Zaslavsky / Sullivan Constructing Knowledge for Teaching Secondary Mathematics

Tasks to enhance prospective and practicing teacher learning
2011
ISBN: 978-0-387-09812-8
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 6, 330 Seiten

Reihe: Mathematics Teacher Education

Teacher education seeks to transform prospective and/or practicing teachers from neophyte possibly uncritical perspectives on teaching and learning to more knowledgeable, adaptable, analytic, insightful, observant, resourceful, reflective and confident professionals ready to address whatever challenges teaching secondary mathematics presents. This transformation occurs optimally through constructive engagement in tasks that foster knowledge for teaching secondary mathematics. Ideally such tasks provide a bridge between theory and practice, and challenge, surprise, disturb, confront, extend, or provoke examination of alternatives, drawn from the context of teaching. We define tasks as the problems or activities that, having been developed, evaluated and refined over time, are posed to teacher education participants. Such participants are expected to engage in these tasks collaboratively, energetically, and intellectually with an open mind and an orientation to future practice. The tasks might be similar to those used by classroom teachers (e.g., the analysis of a graphing problem) or idiosyncratic to teacher education (e.g., critique of videotaped practice). This edited volume includes chapters based around unifying themes of tasks used in secondary mathematics teacher education. These themes reflect goals for mathematics teacher education, and are closely related to various aspects of knowledge required for teaching secondary mathematics. They are not based on the conventional content topics of teacher education (e.g., decimals, grouping practices), but on broad goals such as adaptability, identifying similarities, productive disposition, overcoming barriers, micro simulations, choosing tools, and study of practice. This approach is innovative and appeals both to prominent authors and to our target audiences.

Orit Zaslavsky is an Associate Professor of Mathematics Education at the Department of Education in Technology and Science, Technion - Israel Institute of Technology. She directed several wide scope national professional development projects for secondary mathematics teachers, and teaches undergraduate and graduate courses for prospective and practicing mathematics teachers. Her main research and development projects include: The Interplay between Teachers' Use of Instructional Examples in Mathematics and Students' Learning (3-year study funded by the Israel Science Foundation); and a 5-year national program for promoting excellence and motivating and advancing the potential of high achieving students in learning mathematics (the program focuses on the preparation of secondary teachers to meet these goals).She is currently a guest editor of a special issue of the Journal of Mathematics Teacher Education focusing on the nature and role of tasks for teacher education, and the leading author of a chapter in the 2nd International Handbook Professional Development in Mathematics Education: Trends and Tasks. Peter Sullivan is Professor of Science, Mathematics, and Information Technology Education in Monash University. His main professional achievements are in the field of research. Some major research projects include the Early Numeracy Research Project, and three Australian Research Council (ARC) funded projects: Overcoming barriers in mathematics learning project; the Using interactive media in enhancing teachers' awareness of key characteristics of effecting teaching project; and the Maximizing success in mathematics for disadvantaged students project. He is a member of the Australian Research Council College of Experts for Social Behavioral and Economic Sciences. He is an author of the popular teacher resource Open-ended maths activities: Using good questions to enhance learning that is published in the US as Good questions for math teaching. He is an editor of the Journal of Mathematics Teacher Education.

Autoren/Hrsg.

Sullivan, Peter

Zaslavsky, Orit

Inhaltsverzeichnis

1;Contents;6
2;Contributors;8
3;Setting the Stage: A Conceptual Framework for Examining and Developing Tasks for Mathematics Teacher Education;11
4;Part I ;30
4.1;Instructional Alternatives via a Virtual Setting: Rich Media Supports for Teacher Development;31
4.1.1;Introduction;31
4.1.2;Using Animations in Teacher Preparation;32
4.1.2.1;The “Stories” in the Animated Alternatives;33
4.1.2.1.1;The Alternative “A Correct Solution Method” and Questions for Teacher Discussion;36
4.1.2.1.2;The Alternative “Reasonableness of Answers” and Questions for Teacher Discussion;38
4.1.2.1.3;Comparing the Two Alternatives;39
4.1.3;Mentor/Prospective Teacher Discussion of the Animations;39
4.1.3.1;Discussion of the Alternative “Reasonableness of Answers”;40
4.1.3.2;Discussion of the Alternative “A Correct Solution Method”;42
4.1.4;Concluding Remarks;44
4.1.5;References;44
4.2;Classifying and Characterising: Provoking Awareness of the Use of a Natural Power in Mathematics and in Mathematical Pedagogy;46
4.2.1;Introduction;46
4.2.1.1;Classifying and Characterising Manifested in Natural Language;47
4.2.1.2;Classifying and Characterising in Mathematics;48
4.2.1.3;Classifying and Definitions;49
4.2.1.4;Methods as Classification of Tasks;52
4.2.2;Exploiting Classifying and Characterising in the Classroom;54
4.2.2.1;Sorting and Matching Tasks;56
4.2.2.2;Ordering Tasks;57
4.2.3;Classifying and Characterising Pedagogically;58
4.2.4;Dangers of Classifying and Characterising;59
4.2.5;Conclusion;60
4.2.6;References;61
4.3;Designing Tasks that Challenge Values, Beliefs and Practices: A Model for the Professional Development of Practising Teachers;63
4.3.1;Introduction;63
4.3.2;Four Stages in Professional Development;64
4.3.2.1;Recognising Existing Values, Beliefs and Practices;64
4.3.2.2;Analyse Discussion-Based Practices;67
4.3.2.2.1;Classifying Mathematical Objects;69
4.3.2.2.2;Evaluating Mathematical Statements;70
4.3.2.2.3;Interpreting Multiple Representations;70
4.3.2.2.4;Creating Problems for Others to Solve;71
4.3.2.2.5;Comparing Solution Strategies on Unstructured Problems;72
4.3.2.3;Listening to Students;72
4.3.2.4;A New Classroom Culture;73
4.3.2.5;Suspending Disbelief and Adopting New Practices;74
4.3.2.6;Reflecting on Experience;75
4.3.3;Concluding Remarks;76
4.3.4;References;77
4.4;Pedagogical, Mathematical, and Epistemological Goals in Designing Cognitive Conflict Tasks for Teacher Education;78
4.4.1;Introduction;78
4.4.1.1;The Emergence of Cognitive Conflict as an Instructional Strategy;78
4.4.1.2;Cognitive Conflict in Mathematics and Science Teacher Education;79
4.4.2;Three Examples of Cognitive Conflict Goals;80
4.4.3;A Pedagogical Goal: Trading Places;81
4.4.3.1;Background and Design;81
4.4.3.2;Task Implementation;82
4.4.4;A Mathematical (and Pedagogical) Goal: Father and Son;84
4.4.4.1;Background and Task Design;84
4.4.4.2;Task Implementation;85
4.4.4.2.1;The Country Fair Problem:;86
4.4.5;An Epistemological Goal: The Lemonade Stand;87
4.4.5.1;Background and Task Design;87
4.4.5.2;Task Implementation;89
4.4.5.2.1;The Lemonade Stand Problem:;89
4.4.6;The Nature of Examples: A Comparison;90
4.4.7;Concluding Remarks;91
4.4.8;References;92
4.5;Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics;93
4.5.1;Introduction;93
4.5.2;Some Theoretical Background on Using Tasks to Learn to Teach Mathematics;94
4.5.3;Working with a Line Segment to Think About Shifts of Understanding;97
4.5.4;Working with Mental Calculations to Explore Links Between Algebra and Arithmetic;99
4.5.5;Exploring the Meaning of Algebra;101
4.5.6;Coda;104
4.5.7;References;104
4.6;Didactical Variability in Teacher Education;106
4.6.1;Introduction;106
4.6.2;Teachers’ Variability;108
4.6.3;Consequences for Teacher Training;109
4.6.3.1;Example of an Activity Aiming to Developing Prospective Teacher’ Variability;109
4.6.4;Discussion;111
4.6.4.1;Interpretation of Effects of Variability;111
4.6.4.2;Variability and Teacher Training;115
4.6.5;Appendix: Extract from “Fashion World Magazine”;116
4.6.6;References;117
4.7;Bridging Between Mathematics and Education Courses: Strategy Games as Generators of Problem Solving and Proving Tasks;119
4.7.1;Introduction: The Problem of Bridging Between Mathematics Courses and Education Courses;119
4.7.1.1;Bridging Courses—A Possible Solution;120
4.7.1.1.1;Experimentation;120
4.7.1.1.2;Assessment;120
4.7.1.1.3;Theoretical Anchors;121
4.7.1.1.4;Preparing and Conducting Such Courses;121
4.7.2;The Case of Strategy Games as Generators of Problem Solving and Proving Tasks;122
4.7.2.1;Sample Task 1: Nim Games;122
4.7.2.1.1;Sample Task 1: Student Handout;123
4.7.2.1.2;Sample Task 1: Classroom Management;124
4.7.2.1.3;Related Task-Design Issues for Sample Task 1;126
4.7.2.1.4;Students’ Response to Sample Task 1;126
4.7.2.2;Sample Task 2;126
4.7.2.2.1;Sample Task 2: Student Handout 1 (of 3);127
4.7.2.2.2;Comments;128
4.7.2.2.3;Sample Task 2: Student Handout 2 (of 3);129
4.7.2.2.4;Solutions to the Problems in Student Handout 2;129
4.7.2.2.5;Sample Task 2: Student Handout 3 (of 3);132
4.7.2.2.6;Solutions to the Advanced Problems in Student Handout 3;134
4.7.2.2.7;Sample Task 2: Main Concepts, Skills, and Strategies;136
4.7.2.2.8;Sample Task 2: Classroom Management;137
4.7.2.2.9;Sample Task 2: Related Task-Design Issues;138
4.7.2.2.10;Students’ Response to Sample Task 2;139
4.7.3;Main Points for Students’ Discussion in This Bridging Course;140
4.7.4;Wrapping-up;140
4.7.5;References;142
5;Part II ;143
5.1;Mediating Mathematics Teaching Development and Pupils’ Mathematics Learning: The Life Cycle of a Task;144
5.1.1;Introduction;144
5.1.2;Theoretical Development;146
5.1.2.1;Inquiry Community;146
5.1.2.2;Project Activity;147
5.1.2.3;Using Activity Theory to Address Issues from Activity Using Tasks;149
5.1.3;The Mirror Task;152
5.1.3.1;Didacticians’ Design of the Task;152
5.1.3.2;The Wider Activity Related to the Task;153
5.1.3.3;Discussing and Planning in the Workshop;155
5.1.4;Discussion and Conclusions: Tensions in Mediated Activity;157
5.1.5;References;160
5.2;Learning from the Key Tasks of Lesson Study;162
5.2.1;Introduction;162
5.2.2;Task 1: Development of a Research Theme;164
5.2.3;Task 2: Solve the Mathematical Task in Order to Anticipate Student Thinking;166
5.2.4;Task 3: Development of a Shared Teaching-Learning Plan;168
5.2.5;Task 4: Enactment of the Research Lesson with Data Collection;172
5.2.6;Task 5: Discussion of the Research Lesson;173
5.2.7;How Do the Tasks of Lesson Study Support Teachers’ Learning?;175
5.2.8;References;176
5.3;Mathematical Problem Solving: Linking Theory and Practice;178
5.3.1;Introduction;178
5.3.2;Background to the Tasks;180
5.3.2.1;A Circular Flower Bed;181
5.3.2.2;Solve 4 Problems;182
5.3.3;Development of the Theory and Practice Linkage;182
5.3.3.1;A Circular Flower Bed;182
5.3.3.2;Solve 4 Problems;184
5.3.4;Conclusion;185
5.3.5;Appendix A;186
5.3.6;Appendix B;187
5.3.7;Appendix C;188
5.3.8;References;188
6;Part III ;190
6.1;Guiding Mathematical Inquiry in Mobile Settings;191
6.1.1;Introduction;191
6.1.2;Designing Inquiry Tasks in Mobile Settings: Theoretical Considerations;192
6.1.2.1;Modelling and Representing Real-Life Phenomena in Secondary Mathematics;193
6.1.2.2;Social Interactions and Mathematics Teaching and Learning;194
6.1.2.3;Handheld Devices in Mathematics Education;194
6.1.3;Mathematical Tools and Tasks: Designing a Setting for Mobile Inquiry;195
6.1.3.1;Math4Mobile Applications;195
6.1.3.1.1;SMS Sketching Exercise;196
6.1.3.1.2;In-Class Measuring Task: The Motion of a Toy Car;197
6.1.3.1.3;Exploration Task: Analyzing a Position vs. Time Function;197
6.1.3.1.4;SMS Solving Exercise;198
6.1.3.2;Sequence of Tasks for Secondary Mathematics Teacher Education;198
6.1.3.2.1;Rationale, Goals, and Context;198
6.1.3.2.2;Task Sequence;199
6.1.3.3;Presenting the Task in Teacher-Education Sessions;200
6.1.3.4;Mediating Prospective Teachers’ Mathematical Mobile Discussion on Modelling Task: Summary;202
6.1.4;Opportunities and Challenges;204
6.1.5;References;205
6.2;Technology Integration in Secondary Mathematics: Enhancing the Professionalisation of Prospective Teachers;208
6.2.1;Introduction;208
6.2.2;Conceptual Framework;209
6.2.2.1;Technology as a Cultural Tool;209
6.2.2.2;Learning to Teach in a Community of Practice;210
6.2.3;Program Structure;211
6.2.4;Design of the Task;211
6.2.5;Preparing for the Technology Seminar Task;212
6.2.6;Responses to the Technology Seminar Assessment Task;214
6.2.6.1;Example 1: Unsuccessful Response to the Task (Optimisation Using a Spreadsheet);214
6.2.6.2;Example 2: Task Response Modified During Seminar Presentation (Fitting a Function to Data);216
6.2.6.3;Example 3: Successful Response to the Task (Modelling with a Spreadsheet);217
6.2.7;Teacher Educator Reflections;220
6.2.8;Appendix;221
6.2.9;References;223
6.3;Mathematical Machines: From History to Mathematics Classroom;225
6.3.1;Introduction;225
6.3.2;Some Theoretical Elements;227
6.3.2.1;Mathematical Laboratory and Teachers Education;227
6.3.2.2;Tools of Semiotic Mediation;228
6.3.3;Historic-Epistemological Dimension;231
6.3.3.1;The Background;231
6.3.3.2;Drawings and Texts as Artefacts;232
6.3.3.3;The Task;233
6.3.4;Manipulative Dimension;235
6.3.4.1;The Background;235
6.3.4.2;The Task;236
6.3.5;Digital Dimension;237
6.3.5.1;The Background;237
6.3.5.2;The Task;238
6.3.5.3;First Solution;238
6.3.5.4;Second Solution;239
6.3.6;Concluding Remarks;241
6.3.7;References;242
7;Part IV ;244
7.1;Using Video Episodes to Reflect on the Role of the Teacher in Mathematical Discussions;245
7.1.1;Introduction;245
7.1.2;Exploratory Tasks in the Mathematics Classroom;245
7.1.3;Teacher Education to Transform Classroom Practice;248
7.1.4;The Teacher Education Task and Context;248
7.1.4.1;The General Setting;248
7.1.4.2;The Classroom Situation;249
7.1.4.3;The Workshop Setting;250
7.1.4.4;The Teacher Education Task;251
7.1.5;Discussion;254
7.1.5.1;In Relation to Algebraic Thinking;254
7.1.5.2;Teachers’ Perceptions of Students’ Difficulties and About Classroom Discussions;255
7.1.5.3;The Value of This Teacher Education Activity;255
7.1.6;Conclusion;256
7.1.7;References;256
7.2;Sensitivity to Student Learning: A Possible Way to Enhance Teachers’ and Students’ Learning?;258
7.2.1;Introduction;258
7.2.2;Learning Study: A Systematic Enquiry Approach into Students’ Learning and Understanding;260
7.2.3;A Learning Study About Negative Numbers;261
7.2.3.1;Anticipating Learning Difficulties and Planning of the First Lesson in the Cycle;262
7.2.3.2;Implementing the First Lesson in the Cycle (L1);263
7.2.3.2.1;Analysing the First and Planning the Second Lesson: Subtraction Has a Double Meaning;265
7.2.3.3;Implementing the Second Lesson in the Cycle (L2);266
7.2.3.4;Analysing the Second and Planning the Third Lesson: ‘A View Turn’;266
7.2.3.5;Implementing the Third Lesson in the Cycle (L3);267
7.2.3.6;Analysing the Third and Planning the Last Lesson: The Critical Aspects Emerge;268
7.2.3.7;Implementing the Fourth Lesson in the Cycle (L4);268
7.2.4;Sensitivity and a Systematic Approach to Enhance Student Learning;270
7.2.5;Learning Study: A Possibility for Learning to Teach and for Professional Development?;271
7.2.6;References;272
7.3;Overcoming Pedagogical Barriers Associated with Exploratory Tasks in a College Geometry Course;274
7.3.1;Focus and Significance;274
7.3.2;Theoretical Perspectives;274
7.3.3;Tasks from a College Geometry Textbook;275
7.3.4;Preliminary Pedagogical Considerations;279
7.3.5;Abductive Processes;280
7.3.6;Hindsight and Pedagogical Implications;282
7.3.7;References;284
7.4;Using a Model for Planning and Teaching Lessons as Part of Mathematics Teacher Education;286
7.4.1;Introduction;286
7.4.2;The Research that Informed the Planning and Teaching Model;288
7.4.3;The Elements of the Planning and Teaching Model;288
7.4.3.1;The Classroom Tasks and Their Sequence;288
7.4.3.1.1;Enabling Prompts;291
7.4.3.1.2;Extending Prompts;292
7.4.3.1.3;Explicit Pedagogies;292
7.4.3.1.4;Learning Community;292
7.4.4;An Example of an Implementation of the Planning and Teaching Model;293
7.4.4.1;Lesson One;294
7.4.4.2;Lesson Two;297
7.4.5;The Task for Prospective Teachers;298
7.4.6;References;300
7.5;Building Optimism in Prospective Mathematics Teachers;302
7.5.1;Introduction;302
7.5.2;The Context;303
7.5.3;Theoretically Framing the MCS Pedagogy;303
7.5.3.1;Engaged to Learn Model;304
7.5.3.2;Three-Task Sequence;308
7.5.3.2.1;Task 1 Features and Enactment;308
7.5.3.2.2;Task 2 Features and Enactment;313
7.5.3.2.3;Task 3 Features and Enactment;313
7.5.4;Meta-cognitive Overlay;314
7.5.4.1;Discussion;314
7.5.4.1.1;Mathematical Knowledge;315
7.5.4.1.2;Pedagogical Knowledge;315
7.5.4.1.3;Optimism Building;316
7.5.5;References;317
8;Index;319