Leikin / Zazkis Learning Through Teaching Mathematics

Development of Teachers' Knowledge and Expertise in Practice
1. Auflage 2010
ISBN: 978-90-481-3990-3
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 5, 300 Seiten

Reihe: Mathematics Teacher Education

The idea of teachers Learning through Teaching (LTT) - when presented to a naïve bystander - appears as an oxymoron. Are we not supposed to learn before we teach? After all, under the usual circumstances, learning is the task for those who are being taught, not of those who teach. However, this book is about the learning of teachers, not the learning of students. It is an ancient wisdom that the best way to 'truly learn' something is to teach it to others. Nevertheless, once a teacher has taught a particular topic or concept and, consequently, 'truly learned' it, what is left for this teacher to learn? As evident in this book, the experience of teaching presents teachers with an exciting opp- tunity for learning throughout their entire career. This means acquiring a 'better' understanding of what is being taught, and, moreover, learning a variety of new things. What these new things may be and how they are learned is addressed in the collection of chapters in this volume. LTT is acknowledged by multiple researchers and mathematics educators. In the rst chapter, Leikin and Zazkis review literature that recognizes this phenomenon and stress that only a small number of studies attend systematically to LTT p- cesses. The authors in this volume purposefully analyze the teaching of mathematics as a source for teachers' own learning.

Autoren/Hrsg.

Leikin, Roza

Zazkis, Rina

Inhaltsverzeichnis

1;Contents;6
2;Contributors;8
3;Introduction;10
3.1;References;12
4;Part I Theoretical and Methodological Perspectives on Teachers Learning Through Teaching;14
4.1;Teachers Opportunities to Learn Mathematics Through Teaching;15
4.1.1; Introduction;15
4.1.2; Evidence for Learning Through Teaching;16
4.1.2.1; Theories of Teacher Knowledge and Teaching;16
4.1.2.2; Teachers' Experience and Expertise;17
4.1.2.3; Teaching Experiments and Changing Approaches to Teaching;18
4.1.3; Teachers Knowledge;19
4.1.4; A View of Teaching and Learning;20
4.1.5; Mechanisms of LTT;21
4.1.5.1; Example 1: Learning Mathematics with Students: The Case of Einat;21
4.1.5.1.1; Planning Stage;22
4.1.5.1.2; Interactive Stage;23
4.1.5.1.3; Reflection: Unexpected Questions and Answers;23
4.1.5.2; Interactions as a Source of Einat's Learning;24
4.1.6; What Changes in Teachers Knowledge Occur Through Teaching?;25
4.1.6.1; What Changed in Einat's Knowledge?;25
4.1.6.2; Example 2: Learning from a Student's Mistake: The Case of Lora;26
4.1.6.2.1; What Did Lora Learn from the Above Interaction?;26
4.1.6.3; Example 3: Learning from a Student's Solution: The Case of Shelly;27
4.1.6.3.1; What Did Shelly Learn in This Episode?;27
4.1.6.4; Example 4: Learning from a Student's Question: The Case of Eva;28
4.1.6.4.1; What Did Eva Learn from This Lesson?;28
4.1.6.4.2; Sources of LTT;29
4.1.7; Is This Knowledge New? Is It Mathematics or Pedagogy?;29
4.1.8; The Complexity of LTT: Supporting and Impeding Factors;30
4.1.9;References;31
4.2;Attention and Intention in Learning About Teaching ThroughTeaching;34
4.2.1; Introduction;34
4.2.2; Theoretical Underpinnings;35
4.2.2.1; Metaphors;38
4.2.2.1.1; Cause and Effect;38
4.2.2.1.2; Social Influence as Forces;38
4.2.2.1.3; Learning as Change of State;39
4.2.2.1.4; Human Psyche as Chariot;39
4.2.2.2; Phenomena;40
4.2.2.2.1; Phenomena A (Own World);41
4.2.2.2.2; Phenomenon B (Expressing to Others);41
4.2.2.2.3; Phenomenon C (Training Behavior);41
4.2.2.2.4; Phenomenon D (Hidden Assumptions);42
4.2.2.2.5; General Comment;43
4.2.3; Experience;43
4.2.3.1; Reflection;44
4.2.3.2; Distanciation;44
4.2.3.3; Construal Through Story Telling;45
4.2.4; Attention;46
4.2.4.1; Reacting and Responding; Habit and Choice;48
4.2.4.1.1; Complication;49
4.2.5; Intention and Will;49
4.2.5.1; Disciplined Development;51
4.2.5.2; Maintenance;52
4.2.6; What Can Teachers Do?;53
4.2.6.1; What Can Teachers Do for Students?;53
4.2.6.2; What Can Teachers Do for Themselves?;54
4.2.6.3; What Can Teachers Do for Each Other?;54
4.2.7; Conclusion;54
4.2.8;References;55
4.3;How and What Might Teachers Learn Through Teaching Mathematics: Contributions to Closing an Unspoken Gap;59
4.3.1; Introduction;59
4.3.2; Rationale;59
4.3.3; Accounting for How and What Can Teachers Learn Through Teaching;62
4.3.3.1; Activity--Effect Relationship: Assimilation, Anticipation, and Reflection;62
4.3.3.2; How Might Teachers Learn Through Teaching?;64
4.3.3.3; What Might Teachers Learn Through Teaching?;66
4.3.3.3.1; Teaching Perspectives;66
4.3.3.3.2; A three-Prong Pedagogical Approach;67
4.3.3.3.3; LTT and Mathematics Knowledge for Teaching;68
4.3.3.3.4; LTT and Task/Lesson Design/Adjustment;69
4.3.3.3.5; LTT and Epistemological Paradigm Shift;70
4.3.4; Concluding Remarks;72
4.3.5;References;73
4.4;Learning Through Teaching Through the Lens of Multiple Solution Tasks;78
4.4.1; Introduction;78
4.4.2; Methodology of LTT Research;79
4.4.3; The Case of Rachel;81
4.4.3.1; Five Proofs Presented at the Lesson;82
4.4.3.1.1; Median-Based Proofs;82
4.4.3.1.2; Rotation-Based Proofs;82
4.4.3.1.3; Proof 1.5: We Obtain One More Square Inside;83
4.4.3.2; Not Only the Proofs, Not Only at the Lesson;84
4.4.4; LTT as a Transformation of Solution Spaces for MSTs;86
4.4.5; Using Connecting Tasks to Promote Teachers Learning;88
4.4.5.1; LTT Depended on the Conventionality of the Tasks;90
4.4.6; Conclusions;91
4.4.6.1; MSTs, LTT, and Didactic Situations;91
4.4.6.2; Relationships Between Pedagogy and Mathematics in LTT;92
4.4.7;References;93
5;Part II Examples of Learning Through Teaching: Pedagogical Mathematics;95
5.1; Pedagogical Mathematics;95
5.2; References;97
5.3;What Have I Learned: Mathematical Insights and Pedagogical Implications;98
5.3.1; Disaggregated Perspective on Learning;99
5.3.2; Story 1: Counterintuitive Translation of Parabola (pM);100
5.3.2.1; Teachers' Explanations;101
5.3.2.1.1; Citing Rules;101
5.3.2.1.2; Pointwise Approach;101
5.3.2.1.3; Attending to Zero and ''Making Up'';102
5.3.2.1.4; Transforming Axes;102
5.3.2.1.5; Search for Consistency;102
5.3.2.2; Pedagogical Approach: Rerouting;103
5.3.2.2.1; Translation on a Coordinate Plane;103
5.3.2.2.2; Translating a Parabola;104
5.3.2.2.3; Searching for Consistency Again: Illusion in a Linear Transformation;106
5.3.3; Story 2: Geometry with Affine Coordinates (mM);107
5.3.3.1; Affine Coordinates: An Introduction;107
5.3.3.2; Exploring Medians;108
5.3.3.3; Exploring Tridians;110
5.3.4; Story 3 Bellboy and the Missing Dollar (mP);113
5.3.4.1; Pedagogical Approach Via Mathematical Variation;114
5.3.4.2; Example 1 -- Division with Decimals;114
5.3.4.3; Example 2: ''Big'' Percentage;115
5.3.5; Conclusion;116
5.3.6;References;116
5.4;Dialogical Education and Learning Mathematics Onlinefrom Teachers;118
5.4.1; Introduction;118
5.4.2; From Inter-shaping Relationship to the Notion of Humans-with-Media;119
5.4.3; Making the Risk Zone More Comfortable: Online Courses to Teach How to Use Software in the Mathematics Classroom;120
5.4.4; The Context;122
5.4.5; The Example;124
5.4.6; Online Collaboration to Foster Our Learning of Mathematics;130
5.4.7;References;131
5.5;Role of Task and Technology in Provoking Teacher Change:A Case of Proofs and Proving in High School Algebra;133
5.5.1; The Context of the Present Study;133
5.5.2; Michaels Story;134
5.5.2.1; Some Background;134
5.5.2.2; The xn -- 1 Task;135
5.5.2.3; Our Classroom Observations;137
5.5.2.3.1; Proof 1: A General Approach Based on the Difference of Squares;138
5.5.2.3.2; A Proposed Counterexample Involving the Sum of Cubes;139
5.5.2.3.3; The Counterproposal Containing Seeds of a Generic Proof;139
5.5.2.3.4; Analysis of Proof 1;140
5.5.2.3.5; Proof 2: A Proof Involving Factoring by Grouping;140
5.5.2.3.6; A Student's Query Related to the Factor (x+ 1);141
5.5.2.3.7; Analysis of Proof 2;142
5.5.2.3.8; Proof 3: A New Conjecture Involving xn+ 1 Where n Is an Odd Integer;142
5.5.2.3.9; Analysis of Proof 3;143
5.5.2.3.10; A Few Remarks Regarding the Proving Part of the Activity;144
5.5.2.4; The Subsequent Interview with Michael;145
5.5.2.4.1; His Initial Expectations -- Extract 1;146
5.5.2.4.2; His Changed Views -- Extract 2;146
5.5.2.4.3; Brief Commentary on Extracts 1 and 2;146
5.5.2.4.4; The Tasks and the Technology -- Extract 3;147
5.5.2.4.5; Change in His Teaching -- Extract 4;147
5.5.2.4.6; Pushing Students to Go Farther Mathematically -- Extract 5;148
5.5.2.4.7; Brief Commentary on Extracts 3, 4, and 5;148
5.5.2.4.8; Using Technology to Increase Student Involvement and Promote Learning -- Extract 6;149
5.5.2.4.9; Future Plans -- Extract 7;149
5.5.2.4.10; Brief Commentary on Extracts 6 and 7;150
5.5.3; Analysis and Discussion;150
5.5.3.1; What Changed in Michael;150
5.5.3.1.1; His Knowledge of Mathematics;150
5.5.3.1.2; His Knowledge of Mathematics Teaching and Learning;151
5.5.3.1.3; His Practice in Subsequent Mathematics Classes;152
5.5.3.2; What Enabled These Changes;152
5.5.3.2.1; Access to the Resources and Support Offered by the Research Group;152
5.5.3.2.2; Use of CAS-Supported Tasks Whose Mathematical Content Differed from That Usually Touched upon in Class;153
5.5.3.2.3; Michael's Disposition Toward Student Reflection and Student Learning of Mathematics;154
5.5.3.2.4; The Quality of the Reflections of His Own Students on These Tasks;154
5.5.3.2.5; Michael's Attitude with Respect to His Own Learning;155
5.5.3.3; Reflections on What Changed in Michael and on What Enabled These Changes;155
5.5.4; Concluding Remarks;156
5.5.5;References;157
5.6;Learning Through Teaching, When Teaching Machines: Discursive Interaction Design in Sketchpad;159
5.6.1; Introduction;159
5.6.2; Discursive Roles of the Computer;160
5.6.2.1; Organization of Discourse Around Evaluation;161
5.6.2.2; Breaking Conversational Maxims;162
5.6.2.3; Authority and Imperatives;163
5.6.2.4; Neither Medium Nor Teacher;163
5.6.3; The Sketchpad Trajectory Through Language;165
5.6.3.1; Pre-verbal Origins;165
5.6.3.2; The Emergence of Vocabulary;166
5.6.3.3; From Vocabulary to Grammar; From Grammar to Plot;168
5.6.3.4; Plot Summary and the Moral of the Story;169
5.6.4; Conclusion;172
5.6.5;References;173
5.7;What Experienced Teachers Have Learned from HelpingStudents Think About Solving Equationsin the One-Variable-First Algebra Curriculum;175
5.7.1; Introduction;175
5.7.2; The Difficult Task of Choosing Explanatory Starting Points;175
5.7.3; A One-Variable-First Perspective on Solving Equations and Systems of Equations;176
5.7.4; Context;179
5.7.4.1; The Text;180
5.7.4.2; Our Interview and Classroom Observation;181
5.7.5; Ms. Alley and Ms. Lewis;183
5.7.5.1; On Equations: What Are They? What Is Their Place in the School Algebra Curriculum?;183
5.7.5.2; On Solving: What!s the Name of the Game? Isolate the Variable!;184
5.7.5.3; On Identities and Contradictions: Assigning a Label;187
5.7.5.4; On Solving Equations in Two Variables: Changing the Name of the Game;188
5.7.6; Evidence for Learning Through Teaching;189
5.7.7;References;192
6;Part III Examples of Learning Through Teaching: Mathematical Pedagogy;194
6.1; Mathematical Pedagogy;194
6.2; References;195
6.3;Exploring Reform Ideas for Teaching Algebra: Analysis of Videotaped Episodes and of Conversations About Them;197
6.3.1; The Challenge of Whole-Group Algebra Inquiry;197
6.3.2; Learning Through Teaching: First Round;200
6.3.2.1; Choosing the Video Episodes;200
6.3.2.2; Comparative Analysis of Interaction and Authority;200
6.3.3; Learning Through Teaching: Second Round;205
6.3.3.1; Constructions by Teachers Who Practice Guided Inquiry;205
6.3.3.1.1; How Does One Learn to Discuss?;205
6.3.3.1.2; Who Convinces Whom? Is This the Teacher's or the Students' Job?;206
6.3.3.1.3; Why Is Mathematic Discussion Necessary?;206
6.3.3.1.4; What Does It Mean to ''Be With Me'' in a Discussion?;207
6.3.3.1.5; What Is an Appropriate Task?;207
6.3.3.2; Constructions by Teachers Who Teach Algebra Traditionally;208
6.3.3.3; Learning Through Teaching: Constructions That Uncover Teaching Practices;210
6.3.4; Concluding Remark;211
6.3.5;References;212
6.4;On Rapid Professional Growth: Cases of LearningThrough Teaching;214
6.4.1; Introduction;214
6.4.2; Method;214
6.4.3; The Case of Mary;216
6.4.3.1; Mary's Transformation as a Case of Conceptual Change;217
6.4.4; The Case of Mitchell;219
6.4.4.1; Mitchell's Transformation as a Case of Accommodating Outliers;221
6.4.5; The Case of Danica;223
6.4.5.1; Danica's Transformation as a Case of Reification;224
6.4.6; Discussion on Cases of Rapid Professional Development;225
6.4.7; Discussion on Professional Development Through Teaching;226
6.4.8; Conclusion;228
6.4.9;References;228
6.5;Interactions Between Teaching and Research: Developing Pedagogical Content Knowledge for Real Analysis;231
6.5.1; Introduction;231
6.5.2; My Current Approach to Lecturing;232
6.5.3; Five Current Practices;232
6.5.3.1; Regular Testing on Definitions;233
6.5.3.2; Tasks Involving Extending Example Spaces;234
6.5.3.3; Tasks Involving Constructing and Understanding Diagrams;236
6.5.3.4; Resources for Improving Proof Comprehension;239
6.5.3.5; Tasks Involving Mapping the Structure of the Whole Course;242
6.5.4; Discussion: Interactions Between Teaching and Research;244
6.5.5; Conclusion: Overarching Themes in Teaching;245
6.5.6;References;246
6.6;Teachers Learning from Their Teaching: The Case of Communicative Practices;250
6.6.1; Introduction;250
6.6.2; Background of the Study;252
6.6.3; Insights Related to Oral Language;255
6.6.3.1; I Used to Be a Big Funneler;255
6.6.3.2; Letting Kids Develop Their Own Words for Concepts;256
6.6.4; Insights Related to Mathematical Writing;258
6.6.4.1; Writing Over Time … Because I Actually Saw Growth in the Students;258
6.6.5; Insights Related to Mathematical Reading;259
6.6.5.1; The Reading ''Is Where We Lose a Lot of Kids'';259
6.6.6; Some Broader Changes;260
6.6.6.1; Reading, Writing, and Speaking Really Are Central to Math;261
6.6.6.2; Making the Program ''A Little More Traditional. That''s What I Was Doing.'';261
6.6.7; Discussion;262
6.6.8;References;264
6.7;Feedback: Expanding a Repertoire and Making Choices;266
6.7.1; The Links Lesson;268
6.7.1.1; Ball and Box Activity;271
6.7.1.2; Polyominoes;276
6.7.1.3; Silent Lesson;276
6.7.2; Two Student Teachers Lessons;280
6.7.2.1; Steve's Lesson;280
6.7.2.2; Dianne's Lesson;283
6.7.3; Final Remarks;285
6.7.4;References;287
7;Conclusion;289
7.1; Communication and Interaction;289
7.2; Critical Event and Search for Equilibrium;290
7.3;References;292
8;Name Index;293
9;Subject Index;299

Leseproben

"What Changes in Teachers’ Knowledge Occur Through Teaching? (p. 13-14)

What Changed in Einat’s Knowledge?

The unpredicted turn that the lesson took in relation to the solution of Problem 2 nurtured Einat’s learning of mathematics. According to her plan, Problem 2 was aimed at performing calculations using the Pythagorean theorem. But when a student raised an unforeseen (general) question related to the length of the two paths (Fig. 1d), new mathematical connections were constructed: the paths within the rectangle could be compared using the properties of triangles with equal areas and a constant basis or using the properties of the ellipse.

When Einat moved the internal rectangle from the center of the external one, it became clear to her that the length of the two paths will be different “because the position of the internal rectangle is asymmetric.” This intuitive assumption appeared to be correct for the concrete situation presented in Problem 2, but it was incorrect as a general statement. Einat discovered that not all asymmetrical positions of the internal rectangle resulted in paths of different lengths.

When points E and F are on the ellipse, the paths are equal in length. Thus, an incorrect intuitive assumption was refuted, and incorrect intuitions were replaced with correct mathematical knowledge. The second critical point for Einat’s learning was her intuitive agreement with Opher’s conjecture, which was also refuted. Our additional observation is related to the interrelationship between Einat’s mathematical and pedagogical knowledge. It was her pedagogical sensitivity that encouraged her to formulate new problems that led to mathematical discoveries. At the same time her mathematical knowledge allowed her to evaluate the difficulty of the refutations she had produced, and (again) being attentive to her students she designed a new instructional activity using the Dynamic Geometry software. In sum, in this example, we recognize the development of knowledge in the transformation of intuition into formal knowledge and in the mutual support between pedagogical and subject matter knowledge.

Example 2: Learning from a Student’s Mistake: The Case of Lora
Lora, an experienced instructor in a course for pre-service elementary school teachers, taught a lesson on elementary number theory. The following interaction took place:
Lora: Is number 7 a divisor of K, where K = 34×56? Student: It will be, once you divide by it.
Lora: What do you mean, once you divide? Do you have to divide?
Student: When you go this [points to K] divided by 7 you have 7 as a divisor, this one the dividend, and what you get also has a name, like a product but not a product. . .

Lora’s intention in choosing this example was to alert students to the unique factorization of a composite number to its prime factors, as promised by the Fundamental Theorem of Arithmetic, and to the resulting fact, that no calculation is needed to determine the answer to her question. This later intention is evident in her probing question.

What Did Lora Learn from the Above Interaction? First, she learned that the term “divisor” is ambiguous, and that a distinction is essential between divisor of a number, as a relationship in a number-theoretic sense, and divisor in a number sentence, as a role played in a division situation. She learned further that the student assigned meaning based on his prior schooling and not on his recent classroom experience, in which the definition for “divisor” used in Number Theory was given and its usage illustrated. Before this incident, Lora used the term properly in both cases, but was not alert to a possible misinterpretation by learners. The student’s confusion helped her make the distinction, increased her awareness of the polysemy (different but related meanings) of the term divisor and of the definitions that can be conflicting. This resulted in developing a set of instructional activities in which the terminology is practiced (Zazkis, 1998)."