Mathematical Thinking

The following sections are included:

• Preface to the Book

• Acknowledgments

• Contents

The following sections are included:

• The Teaching Approach as the Result of Lesson Study
  • Learning mathematics by/for themselves
  • The difference between tasks and problems (problematic)
  • Teachers’ questioning, and changing and adding representations
  • Extending the ideas which we have already learned

• Setting the Activities for Explaining, Listening, Reflecting, and Appreciating in Class
  • Structure of Problem Solving Approaches
  • Diversity of solutions and the objective of the class
  • Comparison based on the problematic
  • Using the blackboard for illustrating children’s thinking process

• The Roles of the Curriculum and Textbooks

• Perspectives for Developing Mathematical Thinking
  • Mathematical thinking: a major research topic of lesson study
  • Mathematical thinking: a bird’s-eye view

• References

Please refer to full text.

The following sections are included:

• Developing Children Who Learn Mathematics for Themselves

• Mathematical Thinking as an Ability to Think and to Make Decisions

• The Hierarchy of Ability and Thinking

The following sections are included:

• The Importance of Teaching Mathematical Thinking
  • The driving forces in pursuing knowledge and skills
  • Achieving independent thinking and the ability to learn independently

• Example: How Many Squares Are There?
  • The usual lesson process
  • Problems with this method
  • The preferred method
  • Mathematical thinking is the key ability here

The following sections are included:

• Mathematical Thinking
  • Focus on the mindset: attitude and disposition
  • Three variables for thinking mathematically
  • Importance of Denotative understanding of mathematical thinking
  • Mathematical thinking is the driving force behind knowledge and skills

• Structure of Mathematical Thinking

The following sections are included:

• Inductive Thinking
  • Meaning
  • Examples
  • Important aspects about teaching inductive thinking

• Analogical Thinking
  • Meaning
  • Examples
  • Important aspects about teaching analogical thinking

• Deductive Thinking
  • Meaning
  • Examples
  • Important aspect about teaching deductive thinking

• Integrative Thinking
  • Meaning
  • Type I integration (high-level integration)
  • Type II integration (comprehensive integration)
  • Type III integration (extensional thinking)
  • Example for type I
  • Example 2 for type II
  • Example 3 for type III
  • Important aspects about teaching integrative thinking

• Developmental Thinking
  • Meaning
  • Examples
  • Important aspects about teaching developmental thinking

• Abstract Thinking (Abstraction)
  • Meaning
  • Examples
  • Important aspects about teaching abstract thinking

• Thinking That Simplifies (Simplifying)
  • Meaning
  • Examples
  • Important aspects about teaching simplification

• Thinking That Generalizes (Generalization)
  • Meaning
  • Example
  • Important aspects about teaching generalization

• Thinking That Specializes (Specialization)
  • Meaning
  • Examples
  • Important aspects about teaching specialization

• Thinking That Symbolizes (Symbolization)
  • Meaning
  • Examples
  • Important aspects about teaching symbolization

• Thinking That is Represented by Numbers, Quantities, and Figures (Quantification and Schematization)
  • Meaning
  • Examples
  • Important aspects about teaching quantification and schematization

The following sections are included:

• Idea of Sets
  • Meaning
  • Examples
  • Important aspects about teaching the idea of sets

• Idea of Units
  • Meaning
  • Examples
  • Important aspects about teaching the idea of units

• Idea of Representation
  • Meaning
  • Examples
  • Important aspects about teaching the idea of representations

• Idea of Operation
  • Meaning
  • Examples
  • Important aspects about teaching the idea of operations

• Idea of Algorithms
  • Meaning
  • Example
  • Important aspects about teaching the idea of algorithms

• Idea of Approximations
  • Meaning
  • Examples
  • Important aspects about teaching the idea of approximations

• Idea of Fundamental Properties
  • Meaning
  • Examples
  • Important aspects about teaching the idea of fundamental properties

• Functional Thinking
  • Meaning of functional thinking
  • Coming up with ways to represent functional relationships
  • Educational value of teaching functional thinking

• Idea of Expressions³
  • Idea of representing by expressions
  • Reading the meaning of expressions

The following sections are included:

• Objectifying
  • Meaning
  • Example

• Reasonableness
  • Meaning
  • Example

• Clarity
  • Meaning
  • Examples

• Sophistication
  • Meaning
  • Examples

When teachers try to teach mathematical thinking, they need to think of how they can help children to think mathematically, and appreciate and acquire the ability to use mathematical thinking. When children get stuck, rather than helping them directly with useful knowledge and skills, teachers must prepare a way to teach the mathematical thinking required to attain the knowledge and skill, and moreover to teach the attitude that leads to such thinking methods. Also, this assistance must be of a general nature, and must be applicable to many different situations. Assistance should take a form that is frequently helpful when one focuses upon it. This is because this kind of assistance is useful in many different situations. By repeatedly providing it, a student can grow accustomed to this type of mathematical thinking. This kind of assistance is not something taught directly, but something that should be used by children themselves to overcome problems. Therefore, this assistance takes the form of questions…

Questioning, or question, here is called "Hatsumon" in Japanese. Japanese elementary school teachers use "Hatsumon" in the context of lesson study (Jugyou Kenkyu) and usually plan it before the class for developing children to think by/for themselves. However, the planned Hatsumon is not necessary as same as the actual Hatsumon in a class because a teacher usually changes his/her plan depending on his/her assessment of children's activity in the class. In a classroom, a teacher usually observes children's activity and listens their thinking and ideas. Then, he/she usually makes decision what Hatsumon is necessary to enhance children's thinking based on those monitoring activities. This is the assessment of teaching. Hatsumon is selected to give a feedback to children. One Hatsumon develops a set of inquiring activity by children. It is not the small-step-questions which are usually seen in the textbooks in the U.S. for guiding children to get the same correct answer. In the lesson study, the effect of Hatsumon is usually discussed at the post lesson discussion among observed teachers.

Part I explained the teaching theory of mathematical thinking. Part II will demonstrate how to teach mathematical thinking with assessment in the classroom with 12 examples of exploring “number tables” in which children discover properties of tables and give the reason why these properties are true…

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet
  • Let’s play this game!

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

• Further Development

The following sections are included:

• Types of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

• Further Development

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

• Further Development

The following sections are included:

• Introduction

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Introduction

• Type of Mathematical Thinking to be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Introduction

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation

The following sections are included:

• Introduction

• Type of Mathematical Thinking to Be Cultivated

• Grade Taught

• Preparation

• Overview of the Lesson Process

• Worksheet

• Lesson Process

• Summarization on the Blackboard

• Evaluation
  • Test (after the lesson)

• Further Development