# The Evolution of Mathematical Thinking in Chinese Mathematics Education

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## Abstract

**:**

## 1. Introduction

## 2. Research Method

#### 2.1. Data Selection

#### 2.1.1. Keywords in Chinese Determined

#### 2.1.2. Data Sources

#### 2.1.3. Data Selection

#### 2.2. Data Analysis

Text 1: “basic knowledge means basic concepts, principles, properties, formulas, axioms, theorems and the mathematical thinking and method reflected by mathematical content”.[22] (p. 1)

Text 2:“They may have the following types. The first type is strategical thinking including thinking about transformation, …, etc. The second type is logical thinking including deduction, … etc. The third type is manual thinking including thinking about construction, discriminant of a quadric equation in one unknown, etc.…”.[29]

Text 3MT is reflected in the process of knowledge establishment, development and application, is the abstraction and generalization of mathematical content and method in a higher level, such as abstracting, sorting, generalizing, deducting and modeling.[25] (p. 46)

## 3. Findings

#### 3.1. Earlierst Explanations Given By Ding

#### 3.2. Explanations Given by Researchers from the Persepctives of Methodology and Method (1980–1991)

#### 3.2.1. Mathematical Methodology

Mathematical methodology is a discipline which aims to study and discuss the patterns of mathematical development, method of mathematical thinking and principles of mathematical findings, inventions and innovation.

Letting R represents a group of the relationship structures of original images and suppose the group contains x, which is an original image and to be determined. Then, letting M is a map whose function assumed that R can be mapped into relation structures of mapping image, R*, which contains the mapping image x* whose original image is x. If there was a way to determine x*, x could be known by the inversion, namely the inverse mapping, I = M^{−1}.[28] (p. 25)

^{−1}—the inverse process of moving from the theoretical problems to the practical problems; x*—the mapping image of an unknown x in a practical problem. It can be seen that modeling and abstracting are involved in this process.

#### 3.2.2. Mathematical Thinking Method

In the narrow sense level, mathematical thinking method refers to mathematical argumentation, operations, and thinking, methods and strategies for mathematical application.[29] (pp. 1–2)

In the broad sense level, apart from the considerations in the narrow sense, mathematical objectives, properties, characteristics, roles, development patterns for mathematics (such as concepts, theories, methods and forms) should be included.[29] (pp. 1–2)

#### 3.3. MT and Mathematical Method Stated in the Teaching Syllabus(1992–2001)

The basic knowledge means basic concepts, principles, properties, formulas, axioms, theorems and the mathematical thinking and methods reflected by mathematical contents.[22] (p. 1)

#### 3.3.1. Definition of MT

the space-forms and quantity relationships in real life reflected in human’s consciousness and then processed by human’s thinking, the products of thinking was mathematical thinking.[41]

#### 3.3.2. MT and Mathematical Method

#### 3.3.3. Types of MT

#### 3.4. Mathematical Thinking Method Stated in the Curriculum Standard (2002–2011)

…should obtain the important mathematical knowledge including mathematical facts and activity experience, basic mathematical thinking method and basically mathematical application skills which are necessary for their future society life and further self-development.[25] (p. 3)

#### 3.4.1. Mathematical Thinking Method

^{2}= a

^{2}+2ab+b

^{2}, the algebraic method (by calculating (a + b) (a + b)) can be used to explain the equation. However, if square modeling (by calculating areas of figures) was also used to explain the relationship, it would be much better to understand the formula. This method of using a combination of symbolic and graphic mathematics can help students understand mathematical formulae much better than simply using an algebraic method only. This process of learning mathematical formulae through the use combinations of numerical and graphical methods is an excellent example of mathematical thinking. There is no strict boundary between the two, so the term mathematical thinking method has been used frequently by researchers and teachers [40,49,50].

#### 3.4.2. Major Types of MT

#### 3.4.3. The Teaching of MT

#### 3.5. MT as One of “Four Basics”in Curriculum Standard 2011 (From 2011–2018)

#### 3.5.1. Explanations Given for MT

MT is reflected in the process of knowledge establishment, development and application, is the abstraction and generalization of mathematical content and method in a higher level, such as abstracting, sorting, generalizing, deducting and modeling.[6] (p. 46)

#### 3.5.2. Major Types of MT

#### 3.5.3. Teaching of MT

#### 3.6. Conclusions and Discussions

#### 3.6.1. The Struggling Relationship between MT and Mathematical Methods (methodology)

#### 3.6.2. Involvement in Mathematical Thought Process

#### 3.6.3. Focus on the Classification of MT

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Key Words | Before 1978 | 1979–1991 | 1992–2000 | 2001–2010 | 2011–2019Feb | Total |
---|---|---|---|---|---|---|

Official Documents | 1992 Teaching Syllabus | 2001 Curriculum Standard | 2011 Curriculum Standard | 3 | ||

Journal Articles | 1 | 0 | 8 | 22 | 26 | 57 |

Books | 0 | 3 | 3 | 3 | 1 | 10 |

Total | 1 | 3 | 11 | 25 | 22 | 70 |

Strategical | Logical | Manipulative |
---|---|---|

Thinking about transformation (Hua Gui) | Deducting | Construction |

Abstracting & Briefly summarizing | Categorizing | Substitution |

Thinking about equation & function (It means problem solving with function or equation) | Specializing | Undetermined coefficient method |

Conjecturing | Analogizing | Method of completing the square |

Thinking about symbolic-graphic combination | Generalizing | Parameters |

Holistic & Systematic | Inversion | Discriminant |

Abstracting | Reasoning | Modeling | Aesthetic |
---|---|---|---|

Categorization | Generalization | Simplify | Concise |

Set | Deduction | Quantify | Symmetry |

Constant in changing | Axiomatic | Equation | Uniform |

Symbolic | Number-Graph Combination | Function | Express complex through simple |

Corresponding | Conversion | Optimization | Perceive the essence from appearance |

Finite and infinite | Associate &Analogy | Random | |

Specialization and generalization | Statistic |

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## Share and Cite

**MDPI and ACS Style**

Li, N.; Mok, I.A.C.; Cao, Y.
The Evolution of Mathematical Thinking in Chinese Mathematics Education. *Mathematics* **2019**, *7*, 297.
https://doi.org/10.3390/math7030297

**AMA Style**

Li N, Mok IAC, Cao Y.
The Evolution of Mathematical Thinking in Chinese Mathematics Education. *Mathematics*. 2019; 7(3):297.
https://doi.org/10.3390/math7030297

**Chicago/Turabian Style**

Li, Na, Ida Ah Chee Mok, and Yiming Cao.
2019. "The Evolution of Mathematical Thinking in Chinese Mathematics Education" *Mathematics* 7, no. 3: 297.
https://doi.org/10.3390/math7030297