher in mathematics constitute the lesson?

4.6 Do students and teacher appear to be engaged in the same mathematical activity? What is the activity?

5. ANALYSIS OF THE AFFORDANCES, CONSTRAINTS AND ATTUNEMENTS OF MATHEMATICAL ACTIVITY IN EACH SITUATION

6. INFLUENCE OF STUDENTS’ CONTRIBUTIONS

6.1 Susan’s class on ‘Equations’

7. CONCLUSION

ACKNOWLEDGEMENTS

REFERENCES

Chapter 4 Social Identities As Learners And Teachers Of Mathematics

The situated nature of roles and relations in mathematics classrooms.

1. INTRODUCTION

2. PUPILS’ SOCIAL IDENTITIES AS LEARNERS OF MATHEMATICS

3. LEARNERS’ SOCIAL IDENTITIES

4. FROM IDENTITIES TO RELATIONS

5. CHANGING CLASSROOM RELATIONS

6. CLASSROOM RELATIONS

6.1 Parallel calculation chains

6.2 Solver and recorder

6.3 Clue problems

7. DISCUSSION

8. CONCLUSION

REFERENCES

Chapter 5 Looking For Learning In Practice: How Can This Inform Teaching

1. INTRODUCTION

2. IDENTITY-CHANGING COMMUNITIES OF PRACTICE

3. A TEACHING MOMENT

3.1 Methodology: the teaching moment

3.2 Three stories including the teaching moment

4. A LEARNING EXPERIENCE

4.1 Methodology - the biology story

5. DISCUSSION: WHAT DOES THIS HAVE TO SAY ABOUT TEACHING AND LEARNING?

5.1 The learning of school students

5.2 The conceptualisation of teaching

ACKNOWLEDGEMENTS

REFERENCES

Chapter 6 Are Mathematical Abstractions Situated?

1. INTRODUCTION

1.1 Empiricist views on abstraction

1.2 Situation and context

1.3 Contextual views of abstraction

2. THE STUDY, THE TASKS AND PROTOCOL DATA

2.1 Protocol data

3. ABSTRACTION: MEDIATION, PEOPLE AND TASKS

3.1 Mediation

3.2 People

3.3 Tasks

4. CONCLUSIONS

REFERENCES

Chapter 7 ‘ We Do It A Different Way At My School’

Mathematics homework as a site for tension and conflict

1. INTRODUCTION

2. RYAN, HIS MOTHER AND HIS HOMEWORK

2.1 The practice of homework

2.2 Ryan’s school and home identities

2.3 Ryan’s mother and mathematics

2.4 Tensions and conflicts during the homework event

3. DISCUSSION

ACKNOWLEDGEMENTS

REFERENCES

Chapter 8 Situated Intuition And Activity Theory Fill The Gap

The cases of integers and two-digit subtraction algorithms

1. INTRODUCTION

2. IMPLEMENTING THE INSTRUCTIONAL METHOD

2.1 The teaching of integers

2.2 The teaching of subtraction of two-digit numbers

2.3 Comparison of the 2-digit subtraction and the integer- operations cases

3. SITUATED INTUITION AND AUTHENTIC CLASSROOM LEARNING

3.1 Finally, what can we say about situated cognition and activity theory

REFERENCES

Chapter 9 The Role Of Artefacts In Mathematical Thinking: A Situated Learning Perspective

1. INTRODUCTION

2. RATIONALE FOR THE THEORETICAL OPTIONS AND THE FOCUS OF ANALYSIS

2.1 Why bring activity theory into a situated perspective?

2.2 Why look deeper into the situated role of artefacts?

3. CONCEPTS IN ACTION IN THE ANALYSIS OF THE STUDY

3.1 Activity in activity theory

3.2 Artefacts in activity theory

3.3 Activity from a situated perspective

3.4 Structuring resources and artefacts

4. THE STUDY OF THE ARDINAS’ PRACTICE

5. WHAT EMERGED FROM THE DATA ANALYSIS OF THE ARDINAS’ PRACTICE

5.1 First encounter: what is in a table and what does it tell us?

5.2 Second encounter: the calculator as artefact; what does it tell us?

6. ARTEFACTS AND RESOURCES: HOW THE TECHNOLOGY OF THE PRACTICE PRODUCES A SHARED REPERTOIRE

REFERENCES

Chapter 10 Exploring Connections Between Tacit Knowing And Situated Learning Perspectives In The Context Of Mathematics Education21

1. INTRODUCTION

2. EXPLORING CONNECTIONS BETWEEN SCHOOL MATHEMATICAL PRACTICES AND OTHER SOCIO- CULTURAL ‘ MATHEMATICAL’ PRACTICES

3. TACIT KNOWLEDGE (OR KNOWING) AND VARIATIONS

4. CONNECTING TACIT MATHEMATICAL KNOWING, SITUATED PERSPECTIVES AND SCHOOL MATHEMATICS PRACTICES

4.1 What might a tacit-explicit dimension of school mathematics practice consist of ?

4.2 How does tacit knowledge (or knowing) manifest ‘ psychologically’ in mathematical activities?

4.3 Characterising the site of learning for the jangadeiros

4.4 What pedagogical implications result from connecting tacit mathematical knowing, situated perspectives and school mathematics practices?

5. CONCLUSION

REFERENCES

Chapter 11 Cognition And Institutional Setting

Undergraduates’ understandings of the derivative

1. INTRODUCTION

2. THEORETICAL FRAMEWORK OF THE STUDY

3. THE RESEARCH

4. INSTITUTIONAL CONTEXTS OF THE DEPARTMENTS

4.1 The Mechanical Engineering department

4.2 The Mathematics department

5. RESULTS

5.1 Student tests

5.2 The calculus modules

5.3 Lecturers’ views regarding their teaching practices

6. INSTITUTIONAL SETTINGS: STUDENTS AND LECTURERS

6.1 Calculus modules and students’ developing conceptions

6.2 But why did ‘what you teach’ differ?

6.3 Students’ situated developing conceptions

6.4 How do institutional settings influence lecturers and students?

REFERENCES

Chapter 12 School Practices With The Mathematical Notion Of Tangent Line

1. INTRODUCTION

2. RESEARCH FRAMEWORK

3. MATHEMATICS, SCHOOL MATHEMATICS AND LOCAL COMMUNITIES OF PRACTICE

4. SCHOOL PRACTICES AND SCHOOL MATHEMATICS

4.1 The technical design classroom

4.2 The highway system project classroom

5. THE MATHEMATICAL EXPERIENCES SHARED IN THE CLASSROOMS OBSERVED

5.1 The shared ways of behaving, language, habits, values and tool- use.

5.2 The developing mathematical competences, as recognized within the lessons observed

5.3 The common direction of learning: functioning mathematically, across school classrooms

6. SOME IMPLICATIONS FOR TEACHING

ACKNOWLEDGEMENTS

REFERENCES

Chapter 13 Learning Mathematically As Social Practice In A Workplace Setting

1. INTRODUCTION

2. THEORETICAL FRAMEWORK

3. RESEARCH METHODOLOGY

4. FACTORY PROCESSES

5. STUDENT ACTIVITIES

6. THE MATHEMATICAL DIMENSIONS OF THE WORK PRACTICES ON THE FACTORY FLOOR

7. DATA ANALYSIS

7.1 Adult mentors’ retellings of (mathematical) performance

7.2 Focus group interview with students

8. DISCUSSION

9. CONCLUSIONS

REFERENCES

Chapter 14 Analysing Concepts Of Community Of Practice

1. INTRODUCTION

2. DISTINGUISHING AMONG THE CONCEPTS OF COMMUNITIES OF PRACTICE

2.1 A characterisation of CPT1

2.2 A characterisation of CPT2

2.3 Examples in mathematics education

3. COMPARING AND CONTRASTING CTP1 AND CPT2

3.1 Community of practice: concepts of learning, identity and boundary

3.2 Problems arising from these comparisons

4. DEVELOPING THE CONCEPTS OF COMMUNITY OF PRACTICE

4.1 Sociology of structures

4.2 Discursive practice

4.3 Final remarks

ACKNOWLEDGEMENTS

REFERENCES

Chapter 15 ‘ No Way Is Can’t’: A Situated Account Of One Woman’s Uses And Experiences Of Mathematics

1. INTRODUCTION 1.1 One by One

1.2 Placing

1.3 Measurement

2. WRITING THIS CHAPTER

3. STARTING FROM ZERO

4. THE BLOCK OF FLATS

4.1 Alison’s version

4.2 Sandra’s version

4.3 What’s the situation?

5. PARTICIPATION AND PERSISTENCE

5.1 Peter’s view

6. LEARNING, FORGETTING AND MATHEMATICS

7. THE STORIES WE TELL AND DON’T TELL

8. CONCLUSION

REFERENCES

Acknowledgements

Index of Authors

Index