Chapter 1.
Mathematical Problem Solving Education Theory
Section 1 Understanding Problem Solving 13
1.
The Meaning and Types of Problem Solving/13
2.
A History of Problem-Solving Maps/16
3.
Analysis and Synthesis/18
Section 2: Problem-solving process and problem-raising 23
1.
Problem Solving Process/23
2.
Mathematical Modeling Course/28
3.
Issue/34
4.
The Role of Raising Issues/42
Section 3: Inductive Reasoning and Analogy 45
1.
The Meaning and Role of Inductive Reasoning/45
2.
The Meaning and Role of Analogy/51
3.
Problem-Solving Strategies/54

Chapter 2.
Pedagogical Transformation and Mathematical Communication
Section 1: Pedagogical Situation 71
1.
Pedagogical Situation/71
2.
Developments in Mathematics Pedagogical Situations/75
Section 2: Pedagogical Transformation of Mathematical Knowledge80
1.
Pedagogical Transformation/80
2.
The Process of Knowledge Transformation/82
Section 3 Mathematical Communication 85
1.
Selection and Transmission of Knowledge by Teachers/85
2.
Communication for the Transfer of Mathematical Knowledge/86
3.
Teacher's Note for Expanding Mathematical Thinking/88
4.
Mathematical Expression and Language/92
Section 4: Problems in the Process of Pedagogical Transformation 96
1.
Extreme teaching practices by teachers/96
2.
Cognitive impairment in students/99

Chapter 3.
Learners' Mathematical Learning Characteristics
Section 1: Mathematics Learning Levels and Abilities 107
1.
Math Learning Level/107
2.
Prerequisite Knowledge for Mathematics Learning/111
3.
Math Learning Strategies/112
4.
Students' Misconceptions and Errors/115
Section 2: Characteristics of Mathematics and Thinking Styles 121
1.
Characteristics of Mathematics/121
2. APOS Theory, Conceptual Image, and Concept Definition /124
3.
Mathematical Thinking Styles/136

Chapter 4.
Lesson plan reflecting mathematics education theory
Section 1: Lessons Using Problem-Posting Activities 147
1.
Issue/148
2.
The Reality of Teaching Instruction/150
Section 2: Classes Following Mathematization 157
1.
Realistic Mathematics Education and Mathematization/ 157
2.
The Reality of Teaching Instruction/159
Section 3: Lessons Emphasizing Understanding 165
1.
Skemp and Understanding/165
2.
The Reality of Teaching Instruction/167
Section 4: Classes Where Meaningful Learning Occurs 175
1.
Osubel and Meaningful Receptive Learning/175
2.
The Reality of Teaching Instruction/178

Chapter 5.
Teacher knowledge
Section 1: Background to the Discussion of Teacher Knowledge 187
1.
Shulman and Teacher Knowledge/187
2.
Components of Teacher Knowledge/189
Section 2: Teacher Knowledge in Mathematics Curriculum 193
1.
Ma's PUFM/193
2.
Ball's MKT/194
3. MPCK/196 of TEDS-M
4.
Korean Institute of Curriculum and Evaluation Content Teaching Knowledge/198
Section 3: Reflection and Teacher Knowledge 207
1.
The Meaning of Reflection/207
2.
The Impact of Reflection on Classroom Learning/210
3.
The Role of Reflection/211