Adults and numeracy.

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ADULTS AND NUMERACY

Jeffrey Taylor Evans

A thesis _submitted to the University of London for the degree _{of Doctor of Philosophy}

Department _{of Mathematics,} Statistics _{and Computing} Institute _{of Education}

1993

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ABSTRACT

This _{research critically} develops ideas _about "numeracy", "context" _{and "maths anxiety"} from the _{areas of mathematics} education, adult education _and psychology, Many current views on the "transfer" of school mathematics to practical

contexts are simplistic (e. g. the Cockcroft Report). Many studies largely ignore affect, _{or subordinate} it to the cognitive, or consider only _{expressed anxiety.}

A _{sample of undergraduates} (many _{of them "mature") was} used. In the quantitative phase of the study, the findings

diverge _{somewhat from earlier work.} Gender differences in performance, initially large, decrease in size (sometimes

substantially) after controlling for maths qualifications,

age, and social class. Statistical modelling produces an "inverted U" (rather than _{a simply negative) relationship}

between _{performance and anxiety.} Factor _{analyses question} earlier differentiations of "maths test anxiety" (academic)

items from "numerical _anxiety" (practical) items in the Mathematics Anxiety Rating Scale (MARS).

In the _{qualitative phase,} the interviews _synthesise life history _{and problem solving approaches; multiple readings}

are given to many episodes. Maths is indeed "hot": confidence, anxiety, pleasure and anger are often expressed. Drawing on psychoanalysis, I conceive of affect as emotional charges on ideas. The idea of "exhibited"

(unexpressed) _{anxiety allows} for _{e. g.} the _{operation of} defences. Drawing _{on post-structuralism,} I see numeracy _as

quantitative, spatial aspects of thinking within everyday practices. The context in which a subject addresses

problems, and/or reports anxiety, is described in terms of "positioning" in discursive _practice(s). The analysis _often shows a subject to be "multiply positioned"; thus what

first _seems to be "maths _{anxiety" can} be interpreted sometimes as having other bases. The ability of a signifier to take different _meanings within different practices

suggests for example limits on the possibilities of transfer _{- while at the} same time providing the basis for

any such possibilities.

Overall, the _{quantitative phase} focusses _on general relationships between indicators assumed to be relatively

stable, while basically accepting traditional frameworks

for _{studying cognition and affect.} The qualitative _phase studies the processes of producing thinking and emotion; it explores and challenges many of the basic assumptions on which the quantitative phase rests. Thus, this study shows the _{value of using} both types of research, both for

questioning existing findings and assumptions, and for beginning to formulate _{alternative approaches.}

ACKNOWLEDGEMENTS

One of the messsages of this work is that learning, thinking and performance are social accomplishments: the production

of this thesis is no exception. It is with appreciation _and humility _{that I record my thanks to the following.}

At Middlesex Polytechnic, _now Middlesex University, _the School _of Mathematics, Statistics _and Computing in the Faculty _{of Engineering} Science _{and Mathematics supported my} work over a number of years, with _contibutions to my fees

and a timetable allowance. The institution's Research Committee _{provided a small grant which allowed the} transcribing _{of the} interviews. The DES funded _{a year of}

In-Service Training, _which freed _{me from teaching} duties.

Colleagues Elizabeth Osmon, John Lintott _and Barry Edwards covered teaching for me. Others discussed ideas and methods, or offered _other support, especially Chris Abbess, Bernard Burgoyne, _{Len Doyal,} Hugh Fairweather, Ivan Rappaport, Peter Sneddon, _{Tom Wengraf. My research assistants on the} Access to Higher Education _project, Ross Walton _{and Paul Murphy,} helped _{with coding questionnaires.} Other _colleagues, in _the

Computing Centre, in Media Services, _{and in the Library, all} provided crucial assistance at various stages.

Jeannie Farr, Janet Culliford _and Chris Lord dealt intelligently _{and energetically with the challenging task of} transcribing _recorded interviews, _{and also} in several _cases,

offered suggestions for their interpretation and analysis.

At the Institute _{of Education,} the MSC Dept. (and for the first few years, Curriculum Studies) _{provided a marvellously}

supportive environment. It has been a very exciting place to do this PhD. A number of research _students in MSC and _other

Depts. have shared useful discussions _{about the content, and} also the process, of completing a PhD. Again, I thank the Librarians for _{their commitment both to learning and to}

learners _{such as myself.}

supervisors. Both _eminent in their fields, Harvey _and Valerie _{each helped me to look critically from different}

viewpoints at "big" problems _{such as cognition and affect,}

quantitative and qualitative _{methodology, etc.} I thank them both for their _{generosity in reading many drafts, and for} their _{valuable and encouraging} feedback. In _particular, I

appreciate Harvey's patient, unflappable _{support and his} methodological advice that went far beyond the bounds of the thesis. I thank Valerie for _{her patience} in inducting _{me -}

both through her writings _{and through} discussions _- into _a new and challenging way of thinking. I also appreciate her

generosity in continuing to _{meet me after she} left the Institute in 1988.

Family _and friends have _also been important. _{My parents,}

Isabell _{and the} late Taylor Evans, _{and my} brothers Lorne,

Murray, _and Earl, _all in Canada, have fully _{supported my}

work, _even when they must have sensed that it would

diminish, _{temporarily, the contact} between _us. Jeannie

Billington, _Paul Ernest, Steve Lerman, Ken Menzies _and

Ingrid Thorstad _{offered suggestions and encouragement at}

crucial stages. Chris Stanley has been a supportive friend

in _{all ways.} Finally, Anna Tsatsaroni has _given both

intellectual _{and emotional support} during the latter _stages

of the work: it is to her that I have often turned at those

moments when a recalcitrant paragraph and/or the ideas

behind it _{needed sorting out.} She has truly been _a

"compafiera".

Finally, I appreciate _{the massive support} from _{students of} the Polytechnic: _almost 1000 gave 20 _{minutes each} to complete the questionnaire, and 25 gave up to a further hour each to be interviewed about experiences and thinking about numbers and maths. I hope that their willingness to participate will be repaid, at least in part, by the

contribution of this study, and others like it, to the challenging tasks of making the learning of maths, and the use of numbers, much less mystifying and much more

satisfying.

CONTENTS

Paste

ABSTRACT 2

ACKNOWLEDGEMENTS

3

NOTES ON PRESENTATION 9

LIST OF TABLES 10

LIST OF FIGURES

12

LIST OF APPENDICES 13

CHAPTER 1:

ADULTS AND NUMERACY

_{- AN INTRODUCTION}

15

1.1 General Background to the Study 15

1.2 The Developing Aims of this Study 19

1.3 Overview _{of the} Contents 22

CHAPTER 2:

NUMERACY

AND MATHEMATICS PERFORMANCE

24

2.1 Numeracy _{and Mathematics} Performance _{among Adults} 26 2.1.1 Concepts _{of numeracy and practical maths} in

several contexts 26

2.1.2 _{The context of maths and numeracy:} beginnings 33 2.2 Surveys _{of Numeracy among Adults:} Some Results 37 2.2.1 The ACACE national _survey for Cockcroft 37 2.2.2 The ALBSU analysis _{of the} 4th NCDS follow-up 39 2.3 Mathematics Performance _{and Social} Difference:

Review _{of Selected} Literature 43

2.3.1 Gender differences in mathematics _performance 44 2.3.2 Social _class differences in mathematics

performance 49

2.3.3 Age differences in mathematics _performance 53

2.4 Summary 54

CHAPTER 3: AFFECT, ANXIETY AND MATHEMATICS ANXIETY 58

3.1 Mathematics _{and Affective} Factors 58

3.1.1 Confidence _{and anxiety} 64

3.2 Psychology _{and Anxiety} 65

3.2.1 Conceptions _{of anxiety} 65

3.2.2 Measures _{of anxiety} 68

3.2.3 Anxiety _{and performance} 70

3.2.4 Gender _{and anxiety} 73

3.2.5 Summary and evaluation 75

3.3 The Development _{of Mathematic, Anxiety} 76

3.3.1 _{Two mathematic anxiety scales: the Mathematics}

Anxiety Scale _{and the Mathematics Anxiety Rating Scale 77}

3.3.2 Dimensions _{of mathematics anxiety 81}

3.3.3 Mathematics _{anxiety and performance 84}

3.3.4 Structural _{Bases of maths anxiety 87}

3.4 Summary ₈₉

CHAPTER 4: METHODOLOGY OF THE QUESTIONNAIRE ₉₃

4.1 Conceptual _{Map and Hypotheses 94}

4.1.1 Institutional _{setting and working population 95}

4.1.2 _{A working conceptual map 96}

4.1.3 Research _{questions and hypotheses 99}

4.2 Research Design ₁₀₆

4.2.1 Samples _{of students 106}

4.2.2 Questionnaire design, indicators _{and coding 106}

4.3 Fieldwork 108

4.3.1 Pilot _{tests of the questionnaire 108}

4.3.2 _{The setting of the questionnaire completion 109}

4.3.3 Non-response ₁₁₀

4.4 Summary ₁₁₁

CHAPTER 5: _{ANALYSIS OF QUESTIONNAIRE RESULTS - OVERALL}

SUMMARIES ₁₁₃

5.1 Profiles _{of the Samples 114}

5.2 Levels _{of Performance 116}

5.2.1 Comparison _{of the} Polytechnic _{results with the}

national sample 116

5.2.2 _{The possibility of standardised comparisons 120} 5.3 Performance _{on "School} Maths" _{and "Practical Maths" 120}

5.3.1 Overall _{scores 120}

5.3.2 Sets _{of questions 122}

5.4 Dimensions _{of Anxiety:} Initial Considerations 123 5.5 Gender Differences in Performance _{and Affect} 125 5.5.1 Gender differences _{in performance with controls}

for _{age and qualification in school mathematics 127} 5.6 Social Class Differences in Performance _{and Affect 128} 5.7 Relationships between Performance _{and Maths Anxiety 130}

5.8 Summary

₁₃₀

CHAPTER 6: _{ANALYSIS OF QUESTIONNAIRE RESULTS -}

STATISTICAL MODELS ₁₃₅

6.1 Dimensions _{of Mathematics Anxiety 136}

6.1.1 Factors _{of maths anxiety 136}

6.1.2

_{Aims and techniques}

_{of factor}

_analysis

₁₃₈

6.1.3

Results

_{of factor}

_analyses

₁₄₀

6.2 Modelling _{of Mathematics Anxiety Measures 144} 6.2.1 Multiple _{regression modelling: aims and technical}

issues ₁₄₄

6.2.2 Modelling _{of maths test anxiety and numerical}

anxiety 147

6.3 Modelling

_{of Performance}

_{in School Maths and}

Practical Maths ₁₅₁

6.3.1 Overview _{of the modelling process 152}

6.3.2 Modelling _{of school maths performance 153} 6.3.3 Modelling _{of practical maths performance 160}

6.3.4 Affect, _{anxiety and performance} 162

6.4 Summary 164

INTERLUDE: REFLECTIONS ON THE CONCEPTUALISATION AND THE

RESULTS SO FAR

171

CHAPTER 7: NUMERACY AND CONTEXT 177

7.1 Utilitarian Views 178

7.2 Ethnomathematics 181

7.3 The Brazilian School 164

7.4 Socially Organised Activity 190

7.4.1 _{Task and context:} Michael Cole _{et al.} 190 7.4.2 Activity _{and goal-directed action:} Sylvia

Scribner 192

7.4.3 Arithmetic _{in everyday activity: Jean Lave et al.} 193 7.4.4 _{The structure of everyday activity:} Geoffrey Saxe 203

7.4.5 _{Summary and developments} 205

7.5 Post-structuralism: Valerie Walkerdine ₂₀₆

7.6 Summary 215

CHAPTER 6: TOWARDS AN ALTERNATIVE CONCEPTION OF

MATHEMATICS ANXIETY AND AFFECT 220

8.1 Silence _about Affect in Studies _{of Adult} Cognition 221 8.2 Recent Conceptions _{of Maths} Anxiety _{and Affect}

8.3 Insights from Psychoanalysis

8.3.1

Theoretical

basics:

_{Freud and Lacan}

8.3.2 Empirical _studies done _{within a psychoanalytic}

perspective

8.4 Insights from _{post-structuralism} 8.4.1 Valerie Walkerdine

8.4.2 Affect _{and desire as particular: Nick Taylor and} Wendy Hollway

224 225 225 230 234 234 243

8.5 Summary 252

CHAPTER 9 METHODOLOGY OF THE INTERVIEW 259

9.1 _{Themes - or Foreshadowed} Problems 260

9.2 The Interview Method 265

9.2.1 Developing the Interview Method 265

9.2.2 Areas for _{exploration and questions chosen} 267 9.2.3 Psychoanalytic insights in the interviewing 269 9.2.4 Developing indicators for _{performance and anxiety} 271

9.3 Execution _{of the} Interviews 273

9.3.1 Sampling _{methods and recruitment} 274

9.3.2 _{The setting and conduct of the} interview 274

9.3.3 The general _{reflexive account} 275

9.4 Overview _{of analysis} 277

9.4.1 Cross-subject _{and single-subject analyses} 277 9.4.2 Methods _{of analyses} for _{each theme} 278 9.4.3 Choice _of interviews _{as case studies} 281

9.5 Summary

283

CHAPTER 10 : INTERVIEW RESULTS - CROSS-SUBJECT ANALYSES 285

10.1 Theme 1: Context Understood in terms _of

Positioning in Practices 286

10.1.1 Review _{of theoretical} issues 286

10.1.2 Positioning in the interview 290

10.2 Theme 2: The Inseparability _{of Task and Context} 294 10.3 Theme 3: Gender Differences in Performance Under-

stood as Differences in Positioning within Practices 297 10.4 Theme 4: Numerate Cognition _{as Specific} to

Positioning 302

10.5 Theme 5: Gender Differences in Expressed Anxiety 317

10.6 Summary 322

CHAPTER 11 : INTERVIEW RESULTS - CASE STUDIES 330

11.1 The Remaining Themes 331

11.1.1 Theme 6: Other feelings _{expressed or exhibited} 331 11.1.2 Theme ?: Anxiety _{and other} feelings _{as specific}

to positioning 331

11.1.3 Theme 8: The relationship between _{affect and}

cognition as specific to positioning 332

11.2 Case Studies 333

11.2.1 Ellen 333

11.2.2 Jean 342

11.2.3 Fiona 347

11.2.4 Harriet ₃₅₇

11.2.5 Alan ₃₆₆

11.2.6 Donald ₃₇₂

11.2.7 Peter ₃₈₃

11.3 _{Summary and Conclusions from Case Studies 395}

CHAPTER 12 :

CONCLUSIONS AND CONTRIBUTIONS

₄₀₇

12.1 Conclusions _{and Contributions to Theory 407}

12.1.1 Developing _{ideas 407}

12.1.2 Reconsidering _earlier findings ₄₁₃

12.2 Contributions to Pedagogy _{and Practice} 419

12.3 Contributions _{to Methodology} 423

TABLES

428

FIGURES 463

NOTES

471

BIBLIOGRAPHY

509

APPENDICES 525

NOTES ON PRESENTATION

1. The thesis _{is arranged} in twelve _chapters. Whenever _a section number is given _{- as in Sec.} 2.1 _{or sec.} 2.1.1 _{- the} first _number indicates the _chapter.

2. For technical _{reasons to do with presentation, all} Tables and most Figures are placed after Ch. 12, rather than

integrated into _{the text.}

3. In order to keep the main body of the thesis within word limits, _{some material is presented} in Appendices, following

the Bibliography. These are referred to in the _main text, when relevant.

4. A Glossary _{of the variable names used} in the _statistical analysis of the questionnaire is given in APPENDIX R3.

5. The full transcripts

the basis for the _seven in APPENDICES S1 to S7. augumented by hand

recordings; only in one these _{corrections using}

of the interviews which provide the case studies in Ch. 11 _{are presented}

These transcripts _{were corrected and} while listening to the interview

case was it possible to assimilate a word processor.

6. All _{parts of the thesis are presented single-sided,}

except for the sample questionnaires (APPENDICES Q1 to Q3), and the interview transcripts (APPENDICES Si to S7); _page

references made to interviews _{are to those of} the _original transcripts.

LIST OF TABLES

2.1 Gallup Survey _{for ACACE: Results Analysed by Sex,}

Age and Social Clain 428

2.2 The NCDS 4th Follow-up: _{Results on Literacy and}

Numeracy Questions ₄₂₈

4.1 Presentations _{of the questionnaire: Numbers of}

Respondents ₄₂₉

5.1(a) Profiles _of 1983 Samples _{and Whole Sample:}

Percentages, _{Means and Standard} Deviations ₄₃₀ 5.1(b) Profiles _of 1984 Samples: Percentages, Means

and Standard

Deviations

431

5.1(c) Profiles _of 1985 Samples: Percentages, Means

and Standard Deviations 432

5.2 Comparison _{of Profiles of National} Sample (ACACE)

and Polytechnic Sample 433

5.3

Comparison

_{of Results}

_{of National}

Sample (ACACE)

and Polytechnic Sample 433

5.4(a) Comparison _{of Results of Qu. 10 for} the National

Sample _{and Qu. 6 for} the Polytechnic Sample 434 5.4(b) Comparison _{of Results of Qu. 3 for the} National

Sample and Qu. 18 for the Polytechnic

Sample

434

5.5

Comparison

_{of Performance}

_{among Students}

_across

Questions _{of Several} Types: Percentage Correct 434 5.6 Gender Differences for Main Outcome Variables:

Hypotheses, Summary Statistics _{and Observed} Differences

for Whole Sample 435

5.7 Relationship between Parental Occupation _{(SCP) and}

Own Occupation (SCS): Crosstabulation for 1985 Sample 436 5.8 Parental Occupation (SCP) Differences for Main

Outcome Variables: Hypotheses, Summary Statistics _and

Observed Differences for 1985 Sample 437

5.9 Parental Occupation (SCP) Differences for Selected Outcome Variables by Gender: Summary Statistics for

1985 Sample 438

5.10

Student's

Own (SCS) Differences

for

Main Outcome

Variables: Hypotheses, Summary Statistics _{and Observed}

Differences for 1985 Sample 439

5.11 Age Differences for Main Outcome Variables:

Hypotheses, Summary Statistics _{and Observed} Differences

for Whole Sample 440

5.12

Differences

_related

_{to Maths Qualification}

in

Selected Variables: Summary Statistics for Whole Sample 441

5.13

Relationships

between Performance

_{and Maths}

Anxiety: Hypotheses _{and Correlations} for Whole Sample 442 5.14 Relationships between Performance _{and Selected}

Affective Variables: Hypotheses _{and Correlations} for

Whole Sample 442

6.1 Tentative Classification _{of Polytechnic} Mathematics Anxiety Items _within Maths Test Anxiety _{or Numerical}

Anxiety Dimensions 443

6.2(a) Three-Factor Analysis _{of Mathematics Anxiety} Items: Factor Loadings for Principal _{Factor Analysis}

(PFA), Varimax _{and Oblimin} Rotations ₄₄₄

6.2(b) Three-Factor Analysis _{of Mathematics Anxiety}

Items:

Factor

Loadings

for

_{Maximum Likelihood}

_Factor

Analysis

(MLFA), Varimax and Oblimin Rotations

445

6.2(c) Three-Factor Analysis _{of Mathematics} Anxiety

Items:

Factor

Loadings

for

Alpha Factor

Analysis

(AFA),

Varimax

_{and Oblimin}

Rotations

446

6.2(d) Three-Factor Analysis: Mathematics Anxiety

Items Associated _{with each Factor} 447

6.3(a) Two-Factor Analysis _{of Mathematics} Anxiety

Items: Factor Loadings for Principal Factor Analysis

(PFA), Varimax _{and Oblimin} Rotations 448

6.3(b) Two-Factor Analysis _{of Mathematics} Anxiety

Items:

Factor

Loadings

for

Maximum Likelihood

Factor

Analysis (MLFA), Varimax _{and Oblimin} Rotations 449 6.3(c) Two-Factor Analysis _{of Mathematics} Anxiety

Items: Factor Loadings for Alpha Factor Analysis (AFA),

Varimax _{and Oblimin} Rotations 450

6.4 Models for Maths Test Anxiety (TA) _{and Numerical}

Anxiety (NA): Variables Included, Regression Equations,

and Estimates of Effects for 1985 Sample 451 6.4(a) Models for Maths Test Anxiety (TA) _{and Numerical}

Anxiety

(NA): Variables

Included,

Regression

Equations,

and Estimates of Effects for Wole Sample 452 6.5 Models for School Maths Performance (PERFS):

Variables Included, Regression Equations, _{and Estimates}

of Effects

for Whole Sample and 1985 Sample

453

6.6 Models for Practical Maths Performance (PERFP):

Variables

Included,

Regression

Equations,

_{and Estimates}

of Effects for Whole Sample 454

7.1 Levels _{of Reasoning} Strategy Coded in Best-Buy

Studies _{of Capon and Kuhn} 455

10.1 Positioningi for Interview Qu. 2 and for Interview

Qu. 4: Cross-tabulation _{of Numbers of Subjects} 456 10.2 Positionings for Questionnaire Qu. 18 and for

Interview Qu. 4: Cross-tabulation _{of Numbers of Subjects} 456 10.3(a) Performance _{on Interview} ßu. 2: Cross-tabulation

of Number Correct by Gender and Parental Social Class 457 10.3(b) Performance _{on Interview} ßu. 2: Cross-tabulation

of Number Correct by Gender, Qualification in Maths and

Positioning

457

10.4 Performance _{on Interview} Qu. 3: Cross-tabulation of Number with Both Parts Correct by Gender, Qualification

in Maths _{and Positioning} 458

10.5 Performance _{on Interview} Qu. 4B: Cross-tabulation of Number Correct by Gender, Positioning and Tipping

Rule Enunciated 458

10.6

Performance

_{on Interview}

Qus. 5A and 5B:

Cross-tabulation _{of Number Correct etc.} by Gender _and

Positioning for Both Parts 459

10.7(a) Strategies Used in Solving Best-buy Problems: _a Comparison _{of Results} from Capon and Kuhn, Lave et al.

and this Research 460

10.7(b) Performance _{on Interview} Qu. 6:

Cross-tabulation _{of Strategies} Used and Answer, by

Positioning for Part B and Social Class _{of Parents} 461 10.8 Expressing _{and Exhibiting} Anxiety in the

Interview: Cross-tabulation _{of Numbers with} Gender 462

LIST OF FIGURES

P. _asz 2.1 A Generic _{model for Relating Affect and Outcome 463}

3.1 The Inverted U Relationship _{between Performance}

and Anxiety 71

4.1 Working Conceptual Map for _the Polytechnic _{Survey 463} 5.1(a) Comparison _{of the} Format _{of Qu. 10 on the}

National Survey _{and Qu. 6 on the Polytechnic Survey 464} 5.1(b) Comparison _{of the} Format _{of Qu. 3 on the}

National Survey _{and Qu. 18 on the Polytechnic Survey 464} 5.2(a) Gender Differences in School Maths Performance

(PERFS): Boxplot

for Whole Sample

465

5.2(b) Gender Differences in Practical Maths

Performance (PERFP): Boxplot for Whole Sample 465 5.3(a) Differences in School Maths Performance (PERFS)

by Sex, _{Age and Qualification} in Maths: Plots _{of Means}

of Subgroups

for Whole Sample

466

5.3(b)

Differences

in Practical

Maths Performance

(PERFP) by Sex, Age and Qualification in Maths: Plots

of Means of Subgroups for Whole Sample 467

5.4(a) Relationship between School Maths Performance (PERFS) and Maths Test Anxiety (TA): Plot of Average

PERFS Score for

_{each Decile}

_{of TA for Whole Sample}

468

5.4(b) Relationship between Practical _{Maths Performance} (PERFP) and Numerical Anxiety (NA): Plot of Average

PERFP Score for _{each Decile of NA for} Whole Sample 469 8.1 Lacan's View of Repression: the Signifier Falling

to the Level _{of the} Signified 228

11.1 Chain _{of Signifiers} in Fiona's Associations _with

her Father's Work 354

LIST OF APPENDICES

Page P1 Gallup Survey for ACACE: Interview Schedule _and

Cards 525

P2 The NCDS 4th Follow-up: the Literacy _{and Numeracy}

Questions 528

P3 Fennema - Sherman Mathematics Anxiety Scale (MAS) 529 P4 Mathematics Anxiety Rating Scale _(MARS), 1972

Version 530

Q1 Questionnaire

for Polytechnic

Sample D83

537

Q2 Questionnaire

for Polytechnic

Samples C84 and D84

545

Q3 Questionnaire for Polytechnic Sample B85 553 Q4 Questionnaire Design, Indicators _{and Coding} 561

Q5 Introductory Script for Administering _of

Ques tionnaires, 1983 and 1984 566

Q6 Introductory Script for Administering _of

Ques tionnaires, 1985 567

Q7 The Experience

Scale:

Changes in Affective

Items

between 1983 and 1985 568

Q8 The Performance Scale: Changes in Items between

1983 and 1985 569

Q9 The Situational

Attitude

Scale:

Origins

_{of Items}

570

R1 Codebook for Questionnaire Items, for Sample B85 572 R2 The Performance Scale: Aggregation _of Items into

Indices for School Maths (PERFS), Practical Maths

(PERFP) and ACACE-Comparable Score (PERFA) 574 R3 Glossary _{of Variables used} in Statistical Analyses,

including Scales _{and Categories} Used for Outcome _and

Predictor Variables in Regression Models 575

R4 The Performance Scale: Type of Item, Answer(s) Coded as Correct, Percentage Correct and Some "Characteristic

Errors" 576

R5 Profiles _{of the} Samples 577

R6 Gender Differences in Performance _{and Affect} 579 R7 Social Class Differences in Performance _{and Affect} 579 R8 Age Differences in Performance _{and Affect} 582 R9 Differences _related to Maths Qualification in

Performance _{and Affect} 584

R10 Other Relationships between Performance _{and Affect} 585

I1 Script for Interviews 587

12 Interview Problems to be Solved 592

13 Invitation to Interview 599

14 Respondents Recruited for Interview Categorised by Qualification in Maths, Gender, Age, _{and Parental}

Occupation 600

15 Execution _{of the} Interviews: Details 601

S1 Interview Transcript for Ellen 604

S2 Interview Transcript for Jean 612

S3 Interview Transcript for Fiona 622

S4 Interview Transcript for Harriet 630

S5 Interview Transcript for Alan 642

S6 Interview Transcript for Donald 650

S7

Interview

Transcript

for

Peter

662

U1 Illustration: Sam

₆₇₂

U2 Illustration: Keith 674 U3 Gender Differences _{in Performance:} Results from

Interview Qu. 2 676

V1 Literature Review _{of Gender} Differences in School

Mathematics Performance 681

V2 The Basic Skills Accreditation Initiative (BSAI)

Survey 686

V3 Controversy in Anxiety Research in 1950s and 1960 687 V4 Research into Dimensionality _{of the} MARS 688

Wi

Adults'

Needs for Numerate Skills

in Work Contexts'

691

W2 Ethnomathematics: Paulus Gerdes 692

W3 Specificity _{of Task and Context:} Michael Cole _et

al. 693

W4 Goal-directed Action in the Dairy: Sylvia Scribner 695 W5 Arithmetic in Everyday Activity: Jean Lave et al. 696 W6 The Structure _{of Everyday} Activity: Geoffrey Saxe 699

X1 Maths anxiety

as a mass phenomenon: Sheila

Tobias

700

X2 Inseparability _{of Cognition an Affect:} Case studies 701 X3 Emotion _{as the dynamic of reason:} Laurie Buxton 702

CHAPTER I: _{ADULTS AND NUMERACY - AN INTRODUCTION}

There are indeed many adults in Britain who have the greatest difficulty _{with even such apparently} simple matters as adding up money, checking their change in

shops or working out the cost of five gallons of petrol. Yet these adults are not just the unintelligent

or the uneducated. They come from many walks of life and some are very highly educated indeed _...

(Stringer, 1979)

1.1 General Background to the Study

Concern _over the _{mathematics curriculum and} mathematics teaching has been a constant feature of debates in education

throughout this _century (see e. g. Howson, 1983). One relatively recent example comes from the then Prime Minister

James Callaghan's Ruskin _speech in October 1976 which launched the "Great Debate" in education:

There _{seems to be a need for a more technological} bias in science teaching that _will lead towards practical

applications in industry, rather than towards academic studies. Or to take other examples, why is it that such

a high proportion of girls abandon science before leaving _school

...? (Callaghan, 1976, p. 5)

In response to similar concerns expressed by the Education, Arts _{and Home Office} Sub-Committee _{of the Public} Expenditure

Committee in 1977, the then Labour Government _announced its decision _{to establish an} Inquiry into the teaching _of

intelligibility _{and to the match between the mathematical} curriculum and the skills required in further _education,

employment and adult life generally" (Cockcroft Committee, 1982, _{p. ix).}

These _quotations illustrate _{the traditional view of} "mathematical _ability": it involves _{a set of generally}

applicable cognitive "skills", which can be deployed to perform a range of "tasks", in a variety of "contexts".

These abstract mathematics skills are considered to be applicable in practical contexts through a relatively

unproblematical process of "transfer". In any formal educational system, the issue of transfer of learning is clearly of major importance _{- whether} in the "application"

of school knowledge in contexts outside the school, or in the "harnessing" _{of outside} learning to help _{with school} aims. (Though the latter process has so far received less attention, it is bound to be increasingly crucial in systems where "mature students" or "recurrent education" are

encouraged. ) In this traditional view, performance is often considered to be measurable by the number of correct

responses to a set of test items, such as those often used in _schools. However, the _possibility that _correct

performance could be produced by rote learning has led to the _requirement that _{correct performance} be produced, in a proper way, i. e. through "real understanding" (e. g. Skemp

(1976,1979)). (NOTE 1)

Research _{using the above ideas} in mathematics _{education and} psychology during the last twenty or thirty years has tended to _produce the by _{now well-known} findings to do with differences in performance in school _mathematics (SM): _{e. g.}

gender differences in favour _{of males, and social} class differences in favour _of the _{middle classes.} In the

literature _{on gender} differences, _{performance is normally} explained with reference to cognitive, affective, and social variables (e. g. Fennema, 1979).

Much research _{on affect (and attitudes)} has been carried out in mathematics education since the 1970s, often in order to

help _explain differences _{in "participation" or} course-taking. These differences in turn have been _advanced to _explain the _gender differences _{in SM performance,}

accepted as existing at least by the early teenage _years in the U. S. _and U. K. by _{many, though not all, researchers.}

(There _{has not been as great an emphasis on social class} differences, during _{this period. )}

Much _{of this research on affect} has focussed _{on "maths} anxiety". Anxiety as a concept has its basis in Freud's work which places crucial emphasis on the possibility that

anxiety may be "latent" or unconscious. Psychology took these _concepts, but _{stress on conscious, observable}

phenomena, in mainstream American psychology from the late

1940s. led _{to an emphasis on "manifest anxiety", which was} considered observable and quantifiable. There was some debate _{over whether} the different types _{of anxiety were}

simply "debilitating" of academic (and other) performance,

or whether up to a certain level, anxiety might be "facilitating" _{of performance, as} in the "inverted U" quadratic relationship representing the "Yerkes-Dodson Law",

In the 1970s, because _{of both} the theoretical _{role of maths} anxiety in explaining the supposed performance deficits of certain groups, and (as I will argue) the methodological

limitations _{of much earlier research, maths anxiety came to} be seen as simply debilitating. From _the late 1970s _and

early 1980s, there began to appear reports of work done with college students, on the relationship between maths anxiety and performance. These researches also studied possible

"dimensions" _{of maths anxiety using} factor _analysis:

sometimes two factors of maths anxiety were found (e. g. Rounds and Hendel, 1980), _{which can be seen as parallel} to a distinction between _{school maths and practical maths.}

Much research _{and curriculum} development _{in recent years} has focussed _{on how to promote the application of school maths}

in _{practical contexts;} in _{particular, the report of the} Cockcroft Committee (1982). Cockcroft _used the term

"numeracy" _{to focus attention on the} "mathematical" _aspects of practical _everyday life, basing this in turn on research

which they had commissioned, _{e. g. Sewell} (1981). Of course, this is not to say that _{the interest in "practical maths"} began _{with Cockcroft: there were already a number of} projects with _{a similar} focus, _{such as "Maths} in Work" (NOTE

2) _{and the} CSE Everyday _{Maths syllabus. (NOTE 3) However,} Cockcroft _{opened up a space for the idea of numeracy - by}

inserting it into _{the traditional view. Using the term} signified the Committee's view that the _{use of} basic mathematical operations with confidence in practical

everyday situations was essential to their _{aims -} and also that the "transfer" from abstract _{mathematics to everyday} applications _{was not as straightforward as} the traditional

view might suggest. Interest in out-of-school maths has greatly increased _since the time _{of Cockcroft: studies} have been _{carried out by the Brazilian school of Carraher,}

Carraher _{and Schliemann (e. g. Carraher,} 1988); by "activity

theorists" _{such as} Scribner (1984) _{and Lave (1988); and by} post-structuralists such as Walkerdine (1988).

Two _major developments in _{adult education and further}

education during this period reinforced the interest in

numeracy _- the mushrooming _{of adult} literacy _courses,

following _the BBC _series On _the Move in 1975, _{and the}

development _of "Access" _{courses. The latter aimed to allow}

adults disadvantaged by an _{unsuccessful school career} to

enter a specified course in higher _{education after} an

intensive _{preparatory course, rather than requiring them to}

complete 0- and A-levels; they provided yet _another source

of demand for "maths" qualifications, or "numeracy skills"

(or _sometimes "statistical _{competence"). The adult literacy}

movement led to an awareness, starting amongst tutors, that

a lack of "numeracy" existed as a widespread and important

problem, somewhat independently _of illiteracy. Thus numeracy

became important in _{several contexts of educational}

policy-making. And it is undeniable that _{a lack of numeracy}

can be a great handicap, for individuals _and for _society;

see the quotation at the head _of the _{chapter and Evans}

(1989a) for _{a fuller} discussion.

numeracy what many adults recognise _as "maths". Many students perceive there to be fairly _strong boundaries

between _{the maths that they have met at school or college,}

and the other activities that make up "real life" (or indeed other academic disciplines), even when the latter make substantial use of quantitative, spatial or other "numerate"

ideas _{and strategies. And these people's feelings towards} maths are often rather negative, _{e. g. feelings of} lack of confidence, anxiety and dislike. It is reasonable to suspect that both the _{sense of} "boundaries", _and the _negative

feelings, _{will affect any possibilities of transfer} between academic maths and practical activities. Thus, some recent research has stressed the "discontinuity" between school maths and numerate activity in non-school contexts, e. g. that _of Lave (1988) _{and of Walkerdine} (1988).

Despite _{a wealth of research} being _undertaken, there _still

seemed to be gaps that deserved attention in 1983 when I began this _study. There _{remained problems with the notion of} numeracy, for example, whether it was any different from elementary school maths with "a thin veneer of 'real-world'

associations" (Maier, 1980). There was still relatively

little _{research on adults as compared with} that _{on the} captive populations in compulsory schooling. The latter

research had produced controversial _and sometimes contradictory findings, e. g. on gender differences in maths performance and affect. And the research that there was on

adults tended either not to pay much attention to affect, or else not to focus much on contexts outside the academic.

Thus there _{seemed to be a need for research} that _{would study} cognition and affect, and their inter-relationship, amongst a group of adults who were both involved in academic maths and who were experienced in a wide range of practical

activities that might provide a context for the use of ideas recognisable as "mathematical".

1.2 The Develovinst _{Aims of this} Study

broad _{vroblem: How do adults develop, or fail to develop,}

numeracy skills, and confidence in their _numeracy? I began with a simple model which _{aimed to explain} "difficulties

with numeracy" by several factors _{or processes:}

- socialisation depending on the individual's position in class, gender and other social structures of relations;

- critical incidents in one's history of learning maths; and - affective characteristics (which I saw as representing a sort of internalisation of the pressures from the first two sets of determinants).

I _aimed to _attempt to _produce (or _{gain access} to) information that _{would allow me to describe} the "level" _of numeracy in the population of adults, or a reasonably

representative sub-population; and to relate this to respondents' backgrounds, experiences, and "maths anxiety".

It _{seemed to me at} the time that _{a suitably} designed

questionnaire, analysed using quantitative methods and modelling would meet many of these objectives. And it could be _{combined with} "qualitative" (less _structured)

interviewing _{with several aims: to elicit} descriptions _of

the sorts of everyday activities where subjects might use (or _{not) numerate reasoning;} to describe in more detail

critical incidents and experiences; and to illuminate the individual's _{more fluid perceptions, purposes and interests.}

The interviews _{would also allow me to} describe how _adults might re-emerge from their difficulties with numeracy to have _a "second _chance". Together _{questionnaires and}

interviews _{would provide an appropriate methodology.}

As the _research developed, it became clear that I would need to develop _{my concepts more fully,} in particular: those _of numeracy and "practical maths", along with the implied ideas

about context; "performance" and cognition; and affect,

especially anxiety and "maths anxiety". In response both to these _{conceptual challenges, and to the emerging} limitations

of the questionnaire results, I revised my aims for the quantitative part of the research, so that it would be

focussed _on the _{critical consideration of a number of} earlier findings (mentioned in Sec. 1.1) about mathematics

performance

and anxiety

which are widely

considered

to hold.

The broad _{aim of the thesis} is to understand _{adult numeracy,} and how it is developed. Numeracy is not just low-level,

applied mathematics: it is thinking, problem-solving in a variety of practical contexts, using concepts which are numerical, quantitative, spatial. I argue that, not only must the context be conceived more broadly than in much

earlier work, but indeed it must be rethought as an integral

aspect of activity, of numeracy-in-practice. In this view, the _context is _constituted by discursive _practices, which tend _{to position} the subject to address tasks and to act meaningfully. These discursive practices may be analysed best by focussing _{on language and signification.}

In _order to _{understand numeracy,} I _argue that it is essential to understand affect, emotion, feelings. Using the

idea of affect as a charge connected to a signifier (or a chain of signifiers) in one or several specific discourses,

I further _argue that it is impossible to understand emotions such as anxiety or pleasure without allowing for the operation of unconscious processes. Using this overall

approach, the relationship between the cognitive and the affective may be studied, for particular subjects positioned

in specific discursive practices.

The broad _{aims of the thesis can be related} to a number of objectives as follows. The aim of understanding and developing the idea _{of numeracy} led me:

(i) _{to compare the several notions of "numeracy" relevant} to different _{educational policy contexts;}

(ii) to describe the levels _{of numeracy amongst a selected} population of adult students, and to compare them with those

of the national population surveyed for the Cockcroft Report; _and

(iii) _{to consider critically a number of earlier} findings

about mathematics performance, numeracy and anxiety which are widely considered to be valid; for example, "Males perform better than females.... ", "Women have more maths

Concern _{with a} broader _{conception of context and its} grounding in discursive _{practice pointed me:}

(iv) _{to review the conceptions of "context" used} in mathematics education and psychology, _{and to assess} certain

of these conceptions through _{empirical research;}

(v) to _contribute to the development _{of ways of describing} cognition and affect in context, by proposing _{a methodology}

drawing _on ideas from _{post-structuralism, and aiming to} determine _{the practice(s) called up by subjects when} they

are presented with numerate problems.

The aim of understanding affect led me:

(vi) _{to document the range of affect related} to _mathematics and numbers amongst members of the sample; and

(vii) to illustrate the _{contribution of} insights from psychoanalysis in the study of affect generally, and of

anxiety in particular.

The _{goal of studying} the _{relationship of cognition and} affect led me:

(viii) _{to examine the relationship of cognition and affect,} both _{across samples of adults, and for particular subjects.}

In addition, I aimed:

(ix) to draw conclusions _about the use of quantitative _and qualitative methodologies in educational research; and

(x) _{to point to recommendations} for _{mathematics education} practice, and for the development of numeracy.

1.3 Overview _{of the Contents}

In Ch. 2, i discuss _{the use of the term "numeracy"} in various educational contexts, and the results of several

recent national studies which use different conceptions of numeracy; I also briefly review the literature on gender and social class differences in mathematical performance. Ch. 3 takes _up the discussion _{of affect, and} in _particular

examines the development of psychological notions of anxiety

links between _{anxiety and performance, and on the} "dimensions" _{of maths anxiety.} Ch. 4 _presents hypotheses

based _{on the reviews and discussions in Chs. 2 and 3, and} outlines the methodology for the quantitative _{strand of} the

research. Chs. 5 and 6 summarise the results _{of this strand,} particularly those to do with gender and social class differences in _{anxiety and performance, the dimensions of}

maths anxiety, and the relationship between performance and anxiety; this is done first using descriptive statistics,

and then using statistical modelling. This part of the thesis _{concludes with an Interlude which aims to evaluate} the findings _{of the quantitative strand.}

This discussion _{of the results of} the _{quantitative strand} suggests that the traditional notion of context might need to be overhauled. Therefore, Ch. 7 discusses the various

uses of the ideas of context, and activity-in-context, in mathematics education (and other) research, and their

relationship to the ideas of numeracy and "practical maths". Ch. 8 surveys conceptions of maths anxiety different to those discussed _{so far;} in particular, _{the contribution of} psychoanalysis is discussed, as is its combination with

ideas _{of post-structuralism} in the work of Walkerdine _and others. These two chapters provide the basis for a set of

"foreshadowed _{problems" or} themes _{to be addressed} in the "qualitative" _{strand of the study.} Ch. 9 discusses these themes _{and the methodology} for the interviews _{of members of}

the subsample. In Chs. 10 and 11, the results of this strand are presented, in particular those to do with the context-specificity of numerate cognition (or performance),

affect (especially anxiety), and their inter-relationship _- in the form both of cross-subject analyses (including all of the _{subsample) and} intensive _{case studies.} Finally Ch. 12 presents a summary and conclusions _- contributions to

theory, _{methodology, and pedagogy} / practice.

CHAPTER 2: NUMERACY AND MATHEMATICS PERFORMANCE

I dropped _{mathematics at 12, through some freak} in the syllabus [... ] I cannot deny that I dropped maths with a sigh of relief, for I had always loathed it, always

felt _{uncomprehending even while getting} tolerable

marks, didn't like subjects I wasn't good at, and had no notion of this subject's appeal or significance.

The reason, I imagine, was that, like most girls, I had been badly taught from the beginning: I am not _really

as innumerate as I pretend, and suspect there is little

wrong with the basic equipment but I shall never know. [... ] And that _effectively, though I did _{not appreciate} it _{at the time, closed most careers and half of culture} to me forever.

(Margaret Drabble, _writer,

in The Guardian, _{5 Aug. 1975, p. 16 )}

The broad _{aim of this} thesis is to understand _{numeracy among} adults, and to gain insights into how it is developed. I began by _{seeing numeracy as} thinking, _{problem-solving,}

"performance" in _practical (non-school) _{contexts, using} ideas _{which are numerical, quantitative, spatial.} The idea of context was an important part of my conception, and I

considered that numeracy had affective aspects, too (see Ch. 1)

.

In this thesis, _{adults are understood as people of a range} of ages, including "adolescents", who:

social

relations,

independently

_{of their}

families;

(ii) _{have at least the opportunity for paid or voluntary} work; and

(iii) _{are conscious of having social or political interests.} Thus, in the teaching _{situation, we often hear the question:}

"How is this _{useful to me? ".} (In _{contrast, children's} experiences tend to be more homogeneous). At the beginning,

I was interested in the population _at large _{of adults -} though _{the practical constraints of doing empirical research}

soon led me to choose a population that would allow a compromise between representativeness and convenience; see Ch. 4.

Although there is a great deal _{of information available on} the _{mathematical attainments of school children of various}

ages in the UK (some of which will be reviewed in Sec. 2.3 below), _there is _relatively little _{empirical study of}

adults' "mathematical performance" or adult numeracy. Such studies are done mainly with students in tertiary education

or in adult education, or occasionally with members of the general _public (see Sec. 2.2). In tertiary _education, it is mostly mathematics performance that is studied _{- whereas} in adult education, studies have tended to focus on numeracy;

this is no doubt partly because _{of the attention} focussed _on it by _the Cockcroft Report (1982), _{and partly} because numeracy seems to provide parallels with literacy (NOTE 1).

The term "numeracy" _{was used before the publication of} the Cockcroft Report (1982), but the latter _{opened up a space}

for it _to be discussed _widely inside _{and outside of} education. However, the term has been used in several

different _{ways. I shall examine} these in Section 2.1. To date, _very few _studies have been done on numeracy in the

adult population at large, and those have used different

notions of numeracy. I consider the conceptual basis and results of several studies in Sec. 2.2.

I also aim here to begin to develop _{explanations of} how numeracy (allowing for its different senses) is developed.

differences in _{particular, and also into those on social} class and age differences in mathematical performance,

including _{studies of school-age groups} (NOTE 2). Thus, in this _{chapter and in the discussion of affect} in Ch. 3, I build _{up a} "conceptual _{map" that summarises a set of}

plausible explanatory factors for differences in numeracy and mathematical performance, and that can be used as a basis for designing _{the questionnaire} in Ch. 4.

2.1 Numeracy _{and Mathematical} Performance _{among Adults}

In _sec. 2.1.1, _{I consider} the different _{notions of numeracy} current in different educational settings, and in sec. 2.1.2, I begin to _consider the related _{ways that contexts} of numeracy are conceived.

2.1.1 Concepts _{of numeracy and practical maths in several} contexts

There _{was little research at any age level} based on the idea of numeracy before Cockcroft, though the term was used in adult education. Where it was used, it seems to have

referred to basic numerical skills, and to have been tested by fairly _{abstract questions; e. g.} in Glenn (1978) for

adults, and in Pollock and Thorpe (1979), in a study of school-aged pupils in Scotland.

(a) the Cockcroft Report (1982)

Cockcroft's brief _mentioned the "(mathematical) _skills required in further education, employment and adult life generally" (p. ix), but Ch. 1 of the Report makes no mention of "numeracy". In Ch. 2, the Committee indicates that they

include _among the _{mathematical needs of everyday} life the abilities: to read numbers and to count; to tell the time;

to _pay for _{purchases and} to give _change; to weigh and measure; to understand straightforward timetables and simple