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Education for Mathematics in the Workplace

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Education must achieve a range of goals - it must provide academic knowledge, knowledge and skills which young people need in order to enter the world of work, and must lay the foundations for lifelong learning.

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TABLE OF CONTENTS
ACKNOWLEDGMENT
INTRODUCTION
....................................................................................... xv
Gérard Vergnaud........................................................................................xvii
SECTION 1 MATHEMATICAL KNOWLEDGE IN SCHOOL AND AT WORK ................ 1
PREFACE Jeff Evans..............................................................................................................3
CHAPTER1
1. 2. 3. 4. 5. 6. 7. 8.
THE TRANSFER OF MATHEMATICS LEARNING FROMSCHOOL TO WORK NOT STRAIGHTFORWARD BUT NOT IMPOSSIBLE EITHER! Jeff Evans............................................................................... 5
Introduction...............................................................................................5 Viewsonthe Transfer of LearninginMathematics...................................6 Conceptualising Boundaries and Bridges................................................... 7 Teaching for Transfer................................................................................9 Implications for Research.........................................................................11 Conclusion and Directions for Research..................................................13 References................................................................................................14 Notes.........................................................................................................15
CHAPTER2
WORKING KNOWLEDGE: MATHEMATICSINUSE Richard Noss, CeIia Hoyles and Stefano Pozzi....................17
1.The Study.................................................................................................19 1.1. Documentary Analysis and Interviews...............................................20 1.2. Ethnographic observation...................................................................21
1.3. Simulation interviews ....................................................................... 21 1.4. Questionnaires and teaching experiments ....................................... 22 1.5. Data Collection and Analysis ........................................................... 22 2. Visible Mathematics and Routine Activity ............................................. 23 3. Breakdown Episodes and Underlying Models ........................................ 28 4. Mathematising Workplace Activity ......................................................... 31 5.Reference.s...............................................................................................34 .6Note.s........................................................................................................35
CHAPTER 3
FORMS OF MATHEMATICAL KNOWLEDGE RELATINGTOMEASUREMENTINVOCATIONAL TRAINING FOR THE BUILDING INDUSTRY Madeleine Eberhard............................................................ 37
1.Combining two approaches to knowledge: the construction school........ 37 2.38The central role of measurements in practice........................................ 3. The transfer of measurements as a basic practice for setting out.......... 40 3.1. Dimensions in structural drawings.............................................. 40 3.2. Cumulated dimensions, a technique for transferring dimensions from plans to ground ................................................................... 41 4. An incident at the construction school..................................................... 45 5. Mathematics and knowledge relating to cumulated dimensions at school 47 6. Conclusion ................................................................................................ 48 7. References ................................................................................................ 49 8......................................................................................................... 50. Notes
CHAPTER 4
1.2.3. 4. 5.6. 7. 8.
THE INTEGRATION OF MATHEMATICS INTO VOCATIONAL COURSES John Gillespie..................................................................... 53
Mathematics within vocational education in England and Wales .......... 53 Key Skills54including Application of Number...................................... Ways of providing support for Application of Number........................ 55 Views from two cultures .......................................................................... 55 Adding value to vocational work by integrating mathematics into it.....56 Taking charge of learning...................................................................... 56 Forming hypotheses................................................................................ 57 Integrating or embedding mathematical skills....................................... 57
vii
9. Project examples ...................................................................................... 57 9.1. Examples from Health and Social Care ......................................58 9.2. Examples from Art and Design....................................................60 10. Conclusion and suggestions ....................................................................63 11. References ................................................................................................64
CHAPTER 5
MATHEMATICAL MEANS AND MODELS FROM VOCATIONAL CONTEXTSA GERMAN PERSPECTIVE Rudolf Straesser...................................................................65
1. Vocational Education in Germany: the organisation ................................65 2. Teaching Mathematics in German Vocational Education ........................ 66 2.1. Aims of mathematics education and training ................................66 2.2. Mathematics education in part67time vocational colleges.............. 2.3. Mathematics education in full68time vocational colleges............... 3.69Modelling versus Legitimate Peripheral Participation............................ 3.1. Two pedagogies: modelling versus legitimate peripheral participation ................................................................................. 69 3.2. An aside on ‘transfer’................................................................... 71 3.3. Proposing a way forward.............................................................. 72 4. Mathematical and other means in vocational contexts............................. 73 4.1. Means (‘primary’ artefacts) .......................................................... 73 4.2. Teaching and means (‘secondary’ artefacts) ................................. 74 4.3. Teaching and learning with information technology .................... 75 5. Conclusion............................................................................................... 77 5.1.LESSONSTO LEARN FOR GENERAL MATHEMATICS EDUCATION.... 77 5.2. Speculatingonthe future.............................................................. 78 6. References ............................................................................................... 78
SECTION 2 BRINGING SCHOOL AND WORKPLACE TOGETHER................................... 81
PREFACE Susan L . Forman and Lynn Arthur Steen............................................................83
viii
CHAPTER6
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
WORKING MATHEMATICS FOR LEARNERS WITH LOWER ABILITIES Pieter van der Zwaart..........................................................87
Introduction..............................................................................................87 Designing of the investigation..................................................................88 Exploringthe shopfloor ........................................................................... 90 The mathematicsofthe shop floor........................................................... 92 The translation to worksheets...................................................................94 An example...............................................................................................95 Influenceoncurriculum documents and examination syllabi.................98 Final conclusions......................................................................................99 References................................................................................................99 Notes....................................................................................................... 100
CHAPTER 7
CLASSROOM TEACHERS DOING RESEARCHINTHE WORKPLACE John Hogan and Will Morony ............................................101
1.Introduction............................................................................................101 2.Background.............................................................................................101 2.1. The Key Competencies...............................................................102 2.2. The RIUMIT Project...................................................................103 3.The Workplace Research Methodology.................................................103 3.1.The aim....................................................................................... 104 3.2. The plan...................................................................................... 104 3.3. Data gathering methods..............................................................104 3.4. The trial......................................................................................105 4.Outcomesofthe Research......................................................................108 4.1. The workplace stories.................................................................108 4.2.Reflections by the teachers.........................................................109 5.Conclusion..............................................................................................112 6.Appendix: Education in Australia..........................................................113 7.References..............................................................................................113
CHAPTER 8
ix
MAKING AUTHENTIC MATHEMATICS WORK FOR ALL STUDENTS Susan L . Forman and Lynn Arthur Steen........................... 115
1. The Standards Movement ....................................................................... 117 2. Status, Equity, and Tracking ................................................................... 118 3. Beyond Eighth Grade ............................................................................. 119 4. Authentic Tasks ...................................................................................... 120 5. Challenges .............................................................................................. 123 6. Appendix A: Occupational Skill Standards ........................................... 124 7. Appendix B . Selected Web Sites ........................................................... 125 8. References .............................................................................................. 126
CHAPTER9
MATHEMATICS KNOWLEDGE AS A VOCATIONAL QUALIFICATION Tine Wedege....................................................................... 127
1. Introduction ............................................................................................ 127 2. Some basic concepts ............................................................................... 128 3. Mathematics in the Danish Adult Education System ............................. 129 4. Two different conceptionsofmathematics knowledge .......................... 130 5. Mathematics knowledge as a qualification ............................................. 131 6. The objective and subjective perspective ............................................... 133 7. References .............................................................................................. 135 8. Notes .......................................................................................................136
SECTION 3 EDUCATING FUTURE WORKER.S................................................................ 137
PREFACE Julian Williams................................................................................................. 139
x
CHAPTER 10
GEOMETRY AT WORKEXAMPLES FROM THE BUILDING INDUSTRY Annie Bessot....................................................................... 143
1.Introduction............................................................................................143 2.Geometry and the reading of plans on the buildingsite.........................144 3.Between geometry and space: the drawing as a basis for marking out the building lines .......................................................................................... 144 4.Linking drawings with objects in space: teaching strategies..................146
5.Didactical engineering: basic training procedures for the reading of technical drawings..................................................................................148 5.1. Variables of basic plan reading...................................................149 5.2. A teaching scenario for reading technical drawings within the microspaceofmodels...............................................................149 6.Conclusion..............................................................................................154 7.Appendix: Organizational diagram of the French Education system.....155 8.References..............................................................................................156 9.Notes.......................................................................................................156
CHAPTER11
1. 2. 3. 4. 5. 6. 7.
TEACHING MATHEMATICS TO SHOPASSISTANT APPRENTICES EXPLORING CONTENT AND DIDACTICAL SITUATIONS Corinne Hahn....................................................................159
Mathematics Used by a Jewelry Shop Assistant..................................... The Mathematics Course........................................................................ Students’ Skills With Percentages.......................................................... Suitable Didactical Situations................................................................. Conclusion.............................................................................................. References.............................................................................................. Notes.......................................................................................................
159 160 161 163 165 165 165
CHAPTER 12
xi
DEVELOPING A NEW MATHEMATICS CURRICULUM FOR POSTCOMPULSORY EDUCATION Geoff Wake and Julian Williams......................................... 167
1. Introduction ............................................................................................ 167 2. Prevocational courses ............................................................................ 167 3. General Mathematical Competence ........................................................ 169 4 . Exemplifying the general mathematical competence in students’ work in Science .................................................................................................... 170 5. Specifying new qualifications ................................................................. 173 6. Taxonomy of Learning ........................................................................... 177 7. Further research questions ......................................................................179 8.References..............................................................................................180
SECTION 4 RESEARCH METHODS FOR MATHEMATICS AT WORK....................... 181
PREFACE Robyn Zevenbergen........................................................................................... 183
1. Assumptions about Mathematics in the Workplace ................................ 183 2. Methods for Research in Workplace Mathematics ................................. 184 3. Future Research ...................................................................................... 186 4. References ............................................................................................. 187
CHAPTER 13
THE MATHEMATICAL NEEDS OF ENGINEERING  APPRENTICES Jim Ridgway....................................................................... 189
1. Exploring the Problem: Are the Number SkillsofApprentices in Decline? ............................................................................................... 191 2. What are the Mathematical Challenges of Engineering? .................... 191 2.1. What are the Mathematical ChallengesofSchool Mathematics? ..... .................................................................................................... 192
xii
2.2. How Do Apprentices Perceive Mathematics at School and Work? .. .................................................................................................... 193 3. What Predicts Apprentice Performance? ............................................ 193 4. Mathematics at School versus Mathematics at WorkResolving the Paradox ................................................................................................ 195 5. Some Conclusions .............................................................................. 196 6. References .......................................................................................... 196 7. Notes ..................................................................................................197
CHAPTER 14
IDENTIFICATION OF SOME MATHEMATICAL NEEDS LINKED TO THE USE OF MATHEMATICS AT WORK
AIain Mercier..................................................................... 199
1. Is there a need for mathematics? ........................................................ 200 1.1. Sample 1 . The counting of sardines by Breton fishermen at the beginning of the century ............................................................ 200 1.2. Sample 2 . Presentday use of INRA tables by agricultural technicians for feeding dairy cows ............................................. 202 2.A calculating tool ............................................................................... 203 2.1. Numbers arranged in the form of a diagonal cross serve to calculate proportions in mixtures ............................................... 203 2.2. Working knowledge and basic knowledge ................................. 205 2.3. The ways of change .................................................................... 206 3. Conclusion .......................................................................................... 207 4. References .......................................................................................... 207 5. Notes ................................................................................................... 208
CHAPTER 15
ETHNOGRAPHY AND THE SITUATEDNESS OF WORKPLACE NUMERACY
Robyn Zevenbergen
............................................................209
1. Ethnography and the study of mathematical cultures ......................... 210 2. A Rationale for Using Ethnography ................................................... 211 2.1. Situated learning ......................................................................... 211 2.2. Ethnomathematics ......................................................................213 2.3. Constructivism ............................................................................ 213 2.4. Key Assumptions of Ethnography .............................................. 214 3.Conducting Workplace Ethnographies ............................................... 216
4.DataCollectio.n.................................................................................. 4.1.Participant observation............................................................... 4.2. Interviews................................................................................... 4.3. Artifacts...................................................................................... 5. Conclusion.......................................................................................... 6. References.......................................................................................... 7. Notes...................................................................................................
CHAPTER 16
xiii
217 217 219 221 222 223 224
VISIBILITY OF MATHEMATICAL OBJECTS PRESENT IN PROFESSIONAL PRACTICE
Annie Bessot....................................................................
225
1.Actions and Device, a division between the visible and the invisible 225 2. An a priori analysis of situations in which a wall is constructed ........ 226 2.1. Need for checking ....................................................................... 226 2.2. Casing for a wall perpendicular to an existing wall .................... 227 2.3. When formwork becomes problematic ....................................... 228 2.4. Possible techniques for constructing the filler ............................ 229 2.5. Operating conditions for techniques found on the construction site232
3.Investigation of a construction site training course ............................ 233 3.1. Organisation of the investigation into the construction site training course .......................................................................................... 233 3.2. An episode from the construction site training course or a forgotten rectification ................................................................................. 234 4. The forms taken by the notionofa slope within institutions ............. 235 4.1. The notion of slope within trades in the building industry ......... 235 4.2. The notion of slope in ‘vocational’ teaching .............................. 236 4.3. The notion of slope in maths teaching in technical high schools for the building industry ....................................................................................... 236 5. Conclusion .......................................................................................... 237 6. References .......................................................................................... 237 7. Notes ................................................................................................... 237
CONCLUSION Rudolf Straesser .....................................................................................241
1.
Maths at the work place...................................................................... 241
xiv
2. 3. 4. 5.
Maths for work taught in different settings ........................................ 242 Role of artefacts / technology ............................................................. 244 Comment on research methodology.......................................................245 References ............................................................................................. 246
INDEX OF SUBJECTS..................................................................................247
INDEX OF AUTHORS......................................................................................257
AUTHOR AFFILIATIONS................................................................................261