Assessment Discourses in Mathematics Class- rooms: A Multimodal Social Semiotic Study ii Assessment Discourses in Mathematics Classrooms A Multimodal Social Semiotic Study Lisa Björklund Boistrup iii ©Lisa Björklund Boistrup, Stockholm 2010 Cover illustration: Teacher-student communication in a mathematics classroom, by Anders En- mark from video frame by Lisa Björklund Boistrup ISBN 978-91-7447-116-8 Printed in Sweden by Universitetsservice, US-AB, Stockholm 2010 Distributor: Department of Mathematics and Science Education, Stockholm University iv To David, Moa and Jim v vi Contents Preface .......................................................................................................... xi 1 Assessment in the Mathematics Classroom: Setting the Scene ....1 1.1 To be Curious ......................................................................................................1 1.2 Assessment in Mathematics Classrooms ........................................................2 1.3 Unit of Analysis and Limits of the Data ..........................................................5 1.4 Purpose and Research Questions ....................................................................5 2 Definitions, Previous Research and the Swedish Context ...............7 2.1 Defining Classroom Assessment ......................................................................7 2.2 Previous Research on Classroom Assessment ..............................................9 2.2.1 Frameworks of Classroom Feedback .........................................................10 2.2.2 Students’ Involvement in Classroom Assessment ..................................12 2.2.3 Classroom Assessment and its Relation to Learning ..............................14 2.2.4 Critiques of Research on Classroom Assessment....................................15 2.2.5 Classroom Assessment in Relation to Theories of Learning ..................17 2.2.6 Models of Classroom Assessment Over Time ...........................................18 2.3 Related Studies in Mathematics Education ..................................................20 2.4 The “What” Question in Mathematics Classroom Assessment .................21 2.4.1 The Content of Classroom Mathematics ...................................................21 2.4.2 Processes in Mathematics ............................................................................23 2.5 Studies Addressing the Roles of Semiotic Resources ................................24 2.6 The Classroom in the Institutional Context .................................................26 2.6.1 Classroom Studies Where Social Aspects are Addressed ......................27 2.6.2 Models – With Discourses ............................................................................28 2.7 Assessment in Mathematics in Sweden ........................................................29 2.7.1 The Swedish School System .......................................................................30 2.7.2 Steering Documents in Relation to This Study ........................................30 2.7.3 Critical Issues Concerning the Institutional Framing of Classroom Work in Sweden .......................................................................................................33 3 Theoretical Considerations ...................................................................36 3.1 Overarching Considerations – a Social and Critical Paradigm in Mathematics Education Research .........................................................................37 3.2 Theoretical Framework of the Study .............................................................39 3.2.1 Discursive and Institutional Aspects ..........................................................39 vii 3.2.2 Social Semiotics ............................................................................................40 3.2.3 Semiotic Resources – Actions and Artefacts ............................................41 3.2.4 Assessment in Mathematics as Communicative Acts .............................42 3.2.5 Meta-functions ...............................................................................................43 3.2.6 Discourses and Institutions .........................................................................46 3.2.7 Power and Agency .........................................................................................48 3.2.8 Meaning Making, Learning and Knowing ..................................................49 3.3 Operationalising Theories ...............................................................................50 3.3.1 Assessment Acts in Mathematics Classrooms ..........................................50 3.3.2 Focuses of Assessment Acts ........................................................................51 3.3.3 Semiotic Resources in Classrooms .............................................................53 3.3.4 Discourses of Assessment in Mathematics Classrooms .........................53 4 Methodology ............................................................................................55 4.1 Research Design and Research Methods .....................................................55 4.3 Researcher’s and Participants’ Roles ............................................................58 4.4 Selection of Participants ..................................................................................59 4.5 Data Material .....................................................................................................60 4.5.1 Researcher’s log ............................................................................................60 4.5.2 Video Material and Transcripts ...................................................................60 4.5.3 Written Material .............................................................................................61 4.5.4 Data Loss ........................................................................................................62 4.6 Ethical Considerations .....................................................................................62 4.7 Trustworthiness ................................................................................................64 4.8 Transcribing the Video and Audio material ..................................................65 4.8.1 The Set-up – Videograph .............................................................................66 4.8.2 Synopsis ..........................................................................................................67 4.8.3 Choice of Sequences for Transcription and Coding .................................68 4.8.4 Transcription ..................................................................................................69 4.9 Data Analysis ....................................................................................................71 4.9.1 Analysing the Video and Audio Material ....................................................71 4.9.2 Analysing the Written Material and Researcher’s log .............................73 4.9.3 The Reflective Process ..................................................................................73 4.9.4 Finalising Analyses Into Findings ...............................................................74 5 Assessment Acts in Mathematics Classrooms: Analysis and Outcomes ....................................................................................................75 5.1 Analytical Framework for Assessment Acts .................................................76 5.2 Feed Back, Feed Forward and Feed Up in Mathematics Classrooms ......78 5.2.1 Evaluative Feed Back ....................................................................................78 5.2.2 Descriptive Feed Back ..................................................................................83 5.2.3 Feed Back as Interest and Engagement ...................................................87 5.2.4 Three Kinds of Feed Forward ......................................................................91 5.2.5 Assessment Acts in Relation to Goals .......................................................95 viii 5.2.6 Changes in Assessment Acts .......................................................................98 5.3 Assessment Acts in Mathematics Classrooms: Occurrences and Affordances for Students’ Active Agency ..........................................................100 5.4 Summary and Conclusions Regarding Assessment Acts in Mathematics Classrooms ......................................................................................107 6 Focuses of Assessment Acts in the Mathematics Classroom: Analysis and Outcomes ..........................................................................109 6.1 Analytical Framework for Focuses in the Mathematics Classroom ........111 6.1.1 General Focuses of Assessment Acts ......................................................111 6.1.2 Aspects of Mathematics Competence ......................................................111 6.2 Focuses in the Mathematics Classroom ......................................................112 6.2.1 Focus on the Student as Self ....................................................................113 6.2.2 Focus on Task ..............................................................................................116 6.2.3 Processes Relating to Dealing With Mathematics Notions ...................119 6.2.4 Processes Relating to Applying Mathematics Notions .........................122 6.2.5 Processes Relating