Early algebraization a global dialogue from multiple perspectives
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| 245 | 1 | 0 | |a Early algebraization |b a global dialogue from multiple perspectives |c Jinfa Cai, Eric Knuth editors |
| 264 | 1 | |a Heidelberg ; Dordrecht ; London ; New York |b Springer |c [2011] | |
| 300 | |a XXIII, 623 Seiten |b Illustrationen, Diagramme |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Advances in mathematics education | |
| 500 | |a Literaturangaben | ||
| 650 | 0 | 7 | |a Algebra |0 (DE-588)4001156-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematikunterricht |0 (DE-588)4037949-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Algebra |0 (DE-588)4001156-2 |D s |
| 689 | 0 | 1 | |a Mathematikunterricht |0 (DE-588)4037949-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Cai, Jinfa |d 1963- |0 (DE-588)1084237490 |4 edt | |
| 700 | 1 | |a Knuth, Eric J. |0 (DE-588)1031113002 |4 edt | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |t Early Algebraization |
| 856 | 4 | 2 | |m X:MVB |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=3563512&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-029222517 | |
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| DE-BY-UBG_katkey | 196100206 |
| DE-BY-UBG_media_number | 013108630845 |
| _version_ | 1811360775502036992 |
| adam_text |
IMAGE 1
CONTENTS
PART I: CURRICULAR PERSPECTIVE
PREFACE TO PART I 3
JINFA CAI AND ERIC KNUTH
FUNCTIONAL THINKING AS A ROUTE INTO ALGEBRA IN THE ELEMENTARY GRADES . 5
MARIA L. BLANTON AND JAMES J. KAPUT INTRODUCTION 6
THE CHALLENGE OF CURRICULUM AND INSTRUCTION 6
FUNCTIONAL THINKING AS A ROUTE TO ALGEBRAIC THINKING . . . 7 FUNCTIONAL
THINKING IN THE ELEMENTARY GRADES 8
CHILDREN'S CAPACITY FOR FUNCTIONAL THINKING 9
INTEGRATING FUNCTIONAL THINKING INTO CURRICULUM AND INSTRUCTION . . 16
TRANSFORMING TEACHERS' RESOURCE BASE TO SUPPORT STUDENTS' FUNCTIONAL
THINKING 17
USING CHILDREN'S FUNCTIONAL THINKING TO LEVERAGE TEACHER LEARNING 19
CREATING CLASSROOM CULTURE AND PRACTICE TO SUPPORT FUNCTIONAL THINKING
20
CONCLUSION 20
REFERENCES 21
DEVELOPING STUDENTS' ALGEBRAIC THINKING IN EARLIER GRADES: LESSONS FROM
CHINA AND SINGAPORE 25
JINFA CAI, SWEE FONG NG, AND JOHN C. MOYER INTRODUCTION 26
FEATURES OF THE CHINESE AND SINGAPOREAN CURRICULA 27
ALGEBRA EMPHASES IN THE CHINESE AND SINGAPOREAN CURRICULA 27
THE CHINESE CURRICULUM 28
THE SINGAPOREAN CURRICULUM 32
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1008239607
DIGITALISIERT DURCH
IMAGE 2
XIV CONTENTS
LESSONS FROM CHINESE AND SINGAPOREAN SCHOOL MATHEMATICS . . . 34 WHY
SHOULD CURRICULA EXPECT STUDENTS IN EARLY GRADES TO THINK ALGEBRAICALLY?
35
ARE YOUNG CHILDREN CAPABLE OF THINKING ALGEBRAICALLY? . . 36 HOW CAN WE
HELP STUDENTS TO THINK ARITHMETICALLY AND ALGEBRAICALLY? 37
ARE AUTHENTIC APPLICATIONS NECESSARY FOR STUDENTS IN EARLY GRADES? 38
CONCLUSION 39
REFERENCES 40
DEVELOPING ALGEBRAIC THINKING IN THE CONTEXT OF ARITHMETIC 43 SUSAN JO
RUSSELL, DEBORAH SCHIFTER, AND VIRGINIA BASTABLE UNDERSTANDING THE
BEHAVIOR OF THE OPERATIONS 45
GENERALIZING AND JUSTIFYING 51
1. ARTICULATING GENERAL CLAIMS 51
2. DEVELOPING A MATHEMATICAL ARGUMENT TO JUSTIFY A GENERAL CLAIM 53
3. REPRESENTATION-BASED PROOF: TOOLS FOR PROVING IN THE ELEMENTARY
GRADES 56
EXTENDING THE NUMBER SYSTEM 59
USING NOTATION WITH MEANING 63
CONNECTING ARITHMETIC AND ALGEBRA 67
REFERENCES 68
THE ROLE OF THEORETICAL ANALYSIS IN DEVELOPING ALGEBRAIC THINKING: A
VYGOTSKIAN PERSPECTIVE 71
JEAN SCHMITTAU INTRODUCTION 71
ORIENTING CHILDREN TO THEORETICAL CONCEPTS 74
ROLE OF PSYCHOLOGICAL TOOLS 76
THE PART-WHOLE RELATION 76
CONCLUDING REMARKS 84
REFERENCES 85
THE ARITHMETIC-ALGEBRA CONNECTION: A HISTORICAL-PEDAGOGICAL PERSPECTIVE
87 K. SUBRAMANIAM AND RAKHI BANERJEE INTRODUCTION 87
ARITHMETIC AND ALGEBRA IN THE INDIAN MATHEMATICAL TRADITION . . . 91
BUILDING ON STUDENTS'UNDERSTANDING OF ARITHMETIC 95
THE ARITHMETIC ALGEBRA CONNECTION-A FRAMEWORK 98 REFERENCES 105
SHIKI: A CRITICAL FOUNDATION FOR SCHOOL ALGEBRA IN JAPANESE ELEMENTARY
SCHOOL MATHEMATICS 109
TAD WATANABE SCHOOL ALGEBRA AND ALGEBRA IN EARLY GRADES 110
IMAGE 3
CONTENTS XV
METHODOLOGY I LL
ALGEBRA IN JAPANESE CURRICULUM 112
MATHEMATICAL EXPRESSIONS IN JAPANESE CURRICULUM 114 MATHEMATICAL
EXPRESSIONS IN JAPANESE TEXTBOOKS 114 DISCUSSION 121
REFERENCES 123
COMMENTARY ON PART I 125
JEREMY KILPATRICK ALGEBRA FIRST 126
A CURRICULUM TOPIC 127
NUMERICAL PATTERNS 128
WORD PROBLEMS 128
MULTIPLE PERSPECTIVES 129
REFERENCES 129
PART II: COGNITIVE PERSPECTIVE
PREFACE TO PART II 135
ERIC KNUTH AND JINFA CAI
ALGEBRAIC THINKING WITH AND WITHOUT ALGEBRAIC REPRESENTATION: A PATHWAY
FOR LEARNING 137
MURRAY S. BRITT AND KATHRYN C. IRWIN INTRODUCTION 138
CHILDREN'S UNDERSTANDING OF GENERALITIES FOR OPERATIONS BEFORE SCHOOLING
139
ALGEBRAIC THINKING AND THE NEW ZEALAND NUMERACY PROJECT . . . 140
STUDENTS' ALGEBRAIC THINKING IN THE LAST YEAR OF INTERMEDIATE SCHOOL
(AGE 11-12) 146
THE GROWTH OF ALGEBRAIC THINKING FROM NUMBERS TO SYMBOLS: A LONGITUDINAL
STUDY 147
DISCUSSION 152
A PATHWAY FOR ALGEBRAIC THINKING 153
REFERENCES 157
EXAMINING STUDENTS' ALGEBRAIC THINKING IN A CURRICULAR CONTEXT: A
LONGITUDINAL STUDY 161
JINFA CAI, JOHN C. MOYER, NING WANG, AND BIKAI NIE STANDARDS-BASED AND
TRADITIONAL CURRICULA IN THE UNITED STATES . . 162 LIECAL PROJECT 163
HIGHLIGHTS OF THE DIFFERENCES BETWEEN CMP AND NON-CMP CURRICULA 164
DEFINING VARIABLES 165
DEFINING EQUATIONS 165
INTRODUCING EQUATION SOLVING 166
USING MATHEMATICAL PROBLEMS 168
IMAGE 4
XVI CONTENTS
HIGHLIGHTS OF THE DIFFERENCES BETWEEN CMP AND NON-CMP CLASSROOM
INSTRUCTION 169
CONCEPTUAL AND PROCEDURAL EMPHASES 170
INSTRUCTIONAL TASKS 171
STUDENTS' DEVELOPMENT OF ALGEBRAIC THINKING: METHODOLOGICAL
CONSIDERATIONS 172
THE FOCUS OF ALGEBRAIC THINKING 173
TASKS AND DATA ANALYSIS 174
FINDINGS ABOUT THE DEVELOPMENT OF STUDENTS' ALGEBRAIC THINKING . 174
REPRESENTING SITUATIONS 175
SOLVING EQUATIONS 177
MAKING GENERALIZATIONS 178
CONCLUSIONS AND INSTRUCTIONAL IMPLICATIONS 180
REFERENCES 183
YEARS 2 TO 6 STUDENTS' ABILITY TO GENERALISE: MODELS, REPRESENTATIONS
AND THEORY FOR TEACHING AND LEARNING 1 87
TOM J. COOPER AND ELIZABETH WARREN PERSPECTIVES ON THE MATHEMATICS OF
EARLY ALGEBRA 188
REPRESENTATION AND GENERALISATION 190
MODELS AND REPRESENTATIONS 191
GENERALISATION 191
FOCUS OF EATP 193
FOCUS OF CHAPTER 194
DESIGN OF EATP 194
FINDINGS AND DISCUSSION 196
PATTERNS 197
CHANGE AND FUNCTIONS 198
EQUATIONS AND EQUIVALENCE 201
GENERALISING PRINCIPLES AND ABSTRACT REPRESENTATIONS 204 CONCLUSIONS AND
IMPLICATIONS 206
MODELS AND REPRESENTATIONS 206
GENERALISATION 207
THEORETICAL FRAMEWORK 209
REFERENCES 211
ALGEBRA IN THE MIDDLE SCHOOL: DEVELOPING FUNCTIONAL RELATIONSHIPS
THROUGH QUANTITATIVE REASONING 215
AMY B. ELLIS WHAT IS QUANTITATIVE REASONING? 216
THE IMPORTANCE OF (AND DIFFICULTIES WITH) FUNCTIONAL THINKING . . 218
AN ALTERNATIVE APPROACH TO FUNCTION: QUANTITIES AND COVARIATION . 222 A
FLEXIBLE UNDERSTANDING OF FUNCTIONS 226
COORDINATING COVARIATION AND CORRESPONDENCE APPROACHES . . 226
FLEXIBILITY ACROSS FORMS 230
IMAGE 5
CONTENTS XVII
FOSTERING A FOCUS ON QUANTITIES 234
REFERENCES 235
REPRESENTATIONAL COMPETENCE AND ALGEBRAIC MODELING 239
ANDREW IZSAEK EARLY RESULTS ON STUDENTS' UNDERSTANDINGS OF STANDARD
REPRESENTATIONS IN ALGEBRA 241
THEORETICAL ACCOUNTS OF REASONING WITH EXTERNAL REPRESENTATIONS . 241
STUDENTS' CAPACITIES TO REASON WITH EXTERNAL REPRESENTATIONS . . . 243
FIRST RESULT: CRITERIA FOR EVALUATING EXTERNAL REPRESENTATIONS . . .
244 SECOND RESULT: ADAPTIVE INTERPRETATION 249
CONCLUSION 253
REFERENCES 256
MIDDLE SCHOOL STUDENTS' UNDERSTANDING OF CORE ALGEBRAIC CONCEPTS:
EQUIVALENCE & VARIABLE 259
ERIC J. KNUTH, MARTHA W. ALIBALI, NICOLE M. MCNEIL, AARON WEINBERG, AND
ANA C. STEPHENS INTRODUCTION 260
STUDENT UNDERSTANDING OF EQUIVALENCE & VARIABLE 261 EQUIVALENCE 261
VARIABLE 262
METHOD 262
PARTICIPANTS 262
DATA COLLECTION 263
CODING 264
RESULTS 266
INTERPRETATION OF THE EQUAL SIGN 266
PERFORMANCE ON THE EQUIVALENT EQUATIONS PROBLEM 267 INTERPRETATION OF A
LITERAL SYMBOL 270
PERFORMANCE ON THE WHICH IS LARGER PROBLEM 271
DISCUSSION 273
EQUIVALENCE RESULTS 273
VARIABLE RESULTS 274
CONCLUDING REMARKS 275
REFERENCES 275
AN APPROACH TO GEOMETRIC AND NUMERIC PATTERNING THAT FOSTERS SECOND
GRADE STUDENTS' REASONING AND GENERALIZING ABOUT FUNCTIONS AND
CO-VARIATION 277
JOAN MOSS AND SUSAN LONDON MCNAB INTRODUCTION 277
OUR PROJECT 279
OUR APPROACH: THEORETICAL 279
INSTRUCTIONAL SEQUENCE 281
VISUAL REPRESENTATION: GEOMETRIC GROWING PATTERNS 281
IMAGE 6
XVIII CONTENTS
NUMERIC REPRESENTATIONS: FUNCTION MACHINE 282
INTEGRATION ACTIVITIES: PATTERN SIDEWALK 283
ROLE OF THE TEACHER 284
PROCEDURES AND MEASURES: GRADE 2 INTERVENTIONS 285
RESULTS 285
FINDING RULES FOR PATTERNS AND GENERATING PATTERNS BASED ON GIVEN RULES
286
CONSTRUCTING A PATTERN FROM A RULE: "A 'NUMBER TIMES TWO, PLUS ONE'
PATTERN?" 286
FINDING A RULE FOR A GIVEN PATTERN: "POSITION NUMBER TIMES THREE, PLUS
ONE" 287
STUDENTS'INVENTION OF MULTIPLICATION 288
DECONSTRUCTING MULTIPLICATION: "DOUBLE THE POSITION, PLUS THE POSITION"
289
USING A STRUCTURAL UNDERSTANDING OF MULTIPLICATION TO PREDICT FAR
POSITIONS: "IT'S 40 UP, AND 3 TO THE SIDE" 289
THE DISCOVERY OF ZERO 291
ZERO AS A COEFFICIENT: "ZERO GROUPS OF 4 MILLION IS ZERO" . . 291 ZERO
AS A POSITION NUMBER: "THE ZERO-TH POSITION" 292 TRANSFER OF STRUCTURE
293
CIRCUMVENTING WHOLE OBJECT REASONING 293
INFORMAL ALGEBRAIC EXPRESSIONS OF RULES IN THE SPARKY PROBLEM 294
DISCUSSION 295
THE CURRICULUM WITH ITS FOCUS ON INTEGRATION 296
PRIORITIZING VISUAL REPRESENTATIONS OF PATTERN 297
PEDAGOGY AND STUDENT INVENTIONS 297
CONCLUDING THOUGHTS 298
REFERENCES 298
GRADE 2 STUDENTS'NON-SYMBOLIC ALGEBRAIC THINKING 303
LUIS RADFORD INTRODUCTION 303
EXTENDING SEQUENCES 305
ABSTRACTION 307
THE BOUNDARIES OF ARITHMETIC AND ALGEBRAIC THINKING 308 LAYERS OF
GENERALITY 311
BEYOND INTUITED INDETERMINACY 312
A GENERAL OVERVIEW 316
SYNTHESIS AND CONCLUDING REMARKS 317
REFERENCES 320
FORMATION OF PATTERN GENERALIZATION INVOLVING LINEAR FIGURAL PATTERNS
AMONG MIDDLE SCHOOL STUDENTS: RESULTS OF A THREE-YEAR STUDY . . 323
F.D. RIVERA AND JOANNE ROSSI BECKER ANTICIPATING WHAT IS TO COME:
INITIAL REFLECTIONS ON OUR
THREE-YEAR DATA FROM THE CLINICAL INTERVIEWS 327
IMAGE 7
CONTENTS XIX
COGNITIVE ISSUES SURROUNDING PATTERN GENERALIZATION: WHAT WE KNOW FROM
VARIOUS THEORETICAL PERSPECTIVES AND EMPIRICAL STUDIES 329
CLARIFYING THE DEFINITION OF PATTERN GENERALIZATION 329 TYPES OF
ALGEBRAIC GENERALIZATION INVOLVING FIGURAL PATTERNS 330 METHODOLOGY 331
CLASSROOM CONTEXTS FROM YEARS 1 TO 3 OF THE STUDY 331 NATURE AND CONTENT
OF CLASSROOM TEACHING EXPERIMENTS IN YEARS 1 AND 2 332
NATURE AND CONTENT OF CLASSROOM TEACHING EXPERIMENTS IN YEAR 3 334
NATURE AND CONTENT OF CLINICAL INTERVIEW TASKS FROM YEARS 1 TO 3 335
DATA COLLECTION AND ANALYSIS AND RELEVANT STUDY PROTOCOLS . 335 FINDINGS
AND DISCUSSION PART 1: ACCOUNTING FOR CONSTRUCTIVE AND DECONSTRUCTIVE
GENERALIZATIONS 338
FINDINGS AND DISCUSSION PART 2: UNDERSTANDING THE OPERATIONS NEEDED IN
DEVELOPING A PATTERN GENERALIZATION 342 FINDINGS AND DISCUSSION PART 3:
FACTORS AFFECTING STUDENTS' ABILITY TO DEVELOP CGS 344
FINDINGS AND DISCUSSION PART 4: A THREE-YEAR ACCOUNT OF CLASSROOM
MATHEMATICAL PRACTICES THAT ENCOURAGED THE FORMATION OF GENERALIZATION
AMONG OUR MIDDLE SCHOOL STUDENTS 347
YEAR 1 CLASSROOM PRACTICES: FROM FIGURALLY- TO NUMERICALLY-DRIVEN CSGS
348
YEAR 2 PRACTICE: CONTINUED USE OF NUMERICALLY-DRIVEN CSGS AND A
REFINEMENT IN THE CASE OF DECREASING LINEAR PATTERNS 351
YEAR 3 PRACTICES: A THIRD SHIFT BACK TO FIGURAL-BASED GENERALIZATION AND
THE CONSEQUENT OCCURRENCE OFCSGS,CNGS,ANDDGS 352
FINDINGS AND DISCUSSION PART 5: MIDDLE SCHOOL STUDENTS' CAPABILITY IN
JUSTIFYING CSGS 354
FINDINGS AND DISCUSSION PART 6: MIDDLE SCHOOL STUDENTS' CAPABILITY IN
CONSTRUCTING AND JUSTIFYING CNGS AND DGS . . 357 CONCLUSION 362
REFERENCES 363
COMMENTARY ON PART II 367
BHARATH SRIRAMAN AND KYEONG-HWA LEE INTRODUCTORY REMARKS 367
EARLY ALGEBRAIZATION VERSUS MEANINGFUL ARITHMETIC 368 GENERALIZED
ARITHMETIC, GENERALIZING, GENERALIZATION 369 FROM HAECKEL TO LAMARCK TO
EARLY ALGEBRAIZATION 370
REFERENCES 372
IMAGE 8
CONTENTS
PART III: INSTRUCTIONAL PERSPECTIVE
PREFACE TO PART III 377
ERIC KNUTH AND JINFA CAI
PROSPECTIVE MIDDLE-SCHOOL MATHEMATICS TEACHERS' KNOWLEDGE OF EQUATIONS
AND INEQUALITIES 379
NERIDA F. ELLERTON AND M.A. (KEN) CLEMENTS THE CONTEXT 379
MATHEMATICAL CONSIDERATIONS RELATING TO THE TEACHING AND LEARNING OF
EQUATIONS AND INEQUALITIES 380
STUDENT MISCONCEPTIONS IN REGARD TO QUADRATIC EQUATIONS . . 383 STUDENT
MISCONCEPTIONS WITH REGARD TO LINEAR INEQUALITIES . . 384 THE
PRE-SERVICE TEACHERS INVOLVED, AND TASKS USED, IN THE PRESENT STUDY 386
"CLEVER" TASKS 387
DEVELOPING THE PENCIL-AND-PAPER INSTRUMENTS 389
THE EIGHT EQUATION/ALGEBRAIC INEQUALITY PAIRS 389
STUDY DESIGN, AND RESULTS 395
POPULATION AND SAMPLE CONSIDERATIONS 395
RESULTS 396
CONCLUSIONS IN RELATION TO THE PROSPECTIVE TEACHERS' KNOWLEDGE OF
ALGEBRAIC INEQUALITIES 399
PROSPECTIVE TEACHERS' KNOWLEDGE IN RELATION TO QUADRATIC EQUATIONS 401
BAD NEWS, GOOD NEWS AND SOME CONCLUDING COMMENTS 402 BAD NEWS 402
GOOD NEWS 403
STUDENT CONFIDENCE CONSIDERATIONS 406
CONCLUDING COMMENTS 406
REFERENCES 407
THE ALGEBRAIC NATURE OF FRACTIONS: DEVELOPING RELATIONAL THINKING IN
ELEMENTARY SCHOOL 409
SUSAN B. EMPSON, LINDA LEVI, AND THOMAS P. CARPENTER WHAT IS RELATIONAL
THINKING? 411
USE OF RELATIONAL THINKING IN LEARNING FRACTIONS 413
UNDERSTANDING FRACTIONAL QUANTITIES THROUGH RELATIONAL THINKING 413
USE OF RELATIONAL THINKING TO MAKE SENSE OF OPERATIONS INVOLVING
FRACTIONS 416
DISCUSSION OF CASES 422
A CONJECTURE CONCERNING RELATIONAL THINKING AS A TOOL IN LEARNING NEW
NUMBER CONTENT 423
IMAGE 9
CONTENTS XXI
CONCLUSION 425
REFERENCES 426
PROFESSIONAL DEVELOPMENT TO SUPPORT STUDENTS' ALGEBRAIC REASONING: AN
EXAMPLE FROM THE PROBLEM-SOLVING CYCLE MODEL 429 KAREN KOELLNER,
JENNIFER JACOBS, HILDA BORKO, SARAH ROBERTS, AND CRAIG SCHNEIDER
INTRODUCTION 430
THE PROBLEM-SOLVING CYCLE MODEL OF PROFESSIONAL DEVELOPMENT. . 431 THE
PSC AS IMPLEMENTED IN THE STAAR PROJECT 432 PRIOR RESEARCH ON THE
DEVELOPMENT AND IMPACT OF THE PSC . . 435 IMPACT OF THE PSC ON
INSTRUCTIONAL PRACTICE: A CASE STUDY
ANALYSIS 436
METHODS 436
KEN BRYANT 436
DATA SOURCES 437
DATA ANALYSIS 438
RESULTS AND DISCUSSION 440
PATTERNS DRAWN FROM QMI CODING AND ANALYSIS 440 VIGNETTE ANALYSIS: KEN'S
SKYSCRAPER WINDOWS LESSON . . . 447 CONCLUSIONS 450
REFERENCES 451
USING HABERMAS' THEORY OF RATIONALITY TO GAIN INSIGHT INTO STUDENTS'
UNDERSTANDING OF ALGEBRAIC LANGUAGE 453
FRANCESCA MORSELLI AND PAOLO BOERO INTRODUCTION 453
HABERMAS' CONSTRUCT OF RATIONAL BEHAVIOUR 454
ADAPTATION OF HABERMAS' CONSTRUCT OF RATIONAL BEHAVIOR TO THE CASE OF
THE USE OF ALGEBRAIC LANGUAGE 455
EPISTEMIC RATIONALITY 455
TELEOLOGICAL RATIONALITY 456
COMMUNICATIVE RATIONALITY 456
RELATIONSHIPS WITH OTHER STUDIES ON PROVING AND MODELING AND ON THE
TEACHING AND LEARNING OF ALGEBRA 457
PROVING 457
MODELING 459
TEACHING AND LEARNING OF ALGEBRA 459
DESCRIPTION AND INTERPRETATION OF STUDENT BEHAVIOR 462 HABERMAS'
ANALYTICAL TOOL: EXAMPLES OF ANALYSIS OF STUDENT BEHAVIOR AT DIFFERENT
SCHOOL LEVELS 462
HABERMAS ANALYTICAL TOOL: ANALYSIS OF A TEACHING EXPERIMENT . . . 468
THE CONTEXT OF THE STUDY: DESCRIPTION OF THE RESEARCH PROJECT 468 FIRST
TASK: CHOOSE A NUMBER 469
SECOND TASK: REPRESENTING THE GAME 470
DISCUSSION 477
IMAGE 10
XXII CONTENTS
RESEARCH ADVANCES 477
EDUCATIONAL IMPLICATIONS 478
REFERENCES 479
THEORETICAL ISSUES AND EDUCATIONAL STRATEGIES FOR ENCOURAGING TEACHERS
TO PROMOTE A LINGUISTIC AND METACOGNITIVE APPROACH TO EARLY ALGEBRA 483
ANNALISA CUSI, NICOLINA A. MALARA, AND GIANCARLO NAVARRA INTRODUCTION
483
IN EUROPE 484
FROM TRADITIONAL ALGEBRA TO EARLY ALGEBRA 485
EARLY ALGEBRA AS A META-SUBJECT AND THE ARAL PROJECT 486
SOCIO-CONSTRUCTIVE TEACHING AND TEACHER TRAINING 487 THE ROIE OF THE
TEACHER'S REFLECTION 488
THE ROLE OF THE ARAL GLOSSARY IN TEACHER TRAINING 490
ALGEBRAIC BABBLING 492
ALGEBRAIC BABBLING - ALGEBRA AS A LANGUAGE 493
ALGEBRAIC BABBLING -» SYNTAX, SEMANTICS - BRIOSHI 494 BRIOSHI -
CANONICAL/NON CANONICAL FORM OF A NUMBER - ' =' 495
THE MULTI-COMMENTED TRANSCRIPTS METHODOLOGY (MCTM) 496 FROM THE COMMENTS
TO A CLASSIFICATION OF ATTITUDES 499
EXAMPLE 502
CONCLUDING REMARKS 504
REFERENCES 507
A PROCEDURAL FOCUS AND A RELATIONSHIP FOCUS TO ALGEBRA: HOW U.S.
TEACHERS AND JAPANESE TEACHERS TREAT SYSTEMS OF EQUATIONS 511 MARGARET
SMITH BACKGROUND 512
ALGEBRAIC REASONING 512
TIMSS VIDEO STUDIES 514
DATA 515
ANALYSIS 515
TWO TEACHERS' LESSONS 516
DISCUSSION OF KEY DIFFERENCES 516
CONCLUSIONS 526
REFERENCES 526
TEACHING ALGEBRAIC EQUATIONS WITH VARIATION IN CHINESE CLASSROOM . . .
529 JING LI, AIHUI PENG, AND NAIQING SONG INTRODUCTION 529
THE SOURCE OF THE DATA 531
THEORETICAL FRAMEWORK 531
THE METHOD OF RESEARCH 533
ANALYSIS OF DATA 533
THE INTRODUCTION OF THE CONCEPT OF EQUATION 533
IMAGE 11
CONTENTS XXIII
THE IMPROVEMENT OF UNDERSTANDING OF EQUATION 535
EQUATIONS SOLVING 539
THE APPLICATION OF EQUATIONS 541
DISCUSSION AND CONCLUSION 545
PROCESS OF TEACHING ALGEBRA WITH VARIATION 545
OPERATION OF TEACHING ALGEBRA WITH VARIATION 546
FINAL COMMENTS 548
REFERENCES 555
COMMENTARY ON PART III 557
JOHN MASON INTRODUCTION 557
SYSTEMATICS: STRUCTURE OF ACTIVITY 558
WHAT IS ALGEBRA? 559
WHAT IS AND WHAT COULD BE: TEACHING ALGEBRA AS AN ACTIVITY . . . 560
TRADITIONAL ALGEBRA TEACHING 561
ENVISIONED ALGEBRA TEACHING 563
WHAT MAKES 'ALGEBRA' EARLY? 566
COMPARISONS 568
TRANSFORMING ALGEBRA TEACHING AND LEARNING AS AN ACTIVITY . . . 568 HOW
CAN LOCALLY SUCCESSFUL TEACHING BE ENGINEERED FOR ALL? 569 WHAT IS AND
COULD BE RESEARCHED? 570
WHAT IS REALLY RESEARCHED? 571
CONCLUSIONS 574
REFERENCES 574
OVERALL COMMENTARY ON EARLY ALGEBRAIZATION: PERSPECTIVES FOR RESEARCH
AND TEACHING 579
CAROLYN KIERAN SHAPING THE NOTION OF ALGEBRAIC THINKING WITHIN EARLY
ALGEBRA . . 580 THINKING ABOUT THE GENERAL IN THE PARTICULAR 581
THINKING RULE-WISE ABOUT PATTERNS 582
THINKING RELATIONALLY ABOUT QUANTITY, NUMBER, AND NUMERICAL OPERATIONS
583
THINKING REPRESENTATIONALLY ABOUT THE RELATIONS IN PROBLEM SITUATIONS
585
THINKING CONCEPTUALLY ABOUT THE PROCEDURAL 586
ANTICIPATING, CONJECTURING, AND JUSTIFYING 588
GESTURING, VISUALIZING, AND LANGUAGING 590
THE VIEW OF ALGEBRAIC THINKING THAT EMERGES FROM THIS VOLUME . . 591
REFERENCES 592
AUTHOR INDEX 595
SUBJECT INDEX 609
EDITORS AND CONTRIBUTORS 615 |
| any_adam_object | 1 |
| author2 | Cai, Jinfa 1963- Knuth, Eric J. |
| author2_role | edt edt |
| author2_variant | j c jc e j k ej ejk |
| author_GND | (DE-588)1084237490 (DE-588)1031113002 |
| author_facet | Cai, Jinfa 1963- Knuth, Eric J. |
| building | Verbundindex |
| bvnumber | BV043811269 |
| classification_rvk | SM 617 |
| ctrlnum | (OCoLC)712595645 (DE-599)DNB1008239607 |
| dewey-full | 512.0071 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.0071 |
| dewey-search | 512.0071 |
| dewey-sort | 3512.0071 |
| dewey-tens | 510 - Mathematics |
| discipline | Pädagogik Mathematik |
| format | Book |
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| id | DE-604.BV043811269 |
| illustrated | Illustrated |
| index_date | 2024-09-20T13:29:36Z |
| indexdate | 2024-09-27T16:41:23Z |
| institution | BVB |
| isbn | 9783642177347 3642177344 9783642177354 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029222517 |
| oclc_num | 712595645 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG |
| owner_facet | DE-473 DE-BY-UBG |
| physical | XXIII, 623 Seiten Illustrationen, Diagramme 24 cm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Advances in mathematics education |
| spellingShingle | Early algebraization a global dialogue from multiple perspectives Algebra (DE-588)4001156-2 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4037949-8 |
| title | Early algebraization a global dialogue from multiple perspectives |
| title_auth | Early algebraization a global dialogue from multiple perspectives |
| title_exact_search | Early algebraization a global dialogue from multiple perspectives |
| title_full | Early algebraization a global dialogue from multiple perspectives Jinfa Cai, Eric Knuth editors |
| title_fullStr | Early algebraization a global dialogue from multiple perspectives Jinfa Cai, Eric Knuth editors |
| title_full_unstemmed | Early algebraization a global dialogue from multiple perspectives Jinfa Cai, Eric Knuth editors |
| title_short | Early algebraization |
| title_sort | early algebraization a global dialogue from multiple perspectives |
| title_sub | a global dialogue from multiple perspectives |
| topic | Algebra (DE-588)4001156-2 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
| topic_facet | Algebra Mathematikunterricht |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3563512&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029222517&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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