Chapter 1: Numbers and Operations

1 Teaching Numbers and Operations, Learning Theory
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The significance of number and operation map 2
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The emergence of the concept of number 3
1) Abstraction of the concept of number 3
2) Source of the concept of number 6
3) Operational construction of number concepts 9
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Integers and Rational Numbers 14
1) Historical origin of the concept of negative numbers 14
2) Model 16 for negative maps
3) Principle of formal intransibility 20
4) Map of rational number concepts 22
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Sets and Logic 27
1) Natural number concept and set theory 27
2) Issues related to practical work 30

2 Teaching and Learning Numbers and Operations
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Understanding the Curriculum 34
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Understanding the Textbook 35
1) Negative numbers and integers 36
2) Rational and irrational numbers 38
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Exploring Negative Map Methods 43
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Utilizing Engineering Tools 47
50 Questions to Think About
Reference 51

Chapter 2 Algebra

1 Algebraic Teaching, Learning Theory
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The Significance of Algebraic Instruction 54
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Historical Development of Algebra 54
1) Changes in the meaning of algebra 54
2) Stages of Algebraic Development 57
3) Development of structural algebra59
4) Algebraic Principle 61
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Research on Algebra Teaching and Learning 63
1) Algebraic Language Learning 63
2) Variable concept and cognitive impairment 70
3) Elements related to algebraic thinking 74
4) Problem Solving and Equation 84

2 Algebra professors, learning practices
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Understanding the Curriculum 87
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Understanding Textbooks 88
1) Map of Generalization 88
2) Composition of 'text type' 91
3) Application of algebraic principles 94
4) Case Study of Application of Mathematics Teaching and Learning Theory 96
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Understanding Class 100
1) Map 100 of 'Use and Expression of Letters'
2) Thought Experiment and Classroom Research for Teaching 'Factoring' 102
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Application of Engineering Tools 106
1) Solving equations using a calculator or spreadsheet 106
2) Factoring with an engineering calculator 108
3) Solving Problems with Math Programs 109
4) Visualizing the 'domain of inequality' in a math program 110
Questions to Consider 111
Reference 112

Chapter 3 Functions

1 Functional Teaching, Learning Theory
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The significance of function maps 116
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Historical Development of Functions 117
1) Pre-function step 118
2) Geometric function step 120
3) Algebraic function step 122
4) Logical function step 124
5) Collective function step 125
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Research on Function Teaching and Learning 127
1) Introduction to various aspects of functions and functions 127
2) Functional guidance according to Freudenthal's didactic phenomenology 130
3) Function graph map according to Krabbendam's qualitative approach 135
4) Function map according to Janvier's translation activities 138
5) Epistemological obstacles to function learning 140

2 Function teaching, learning practice
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Understanding the Curriculum 146
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Understanding the Textbook 147
1) Preliminary steps of function mapping 147
2) Function concept map 148
3) Function types and context 150
4) Function expression and translation 157
5) Function Operation 161
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Understanding the Class 163
1) Lesson Plan 163
2) Expected student reactions 165
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Application of Engineering Tools 170
1) Use of graphic calculators and computer software presented in textbooks 170
2) Understanding the properties of functions using Excel 172
3) Mathematical Modeling Using Excel 173
Question to Consider 178
Reference 179

Chapter 4: Geometry and Proof

1 Geometry and Proof Teaching, Learning Theory
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The Significance of Geometry and Proof Maps 184
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The Historical Development of Geometry 185
1) The Origin of Geometry 185
2) Euclidean geometry 185
3) Interpretive Geometry 198
4) Non-Euclidean Geometry 201
5) Transformation Geometric Perspective 203
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Research on Geometry and Proof Teaching and Learning 205
1) Understanding and applying geometric concepts 205
2) van Hieles's theory of geometric thinking levels 210
3) Freudenthal's theory of teaching and learning geometry 215
4) The meaning of proof from a mathematical philosophical perspective 219

2 Geometry and Proof Teaching and Learning Practice
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Understanding the Curriculum 228
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Students' Proof Learning Status 230
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Directions for Improving Teaching and Learning 234
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Application of Engineering Tools 239
1) Computer-assisted mapping of the interior angles of triangles 239
2) Exploring the incenter of a triangle using a computer 242
Question to Consider 243
Reference 244

Chapter 5 Differentiation and Integration

1 Teaching Differentiation and Integration, Learning Theory
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The Significance of Teaching Differentiation and Integration 248
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Historical Development of Differentiation and Integration 249
1) Archimedes' quadrature and equilibrium 249
2) Exploring the ideas of differentiation and integration through the volume of Kepler's wine barrel 253
3) Cavalieri's indivisible method 254
4) Newton and Leibniz's Calculus 255
5) Calculus after the 18th century 258
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Research on Teaching and Learning Differential and Integral Calculus 259
1) Definition and conceptual image of limits and continuity 259
2) APOS theory 262
3) Historic Principle 264

2 Teaching and Learning Differentiation and Integration
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Understanding the Curriculum 270
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Differentiation and Integration in Textbooks 271
1) Sequence 272
2) Infinity concept 273
3) Fundamental Theorem of Calculus 277
4) The base of the natural logarithm is 281
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Directions for Improving Calculus Teaching and Learning 284
1) Instructional method for limit values ​​of sequences and functions 284
2) Various methods for introducing differentiation 287
3) Tangential concept map based on Lakatos's semi-empiricalism 289
4) Application of Cavalieri's indivisible method 293
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Application of Engineering Tools 295
1) Exploring the graph of a function using a computer program 295
2) Understanding the division method using a computer program 296
3) Integral calculation using computer program 298
300 Questions to Think About
References 301

Chapter 6 Probability and Statistics

1. Teaching Probability and Statistics, Learning Theory
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The Significance of Probability and Statistics Maps 304
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The Historical Development of Probability and Statistics 305
1) Dice and Probability 305
2) Systematic review of the types and sizes of possibilities 306
3) Development of combinatorial thinking 308
4) Connection between probability theory and statistics 310
5) Changes in probability due to additional information 312
6) Axiomatization of Probability 313
7) Errors and Statistical Analysis 314
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Research on Probability and Statistics Teaching and Learning 315
1) The Role of Intuition in Probability Education 315
2) Judgment Strategies and Probability Education 316
3) Paradox and Probability Education 317
4) Developmental level of probabilistic thinking 320
5) Conditional Probability Concept Map 321
6) Data-driven probability education 323
7) Exploratory Data Analysis and Statistical Problem Solving 324
8) Statistical Literacy Education 325
9) Informal Statistical Reasoning Education 328
10) Data-driven statistical education 329

2 Probability and Statistics Teaching and Learning Practice
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Understanding the Curriculum 330
1) Probability and Statistics 330 in our country's curriculum.
Understanding Textbooks 331
1) Recognizing the usefulness of statistics 332
2) Map of probability concepts 334
3) Map of the normal distribution 337
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Understanding Class 339
1) The Role of the Teacher and the Extreme Teaching Phenomenon 339
2) De-backgrounding and de-personalization through meta-level learning 341
3) Variations of textbook assignments and teaching examples related to representative values ​​344
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Utilizing Engineering Tools 346
Problem to Think About 352
Reference 353

Search 357