Mathematical encounters and pedagogical detours: stories of disturbance and learning opportunities in teacher education
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham
Springer International Publishing
2021
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xii, 210 Seiten Illustrationen, Diagramme |
| ISBN: | 9783030584337 9783030584368 |
Internformat
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Datensatz im Suchindex
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| adam_text | Contents 1 Disturbance as a Driving Force.................................................................. Form and Sources of Disturbance..................................................... A Framework of Professional Responding to Disturbing Encounters........................................................................................... 1.2.1 Responding to Disturbance as a Problem Solver................. 1.2.2 Responding to Disturbance as a Teacher in Search for a Good Explanation......................................................... 1.2.3 Responding to Disturbance as a Teacher in Search of Ways of Teaching Difficult Problems While Preserving Student Autonomous Learning......................... 1.2.4 Responding to Disturbance as a Teacher-Educator Teaching Mathematics........................................................... 1.2.5 Responding to Disturbance as a Teacher-Educator Teaching How to Teach......................................................... 1.2.6 Coda: Response to Disturbance as a TeacherEducator-Researcher ............................................................. References...................................................................................................... 1.1 1.2 2 A Fictional Dialogue on Infinitude of Primes.......................................... 2.1 Introduction.......................................................................................... 2.2 Theoretical Underpinnings................................................................... 2.2.1
Duoethnography..................................................................... 2.2.2 Virtual Monologue................................................................. 2.3 Introducing and Exemplifying Virtual Duoethnography.................. 2.3.1 What is Virtual Duoethnography?........................................ 2.3.2 Lakatos’ Dialogue as an Example of Virtual Duoethnography...................................................................... 2.4 Virtual Duoethnography: Situating a Fictional Dialogue............... 2.4.1 Learning and Teaching by Scripting Instructional Interactions.............................................................................. 2.4.2 Background and Task............................................................. 1 1 2 3 5 8 11 11 14 15 17 18 19 19 20 21 21 21 22 22 23 ix
Contents x 3 2.5 Fictional dialogue on Infinity of Primes........................................... 2.5.1 Same Theorem?....................................................................... 2.5.2 On Number and Unit............................................................... 2.5.3 On “Measured”....................................................................... 2.5.4 Prime or Not Prime................................................................. 2.5.5 Absurd?.................................................................................... 2.5.6 The essence and the difference............................................. 2.5.7 The Gap.................................................................................... 2.6 Discussion........................................................................................... References...................................................................................................... 25 26 26 27 28 30 31 33 34 36 Encounters with Euclidean Propositions................................................. 39 39 39 41 42 42 Introduction.......................................................................................... How It Started...................................................................................... Leaving the Choice to Students.......................................................... On Continued Proportions.................................................................. 3.4.1 Warm-Up Tasks...................................................................... 3.4.2 Proposition 4 from Book
IX, Its Proof, and Students’Script................................................... 43 3.4.3 Solid Numbers and Similar Solid Numbers.......................... 3.4.4 On Mean Proportional Numbers........................................... 3.4.5 Interpreting Euclid’s Proof of Proposition IX.4................... 3.4.6 Geometric Sequencesand Geometry...................................... 3.5 New Proposition VIII.6....................................................................... 3.6 On the Euclidean Algorithm.............................................................. 3.6.1 Proposition X.2 and Mike’s Script....................................... 3.6.2 Which Proposition Is Euclid’s Algorithm?......................... 3.6.3 Detour on Applicability......................................................... 3.6.4 On Rational and Irrational.................................................... 3.7 Concluding Remarks.......................................................................... References...................................................................................................... 3.1 3.2 3.3 3.4 4 Stories on Problem-Solving Instruction................................................... 4.1 4.2 4.3 Introduction......................................................................................... Explore - Launch - Re-explore - Relate........................................... 4.2.1 Explore.................................................................................... 4.2.2
Launch.................................................................................... 4.2.3 Explore.................................................................................... 4.2.4 Relate...................................................................................... Launch - Explore - Re-launch - Re-explore - Relate..................... 4.3.1 Launch.................................................................................... 4.3.2 Explore.................................................................................... 4.3.3 Re-launch................................................................................ 4.3.4 Re-explore.............................................................................. 4.3.5 Relate...................................................................................... 45 46 48 51 52 57 58 61 65 66 70 71 73 73 75 75 76 76 80 80 80 81 81 84 85
Contents 5 6 xi 4.4 Launch - Get Stuck - Intervene - Relate........................................ 4.4.1 Background............................................................................. 4.4.2 Launch..................................................................................... 4.4.3 Get Stuck................................................................................. 4.4.4 Intervene................................................................................. 4.4.5 Relate........................................................................................ 4.4.6 Follow-Up............................................................................... 4.5 Launch - Intervene - Re-launch - Explore - Relate........................ 4.5.1 Background............................................................................. 4.5.2 Launch...................................................................................... 4.5.3 Intervene (Formalize).............................................................. 4.5.4 Detour: Types and Effects of Teacher Interventions........... 4.5.5 Re-launch................................................................................. 4.5.6 Explore...................................................................................... 4.5.7 Relate........................................................................................ 4.6 Concluding Reflections......................................................................
References...................................................................................................... 86 86 87 88 89 91 92 93 93 94 95 96 98 99 100 102 103 Encounters with Cardano’s Method........................................................ 5.1 Introduction.......................................................................................... 5.2 Background.......................................................................................... 5.3 Collaboratively Re-telling the Story.................................................. 5.4 The Initial Engagement and Structuring the Method........................ 5.5 Detour: Reverse Heuristic Engineering............................................. 5.6 How Could Such a Clever Solution Be Found?................................. 5.7 Can the Law of Transitivity Fail?........................................................ 5.7.1 Reversible or Not and Does It Matter?................................. 5.8 Reverse Heuristic Engineering and Problem Posing........................ 5.8.1 Detour: The Variation Theory of Learning as a Task-Design Heuristics................................................... 5.9 Explorations Based on the Examples Generated by Students......... 5.9.1 “Nice” Roots - Difficult Solutions....................................... 5.9.2 “Nice” Roots - “Nice” Solutions............................................ 5.9.3 Detour: “Niceness” and Artificiality.................................... 5.9.4 Appearance of “Nice Monsters”........................................... 5.10 Concluding
Remarks......................................................................... References...................................................................................................... 105 105 106 107 109 Ill 112 115 117 122 Synthesizing Exception-Barring and What-If-Not: If Not, What Yes?............................................................................... 145 6.1 Introduction........................................................................................ 6.2 Background........................................................................................ 6.3 “If Not, What Yes?”: An Instructional Aspect................................. 6.3.1 The Account............................................................................ 6.3.2 Illustration in the Context of Calculus.................................. 124 125 128 129 136 139 143 144 145 146 147 147 148
Contents xii Detour: A Happy End and (Somewhat Awkward) Follow-Up of the Story............................................. 155 6.4 “If Not, What Yes?” A Design Aspect............................................... 6.4.1 Student-Generated Example in the Context of Calculus.................................................................. 160 6.4.2 Student-Generated Example in the Context of Number Theory..................................................... 161 References...................................................................................................... 6.3.3 7 From “Obviously Wrong” Methods to Surprisingly Correct Answers................................................................................. 165 7.1 Introduction......................................................................................... 7.2 Tax or Discount?................................................................................. 7.3 Surprises with Order of Operations: A Story in Three Parts........... 7.3.1 Part 1 : Mathematical Conventions Task................................ 7.3.2 Part 2: Convention: Order of Operations.............................. 7.3.3 Part 3: Order of Operation Tasks........................................... 7.3.4 Order of Operations: Arbitrary Choice?.............................. 7.4 Modeling and Conversion of Square Units...................................... 7.4.1 Part 1 : Unexpected Solution................................................. 7.4.2 Part 2: Scripting on Unexpected Solution............................ 7.4.3 Towards a Conventional
Approach....................................... 7.4.4 Conversion Motivated............................................................ 7.4.5 Attending to a Variety of Shapes........................................... 7.4.6 Computation “Works”............................................................ 7.4.7 Towards a Geometric Explanation......................................... 7.4.8 Modeling as Scaling: Mathematical Notes.......................... 7.5 Familiar Formula in Unfamiliar Situation......................................... 7.5.1 Affine Coordinates: An Introduction..................................... 7.5.2 Exploring Medians.................................................................. 7.5.3 Eureka! Mathematical Resolution......................................... 7.5.4 Further Applicability.............................................................. 7.6 Concluding Remarks.......................................................................... References...................................................................................................... 160 164 165 166 167 168 169 174 179 180 180 182 183 184 185 187 189 190 192 192 195 199 201 202 203 Epilogue................................................................................................................ 205 Index....................................................................................................................... 207
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| adam_txt |
Contents 1 Disturbance as a Driving Force. Form and Sources of Disturbance. A Framework of Professional Responding to Disturbing Encounters. 1.2.1 Responding to Disturbance as a Problem Solver. 1.2.2 Responding to Disturbance as a Teacher in Search for a Good Explanation. 1.2.3 Responding to Disturbance as a Teacher in Search of Ways of Teaching Difficult Problems While Preserving Student Autonomous Learning. 1.2.4 Responding to Disturbance as a Teacher-Educator Teaching Mathematics. 1.2.5 Responding to Disturbance as a Teacher-Educator Teaching How to Teach. 1.2.6 Coda: Response to Disturbance as a TeacherEducator-Researcher . References. 1.1 1.2 2 A Fictional Dialogue on Infinitude of Primes. 2.1 Introduction. 2.2 Theoretical Underpinnings. 2.2.1
Duoethnography. 2.2.2 Virtual Monologue. 2.3 Introducing and Exemplifying Virtual Duoethnography. 2.3.1 What is Virtual Duoethnography?. 2.3.2 Lakatos’ Dialogue as an Example of Virtual Duoethnography. 2.4 Virtual Duoethnography: Situating a Fictional Dialogue. 2.4.1 Learning and Teaching by Scripting Instructional Interactions. 2.4.2 Background and Task. 1 1 2 3 5 8 11 11 14 15 17 18 19 19 20 21 21 21 22 22 23 ix
Contents x 3 2.5 Fictional dialogue on Infinity of Primes. 2.5.1 Same Theorem?. 2.5.2 On Number and Unit. 2.5.3 On “Measured”. 2.5.4 Prime or Not Prime. 2.5.5 Absurd?. 2.5.6 The essence and the difference. 2.5.7 The Gap. 2.6 Discussion. References. 25 26 26 27 28 30 31 33 34 36 Encounters with Euclidean Propositions. 39 39 39 41 42 42 Introduction. How It Started. Leaving the Choice to Students. On Continued Proportions. 3.4.1 Warm-Up Tasks. 3.4.2 Proposition 4 from Book
IX, Its Proof, and Students’Script. 43 3.4.3 Solid Numbers and Similar Solid Numbers. 3.4.4 On Mean Proportional Numbers. 3.4.5 Interpreting Euclid’s Proof of Proposition IX.4. 3.4.6 Geometric Sequencesand Geometry. 3.5 New Proposition VIII.6. 3.6 On the Euclidean Algorithm. 3.6.1 Proposition X.2 and Mike’s Script. 3.6.2 Which Proposition Is Euclid’s Algorithm?. 3.6.3 Detour on Applicability. 3.6.4 On Rational and Irrational. 3.7 Concluding Remarks. References. 3.1 3.2 3.3 3.4 4 Stories on Problem-Solving Instruction. 4.1 4.2 4.3 Introduction. Explore - Launch - Re-explore - Relate. 4.2.1 Explore. 4.2.2
Launch. 4.2.3 Explore. 4.2.4 Relate. Launch - Explore - Re-launch - Re-explore - Relate. 4.3.1 Launch. 4.3.2 Explore. 4.3.3 Re-launch. 4.3.4 Re-explore. 4.3.5 Relate. 45 46 48 51 52 57 58 61 65 66 70 71 73 73 75 75 76 76 80 80 80 81 81 84 85
Contents 5 6 xi 4.4 Launch - Get Stuck - Intervene - Relate. 4.4.1 Background. 4.4.2 Launch. 4.4.3 Get Stuck. 4.4.4 Intervene. 4.4.5 Relate. 4.4.6 Follow-Up. 4.5 Launch - Intervene - Re-launch - Explore - Relate. 4.5.1 Background. 4.5.2 Launch. 4.5.3 Intervene (Formalize). 4.5.4 Detour: Types and Effects of Teacher Interventions. 4.5.5 Re-launch. 4.5.6 Explore. 4.5.7 Relate. 4.6 Concluding Reflections.
References. 86 86 87 88 89 91 92 93 93 94 95 96 98 99 100 102 103 Encounters with Cardano’s Method. 5.1 Introduction. 5.2 Background. 5.3 Collaboratively Re-telling the Story. 5.4 The Initial Engagement and Structuring the Method. 5.5 Detour: Reverse Heuristic Engineering. 5.6 How Could Such a Clever Solution Be Found?. 5.7 Can the Law of Transitivity Fail?. 5.7.1 Reversible or Not and Does It Matter?. 5.8 Reverse Heuristic Engineering and Problem Posing. 5.8.1 Detour: The Variation Theory of Learning as a Task-Design Heuristics. 5.9 Explorations Based on the Examples Generated by Students. 5.9.1 “Nice” Roots - Difficult Solutions. 5.9.2 “Nice” Roots - “Nice” Solutions. 5.9.3 Detour: “Niceness” and Artificiality. 5.9.4 Appearance of “Nice Monsters”. 5.10 Concluding
Remarks. References. 105 105 106 107 109 Ill 112 115 117 122 Synthesizing Exception-Barring and What-If-Not: If Not, What Yes?. 145 6.1 Introduction. 6.2 Background. 6.3 “If Not, What Yes?”: An Instructional Aspect. 6.3.1 The Account. 6.3.2 Illustration in the Context of Calculus. 124 125 128 129 136 139 143 144 145 146 147 147 148
Contents xii Detour: A Happy End and (Somewhat Awkward) Follow-Up of the Story. 155 6.4 “If Not, What Yes?” A Design Aspect. 6.4.1 Student-Generated Example in the Context of Calculus. 160 6.4.2 Student-Generated Example in the Context of Number Theory. 161 References. 6.3.3 7 From “Obviously Wrong” Methods to Surprisingly Correct Answers. 165 7.1 Introduction. 7.2 Tax or Discount?. 7.3 Surprises with Order of Operations: A Story in Three Parts. 7.3.1 Part 1 : Mathematical Conventions Task. 7.3.2 Part 2: Convention: Order of Operations. 7.3.3 Part 3: Order of Operation Tasks. 7.3.4 Order of Operations: Arbitrary Choice?. 7.4 Modeling and Conversion of Square Units. 7.4.1 Part 1 : Unexpected Solution. 7.4.2 Part 2: Scripting on Unexpected Solution. 7.4.3 Towards a Conventional
Approach. 7.4.4 Conversion Motivated. 7.4.5 Attending to a Variety of Shapes. 7.4.6 Computation “Works”. 7.4.7 Towards a Geometric Explanation. 7.4.8 Modeling as Scaling: Mathematical Notes. 7.5 Familiar Formula in Unfamiliar Situation. 7.5.1 Affine Coordinates: An Introduction. 7.5.2 Exploring Medians. 7.5.3 Eureka! Mathematical Resolution. 7.5.4 Further Applicability. 7.6 Concluding Remarks. References. 160 164 165 166 167 168 169 174 179 180 180 182 183 184 185 187 189 190 192 192 195 199 201 202 203 Epilogue. 205 Index. 207 |
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| spelling | Koichu, Boris ca. 20./21. Jh. Verfasser (DE-588)1281140813 aut Mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education Cham Springer International Publishing 2021 xii, 210 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematiklehrer (DE-588)4037946-2 gnd rswk-swf Mathematikunterricht (DE-588)4037949-8 gnd rswk-swf Mathematiklehrer (DE-588)4037946-2 s Mathematikunterricht (DE-588)4037949-8 s DE-604 Zazkis, Rina Verfasser (DE-588)106949691X aut Erscheint auch als Online-Ausgabe 978-3-030-58434-4 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032535001&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Koichu, Boris ca. 20./21. Jh Zazkis, Rina Mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education Mathematiklehrer (DE-588)4037946-2 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
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| title | Mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education |
| title_auth | Mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education |
| title_exact_search | Mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education |
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| title_full_unstemmed | Mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education |
| title_short | Mathematical encounters and pedagogical detours |
| title_sort | mathematical encounters and pedagogical detours stories of disturbance and learning opportunities in teacher education |
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