The Pedagogy of Secondary-School Mathematics [1st ed. 2023] 9789819912476, 9789819912483, 9819912474

Table of contents :
Textbook for Normal Colleges and Universities
Contents
1 Introduction
1.1 Scientific Mathematics and Mathematics as an Educational Science
1.1.1 Scientific Mathematics
1.1.2 Mathematics as an Educational Science
1.2 Emergence and Development of Mathematics Education
1.3 Nature, Task, and Significance of Mathematics Pedagogy
1.3.1 Nature of Mathematics Pedagogy
1.3.2 Task of Mathematics Pedagogy
1.3.3 Significance of Mathematics Pedagogy
2 Secondary School Mathematics Logic
2.1 Secondary School Mathematics Concepts
2.1.1 Meaning of Concept
2.1.2 Structure of Concept
2.1.3 Relationship Between Concepts
2.1.4 Definition of Concept
2.1.5 Series of Concept
2.1.6 Classification of Concepts
2.2 Secondary School Mathematics Propositions
2.2.1 Judgement
2.2.2 Proposition
2.2.3 Generation of Reverse Proposition
2.2.4 Identity Principle of Propositions
2.3 Basic Laws of Formal Logic
2.3.1 Law of Identity
2.3.2 Law of Contradiction
2.3.3 Law of Excluded Middle
2.3.4 Law of Sufficient Reason
2.4 Secondary School Mathematics Inference
2.4.1 Inductive Inference
2.4.2 Analogical Inference
2.4.3 Deductive Inference
2.5 Secondary School Mathematics Proof
2.5.1 Analysis Method and Synthesis Method
2.5.2 Direct and Indirect Proof
2.6 Teaching of Secondary School Mathematics Concepts and Propositions
2.6.1 Teaching of Mathematical Concepts
2.6.2 Teaching of Mathematical Propositions
3 Secondary School Mathematics Thinking
3.1 Meaning of Mathematical Thinking
3.2 Methods of Secondary School Mathematics Thinking
3.2.1 Observation and Experimentation
3.2.2 Analysis and Synthesis
3.2.3 Comparison and Classification
3.2.4 Abstraction and Generalization
3.2.5 Concretization, Specialization, and Systematization
3.2.6 Analogy, Induction, and Deduction
3.2.7 Imagination and Intuition
3.3 Quality of Secondary School Mathematics Thinking
3.3.1 Broadness of Thinking
3.3.2 Profoundness of Thinking
3.3.3 Criticalness of Thinking
3.3.4 Flexibility of Thinking
3.3.5 Sense of Organization of Thinking
3.3.6 Creativity of Thinking
3.4 Cultivation of Mathematical Thinking Ability in Secondary Schools
3.4.1 Divergent Thinking and Its Basic Approach to Development
3.4.2 Reverse Thinking and Its Basic Approach to Development
3.4.3 Creative Thinking and Its Basic Approach to Development
4 Secondary School Mathematics Methods
4.1 Meaning of Mathematical Methods
4.1.1 Meaning of Mathematical Methods
4.1.2 Approaches to Studying Mathematical Methods
4.1.3 Significance of Studying Mathematical Methods
4.2 Reduction Method
4.2.1 Overview of Reduction Method
4.2.2 Direction of Reduction
4.2.3 Methods of Reduction
4.2.4 Understanding Reduction Dialectically
4.3 Discovery Methods
4.3.1 Analogy Method
4.3.2 Induction Method
4.3.3 Association Method
4.3.4 Intuitive Method
4.3.5 Aesthetic Method
4.4 Demonstration Methods
4.5 Test Method
4.5.1 Basic Idea of Mathematical Test Method
4.5.2 Test and Conjecture
4.5.3 Non-standard Problems
4.6 Teaching of Secondary School Mathematics Thoughts and Methods
4.6.1 Secondary School Mathematics Thoughts
4.6.2 Teaching of Mathematical Thoughts and Methods in Secondary Schools
5 Mathematical Ability in Secondary Schools
5.1 Significance of Mathematical Ability
5.1.1 Relationship Between Knowledge and Ability
5.1.2 Significance of Cultivating Mathematical Ability in Secondary Schools
5.1.3 Basic Approaches to Cultivating Mathematical Ability in Secondary Schools
5.2 Cultivation of Operational Ability
5.2.1 Operational Ability of Secondary School Mathematics
5.2.2 Basic Approaches to Cultivating Operational Ability
5.3 Cultivation of Logical Thinking Ability
5.3.1 Logical Thinking Ability of Secondary School Mathematics
5.3.2 Basic Approaches to Cultivating Logical Thinking Ability
5.4 Cultivation of Spatial Imagination Ability
5.4.1 Spatial Imagination Ability of Secondary School Mathematics
5.4.2 Basic Approaches to Cultivating the Spatial Imagination Ability
5.5 Cultivation of Problem Solving Ability
5.5.1 Basic Knowledge of Problem Solving
5.5.2 Thinking Process of Problem Solving
5.5.3 Cultivation of Problem Solving Ability
5.6 Problem Solving and Its Teaching
5.6.1 Meaning of Mathematical Problems
5.6.2 Problem Solving
5.6.3 Teaching of Mathematical Problem Solving
6 Secondary School Mathematics Learning
6.1 Characteristics of Secondary School Mathematics Learning
6.1.1 Associationistic View of Learning
6.1.2 Cognitive View of Learning
6.2 Process of Secondary School Mathematics Learning
6.2.1 Cognitive Structure
6.3 Intelligence and Non-intelligence Factors
6.3.1 Intelligence Factors
6.3.2 Non-intelligence Factors
6.4 Teaching in Conformity with the Laws of Psychological Activities
7 Secondary School Mathematics Curriculum Standard
7.1 Curriculum Standards and Their Significance
7.1.1 Curriculum Standards
7.1.2 Syllabus
7.1.3 Significance of Curriculum Standard
7.2 Introduction to Mathematics Curriculum Standard for Compulsory Education
7.2.1 Nature and Basic Concepts of Mathematics Curriculum
7.2.2 Objectives of Mathematics Curriculum
7.2.3 Content of Mathematics Curriculum
7.2.4 Evaluation of Mathematics Learning
7.3 Introduction to the General High School Mathematics Curriculum Standard
7.3.1 Nature and Basic Concepts of Mathematics Curriculum
7.3.2 Basic Framework of Mathematics Curriculum
7.3.3 Objectives of Mathematics Curriculum
7.3.4 Content of Mathematics Curriculum
7.3.5 Suggestions for Teaching of Mathematics Curriculum
8 Teaching Form and Means of Secondary School Mathematics
8.1 Teaching Form of Secondary School Mathematics
8.1.1 Overview of Teaching Form
8.1.2 Main Work of Mathematics Lesson
8.1.3 Type and Structure of Mathematics Lesson
8.2 Common Teaching Means of Secondary School Mathematics
8.2.1 Mathematical Language
8.2.2 Mathematics Blackboard-Writing
8.2.3 Mathematical Teaching Aids
8.3 Modern Teaching Means of Secondary School Mathematics
8.3.1 Teaching with Slides and Projection
8.3.2 Teaching Means of Videoplaying and Video Shooting
8.3.3 Multimedia Teaching Means
8.3.4 Network Teaching Means
9 Teaching Principles and Methods of Secondary School Mathematics
9.1 Teaching Principles of Secondary School Mathematics
9.1.1 Principle of Combining the Concrete with the Abstract
9.1.2 Principle of Combining Theory with Practice
9.1.3 Principle of Combining Rigorousness with Being Realistic
9.1.4 Principle of Combining Imparting Knowledge with Cultivating Ability
9.1.5 Principle of Combining Shape with Number
9.2 Teaching Methods of Secondary School Mathematics
9.2.1 Heuristic Teaching Method
9.2.2 Several Common Teaching Methods
9.2.3 Several New Teaching Methods
9.3 Determining Teaching Principles and Selecting Teaching Methods
9.3.1 Application of Secondary School Mathematics Teaching Principles
9.3.2 Selection of Secondary School Mathematics Teaching Methods
10 Teaching Work of Secondary School Mathematics
10.1 Lesson Preparation of Secondary School Mathematics
10.1.1 Develop a Teaching Work Plan
10.1.2 Prepare Lessons for the Class Hour
10.2 Giving Lessons of Secondary School Mathematics
10.2.1 Organize Classroom Teaching Well
10.2.2 Pay Attention to the Use of Textbooks
10.2.3 Be Particular About Classroom Questioning
10.2.4 Strengthen the Teaching of Mathematical Thinking Methods
10.2.5 Handle Several Relationships Properly
10.3 Extracurricular Work of Secondary School Mathematics
10.3.1 Correct Homework in a Timely and Serious Manner
10.3.2 Strengthen Extracurricular Tutoring
10.3.3 Carry Out Extracurricular Activities of Mathematics Actively
10.4 Performance Appraisal in Secondary School Mathematics
11 Teaching Research of Secondary School Mathematics
11.1 Teaching Research Methods of Secondary School Mathematics
11.1.1 General Methods of Secondary School Mathematics Teaching Research
11.1.2 Specific Methods of Secondary School Mathematics Teaching Research
11.2 Teaching Research Work of Secondary School Mathematics
11.2.1 Routine Research Work
11.2.2 Special Research Work
11.3 Teaching Reform of Secondary School Mathematics
11.3.1 A Brief Historical Review
11.3.2 Modernization Movement
11.3.3 Reform of Secondary School Mathematics Education in China
12 Educational Measurement of Secondary School Mathematics
12.1 Meaning of Educational Measurement of Secondary School Mathematics
12.1.1 Birth and Development of Educational Measurement of Secondary School Mathematics
12.1.2 Characteristics of Educational Measurement of Secondary School Mathematics
12.1.3 Significance of Educational Measurement of Secondary School Mathematics
12.2 Test Question Setting of Secondary School Mathematics
12.2.1 Types of Secondary School Mathematics Test Questions
12.2.2 Principles of Test Question Setting of Secondary School Mathematics
12.2.3 Standards for Test Question Setting of Secondary School Mathematics
12.2.4 Steps of Test Question Setting of Secondary School Mathematics
12.2.5 Methods of Test Question Setting of Secondary School Mathematics
12.3 Standardized Test Questions and Standardized Tests
12.3.1 Standardized Test Questions
12.3.2 Formulation of Choice Questions
12.3.3 Methods of Solving Choice Questions
12.3.4 Standardized Tests
12.4 Sorting and Analysis of Test Scores
12.4.1 Sorting of Test Scores
12.4.2 Item Analysis of Test Scores
12.4.3 Overall Analysis of Test Scores
12.4.4 Standard Score
13 Teaching and Research Practice of Secondary School Mathematics
13.1 Teaching Skill Training of Secondary School Mathematics
13.1.1 Teaching Skills of Secondary School Mathematics
13.1.2 Training of Teaching Skills
13.2 Basic Teaching Skills of Secondary School Mathematics
13.2.1 Basic Skill of Organizing Textbooks
13.2.2 Basic Skill of Solving Mathematical Problems
13.2.3 Basic Skill of Mathematical Language
13.2.4 Basic Skill of Mathematical Blackboard Writing
13.2.5 Basic Skill of Organizing Teaching
13.3 Teaching Practice of Secondary School Mathematics
13.3.1 Purpose of Mathematics Teaching Practice
13.3.2 Tasks of Mathematics Teaching Practice
13.3.3 Process of Mathematics Teaching Practice
13.3.4 Some Points for Attention During Teaching Practice
13.4 Writing of Papers on Secondary School Mathematics Education
13.4.1 Determination of the Subject of Papers on Mathematics Education
13.4.2 Components of a Paper on Mathematics Education
13.4.3 Process of Writing a Paper on Mathematics Education
Appendix A Examples of Teaching Plan of Secondary School Mathematics
Appendix B Examples of Lesson Presentation of Secondary School Mathematics
Appendix C Examples of Secondary School Mathematics Research

Citation preview

Shizao Zhang

The Pedagogy of Secondary-School Mathematics Translated by Youben Yao · Hailin Pan

The Pedagogy of Secondary-School Mathematics

Shizao Zhang

The Pedagogy of Secondary-School Mathematics

Shizao Zhang School of Mathematics and Statistics Yancheng Teachers University Yancheng, Jiangsu, China Translated by Youben Yao Nanjing Vocational University of Industry Technology Nanjing, Jiangsu, China

Hailin Pan Yancheng Teachers University Yancheng, Jiangsu, China

ISBN 978-981-99-1247-6 ISBN 978-981-99-1248-3 (eBook) https://doi.org/10.1007/978-981-99-1248-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

(江苏版 1991 年, 高教版 2009 年, 施普林格版 2023 年) (Jiangsu edition 1991, Higher Education Press edition 2009, Springer edition 2023)

祝贺佳作 获得多次国际书展好评 体现学科建设中国特色 施普林格社出版发行 田 刚 曹一鸣 范良火 贺 2022.10.30

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(东南大学版 2009 年, 北师大版 2018 年, 施普林格版 2024) (Southeast University edition 2009, Beijing Normal University edition 2018, Springer edition 2024)

(南京大学版 2009 年初版, 2014 修订版, 施普林格版 2024) (First edition 2009 and revised edition 2014 by Nanjing University Press, Springer edition 2024)

Textbook for Normal Colleges and Universities

As an important basic specialized course of departments of mathematics of normal colleges and universities, Didactics of Mathematics centers on secondary school mathematics education and serves the training of qualified secondary school mathematics teachers directly. In the past 30 years, with the deepening of mathematics education reform and the increasing development of theories on mathematics education, the task of establishing mathematics pedagogy has been brought forward on the basis of didactics of mathematics, and a variety of academic works and textbooks on mathematics pedagogy have been published at home and abroad. As a basic textbook of mathematics pedagogy, this book first describes the nature, tasks, and significance of mathematics pedagogy, introduces the basic knowledge and basic skills of secondary school mathematics logic, thinking, methods, and ability training, introduces secondary school mathematics learning, mathematics curriculum standards, teaching forms and means, teaching principles and methods, teaching work, teaching research and reform, then introduces the educational measurement of secondary school mathematics, secondary school mathematics teaching and research practice, etc., and finally selects several teaching plans and teaching research papers as examples for reference; at the same time, an appropriate number of review questions and exercises is provided at the end of each chapter for the purpose of teaching; and reference keys to the exercises are provided for the sake of self-study. The selection and arrangement of the above contents are based on the following considerations: First, in terms of its content, it involves mathematics, mathematics history, curriculum theory, teaching theory, learning theory, thinking theory, methodology, teacher theory, logic, technology science, examination science, comparative education, fundamentals of modern mathematics, etc., which not only reflects a high degree of comprehensiveness, but also takes the balanced development of various disciplines into consideration, so as not to emphasize one thing at the expense of another, and which is in line with the overall policy of discipline construction. Second, in terms of its system, it initially highlights the common thread of mathematics. Based on secondary school mathematics logic, thinking, method, and vii

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ability, and taking the study of secondary school mathematics teaching as the core, it discusses the problem of improving secondary school mathematics teaching and research ability. Here, the train of thought of logic—thinking—method—ability can easily be mathematized obviously, and when discussing teaching and cultivating ability from the perspectives of learning, lesson preparation, teaching, examinations, and teaching research, the specific mathematical problems are linked as much as possible. Third, this is a course of “assembly” nature. Teaching needs necessary basic knowledge of mathematics. At the same time, the students are used to the learning method of grasping the concepts and methods before doing exercises, and in addition, they lack due understanding of the importance of this course. Therefore, if we introduce the content of curriculum theory and teaching theory first, the students will not pay attention to it even if the teacher plays all his/her cards to solve the problem. Therefore, the content of improving mathematical accomplishment comes first, and the content of cultivating mathematics teaching ability comes next. This arrangement takes students’ mathematical basis and learning habits into consideration, which is conducive to arousing students’ enthusiasm for learning and enhancing the effect of teaching. Fourth, where there is teaching, there should be teaching methods. As far as mathematics pedagogy is concerned, there should also be corresponding primary school, secondary school, and college mathematics pedagogies. From the actual point of view of the current discipline construction, the focus is actually on secondary school mathematics pedagogy, and at the same time in terms of applying the principles and methods of adjacent disciplines, there should also be differences between primary schools, secondary schools, and universities. This book is hence called The Pedagogy of Secondary-School Mathematics, to clarify the focus of materials and research, so as to help achieve the purpose of pertinence and practicality. Fifth, in the selection of the above content and the arrangement of the system, it also includes mathematics curriculum standards, examination science, and secondary school mathematics teaching research and reform, secondary school mathematics teaching, and research practice and other chapters, which takes not only the need for normal colleges and universities to cultivate new teachers at present into consideration, but also the need of teachers’ continuing education for further improvement, and can serve the mathematics teaching and reform in secondary schools, so that it is in line with the tasks of this discipline. The above content covers a wide range, with mathematical, pedagogical, and psychological knowledge as the main body, secondary school mathematics logic, thinking, method, and ability as the foundation, the research of secondary school mathematics teaching as the core, and the improvement of secondary school mathematics teaching and research ability as the purpose. It strives to integrate theories, methods with skills, which are interrelated and emphasized differently, to form an organic whole; it strives to reflect novel ideas, comprehensive discussions, refined materials, concise descriptions, to be popular and practical, and reflect all aspects of

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secondary school mathematics teaching as much as possible to make it more scientific, theoretical, and practical; it not only pays attention to reflect the new achievements in secondary school mathematics teaching research at home and abroad, but also closely links the current teaching practice of mathematics teaching in secondary schools and didactics courses in normal colleges and universities, so as to make it more suitable for the teaching need of didactics course of mathematics departments in normal colleges and universities at present, and serve the current secondary school mathematics teaching, teaching reform practice, and teaching quality improvement as well. This book can be used as a textbook for undergraduates (college students) of normal colleges and universities and can also be used as a textbook for mathematics education courses in colleges of education, schools for teachers’ advanced studies, correspondence universities, etc., as well as for advanced studies or reference for the broad masses of secondary school mathematics teachers. During the process of compiling and translating this book, I have received strong support from the Mathematical Education Committee of Chinese Mathematical Society, Jiangsu Education Department, and Yancheng Teachers University, the guidance and help from Ma Zhonglin, Zhang Dianzhou, Wang Zikun, Cheng Changchun, Dai Binrong, Yu Jianjiang, Jiang Haibo, Fan Lianghuo, Cao Yiming, Wang Yuheng, Cui Hui, and other professors and experts, as well as the assistance from Pan Hailin, Zhang Wenfei, You Hua, Ye Wen, Bian Yujie, Sun Yuanyuan. Mr. Yao Youben from Nanjing Vocational University of Industry Technology and Pan Hailin, etc. translated the whole book into English. It is a great honor for its translated version to have the preface by Mr. Tian Gang, a famous mathematician and academician, and inscriptions by academicians Li Daqian and Zhang Jingzhong. I would like to take this opportunity to express my heartfelt thanks and pay high tribute to them. March 2007

Shizao Zhang

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Scientific Mathematics and Mathematics as an Educational Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Scientific Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Mathematics as an Educational Science . . . . . . . . . . . . . . 1.2 Emergence and Development of Mathematics Education . . . . . . . 1.3 Nature, Task, and Significance of Mathematics Pedagogy . . . . . . 1.3.1 Nature of Mathematics Pedagogy . . . . . . . . . . . . . . . . . . . 1.3.2 Task of Mathematics Pedagogy . . . . . . . . . . . . . . . . . . . . . 1.3.3 Significance of Mathematics Pedagogy . . . . . . . . . . . . . .

1 1 1 3 4 7 7 8 9

Secondary School Mathematics Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Secondary School Mathematics Concepts . . . . . . . . . . . . . . . . . . . . 2.1.1 Meaning of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Structure of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Relationship Between Concepts . . . . . . . . . . . . . . . . . . . . . 2.1.4 Definition of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Series of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Classification of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Secondary School Mathematics Propositions . . . . . . . . . . . . . . . . . 2.2.1 Judgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Generation of Reverse Proposition . . . . . . . . . . . . . . . . . . 2.2.4 Identity Principle of Propositions . . . . . . . . . . . . . . . . . . . 2.3 Basic Laws of Formal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Law of Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Law of Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Law of Excluded Middle . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Law of Sufficient Reason . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Secondary School Mathematics Inference . . . . . . . . . . . . . . . . . . . . 2.4.1 Inductive Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 14 15 17 19 19 20 20 24 28 29 31 31 32 32 33 34 34

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2.5

2.6

3

4

2.4.2 Analogical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Deductive Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Secondary School Mathematics Proof . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Analysis Method and Synthesis Method . . . . . . . . . . . . . . 2.5.2 Direct and Indirect Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . Teaching of Secondary School Mathematics Concepts and Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Teaching of Mathematical Concepts . . . . . . . . . . . . . . . . . 2.6.2 Teaching of Mathematical Propositions . . . . . . . . . . . . . .

36 37 37 38 39 43 43 47

Secondary School Mathematics Thinking . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Meaning of Mathematical Thinking . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Methods of Secondary School Mathematics Thinking . . . . . . . . . 3.2.1 Observation and Experimentation . . . . . . . . . . . . . . . . . . . 3.2.2 Analysis and Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Comparison and Classification . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Abstraction and Generalization . . . . . . . . . . . . . . . . . . . . . 3.2.5 Concretization, Specialization, and Systematization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Analogy, Induction, and Deduction . . . . . . . . . . . . . . . . . . 3.2.7 Imagination and Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quality of Secondary School Mathematics Thinking . . . . . . . . . . . 3.3.1 Broadness of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Profoundness of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Criticalness of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Flexibility of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Sense of Organization of Thinking . . . . . . . . . . . . . . . . . . 3.3.6 Creativity of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Cultivation of Mathematical Thinking Ability in Secondary Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Divergent Thinking and Its Basic Approach to Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Reverse Thinking and Its Basic Approach to Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Creative Thinking and Its Basic Approach to Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 55 55 56 57 59

Secondary School Mathematics Methods . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Meaning of Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Meaning of Mathematical Methods . . . . . . . . . . . . . . . . . . 4.1.2 Approaches to Studying Mathematical Methods . . . . . . . 4.1.3 Significance of Studying Mathematical Methods . . . . . . 4.2 Reduction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Overview of Reduction Method . . . . . . . . . . . . . . . . . . . . . 4.2.2 Direction of Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Methods of Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 89 91 92 93 93 94 97

62 66 67 69 69 70 71 72 72 73 75 75 77 80

Contents

4.2.4 Understanding Reduction Dialectically . . . . . . . . . . . . . . . Discovery Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Analogy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Induction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Association Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Intuitive Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Aesthetic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Demonstration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Basic Idea of Mathematical Test Method . . . . . . . . . . . . . 4.5.2 Test and Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Non-standard Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teaching of Secondary School Mathematics Thoughts and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Secondary School Mathematics Thoughts . . . . . . . . . . . . 4.6.2 Teaching of Mathematical Thoughts and Methods in Secondary Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102 104 104 109 115 120 124 128 133 133 135 137

Mathematical Ability in Secondary Schools . . . . . . . . . . . . . . . . . . . . . . 5.1 Significance of Mathematical Ability . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Relationship Between Knowledge and Ability . . . . . . . . 5.1.2 Significance of Cultivating Mathematical Ability in Secondary Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Basic Approaches to Cultivating Mathematical Ability in Secondary Schools . . . . . . . . . . . . . . . . . . . . . . . 5.2 Cultivation of Operational Ability . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Operational Ability of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Basic Approaches to Cultivating Operational Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Cultivation of Logical Thinking Ability . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Logical Thinking Ability of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Basic Approaches to Cultivating Logical Thinking Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Cultivation of Spatial Imagination Ability . . . . . . . . . . . . . . . . . . . . 5.4.1 Spatial Imagination Ability of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Basic Approaches to Cultivating the Spatial Imagination Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Cultivation of Problem Solving Ability . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Basic Knowledge of Problem Solving . . . . . . . . . . . . . . . . 5.5.2 Thinking Process of Problem Solving . . . . . . . . . . . . . . . . 5.5.3 Cultivation of Problem Solving Ability . . . . . . . . . . . . . . 5.6 Problem Solving and Its Teaching . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 151

4.3

4.4 4.5

4.6

5

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140 140 144

152 153 155 155 156 159 159 160 164 164 164 167 167 169 171 180

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5.6.1 5.6.2 5.6.3 6

7

8

Meaning of Mathematical Problems . . . . . . . . . . . . . . . . . 180 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Teaching of Mathematical Problem Solving . . . . . . . . . . 181

Secondary School Mathematics Learning . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Characteristics of Secondary School Mathematics Learning . . . . 6.1.1 Associationistic View of Learning . . . . . . . . . . . . . . . . . . . 6.1.2 Cognitive View of Learning . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Process of Secondary School Mathematics Learning . . . . . . . . . . . 6.2.1 Cognitive Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Intelligence and Non-intelligence Factors . . . . . . . . . . . . . . . . . . . . 6.3.1 Intelligence Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Non-intelligence Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Teaching in Conformity with the Laws of Psychological Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 187 188 190 190 194 194 197

Secondary School Mathematics Curriculum Standard . . . . . . . . . . . . 7.1 Curriculum Standards and Their Significance . . . . . . . . . . . . . . . . . 7.1.1 Curriculum Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Significance of Curriculum Standard . . . . . . . . . . . . . . . . 7.2 Introduction to Mathematics Curriculum Standard for Compulsory Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Nature and Basic Concepts of Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Objectives of Mathematics Curriculum . . . . . . . . . . . . . . . 7.2.3 Content of Mathematics Curriculum . . . . . . . . . . . . . . . . . 7.2.4 Evaluation of Mathematics Learning . . . . . . . . . . . . . . . . . 7.3 Introduction to the General High School Mathematics Curriculum Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Nature and Basic Concepts of Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Basic Framework of Mathematics Curriculum . . . . . . . . 7.3.3 Objectives of Mathematics Curriculum . . . . . . . . . . . . . . . 7.3.4 Content of Mathematics Curriculum . . . . . . . . . . . . . . . . . 7.3.5 Suggestions for Teaching of Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 203 205 209

Teaching Form and Means of Secondary School Mathematics . . . . . 8.1 Teaching Form of Secondary School Mathematics . . . . . . . . . . . . 8.1.1 Overview of Teaching Form . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Main Work of Mathematics Lesson . . . . . . . . . . . . . . . . . . 8.1.3 Type and Structure of Mathematics Lesson . . . . . . . . . . . 8.2 Common Teaching Means of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Mathematical Language . . . . . . . . . . . . . . . . . . . . . . . . . . .

199

211 211 215 219 222 222 224 225 226 227 231 235 235 235 237 237 239 239

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8.3

9

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8.2.2 Mathematics Blackboard-Writing . . . . . . . . . . . . . . . . . . . 8.2.3 Mathematical Teaching Aids . . . . . . . . . . . . . . . . . . . . . . . Modern Teaching Means of Secondary School Mathematics . . . . 8.3.1 Teaching with Slides and Projection . . . . . . . . . . . . . . . . . 8.3.2 Teaching Means of Videoplaying and Video Shooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Multimedia Teaching Means . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Network Teaching Means . . . . . . . . . . . . . . . . . . . . . . . . . .

Teaching Principles and Methods of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Teaching Principles of Secondary School Mathematics . . . . . . . . 9.1.1 Principle of Combining the Concrete with the Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Principle of Combining Theory with Practice . . . . . . . . . 9.1.3 Principle of Combining Rigorousness with Being Realistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Principle of Combining Imparting Knowledge with Cultivating Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Principle of Combining Shape with Number . . . . . . . . . . 9.2 Teaching Methods of Secondary School Mathematics . . . . . . . . . 9.2.1 Heuristic Teaching Method . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Several Common Teaching Methods . . . . . . . . . . . . . . . . . 9.2.3 Several New Teaching Methods . . . . . . . . . . . . . . . . . . . . . 9.3 Determining Teaching Principles and Selecting Teaching Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Application of Secondary School Mathematics Teaching Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Selection of Secondary School Mathematics Teaching Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Teaching Work of Secondary School Mathematics . . . . . . . . . . . . . . . . 10.1 Lesson Preparation of Secondary School Mathematics . . . . . . . . . 10.1.1 Develop a Teaching Work Plan . . . . . . . . . . . . . . . . . . . . . 10.1.2 Prepare Lessons for the Class Hour . . . . . . . . . . . . . . . . . . 10.2 Giving Lessons of Secondary School Mathematics . . . . . . . . . . . . 10.2.1 Organize Classroom Teaching Well . . . . . . . . . . . . . . . . . . 10.2.2 Pay Attention to the Use of Textbooks . . . . . . . . . . . . . . . 10.2.3 Be Particular About Classroom Questioning . . . . . . . . . . 10.2.4 Strengthen the Teaching of Mathematical Thinking Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 Handle Several Relationships Properly . . . . . . . . . . . . . . . 10.3 Extracurricular Work of Secondary School Mathematics . . . . . . . 10.3.1 Correct Homework in a Timely and Serious Manner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Strengthen Extracurricular Tutoring . . . . . . . . . . . . . . . . .

243 245 246 246 247 247 249 253 253 253 255 258 259 262 264 264 266 269 273 273 274 277 277 277 278 282 283 283 284 285 286 290 290 291

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10.3.3 Carry Out Extracurricular Activities of Mathematics Actively . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10.4 Performance Appraisal in Secondary School Mathematics . . . . . . 293 11 Teaching Research of Secondary School Mathematics . . . . . . . . . . . . 11.1 Teaching Research Methods of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 General Methods of Secondary School Mathematics Teaching Research . . . . . . . . . . . . . . . . . . . . 11.1.2 Specific Methods of Secondary School Mathematics Teaching Research . . . . . . . . . . . . . . . . . . . . 11.2 Teaching Research Work of Secondary School Mathematics . . . . 11.2.1 Routine Research Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Special Research Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Teaching Reform of Secondary School Mathematics . . . . . . . . . . 11.3.1 A Brief Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Modernization Movement . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Reform of Secondary School Mathematics Education in China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Educational Measurement of Secondary School Mathematics . . . . . 12.1 Meaning of Educational Measurement of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Birth and Development of Educational Measurement of Secondary School Mathematics . . . . . . 12.1.2 Characteristics of Educational Measurement of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . 12.1.3 Significance of Educational Measurement of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . 12.2 Test Question Setting of Secondary School Mathematics . . . . . . . 12.2.1 Types of Secondary School Mathematics Test Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Principles of Test Question Setting of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Standards for Test Question Setting of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Steps of Test Question Setting of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.5 Methods of Test Question Setting of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Standardized Test Questions and Standardized Tests . . . . . . . . . . . 12.3.1 Standardized Test Questions . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Formulation of Choice Questions . . . . . . . . . . . . . . . . . . . 12.3.3 Methods of Solving Choice Questions . . . . . . . . . . . . . . . 12.3.4 Standardized Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Sorting and Analysis of Test Scores . . . . . . . . . . . . . . . . . . . . . . . . .

297 297 298 299 301 301 311 313 313 314 316 321 321 321 322 324 325 325 325 327 328 329 332 332 332 335 338 339

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12.4.1 12.4.2 12.4.3 12.4.4

Sorting of Test Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Item Analysis of Test Scores . . . . . . . . . . . . . . . . . . . . . . . Overall Analysis of Test Scores . . . . . . . . . . . . . . . . . . . . . Standard Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 Teaching and Research Practice of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Teaching Skill Training of Secondary School Mathematics . . . . . 13.1.1 Teaching Skills of Secondary School Mathematics . . . . . 13.1.2 Training of Teaching Skills . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Basic Teaching Skills of Secondary School Mathematics . . . . . . . 13.2.1 Basic Skill of Organizing Textbooks . . . . . . . . . . . . . . . . . 13.2.2 Basic Skill of Solving Mathematical Problems . . . . . . . . 13.2.3 Basic Skill of Mathematical Language . . . . . . . . . . . . . . . 13.2.4 Basic Skill of Mathematical Blackboard Writing . . . . . . 13.2.5 Basic Skill of Organizing Teaching . . . . . . . . . . . . . . . . . . 13.3 Teaching Practice of Secondary School Mathematics . . . . . . . . . . 13.3.1 Purpose of Mathematics Teaching Practice . . . . . . . . . . . 13.3.2 Tasks of Mathematics Teaching Practice . . . . . . . . . . . . . 13.3.3 Process of Mathematics Teaching Practice . . . . . . . . . . . . 13.3.4 Some Points for Attention During Teaching Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Writing of Papers on Secondary School Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Determination of the Subject of Papers on Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Components of a Paper on Mathematics Education . . . . 13.4.3 Process of Writing a Paper on Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

340 342 344 348 353 353 353 355 357 357 358 358 359 359 360 361 361 363 364 365 365 367 369

Appendix A: Examples of Teaching Plan of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Appendix B: Examples of Lesson Presentation of Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Appendix C: Examples of Secondary School Mathematics Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Chapter 1

Introduction

1.1 Scientific Mathematics and Mathematics as an Educational Science 1.1.1 Scientific Mathematics Since ancient times, people have been seeking to understand the essence of mathematics, and there have been many expositions. As far back as 1878, Engels gave a classic definition of mathematics in his book Anti-Dühring: “Mathematics is a science that studies quantitative relations and spatial forms in reality”, which has always been adopted in Chinese textbooks. With the flourishing development of mathematics, people’s views on the essence of mathematics are not completely consistent. “Quantity” is not only a real number, but has developed into objects of any kind, such as vectors, tensors, matrices, operators, and even elements in abstract sets with algebraic structures; “quantitative relation” has developed into a structural relation (sequential structure, algebraic structure, topological structure); and as far as “spatial form” is concerned, Lobachevskian space, Riemannian space, projective space, topological space, etc. have come into being after Euclidean space. If these relations and forms are understood as “quantitative relations and spatial forms” in a broad sense, then Engels’ classical exposition on mathematics above can still be used to depict modern mathematics. In the long course of human history, the development of mathematics has gone through a long stage. First, for the needs of calculation, positive integers and positive fractions came into being; for the needs of measurement and calculation, geometry came into being. The mathematics before the seventeenth century is referred to as mathematics of constants or elementary mathematics. After the sixteenth and seventeenth centuries, with the rise of capitalism in the West, production practice puts forward many new research topics to the natural sciences, and there was an urgent need for mechanics, astronomy, and other basic disciplines to help to resolve the problems, thus resulting in a need for mathematics to put forward new concepts and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_1

1

2

1 Introduction

methods correspondingly. Under such historical conditions, Descartes introduced variables into mathematics and then established calculus. Since the 1940 and 1950s, with the further economic and technological development, mathematics has entered a new period of development, not only forming a multitude of theories, but also producing many emerging disciplines, so mathematics has entered the modern development period. In this way, mathematics has experienced four stages of development: birth period (from ancient times to the fifth and sixth centuries), elementary mathematics period (before the seventeenth century), classical advanced mathematics period (before the twentieth century), and modern mathematics period (from the beginning of the twentieth century to now). Generally speaking, mathematics can be divided into three parts: basic mathematics, applied mathematics, and computational mathematics. As the core of mathematics, basic mathematics is also the purest and most abstract part, so it is also called pure mathematics. The birth and development of mathematics has always been deepening and evolving around two basic concepts: number and shape. Therefore, generally speaking, those that study numbers and their relations are classified into the field of algebra; those that study shapes and their relations are classified into the field of geometry. But shape and number are interconnected whole, and analytical mathematics was born with the emergence of variables in the seventeenth century, so it is safe to say that basic mathematics is composed of three main parts: algebra, geometry, and analytics. These three parts intersect and interpenetrate one another, resulting in a series of new disciplines such as analytic geometry, analytic number theory, and algebraic geometry. Applied mathematics studies specific mathematical problems in reality. It not only makes use of the results of basic mathematics, but in turn extracts problems from practice and explores new ideas and methods to enrich basic mathematics. Although the application field of mathematics is boundless, it can be roughly divided into three aspects: economic construction (industry, agriculture, business, etc.), science and technology (especially high-tech), and military and national defense. Operations research, cybernetics and mathematical statistics, and other disciplines are mostly applied mathematics, while economic mathematics and biomathematics, etc. are relatively standard disciplines of applied mathematics. Computational mathematics focuses on thousands of calculations. The early computational mathematics was devoted to finding numerical solutions to various equations (algebraic equations, differential equations, integral equations, etc.). The past 50 years has witnessed extremely rapid development of the computational mathematics, and people have introduced computation into the scientific world as a third scientific method independent of theory and experiment. For a long time, people think that mathematics is characterized by high degree of abstraction, preciseness of conclusion, and pervasiveness of application. However, after mathematics entered the modern development period, it has shown its remarkable characteristics of high differentiation and high unity. At present, mathematics has entered a new stage of development. Not only the number of branch disciplines is increasing, but also various sciences are “mathematized”, and various basic disciplines interpenetrate one another as well. For

1.1 Scientific Mathematics and Mathematics as an Educational Science

3

example, in addition to the aforementioned biomathematics and economic mathematics, there are also interdisciplinary subjects and a series of mathematical branches such as computational physics, computational mineralogy, computational chemistry, medical mathematics, management mathematics, quantitative genetics, and numerical taxonomy. During the process of differentiation, mathematics is also undergoing a process of unity, which is the opposite of differentiation, a process of interpenetration of methods and ideas in different fields of mathematics. The idea of set theory has become the basic idea of unity, and the unity realized in the works of the school of Bourbaki is based on the so-called basic structure (sequential structure, algebraic structure, and topological structure). They hold that any mathematical structure is a composite of this basic structure. With the deepening of mathematical research, this feature of high differentiation and high unity will become more prominent. Mathematics has been playing an increasingly important role in contemporary science and technology, its range of application is getting wider and wider, and its requirements are getting higher and higher, which completely bears out the famous remark of Lafrague in Reminiscence of Marx: “In the opinion of Marx, only when science succeeds in using mathematics can it be truly be perfected”. Looking ahead, mathematics will be deeply involved in the vast universe, tiny particles, speedy rockets, mysterious organisms, ingenious chemical engineering, changing earth, and multifarious commodities, and new aspects that we can’t predict may also emerge, thus bringing science and technology to move to a new level of development.

1.1.2 Mathematics as an Educational Science The so-called mathematics as an educational science, simply put, is mathematics that is used as teaching materials and teaching contents for the purpose of teaching. It is not the same as scientific mathematics. (1) Mathematics as an educational science is taken from some disciplines or branches of scientific mathematics, it is the most important basic knowledge of scientific mathematics that have matured and been theoretically perfected, and it is the content that has both theoretical and practical value and broad prospect for development as well. The selected materials are processed and adapted according to the principles of educational science. (2) Scientific mathematics is the accumulation of pure mathematics achievements, independent of the historical process of its discovery and invention, and different doctrines and different schools can coexist at the same time. However, mathematics as an educational science often takes into account both the laws of people’s cognition and the historical discovery and invention process as well, which runs through the whole or a certain stage of teaching process. (3) Scientific mathematics generally adopts axiomatic approach and has a strict logic system and a solid theoretical foundation. However, because of restricted

4

1 Introduction

conditions, as far as the mathematics as an educational science is concerned, the requirements for its theoretical logic system are appropriately reduced and often introduced by certain experimental means in combination with people’s production and life experience on the premise of not departing from the truth. (4) With the emergence of new concepts and the introduction of new methods, scientific mathematics can deal with traditional problems more precisely and simply. But the mathematics as an educational science can only select or partially improve the traditional content within a certain range, to create conditions for further learning new concepts and new methods. (5) Generally, scientific mathematics only pays attention to the progressive introduction from the shallower to the deeper and from the easy to the difficult. Mathematics as an educational science must also consider how to make necessary preparations for introduction, introduce understanding and memory methods, introduce practical applications, cultivate the ability to analyze and solve problems, and conduct necessary ideological and moral education and so on. It is thus evident that the mathematics as an educational science cannot exist without scientific mathematics; the talents who have mastered advanced mathematics of educational science can, in turn, carry out the research on scientific mathematics and promote its development better. They always complement each other, promote each other, and are closely linked. In this regard, some people at home and abroad also use “elementary mathematics” and “basic mathematics” to represent mathematics as an educational science. Elementary mathematics originally refers to the sum of the entire mathematics before the seventeenth century. The so-called elementary mathematics here is actually a kind of modern elementary mathematics (the elementary mathematics in the past is called traditional elementary mathematics correspondingly). It has two meanings: firstly, it is the “preliminary” basis of modern scientific mathematics; secondly, it is the “elementary” part that is relatively simple and plain in modern scientific mathematics and can be accepted by most secondary school students. That is to say, the concepts of modern elementary mathematics and traditional elementary mathematics are in an overlapping relationship. Modern elementary mathematics is composed of relatively preliminary and basic contents in traditional elementary mathematics, classical advanced mathematics, and modern mathematics, and it develops with the development of mathematics.

1.2 Emergence and Development of Mathematics Education Mathematics is an ancient subject. With the emergence and development of mathematics, mathematics education also emerged and developed continuously.

1.2 Emergence and Development of Mathematics Education

5

Mathematics education has a long history in our country. According to the record, the mathematical knowledge was in bud as far back as five or six thousand years ago, and its imparting was closely linked with production and life. It was impossible for people to separate mathematics education consciously and conduct research independently. The embryonic form of mathematics education emerged about three or four thousand years ago. Among the teaching content of six classical arts of “rites, music, archery, horse-riding, calligraphy and counting” in the Zhou Dynasty, “counting” refers to mathematics, which may be the earliest molding mathematics education in our country. The Nine Chapters on the Mathematical Art and Zhou Bi Suan Jing, which were completed around the first century, especially The Nine Chapters on Mathematical Art, are recognized as enduring masterpieces in the history of mathematics and become one of the sources of mathematics in the world. In the Qin and Han Dynasties of our country, primary schools were established, and mathematics was defined as a school subject. During the Wei, Jin, Southern and Northern Dynasties, the foundation of mathematics education is initially formed, with The Nine Chapters on Mathematical Art as the main content, and mathematicians such as Zhao Shuang, Liu Hui, and Zu Chongzhi came along, resulting in Mathematical Classic on Sea Islands, Mathematical Classic of Sun Zi, Mathematical Classic of Xiahou Yang, Mathematical Classic of Zhang Qiujian, Mathematical Summary, and other monographs. The imperial examination system was initiated in the Sui and Tang Dynasties. The central department had arithmetic and specialized schools to select and train mathematical professionals. Mathematicians such as Wang Xiaotong, Seng Yixing, and Li Chunfeng came along, resulting in Mathematical Classic of Jigu, Dayan Calendar, Mathematical Art of Han Yan and other monographs. During the Song and Yuan Dynasties, mathematics education was further developed and popularized, and mathematicians such as Jia Xian, Shen Kuo, Qin Jiushao, Li Ye, Yang Hui, Guo Shoujing, and Zhu Shijie came along, resulting in Outline of The Nine Chapters on the Mathematical Art, Dream Pool Essays, Mathematical Treatise in Nine Chapters, Sea Mirror of Circle Measurement, Detailed Annotation of The Nine Chapters of Mathematical Art, Shoushi Calendar, Introduction to Mathematics, Precious Mirror of the Four Elements, and other monographs. Mathematical education during the Ming and Qing Dynasties was once popular and once stagnant. Mathematicians such as Cheng Dawei, Mei Wending, Xu Guangqi, and Li Shanlan came along, who edited or translated Complete Collection of Algorithm, Chishui Collections, Tongwen Shunzhi, Shuli Jingyun, Euclid’s Elements, and other monographs. Later, with the introduction of Western mathematics and the abolition of the imperial examination system, secondary schools and institutions of higher education were generally established in various places, where mathematics became an important course. After the Revolution of 1911 and the May Fourth Movement, mathematics education was further promoted and popularized, and especially after the founding of the People’s Republic of China, mathematics education has entered a new period of vigorous development in our country. In foreign countries, in Egypt about four or five thousand years ago, there were already mathematics books on arithmetic and geometry, and the form of mathematics education appeared. The mathematics education in Greece in the third century BC has

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already reached a certain scale and level, and its greatest achievement was Euclid’s Elements, whose far-reaching consequence and scope of influence were unmatched by any other mathematics book up to now. The rise of Western capitalism in the middle of the seventeenth century promoted the development of mathematics education. Later, the mathematics education systems with different characteristics have formed in the early twentieth century after various reforms. It is thus clear that with a long history, mathematics education originates from ancient times. Of course, in the long course of human history, mathematics education can only be preliminary and scattered at first. Later, with the development of production and economy, the expansion of mathematics application, and the establishment of its position in schools, mathematics education can be further developed. At the same time, people have paid more attention to the research on the regularity of mathematics education itself. In foreign countries, the term “didactics of mathematics” first appeared in the book “An Intuitive Theory on Numbers” by Swedish educator Pestalozzi in 1803. In our country, the didactics of mathematics emerged, formed, and developed with the rise of normal education. But as a science, didactics of mathematics started from the beginning of the nineteenth century and developed rapidly over the past century. As stipulated in the Unified Statute of Primary Normal Schools promulgated by the Qing government in 1904, arithmetic and the sequence method of geometric algebra were included in mathematics teaching. As further stipulated in the Unified Statute of Higher Normal Schools promulgated in the same year, various didactics including arithmetic didactics were listed as compulsory courses. After the Revolution of 1911, with the development of normal education, didactics of mathematics became an independent subject. Liu Kaida compiled Secondary school Mathematics Didactics, the earliest mathematics didactics textbook in our country, in 1947. After the founding of the People’s Republic of China, Northeast Normal University took the lead-in offering this course, and later normal colleges and universities followed suit. After the Cultural Revolution, with the promulgation of the teaching syllabus, the textbook of Teaching Materials and Methods for Secondary School Mathematics co-edited by thirteen normal colleges was published, the class hours were also guaranteed, and the teaching requirements tended to be unified basically, and a variety of textbooks of this kind have been published one after another. Based on the teaching objectives proposed by a certain society for a certain period, the didactics of mathematics studies the regularity of mathematics teaching in a certain development period of mathematics, with teaching materials and teaching methods as its core. This is the reason why this course has always been called teaching materials and methods of mathematics or didactics of mathematics. Due to the regularity of mathematics teaching, on the one hand, it is bound to be constrained by and fits in with the development of mathematics itself and society; on the other hand, it is bound to develop with the development of pedagogy, psychology, philosophy, technology, etc. and the deepening of people’s understanding of the process of mathematics teaching. With increasingly higher requirements for mathematics teaching and the need to scientifically understand the laws

1.3 Nature, Task, and Significance of Mathematics Pedagogy

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of mathematics teaching, the task of establishing mathematics pedagogy has been proposed internationally on the basis of didactics of mathematics since 1980s.

1.3 Nature, Task, and Significance of Mathematics Pedagogy 1.3.1 Nature of Mathematics Pedagogy With mathematics teaching as its research object, mathematics pedagogy is a science that studies the process of mathematics teaching. Modern teaching theory believes that mathematics teaching is the teaching of mathematical activities (thinking activities), rather than the teaching of results of mathematical activities (mathematical knowledge). Mathematics teaching is both a science and an art; it is not only an information transfer process, but also a complex control process. In addition, mathematics teaching must also pay attention to cultivating students’ thinking ability and must take into account the laws of people’s cognition and the characteristics of psychological activities. The nature of a subject is the key to determining the task and system of this subject. It is a global and fundamental issue. Regarding the nature of secondary school mathematics pedagogy, people tend to believe that it is related to the needs of society and contemporary scientific, technological, and educational development and is related to mathematics, philosophy, pedagogy, psychology, logic, technology, epistemology, information theory, cybernetics, system theory, and other disciplines, and it is a science that summarizes the regularity of secondary school mathematics education, starting from its own research object and applying the principles of the above disciplines and methods. At the same time, the theory of secondary school mathematics teaching, the teaching content, methods, means, and other issues it studies are also constantly developing and changing with the passage of time. In a certain period of time, it may have a gradually improving system, but it is difficult to have a finally perfect model. It is not a summary of specific experience in the teaching of secondary school mathematics, but a theoretical subject with extremely rich content. Therefore, secondary school mathematics pedagogy is a comprehensive and independent interdisciplinary subject and a relatively developing applied theoretical subject with strong practicality. Mathematical pedagogy is an interdisciplinary subject of many disciplines, but it is more of an interdisciplinary subject of mathematics and pedagogy. Some people say that there is a subdiscipline of teaching materials and methods in pedagogy, so in terms of theory, it seems that secondary school mathematics pedagogy is more likely to fall under pedagogy. Regarding the affiliation of this discipline, we believe that we must not only look back, but also confront the present and the future; we must consider it from the scope of old theories and analyze it based on the actual situation today as well.

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Under the historical conditions in the past, the mathematics didactics was brought into the category of pedagogy to study the general laws and methods of mathematics teaching in secondary schools, which fit in with the development condition of mathematics and pedagogy at that time and the requirements of secondary school mathematics teaching. However, there is a trend of “mathematization” of various sciences today, mathematics itself is facing a major breakthrough, mathematical science is becoming more and more important, and the requirements for mathematics teaching are also getting higher and higher. The mathematics didactics is not only mathematics didactics in secondary schools, but also includes mathematics didactics in preschools, secondary schools, universities, vocational education, and continuing education. As it were, there should be mathematics didactics in any field of mathematics teaching. The research on mathematics education today can no longer only take the general laws put forward in teaching theory as the problems to be studied. At the same time, as a mathematics educator today, one should be a mathematician first, not just an educator. Likewise, the teaching and research of mathematics didactics cannot be undertaken by teachers of pedagogy not only now but also in the future. Therefore, in terms of the development trend of this subject and the quality of the talents responsible for teaching and research of this subject, mathematics pedagogy, an interdisciplinary subject of various subjects, is not more likely to fall under pedagogy, but in essence it should be more likely to fall under mathematics.

1.3.2 Task of Mathematics Pedagogy What is the task of secondary school mathematics pedagogy? As pointed out in the syllabus of Teaching Materials and Methods for Secondary School Mathematics for normal colleges promulgated by the Ministry of Education in May 1980, “students are enabled to be familiar with the Secondary school Mathematics Syllabus, clarify the purpose of secondary school mathematics teaching, preliminarily grasp the general rules that should be followed in secondary school mathematics teaching, understand the routine work of secondary school mathematics teachers (including formulating teaching work plans, preparing lessons, giving lessons, tutoring, examination, extracurricular activity guidance, etc.), have the preliminary ability to analyze and process textbooks, and lay the necessary foundation for future secondary school mathematics teaching work”. Indeed, the above formulation is correct and necessary. However, it is still necessary to further emphasize the status and role of secondary school mathematics pedagogy in the teaching of mathematics departments in normal colleges and universities. Here, some people have got the point. If we compare secondary school mathematics teachers trained by the mathematics departments of normal colleges and universities to a product, then the teaching of various basic and specialized courses offered by the mathematics departments can be compared to all parts of this product, and the teaching of mathematics pedagogy is to assemble these parts to make it a qualified product. In other words, secondary school mathematics pedagogy is to associate all

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the basic theories, specialized knowledge, and basic skills that students have learned with the actual need of secondary school mathematics teaching, so that they can all serve secondary school mathematics teaching. This is the embodiment of comprehensiveness of this subject, which is the reason why this subject must be offered for senior students. It is thus evident that this subject is very important. As it turns out, a graduate from mathematics department of a normal university, only equipped with a wealth of specialized knowledge and at the same time with certain secondary school mathematics education accomplishment, teaching, and research skills, can make a truly qualified secondary school mathematics teacher. However, we have to admit that the ministerial syllabus was promulgated a long time ago, new content needs to be added, especially in cultivating qualified secondary school mathematics teachers, and it is necessary to add the requirements for training new force responsible for the reform of secondary school mathematics teaching. One of the important reasons for the long-term backwardness of science of mathematics education in our country, the confusion of mathematics teaching theories, and the singleness of mathematics teaching skills is that there is not a mathematics teacher team with good quality and high accomplishment. This team needs not only senior talents specializing in the teaching and research of science of mathematics education, but also a large new force in the front line of teaching reform. The current mathematics teachers in universities, secondary, and primary schools in our country cannot meet the requirements of the new situation, so they need to be cultivated and improved. If the graduates of our mathematics department are all new forces in the educational reform and their quality has generally been improved, our career development can be guaranteed. Therefore, secondary school mathematics pedagogy is an applied theoretical subject that studies the characteristics and laws of secondary school mathematics teaching. Its task is as follows: Guided by dialectical materialism, summarizing the daily work experience of secondary school mathematics teaching, to enable students to be preliminarily armed with the ability to engage in secondary school mathematics teaching and research after graduation, lay the necessary foundation to be competent for secondary school mathematics teaching, be responsible for the reform of secondary school mathematics teaching, and to meet the needs of “three orientations”.

1.3.3 Significance of Mathematics Pedagogy (1) In terms of the rapid development of science of mathematical, it helps to clarify the importance and consciousness of learning and researching the science of mathematics education. At present, the world is in a new era of rapid scientific and technological advancement, and the scientific and technological update is getting faster and faster, particularly in mathematics. How we can better adapt to the requirements of this new

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situation so that people can master the knowledge and ability of modern mathematics more quickly, efficiently, and economically has become a sharp challenge for us. To this end, one is to learn the best mathematics through the best way, which is not impossible from a holistic point of view; the other is to improve ability, develop intelligence, and give the learners a key to open the door to mathematics. Learning and researching the science of mathematics education can help people to achieve the above goals. In the past 70 years, there has been an international upsurge in the modernization movement of mathematics education, which was born to fit in with this need. Therefore, it is safe to say that the modernization of a country must go hand in hand with modern science and technology, modern science and technology must go hand in hand with modernized mathematics, and modernized mathematics must go hand in hand with modernized science of mathematics education. (2) In terms of the current situation and training goals of normal colleges, it helps to recognize the urgency and pertinence of learning and researching the science of mathematics education. After the “Cultural Revolution”, the situation of mathematics teaching in secondary schools in our country has fundamentally taken a turn for the better after bringing order out of chaos. However, for a number of reasons, the educational ideology, teaching content, and teaching methods are all divorced from reality to varying degrees, and the “tactic of excessive assignments” and the phenomena of “more lectures, more exercises, and more tests” still exist; the phenomena of teaching in an improper way, students being overburdened, high scores but low qualities, etc. can be found everywhere. This shows that we have not been able to act in accordance with the laws of education at all, and popularizing the knowledge of science of mathematics education has become a very urgent task at present. The training objects of normal colleges are secondary school teachers. As pointed out by the Central Committee of the Communist Party of China at the National Teachers’ Work Conference in June 1980, “teachers must have noble sentiments, profound knowledge, and understand the laws of education”. As also pointed out in the Program for the Reform and Development of Education in China promulgated by the Central Committee of the CPC and the State Council, “as engineers of the human soul, teachers must strive to improve their ideological and political quality and professional skill; love the cause of education, impart knowledge and cultivate people, and be a model for others; organize teaching carefully, take an active part in educational reform, and improve teaching quality continuously”. At present, with the implementation of the curriculum standards of primary and secondary school mathematics, fundamental changes have taken place in the teaching of mathematics in primary and secondary schools in our country. For students in normal colleges and universities, they will soon graduate from colleges and switch from students to teachers, and it is of special significance and great urgency for them to learn and

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research the basic knowledge related to science of mathematics education so as to adapt to this switch and be competent for the important task of teaching in the future. (3) In terms of the characteristics and current situation of mathematics pedagogy, it helps to experience the persistence and difficulty of learning and researching science of mathematics education. As mentioned above, mathematics pedagogy is a comprehensive and independent interdisciplinary subject and a relatively developing applied theoretical subject with strong practicality as well. It is not only restricted by the development of other subjects, but also subject to further maturity of this discipline. It should be mentioned that the ideal mathematics pedagogy should have scientific concepts, categories and systems, and scientific epistemology and methodology as well. Obviously, it has not yet met such requirements. At the same time, due to its practicality, the current international secondary school mathematics education is in a period of major changes, and many major issues need to be studied by people urgently to fill in the gap. A large amount of educational practical experience needs to be advanced to theory urgently and also needs to be guided by correct theories urgently. It should be noted that the science of mathematics education is still relatively backward in our country, mathematics educators have poor educational science accomplishments, they need to brush up on new educational theories, new curriculum standards and teaching materials, new teaching means and methods, etc., and we need to continue practicing and exploring. Therefore, according to the characteristics and current situation of science of mathematics education, it is a long-term and arduous task to learn and research science of mathematics education. Review Questions 1. What is scientific mathematics, what is mathematics as an educational science, and what are the differences and connections between them? 2. How did mathematics education come into being and develop, and what were the characteristics of mathematics education in ancient China? 3. How did mathematics pedagogy come about and how does it relate to mathematics didactics? 4. What is the nature and task of mathematics pedagogy and why? 5. What is the profound and practical significance of studying the science of mathematics education, and what attitude should we take to study it?

Chapter 2

Secondary School Mathematics Logic

As once pointed out by Engels, “elementary mathematics, that is, mathematics of constants, is active within the scope of formal logic, at least this is the case in general”. Formal logic is a science that studies forms of thinking (concept, judgement, inference, proving) and their laws (law of identity, law of contradiction, law of excluded middle, law of sufficient reason). Mathematics has its own unique logic system, with rigorous logicality. For secondary school mathematics teachers, first of all, they should master the basic knowledge related to logic of secondary school mathematics.

2.1 Secondary School Mathematics Concepts 2.1.1 Meaning of Concept Mathematical concepts are the reflections of spatial forms and quantitative relations in the real world and their essential attributes in thinking. As stated by Engels, “In a certain sense, the content of science is the system of concepts”. Some modern scholars believe that “the learning process of mathematics is the process of constantly establishing various mathematical concepts”. As we know, people’s knowledge of objective things is generally to form ideas (representations) through senses and perception, which is the stage of perceptual cognition. Then through a series of thinking activities such as analysis, comparison, abstraction, generalization, etc., people come to understand the essential attributes of things and form concepts, which is the stage of rational cognition. The rational cognition is continuously deepened on the basis of practice, and the concepts are further developed accordingly. Some mathematical concepts directly derive from the spatial forms and quantitative relations reflecting objective things. For example, such concepts as natural © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_2

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number, point, line, plane, and solid are the cases. However, most mathematical concepts are formed and developed after many abstraction and generalization processes on the basis of some mathematical concepts. For example, the concepts of irrational number and complex number are generated on the basis of the rational number system and the real number system, respectively; and the generation and development process of such concepts as relation, mapping, group, ring, and field is more complicated. Mathematical concepts have the following characteristics: First, mathematical concepts are abstract and concrete. This is because mathematical concepts represent essential attributes of a kind of things, resulting in its abstractness, which is far away from concrete reality, and the higher the level of abstraction, the farther away from reality. But no matter how abstract it is, highlevel abstraction always takes low-level things as concrete content, and mathematical concepts are always the basic components of mathematical propositions and mathematical reasoning, so they must be implemented into concrete numbers, formulas, and shapes. Second, mathematical concepts are relative and developmental. Within a certain scientific system or a specific research field, the meaning of mathematical concepts is always consistent. For example, numbers in elementary schools always refer to positive rational numbers; straight lines in middle schools always refer to planar straight lines. However, concepts such as number and shape are constantly developing. For example, natural numbers → rational numbers → real numbers → complex numbers; points in a straight line → points in a plane → points in a space → points in n-dimensional space; acute angle → arbitrary angle → space angle, etc. Third, being “three-in-one”, the definitions, terms, and symbols of mathematical concepts are in a complete scientific system, for example, triangle “”, parallel “”,  differential “ dx”, and integral “ ”. In addition to specific definitions, they also have corresponding specific terms and symbols, being “three-in-one” of term, definition and symbol, which is unmatched by other sciences.

2.1.2 Structure of Concept In a scientific system, any concept reflects a certain range of things and the common essence of things within this range. The range (or set) of things reflected by a concept is called the extension of the concept; the sum (or set) of the essential attributes of these things is called the connotation of the concept, and they are the description of the quantity and quality of the set of things, respectively. For example, the extension of the concept of positive even number is the set { 2, 4, 6, 8, …, 2n, …}, and the connotation is the attribute of “positive integers being divisible by 2”; the extension of the concept of triangle is various triangles, and its connotation is a closed figure composed of three line segments. In concepts of the same type, there is an inverse relationship between extension and connotation. That is, when the extension expands (or reduces), the connotation

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reduces (or expands) instead. For example, in terms of extension, {quadrilateral} ⊃ {parallelogram} ⊃ {rectangle} ⊃ {square}; {real number} ⊃ {rational number} ⊃ {natural number}, but in terms of connotation, they expand in turn. In mathematics, in order to deepen the understanding of a certain concept, or to use a more general concept to explain a special concept, the method of gradually increasing the connotation of the concept so as to reduce the extension of the concept is often adopted to obtain a series of concepts with subordination relationship, and this method is called limitation of concept. For example, if a parallelogram is added with the property that “one interior angle is a right angle”, it becomes a rectangle. On the contrary, in order to understand general concepts from some special concepts, or in order to understand the common property of the concepts of the same type, sometimes the connotation of a concept is gradually reduced to enable the extension of the concept to gradually expand, so as to obtain a series of concepts with subordination relationship, and this method is called generalization of concept. For example, irrespective of specific meanings of the elements in various number systems, when only their operational properties are taken into account, they can be generalized into such concepts as group, ring, and field. The method of limitation and generalization of concept is often adopted in mathematics to produce new concepts.

2.1.3 Relationship Between Concepts 1. Identical relationship The relationship between two concepts with exactly the same extension is called identical relationship, which can be shown in Fig. 2.1, and the conjunctions “namely”, “that is”, and so on are commonly used descriptively. In a judgement process, two concepts with identical relationship can replace each other. For example, an equilateral triangle and a regular triangle, the height on the base, the bisector of the vertex angle, and the median of the base in an isosceles triangle are all the same concept, and they can replace each other and reinforce each other in judgement. 2. Cross-relationship The relationship between two concepts with overlapping extension is called the crossrelationship, as shown in Fig. 2.2, and “some” and “somewhat”, etc. are often used descriptively. For example, isosceles triangles and right triangles, natural numbers and positive integers, etc. are all cross-relationships. Whether an equation set is Fig. 2.1 . A=B

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Fig. 2.2 . B

A

Fig. 2.3 . B A

solvable or not is to determine whether the solution set of each equation has crossrelationship. 3. Subordination relationship The relationship between two concepts whose extensions have containment relation is called subordination relationship, as shown in Fig. 2.3. Among them, concept A with a larger extension range is called superordinate concept or species concept, and concept B with a smaller extension range is called subordinate concept or class concept. For example, equality and equation, equation and integral equation are all subordinate relationships. Among them, equality is the species concept of equation, and equation is the species concept of integral equation; equation is the class concept of equality, and integral equation is the class concept of equation. 4. Contradictory relationship The relationship between two concepts whose extensions are mutually exclusive but the sum of whose extensions is equal to the extension of the most adjacent species concept is called contradictory relationship, as shown in Fig. 2.4. For example, rational numbers and irrational numbers, right triangles and non-right triangles, intersecting lines and parallel lines in a plane are contradictory relationships. 5. Opposite relationship The relationship between two concepts whose extensions are mutually exclusive but the sum of whose extensions is smaller than the extension of the most adjacent species concept is called opposite relationship, as shown in Fig. 2.5. For Fig. 2.4 . A

B

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Fig. 2.5 . A

B

example, positive numbers and negative numbers, acute triangles and right triangles, intersecting lines and parallel lines in the space are all opposite relationships.

2.1.4 Definition of Concept Using known concepts to understand unknown concepts to transform unknown concepts into known concepts is called defining a concept. The definition of a concept is composed of two parts: the defining concept (known concept) and the defined concept (unknown concept). For example, rational numbers and irrational numbers (defining concept) are collectively referred to as real numbers (defined concept); a parallelogram (defined concept) is a quadrilateral with two pairs of parallel sides (defining concept). There are several definition methods as follows. 1. Definition method of “species + class difference” This is a definition method that defines the most adjacent species concept in the superordinate concepts of the defined concept according to the subordination relationship of the concepts and then points out the essential attribute of the defined concept to distinguish from other class concepts in its species concept. For example, in the above definition of parallelogram, quadrilateral is its most adjacent species concept; the class difference is the essential attribute of “two pairs of parallel sides”. As the class difference is not unique, the definition made by this method is generally not unique. For example, a parallelogram can also be defined by taking “two pairs of congruent sides”, “one pair of opposite sides that are both parallel and congruent”, “two diagonals bisecting each other”, etc. as class difference, and they are all equivalent to each other. This definition method reveals the connotation of concept accurately and clearly, helping to establish relationship between concepts and make knowledge systematized. Therefore, it is widely used in defining secondary school mathematical concepts. 2. Method of genetic definition As a special “species + class difference” definition method, it takes the genesis and formation characteristics that only belong to the defined things rather than other things as the class difference. For example, the locus of a point equidistant from a fixed point in a plane (space) is called a circle (sphere). In addition, the method of

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genetic definition is also adopted for the concepts of cylinder, cone, circular truncated cone, differential, integral, coordinate system, etc. in secondary school mathematics. 3. Method of inverse definition This is a definition method that gives the concept extension, also known as inductive definition method. For example, integers and fractions are collectively called rational numbers; sine, cosine, tangent, and cotangent functions are called trigonometric functions; ellipse, hyperbola, and parabola are called conics; logical sum, negation, and product operations are called logical operations, etc., which all adopt this definition method. 4. Method of stipulative definition It is a special method of inverse definition, and the stipulation method is employed for definition to meet some special needs. For example, a 0 = 1 (a = 0), 0! = 1, C0n = 1, and numbers in the form of a + bi are called complex numbers, etc., which all adopt the method of stipulative definition. The necessity and rationality of this definition method should be clarified in mathematics teaching. In addition, methods of descriptive definition (such as the definitions of equality and limit in current secondary school mathematics) and recursive definition (such as the definitions of n-order determinant, n-order derivative and n-tuple integral) are also employed in secondary school mathematics. They give a definition by virtue of another object (e.g., the concept of logarithm is defined by resorting to the concept of exponent). In order to define a concept correctly, the definition should meet the following basic requirements: (1) A definition should be proportionate, that is, the extension of the defining concept must be the same as that of the defined concept, which can neither be expanded nor reduced. That means being appropriate, neither too wide nor too narrow. For example, infinite non-recurring decimals are called irrational numbers, so it is obviously wrong to define irrational numbers in terms of infinite decimals (too wide) or in terms of surds (too narrow). (2) Definitions cannot be cycled, that is, in the same scientific system, concept A cannot be defined in terms of concept B, while at the same time, concept B is defined in terms of concept A. For example, an angle of 90° is called a right angle, and one-ninetieth of a right angle is called 1°, where a cycle occurs. (3) A definition should be clear and concise, generally without negative forms and unknown concepts. For example, a line as straight as a ramrod is called a straight line (unclear); a planar parallelogram with two pairs of opposite sides parallel to each other (unconcise); numbers that are not rational numbers are called irrational numbers (negative form). For middle school students, the complex numbers a + bi with imaginary part b = 0 are called real numbers (using unknown concepts), etc. These are all improper.

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2.1.5 Series of Concept In a scientific system, some concepts may form a logical chain in a certain order, forming a series of concept. For example, Quantities : ⎧ ⎪ Quantities with opposite meanings − Positive and negative numbers ⎪ ⎪ ⎨ −Rational number − Real number ⎪ Multidirectional quantities − Complex number ⎪ ⎪ ⎩ −Modulus and argument of complex number Limit − Derivative − Differential − Gradient

The undefined concepts are called primitive concepts. For example, number, quantity, point, line, plane, solid, set, element, correspondence, etc. are all primitive concepts. Descriptive method or abstraction method is employed for some primitive concepts, and visual description or designation of object is employed for others to make clear their meanings in mathematics.

2.1.6 Classification of Concepts The classification of concepts is a logical method to reveal the concept extension. Generally, the classified concept is regarded as species concept, and its extension is divided into several opposite class concepts according to a certain attribute. That attribute is called the basis for classification. Those classified according to one or two attributes are called single classification and classification by dichotomy, respectively. For example, triangles are classified into acute triangles, right triangles, and obtuse triangles in terms of angle, and they are classified into scalene triangles and isosceles triangles in terms of side. Dichotomy is often used for continuous classification in mathematics. For example, ⎧  ⎧ ⎧ ⎪ Positive integer ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Interger ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ Non - positive integer ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ Rational number⎪ ⎪ ⎪ Positive fraction ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Real number ⎩ Fraction Negative fraction ⎨ ⎪ ⎪ ⎪  Complex number ⎪ ⎪ ⎪ ⎪ Positive irrational number ⎪ ⎪ ⎪ ⎪ ⎩ Irrational number ⎪ ⎪ ⎪ Negative irrational number ⎪ ⎪  ⎪ ⎪ ⎪ Pure imaginary number ⎪ ⎪ Imaginary number ⎩ Non - pure imaginary number

A correct classification should meet the following requirements:

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(1) The same standard should be followed for classification; (2) classification should be carried out step by step. (3) No repetition or omission is allowed in classification; (4) the subclasses obtained after classification should be incompatible. It is a key to solving mathematical problems to understand and apply mathematical concepts correctly, and it is also very important to apply classification methods correctly. Example Given x ∈ R, prove: f (x) = x 2006 − x 2005 + x 2004 − x 2003 + 1 > 0. Proof It is difficult to prove this inference in a usual way. But it’s not hard to find that when x = 0, f (x) > 0. Therefore, it is natural to think about the value of f (x) when x = 0. Obviously, f (x) > 0 when x < 0, and then think about the value of f (x) when x > 0. What’s more, f (x) > 0 when x = 1, and then think about the cases when x = 1. When x > 1, x 2006 > x 2005 and x 2004 > x 2003 , so f (x) > 0. When 0 < x < 1, x 2006 > 0, −x 2005 + x 2004 > 0 and −x 2003 + 1 > 0, therefore f (x) > 0. In this way, the value of x is continuously classified four times (dichotomy), namely ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x =0 x 0 ⎪ x >1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ x = 1 0 ∠B AB > AC A respectively assertive proposition also has the important property of being equivalent to its converse proposition. Suppose a respectively assertive proposition N contains n propositions, whose conditions and conclusions are respectively Ai and Bi (i = 1, 2…n), and Ai → Bi , let’s prove Bi → Ai . Now take n − 1 propositions from this respectively assertive proposition N, for example. Aj → Bj ( j = 2, 3 . . . n). Because of the property of respectively assertive propositions, these n − 1 propositions are simultaneous, which is tantamount to saying, A1 → B 1 , Since propositions that are mutually contrapositive are equivalent, we can get B1 → A1 from the above formula,

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In the same way, Bk → Ak (k = 2, 3… n). Therefore, Bi → Ai (i = 2, 3… n). Therefore, if a respectively assertive proposition is correct, then its converse proposition (which is also a respectively assertive proposition) must be correct and can be directly employed as a converse theorem. In secondary school mathematics, there are quite a few respectively assertive propositions, such as the determination theorem for roots of quadratic equations with one unknown, the theorem of perpendicular and oblique lines of a straight line, the theorem of positional relations between a point (or line) and a circle, the theorem of positional relations between two circles, and so on.

2.3 Basic Laws of Formal Logic The basic laws of formal logic, namely law of identity, law of contradiction, law of excluded middle, and law of sufficient reason, must be mastered in order to reason or prove by taking advantage of concepts and judgements accurately.

2.3.1 Law of Identity The content of the law of identity is that during the process of the same thinking at the same time in the same place, the concepts and judgements used must be determinate and consistent. The formula of the law of identity: A → A, that is, A is A. It is thus clear that based on the content of the law of identity, it has two specific requirements, which are: First, the object of thinking should remain the same. That is to say, the object to be examined in the course of thinking must be determinate, consistent, and cannot be changed halfway. Second, the concept of the same thing should remain the same. That is to say, in the course of thinking, the same object of thinking should be represented by the same concept, and the same thing cannot be represented by different concepts, nor can different things be confused and represented by the same concept. The principle manifestation of errors of violating the law of identity is disguised replacement of concept or the use of unclear concept and so on as far as concept is concerned and unclear thesis or clandestine change of thesis in reasoning. Example 1 Organizations of Young Pioneers should be established in a school. A normal college is a school. Organizations of Young Pioneers should be established in a normal college. Example 2 If x = ay, then x n − a n = a n (y n − 1) and a n−1 (x − a) = a n (y − 1).

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Since (yn − 1) is divisible by (y − 1), a n (y n − 1) is also divisible by a n (y − 1), namely the value of can be divisible by the value of a n−1 (x − a) for any positive integer n and for any real number x and a. The “school” in the first sentence is not the same concept as that in the second sentence in Example 1, which is a mistake of disguised replacement of concept. Through checking calculation, it is not difficult to find the conclusion of Example 2 is also wrong. For example, supposing x = 3, a = 2, n = 2, then x n − a n = 5, a n−1 (x − a) = 2, but 5 cannot be divisible by 2. The error lies in the referenced concept of “divisible”, which involves polynomials in the first part of the reasoning and involves natural numbers in the last part, and they are not the same. If a polynomial is divisible by another polynomial and the residue is zero, it does not mean that the coefficient of the quotient is all integers. For example, x 2 − 22 is divisible by 2 (x − 2), and its quotient is x2 + 1. As the concept of “divisible” is changed clandestinely here, it violates the law of identity, thus leading to an error.

2.3.2 Law of Contradiction The content of the law of contradiction is that during the process of the same thinking at the same time in the same place, we cannot affirm what it is and negate what it is as well. That is to say, two contradictory judgements in the course of the same thinking cannot both be true, one of which must be false. The formula of the law of contradiction is A ∧ A, that is, A is not A. The law of contradiction is virtually the law of non-contradiction, it is the extension of the law of identity, expressing the content of the law of identity in a negative form. The law of contradiction is the logical foundation of negative judgement, its function is to eliminate self-contradiction in thinking, and keep non-contradictoriness of thinking. The contradiction in thinking mentioned here is the logical contradiction that occurs when people’s minds fall into confusion or they play with sophistry deliberately, which is different from the contradiction existing in objective things themselves. Two contradictory judgements cannot both be true, but can both be false. For example, that ABC is an acute triangle and that ABC is an obtuse triangle are two contradictory judgements, one of which is right, and the other of which must be wrong; but if one of which is wrong, the other of which may not be right. This is because there is a third case where ABC is a right triangle.

2.3.3 Law of Excluded Middle The content of the law of excluded middle is that during the process of the same thinking at the same time in the same place, a clear positive or negative judgement

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must be made for the same object. Namely, in the course of the same thinking, two contradictory concepts or judgements cannot both be false, and one of them must be true, to exclude a third possibility. The formula of the law of excluded middle is A ∨ A, that is, A or A. The law of excluded middle requires people’s thinking√ to be definite, which is the 2 is an irrational number logical basis of proof by contradiction. For example, that √ and that 2 is a rational number are two contradictory judgements, they cannot exist at the same time, and one and only one of which must be true. The law of excluded middle and the law of contradiction are both connected and different. The connection lies in that they are both about two contradictory judgements, pointing out that two contradictory judgements cannot coexist at the same time, one of which must be false. However, neither of them is capable of further determining which is true and which is false, and this question can only be solved through specific analysis with the help of other knowledge. The difference lies in that the law of contradiction states that two contradictory judgements cannot both be true, one of which must be false, while the law of excluded middle states that two contradictory judgements cannot both be false, one of which must be true. By virtue of the law of contradiction, we can only deduce false judgements from true judgements, not the other way around. However, by virtue of the law of excluded middle, we can deduce false judgements from true judgement, and the other way around as well, so the law of contradiction is the logical basis of negative judgement, and the law of excluded middle is the logical basis of proof by contradiction.

2.3.4 Law of Sufficient Reason The content of the law of sufficient reason is any true judgement must have sufficient reasons, that is, the affirmation or negation of anything must have sufficient reasons and basis. The law of sufficient reason can be expressed as: if there is B, there must be A, from which B can be deduced from A. The law of sufficient reason is the logical basis for reasoning and proof, and it is closely related to judgement. For example, in mathematical propositions, sufficient conditions as well as necessary and sufficient conditions can be taken as sufficient reasons for a conclusion, and the original theorem can be taken as sufficient reasons for its contrapositive proposition, and so on. The law of sufficient reason is closely related to the first three laws. The law of identity, law of contradiction, and law of excluded middle serve to maintain the determinacy and non-contradictoriness of the same judgement (or concept). The law of sufficient reason, on the other hand, aims to keep the connection between judgements well-grounded and convincing. Therefore, in the course of thinking, if the law of identity, law of contradiction, and law of excluded middle are violated, the law of sufficient reason will inevitably be violated.

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In a word, for mathematical reasoning and proof, the objects must be determinate (law of identity), the judgements must not be self-contradictory (law of contradiction), not be ambiguous (law of excluded middle), but be well-grounded (law of sufficient reason). In mathematics teaching, we should attach importance to cultivating the students’ habit of thinking in strict accordance with these logical laws, to improve students’ logical thinking abilities.

2.4 Secondary School Mathematics Inference The thinking process of making a new judgement from one or several existing judgements according to the relations between them is called inference. Its structure includes premise and conclusion, the based existing judgement is called premise of inference, and the new judgement made is called conclusion of inference. Correct inference shall be in line with logical forms and comply with the rules of inference. The types of inference include of direct inference and indirect inference. Direct inference has only one premise, so it is relatively simple. Indirect inference consists of two or more premises and can be further divided into inductive inference, analogical inference, and deductive inference.

2.4.1 Inductive Inference Inductive inference is a kind of inference from the specific to the general, that is, the thinking process of extending the judgement from individual or special things to that of general things of the same kind. Based on whether the scopes of the judgements in premise and conclusion are the same or not, it can be divided into complete induction and incomplete induction. 1. Complete induction If the sum of one or more judgement scopes in the premise of inductive inference is equal to the scope of the judgement in the conclusion, this inductive inference is called complete induction. Its representation is: are all objects of the things of type A. has (or does not have) P. has (or does not have) P. ... ... ... ... has (or does not have) P. ------------------------------------------------------The things of type A have (or do not have) P.

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For example, to prove that three heights or their extension lines of a triangle are concurrent, we can prove that three heights or their extension lines of an acute triangle, a right triangle, and an obtuse triangle are concurrent, and then we can infer the conclusion that three heights or their extension lines of any triangle are concurrent. For another example, to derive the formula for the distance between two points, we can discuss each case of the two points in each quadrant and coordinate axis, respectively. All of the inference methods above are complete induction. Since the complete induction has made a judgement on the judgement scope of the conclusion in the premise judgement, if they are all true, then the conclusion obtained is completely reliable, so the complete induction can be used as a strict mathematical inference method. However, in application, it should be noted that the scope of premise judgement can neither be repeated nor omitted, that is, the sum of the scope of premise judgement cannot be less than the scope of conclusion judgement. 2. Incomplete induction If the sum of the scope of the premise judgement of inductive inference is less than the scope of the conclusion judgement, such inductive inference is called incomplete induction. Its representation is: are Part of the objects of things of type A. has (or does not have) P. has (or does not have) P. ... ... ... ... has (or does not have) P. ------------------------------------------------------The things of type A have (or do not have) P.

For example, in secondary school mathematics, the inference of generalizing the operational rules of real numbers and exponential operation properties from the operations of concrete real numbers, etc. is incomplete induction. Some meteorological proverbs, agricultural proverbs, regimes, etc. are also obtained by incomplete induction. It must be noted that a conclusion derived from incomplete induction may be true or false. Therefore, incomplete induction cannot be used as a strict inference method in mathematics, but it can be helpful to put forward hypotheses or conjectures in scientific research and facilitate the discovery of laws in solving problems and inspire thinking. During teaching, in order to explain the correctness of some theorems, formulas, and properties, it is often explained by means of some special examples, which is to use examples for verification in essence, and can also be considered as an inference with incomplete induction.

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2.4.2 Analogical Inference Analogical inference is a kind of inference from the specific to the specific, that is, judging that they may have other identical or similar properties based on some identical or similar properties of two things (or two kinds of things). Its representation is: The things of type A have properties a, b, c, d. The things of type B have properties a, b, c. -------------------------------------------------------------The things of type B may have property d.

For example, in algebra, both algebraic fractions and fractional numbers have the same form of numerator and denominator, so it can be inferred that algebraic fractions can be simplified and calculated just like fractional numbers; from the relationship between lines in a plane, we can infer the relationship between planes in the space, etc., these are all analogical inference. It must be noted that the conclusion obtained by analogical inference may not be true, and it has only a certain degree of reliability. Some conclusions have yet to be proved by practice and theory. For example, without using any other mathematical notations, the maximum numbers that is made up of three ones, three twos, and three threes are 111 , 222 , and 333 , respectively. However, the maximum number that 4 is made up of three fours is not 444 by analogy, but 44 . Generally speaking, if the properties common to two kinds of things are closely related to the property inferred, the conclusion is rather reliable. The more properties the two kinds of things have in common, the more reliable the conclusion inferred is. Although the conclusions obtained by analogical inference are not always true, it still makes a positive difference in people’s cognitive activities. For example, many important hypotheses in science are put forward by analogical inference. The clues to many important discoveries and even solution methods in mathematics are provided by analogical inference. Many inventions and creations in production practice and scientific experiments are also inspired by analogical inference. Therefore, analogical inference is still a tool for acquiring new knowledge. However, we should avoid making mistakes by abusing analogical inference. For example, some students make the following mistakes by making an analogy between loga (x + y), sin(x + y) and a(b + c): loga (x + y) = loga x + loga y sin(x + y) = sin x + sin y

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2.4.3 Deductive Inference Deductive inference is a kind of inference from the general to the specific, that is, a thinking form that takes the general judgement of a certain kind of things as the premise to make the judgement of individual and specific things of this kind of things. There is an inevitable connection between the premises and the conclusions of deductive inference. As long as the premises are true and the inference is logical, we are sure to obtain a correct conclusion. Therefore, deductive inference can be used as a strict inference method in mathematics. Simple deductive inference is often realized in the form of syllogism. Its representation is: Elements in set M have (or do not have) P. x M. -----------------------------------------------------x also has (or does not have) P.

The structure of syllogism includes three judgements: major premise—the judgement reflecting the general principle, minor premise—the judgement reflecting the relation of individual object to the general principle, and the conclusion. If both the major premise and the minor premise are correct, the conclusion must be correct. For example, because negative numbers have no logarithms (major premise), − 1 is a negative number (minor premise), Therefore, −1 has no logarithm (conclusion). For another example, because the diagonals of a parallelogram bisect each other (major premise), ABCD is a parallelogram (minor premise), Therefore, diagonals AC and BD of ABCD bisect each other (conclusion). However, in practical application, the major premise or minor premise can be omitted, or even only the conclusion is required.

2.5 Secondary School Mathematics Proof Mathematical proof is an inference process where the axioms, theorems, definitions, formulas, properties, and other mathematical propositions that have been confirmed to be true are used to demonstrate a certain mathematical proposition. The process of mathematical proof usually manifests as a series of inference.

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Any logical proof is composed of thesis, argument, and demonstration. Thesis is a judgement whose truth needs to be proved, arguments are the judgements referenced to prove the truth of the thesis, and demonstration is the process to prove the truth of the thesis by a series of inference from the argument. Mathematical proof is conventionally written in three parts: given, prove, and proof. Among them, arguments include the conditions given by a thesis and those arguments referenced when proving the thesis, as well as known axioms, theorems, formulas, definitions, laws, properties, and other propositions. Proving is the conclusion of the thesis, that is, the proposition that needs to be proved true; proof is demonstration, that is, the inference process of proving the truth of the thesis. There are two commonly used proof formats: combination type and advancing type. The combination type is a writing format where “because, so” is used in combination to represent inference relation (see Examples 1–5 in this section), and the advancing type is a writing format where implication relation or inference relation is represented by symbols “⇒”, and they consist of two basic forms: horizontal type or vertical type. Generally speaking, the writing format of combination type can omit many repeated words, which is relatively concise and people are used to. However, it is easy to confuse causality and feels like “chaos”. The writing format of advancing type can ensure clear causality and coherent inference. However, as all the premises need to be listed for each step of inference, the same premise sometimes needs to be repeated several times, which is usually more “complicated” than the combination type. The inductive inference, analogical inference, and deductive inference introduced above are virtually inductive method, analogical method, and deductive method in proof. Now several commonly used proof methods are briefly introduced below.

2.5.1 Analysis Method and Synthesis Method In the mathematical proof, if the direction of inference is tracing proving back to known conditions or from the unknown to the known, this method of thinking is called analysis method, simply called “finding the cause from the result”. On the contrary, if the direction of inference is from the known to proving or from the known to the unknown, this method of thinking is called synthesis method, simply called “obtaining the result from the cause”. Example 1 Given a, b are unequal positive numbers, prove that a 3 +b3 > a 2 b+ab2 . Proof 1: Analysis method To prove a 3 + b3 > a 2 b + ab2, we only need to prove (a + b) a 2 − ab + b2 > ab(a + b), since a > 0, b > 0, a + b > 0,  then we only need to prove a 2 − ab + b2 > ab, a 2 − 2ab + b2 > 0.

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that is (a − b)2 > 0, which is obviously true. Therefore, a 3 + b3 > a 2 b + ab2 . Proof 2: Synthesis method Since a = b, we have a − b = (a − b)2 >  0, 0, 2 2 2 that is a − 2ab + b > 0, a − ab + b2 > ab, and a > 0, b> 0, a + b > 0, then (a + b) a 2 − ab + b2 > ab(a + b), Therefore, a 3 + b3 > a 2 b + ab2 .

2.5.2 Direct and Indirect Proof In mathematical proof, the method of proving the truth of a proposition in a positive way is called direct proof. Any proof that proves the truth of a proposition by deductive method is direct proof. It is a commonly used proof method in secondary school mathematics. The proof method that affirms the truth of a thesis by proving that its negative thesis is not true or by proving that its equivalent proposition is established instead of proving the truth of a thesis directly is called indirect proof. Indirect proof mainly consists of proof by contradiction and identity method. 1. Proof by contradiction To prove the proposition “A → B” is true, we can start with the contrary by proving that its converse proposition “ A → B” is false, so as to affirm that “A → B” is true; or prove its equivalent proposition “B → A” to be true. This proof method is called proof by contradiction, which includes reduction to absurdity and method of exhaustion. The general steps of proof by contradiction are as follows: (1) Assume that the conclusion of the proposition is not true (that is, the negation of the conclusion is true); (2) Proceed from the negative conclusion, make inference step by step, and obtain a self-contradiction with axioms, aforesaid theorems, definitions, supposed conditions, provisional assumptions, etc. (which reveals that the conclusion cannot be negated); (3) Based on the law of excluded middle, finally affirm the original proposition to be true. In the application of proof by contradiction, if there is only one possible case in the negative aspect of a proposition conclusion, then as long as this case is overturned, the conclusion can be affirmed to be true, which is called reduction to absurdity (see Example 2). If there is more than one case in the negative aspect of a proposition conclusion, then it is necessary to refute all the possible cases in the negative aspect in order to affirm the conclusion, and this proof by contradiction is called method of exhaustion (see Example 3).

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Fig. 2.7 .

Example 2 Prove: cos10° is an irrational number. Proof Assume that cos10° is a rational number, denote it as cos 10◦ = qp (p and q

3

 are coprime numbers), then cos 30◦ = 4 cos3 10o − 3 cos 10◦ = 4 qp − 3 qp is √

a rational number. But cos 30° = 23 is an irrational number, which contradicts the aforesaid assumption. Therefore, cos10° is an irrational number. Example 3 As shown in Fig. 2.7, in ABC, it is known that BE and CF are the bisectors of B and C, respectively, and ∠BE = ∠CF. Prove: AB = AC. Proof If AB = AC, then AB > AC or AB < AC. Make 口BEGF and join CG. (1) Assume AB > AC, then ∠ACB > ∠ABC, ∠BCF > ∠CBE, BF > CE, Since BF = EG, we know EG > EC, ∠ECG > ∠EGC. And ∠FCG = ∠FGC, so ∠FCE < ∠FGE = ∠FBE. Then ACB < ABC (contradicting the assumption), that is, AB > AC is impossible. (2) Similarly, we can prove that AB < AC is also impossible. Therefore, AB = AC. 2. Identity method As mentioned above, two propositions that are mutually converse or negative are not necessarily equivalent, a proposition is equivalent to its converse (or negative) proposition only when its conditions and conclusions are uniquely existent and the concepts they refer to are the same, and this principle is called the identity principle. For a proposition conforming to the identity principle, when it is difficult to prove it directly, its equivalent converse proposition can be proved, and this method of proof is called the identity method. The general steps of the identity method are as follows: (1) When the items contained in the conditions and conclusion of a proposition are uniquely existent, draw (set) a figure (or equation) that conforms to the conclusion of the proposition; (2) Prove that the figure drawn (or the equation set) conforms to the known conditions;

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(3) Based on the uniqueness, determine that the figure drawn (or the equation set) coincides with the known figure (or is identical with the known relation expression); (4) Finally affirm the original proposition is true.

√ √  Example 4 Prove: lg 3 + 5 + 3 − 5 = 21 . Proof Since both sides of the equation are uniquely existent real numbers, we can consider using the identity method.

√ √  Let lg 3 + 5 + 3 − 5 = t, √ √ then 10t = 3 + 5 + 3 − 5. √ Square both sides, we get 102t = 6 + 2 9 − 5 = 10. and 2t = 1, t = 21 ,

√ √  Therefore, lg 3 + 5 + 3 − 5 = 21 . Example 5 As shown in Fig. 2.8, it is known that the orthocenter of  ABC is H and the extension line of median AM cuts  HBC at G. Prove: M is the midpoint of AG. Analysis Since the other common point G of straight line AM and HBC is unique except for the common point K and point G in ray AM and making MG = AM is also unique, then the identity method can be used. Pick a point G in ray AM to make MG = AM, so now we need to prove that G is coincident with G. Proof Pick a point G in ray AM so that MG = AM, and join G B and G C. Since MB = MC, ABG C is a parallelogram, ∠BG C = ∠BAC. Since A, F, H, and E are concyclic, ∠BAC + ∠BHC = 180°, ∠BG C + ∠BHC = 180°. Therefore, B, G , C, and H are also concyclic, that is, point G is in HBC. And the other common point G of straight line AM and HBC is unique except for common point K, so point G coincides with G. Therefore, M is the midpoint of AG. Fig. 2.8 .

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The proof by contradiction and the identity method are all indirect proof. The main differences between them are: (1) The methods are different. For the proof by contradiction, first negate the conclusion, and then give refutation. For the identity method, first draw (set) a figure (or equation) that conforms to the propositional conclusion and then deduce that the figure (or equation) made is the same as the known figure (or relation expression). (2) The bases are different. The logical basis of proof by contradiction is the law of excluded middle, using the equivalence of the original proposition and its contrapositive proposition for proof; the logical basis of the identity method is the law of identity, using the equivalence of the original proposition and its converse proposition for proof. (3) The scopes of application are different. The proof by contradiction starts with negating the conclusion of a proposition to simply infer a contradiction, and this contradiction is not necessarily caused by the “unique existence” of a figure (or a relation expression). Thus, proof by contradiction applies to all kinds of propositions, while the identity method applies only to propositions that conform to the law of identity. To sum up, Methods of inference and proof : ⎧

⎧ ⎪ Complete induction ⎪ ⎪ ⎪ ⎪ Inductive method ⎨ ⎪ ⎪ Incomplete induction ⎪ ⎪ Modes of inference ⎪ ⎪ ⎪ Analogical method ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ Deductive method ⎪ ⎪

⎨ Analysis method Train of thought ⎪ Synthesis method ⎪ ⎪ ⎧ ⎪ ⎪ Direct proof ⎪ ⎪ ⎪ ⎪

⎧ ⎪ ⎨ ⎪ ⎪ ⎨ Proof by contradiction Reduction to absurdity ⎪ ⎪ Methods of proof ⎪ ⎪ ⎪ Indirect proof Method of exhaustion ⎪ ⎪ ⎪ ⎩ ⎩ ⎩ Identity method

Finally, it should be noted that the branch of researching logic mathematically is called mathematical logic or symbolic logic, which is modern formal logic and has four main branches: deductive logic, taking deductive inference as the research object; probabilistic logic, taking inductive inference as the research object; proof theory, taking concepts, propositions, inferences, and proofs in mathematics as research objects; and algorithm theory, taking the calculations in mathematics as the research object. The branch that researches logic under the guidance of dialectics is called dialectical logic, which takes practicality, materialism, development, and comprehensiveness as its basic viewpoints. The logic branch that researches the forms, structures, and laws of thinking is called formal logic, which is an instrumental discipline similar to grammar. The relationship between dialectical logic and formal logic is like that between advanced mathematics and elementary mathematics.

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Based on formal logic, the logic of secondary school mathematics absorbs the related content of mathematical logic and dialectical logic properly, which is the need of the secondary school mathematics teaching reform.

2.6 Teaching of Secondary School Mathematics Concepts and Propositions 2.6.1 Teaching of Mathematical Concepts Mathematics is a knowledge system composed of concepts and propositions. Concepts can be regarded as the cells of thinking. Understanding and mastering mathematical concepts is the key to acquiring basic mathematical knowledge and improving mathematical ability. It has always been an important task to strengthen the teaching of concepts in mathematics teaching in secondary schools. As pointed out in Secondary School Mathematics Curriculum Standards, “in concept teaching, students should be gradually guided from actual cases or students’ existing knowledge to abstract and grasp the meaning of concepts. For concepts that are easily confusing, students should be guided to make clear their differences and relations by comparison”. The basic requirements for mathematical concept teaching are as follows: teachers should be able to accurately reveal the connotation and extension of concepts, as well as the relationship between concepts, so that students have a deep understanding of concepts, and be able to make flexible use of concepts to solve all kinds of problems, to achieve the goal of understanding, consolidating, and using concepts systematically. It should be pointed out here that we should not go to extremes while attaching great importance to the teaching of concepts. We should not pursue perfection in form, define everything, and require students to master everything for the sake of so-called rigor. Based on the actual situation in current secondary schools, such concepts as equality, equation, and same solution can be downplayed. It is one of the internationally general teaching requirements to moderately downplay the concepts. To learn mathematics well, the key is to grasp the spirit of mathematics, form mathematical concepts, master mathematical skills, and understand its mathematical thinking. We should not waste energy on unimportant concepts and seemingly profound problems. However, in the current secondary school mathematics teaching, the teaching of concepts is not satisfactory. Some teachers do not pay attention to or even be bad at teaching mathematical concepts. Some teachers make no distinction between the major and the minor ones or make improper requirements, resulting in the students’ problems, such as failure to make sense of concepts, inaccurate calculation, inexact reasoning, unclear drawing, and failure to use the concepts directly to solve problems and so on.

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To this end, in the teaching of mathematical concepts, we should make efforts to: (1) Attach importance to the introduction of concepts—the principle of reality Secondary school mathematics concepts actually have their specific content and realistic prototypes no matter how abstract they are. In teaching, we should not only pay attention to introducing concepts from students’ life experience (such as negative number, number axis, symmetry, tangent concepts, etc.) but also pay attention to introducing concepts from solving internal operation problems in mathematics (such as negative number, irrational number, complex number concepts). In this way, we can provide students with perceptual materials from their familiar languages and examples and guide them to abstract out corresponding mathematical concepts to enable them to master the essence of mathematical concepts well. (2) Reveal the extension and connotation of concepts—the principle of scientificity In order to understand concepts accurately and profoundly, teachers must make dialectical analysis on the basis of providing perceptual knowledge, revealing the essence of different concepts in different ways. For example, for the definition of “species + class difference”, teachers should reveal the species concept and class difference to enable the students to understand the defined concept has both the general attributes of its species concept and its own unique characteristics. Teachers should clarify the true meaning of every word of the concept, specific pronunciation, and usage of corresponding symbols at the same time. In this way, by grasping the extension and connotation of concepts and strengthening the “three-in-one” teaching, teachers can then further grasp the essence of concepts. (3) Expound the ins and outs of concepts—the principle of systematicness Mathematical concepts develop continuously with the development of mathematical knowledge, so the understanding of mathematical concepts should be deepened in the mathematical knowledge system. The understanding of the concepts learnt can be increased by learning the concepts from the relationship between mathematical concepts. For example, there are inherent relations between such concepts as factor  common factor  factorization  fraction simplification  fractional operation  solution of fractional equation; linear function  quadratic function  rational fraction function  exponential function  logarithmic function  trigonometric function  inverse trigonometric function, etc. It is conducive to deepening the understanding of relevant concepts and facilitating students to memorize them to clarify systematicness of concepts. (4) Pay attention to the comparison between concepts—the principle of comparison Some concepts come in pairs, and the two concepts fall under the same species concept and are in contradictory state (such as positive number and negative number, power and extraction of root). Some concepts derive from the converse relationship of concepts (such as exponent and logarithm, derivative and primitive function concepts). Some concepts are obtained by gradually generalizing and extending a

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certain concept (e.g., trigonometric functions of arbitrary angle are generalized from trigonometric functions of acute angle, etc.). We should pay attention to the comparison between similar, opposite and derivative concepts, and correct students’ mistakes in understanding concepts through counterexamples in particular, which is helpful for students to understand concepts accurately. (5) Strengthen the application of concepts—the principle of application The operations, reasoning, proof, and so on in secondary school mathematics are based on relevant concepts. In teaching, teachers should strengthen the application of concepts in operations, reasoning, and proof. Sometimes, students should be equipped with a variety of exercises around a concept so that they can apply concepts from multiple perspectives and at multi-levels, from retrospective application to comprehensive application, to achieve the purpose of grasping mathematical concepts in the application. At the same time, the following points should be paid attention to as far as teaching methods are concerned: ➀ Realize the importance of concepts and strengthen the teaching of concepts conscientiously Mathematical concepts are the basis of mathematical learning, and they are used all the time in solving concrete problems such as calculation, proof, and drawing. For example, if you don’t understand the concept of quadratic radical, it is impossible for you to simplify the quadratic radical (x − 5)2 − (1 − x)2 . If you don’t understand the concepts of right triangle, hypotenuse, height on the hypotenuse, projection of side on a straight line, and geometric mean, etc., it will also be difficult to demonstrate that “in a right triangle, the height on the hypotenuse is the geometric mean of the projections of two legs on the hypotenuse”. ➁ Attach importance to situational design of problems, and provide realistic prototype of concepts During teaching, concepts are usually described in an abstract way and the realistic materials presented are relatively singular. Only through the display of a large number of vivid background materials can teachers make it easier for students to analyze, compare, abstract and generalize, and make clear their essential properties. For some concepts that are derived on the basis of original concepts, such as straight angle, round angle, ellipse, hyperbola, teachers should be good at demonstrating the graphical changes presented by teaching aids or multimedia to produce intuitive and visual effect, conducive to students’ observation. For other concepts, such as logarithm and inverse function, teachers should be good at taking advantage of the logical relations of concepts to derive logarithm operations from exponential operations and derive inverse function concept from function concept. ➂ Reveal the scientific connotation of concepts through variants, positive examples, and counterexamples

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In concept teaching, the use of variants is very important. For example, for the concept of equation, various variants should be taken advantage of to enable students to understand the two key characteristics of “containing unknown number” and “equality”. The height of a triangle should be understood through different shapes of acute, right, and obtuse triangles. We should reveal the concepts through positive examples, eliminate interference of non-essential features through counterexamples at the same time, and explain the normative name, symbol, representation method, and relationship between the concepts. Only in this way can we make students grasp the scientific connotation of the concepts. ➃ Grasp primary concepts and choose the key points to explain. There are primary concepts and secondary concepts, so teachers should grasp the primary concepts to explain. For example, the concept of proportionality should be grasped when learning the concepts of proportionality, extreme terms of proportion, internal terms of proportion, and mean term of proportion. Another example, the concept of function includes the concepts of constant, variable, functional relationship, domain of definition, codomain, corresponding rule, etc., but we should grasp two primary concepts of “functional relationship” and “domain of definition”. At the same time, attention should be paid to the selection of key points. For example, when teaching “three lines and eight angles”, the concept of corresponding angle should be selected as the focus of explanation; when teaching trigonometric functions and inverse trigonometric functions, the concepts of sine and arcsine functions should be chosen as the focus of explanation. ➄ Adopt different teaching methods for different definitions. For example, for the descriptive definition method such as equality and limit, we should give as many examples as possible, enabling the students to abstract and generalize to evolve to concepts through examples, and then to seek for applications in practice. For the genetic definition method of circle, sphere, coordinate system, etc., the formation process should be explained clearly through demonstration or description. For the stipulative definition method such as, a 0 = 1(a = 0), 0! = 1, Cn0 = 1, etc., its necessity and rationality should be pointed out. For the definition method of “species + class difference” such as parallelogram and regular prism, the species-class relation and class difference, etc. should be revealed clearly. ➅ Inspire the interest in learning, and focus on the development of mathematical ability. Because mathematical concepts generally come before learning mathematics operations, students do not realize the purpose and importance of grasping them. Besides, the mathematical concepts themselves are relatively abstract and boring, and the students tend to show a lack of enthusiasm for learning, so teachers should make great efforts to stimulate the students’ enthusiasm for learning mathematics concepts by means of history, affection, words, doubt, change, and beauty

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in combination with production and life examples. At the same time, learning concepts should be based on improving students’ consciousness of mathematics and mathematical ability. To this end, it is also very important to analyze the reasons why students are not good at using concepts to solve problems, resulting in inaccurate calculation, inexact reasoning, and unclear drawing, so as to achieve the goal of improving students’ enthusiasm for learning mathematical concepts and improving mathematical ability.

2.6.2 Teaching of Mathematical Propositions Secondary school mathematics is a rigorous logical system composed of concepts, axioms, theorems, and formulas. Propositions (axioms, theorems, formulas, etc.) are combinations of concepts. Obviously, if you cannot master the propositions of secondary school mathematics conscientiously, you cannot learn secondary school mathematics well. Therefore, strengthening the teaching of secondary school mathematics propositions has always been a very important task of secondary school mathematics teaching. The basic requirements for secondary school mathematics propositions teaching are as follows: enabling students to understand the meaning of mathematics propositions deeply, make clear their derivation process and application scope, and have the ability to use mathematics proposition flexibly to solve problems. 1. Teaching of mathematical axioms. Since mathematics establishes knowledge system by means of formal logic, every true proposition is derived from known true propositions. If we trace it back in turn in this way, there are certain true propositions which cannot be derived from other mathematical true propositions, and these propositions are called axioms. The socalled axiom refers to the universal principles that are essential to any mathematical disciplines. This axiomatic research method of mathematics originated from ancient Greece, marked by Euclid’s Elements in the third century B.C. In the nineteenth century, the appearance of non-Euclidean geometry promoted the perfection of axiomatic method. It is required that the selected axiom system should have “three properties”, namely: Non-contradiction: It requires that contradictory conclusions can never be drawn starting from the axiomatic system, no matter how far the reasoning is. Independence: It requires that any axiom in the axiomatic system cannot be derived by logical means with the aid of other axioms. Completeness: It requires that there should be no need to add any new axioms in the use of axiomatic systems. Among the above “three properties”, the non-contradiction is the most fundamental one. However, in the secondary school teaching, considering the students’ receptivity, the limitation of teaching content and time, the requirements are not so

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strict, the scope of axioms is often expanded, and at the same, time no too high requirements for independence and completeness are raised. For example, the midpoint of line segment, the uniqueness of angle bisector in the plane geometry, and the decision theorems for congruent triangles are all raised as axioms, which are arranged according to the actual conditions of teaching. In teaching axioms, we should guide the students to abstract the content of related axioms from their life experience. At the same time, since axioms are subject to objective tests, we should guide students to verify axioms with concrete examples and gradually apply axioms into proving mathematical propositions or solving practical problems. 2. Teaching of mathematical theorems. Firstly, the idea of proof should be made clear. Mathematics is characterized by logical rigor. The conclusions in mathematics are often guaranteed by logical reasoning and based on evidence. However, middle school students are not used to it and do not realize the importance of proof, let alone specific proving. In teaching, teachers should attach importance to the cultivation of the idea of proof and further master the writing format of proof. Only through rigorous training to develop a habit of proof can we lay a solid foundation for learning mathematical theorems well. Secondly, in terms of specific methods, attention should be paid to the following points: (1) Distinguish the conditions and conclusions of the theorem; master the content and forms of expression of the theorem. After a proposition is introduced, teachers should guide the students to distinguish its conditions and conclusions conscientiously, to rewrite the proposition in words into expressions with mathematics notations, and draw a figure according to the question. At the same time, teachers should guide students to grasp the content and expression form of the theorem fully and accurately and be able to describe it in verbal language and mathematical language, respectively, instead of simplifying it arbitrarily. For example, it is inappropriate to simply the Pythagorean Theorem as “if the shorter leg is 3 and the longer leg is 4, the hypotenuse will be 5”, “a 2 + b2 = c2 ”, “AB2 + AC2 = BC2 ”, “longer leg2 + shorter leg2 = hypotenuse2 ”, and so on. (2) Analyze the train of thought of proving theorems and master the methods of proving theorems. The proof method of theorems is usually demonstrative and typical. Mastering the proof method of theorems has a great influence on improving the proving capacity. The key to mastering the proof method is train of thought. At the beginning of proving theorems, students should write down the exact basis for each step of reasoning and gradually simplify it as the proficiency improves. Generally, the synthesis method is used to write the proving process of new propositions, and the analysis method is used gradually as the content of propositions deepens, and the method of combining analysis and synthesis

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can be used to seek the method of proof at the same time. In addition, attention should also be paid to guiding students to learn and master common methods of proving theorems, such as proof by contradiction, identity method, problem changing method, patching method, and geometric transformation method. (3) Understand the internal relations between theorems and other knowledge to systematize knowledge. During teaching, teachers should teach the students to systematize the theorems and formulas that they have learned, making vertical organization according to their logical relation, or making horizontal organization according to their application, and finding out whether the theorem has a converse theorem. Only by forming a network of related propositions to systematize and methodize knowledge can we further master theorems deeply. (4) Strengthen the application of theorems and improve the ability to use the theorems to solve problems. The ultimate goal of learning theorems is for the purpose of application. In the teaching, teachers should introduce the application of theorems and the scope of application in time and carefully arrange exercises, so that students can further practice the application of theorems purposefully and systematically, learn to analyze and synthesize them through application, learn to apply theorems after transforming problems, and master the method of adding auxiliary lines to solve geometric problems. Only through proper repeated practice, can students deepen their understanding of the theorems and improve their ability to use the theorems to solve practical problems. In addition, during the process of learning the theorems, we should also understand whether the theorem is a property theorem or a decision theorem, the status, and role of the theorem in theory and practice, as well as the changes in the conclusion when the conditions of the theorem are enhanced or weakened. Only in this way can students achieve a deep understanding of axioms, theorems, and formulas gradually and achieve the purpose of flexible use of them. Review Questions and Exercises (I) 1. What is a mathematical concept, its extension, and connotation? Give an example. 2. What are the relationships between mathematical concepts? Give an example. 3. What are the commonly used definition methods for mathematical concepts? What requirements should be met for correct definitions? 4. What is the significance of classifying mathematical concepts, and what requirements should be met for correct classification? 5. What is a mathematical proposition and its structure? Give an example. 6. What are the basic forms of propositions, and how do they relate to each other? Give an example. 7. What are the basic laws of formal logic, and how should they be used in reasoning and proving?

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8. What are inductive inference, analogical inference, and deductive inference, and how to use them correctly? Give an example. 9. What is the analysis method and synthesis method? What is the proof by contradiction and the identity method, and what is the relationship between the proof by contradiction and the identity method? 10. What are the primary concepts in secondary school mathematics, and how are they defined? 11. What are the primary theorems in secondary school mathematics? And which of these primary theorems are property theorems, and which are decision theorems? 12. Give examples to illustrate how to teach concepts and propositions, and what problems should be paid attention to in teaching? 13. Select a concept and a theorem in secondary school mathematics textbook for lesson preparation, and conduct trial teaching and evaluation in the group. 14. Use truth table to prove that the following propositions are true: (1) [( p → q) ∧ p] → q = 1; (2) p ∧ ( p ∨ ( p → q)) = 0; (3) ( p → q) ∧ (r → q) = ( p ∨ r ) → q. 15. (1) Write out the converse theorem of the theorem “In a right triangle, if an acute angle is equal to 30°, then its opposite side is half of the hypotenuse”; (2) As shown in Fig. 2.9, in trapezoidal ABCD, if AD//BC, AB = AD + BC, and the bisectors of ∠ A and ∠ B intersect CD at E, then CE = DE. Try to write out the converse propositions of this proposition, and explain which of them are true. 16. Point out the logical errors in the following reasoning, and analyze the reasons for them: (1) Real numbers correspond to the √points on the number axis on a one-to-one √ basis, 3 is a real number, so 3 corresponds to the points on the number axis √ √ on a one-to-one basis. 5 is an irrational number, and 5 is also a rational number. (2) (3) Two √ straight lines in√ the space √ are intersecting and parallel. (4) 6 = (−4)(−9) = −4 · −9 = 2i · 3i = 6i 2 = −6. (5) As shown in Fig. 2.10, ABC is known. Prove AB = AC. Fig. 2.9 .

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Fig. 2.10 .

Proof Draw the perpendicular bisector of BC, which intersects the bisector of ∠ A at D, and then draw perpendiculars passing through D to meet AB and AC at E and F, respectively. Then: In right triangles  BDE and  CDF, since BD = CD, DE = DF, BDE ∼ =  CDF, then ∠ 1 = ∠ 2,∠ 3 = ∠ 4, AD = AO, and then  ABD ∼ =  ACD. Therefore, AB = AC. 17. Are the following procedures correct and why? √ (1) Solve the inequality x 2 − 2x + 1 > 2x. Solution Square both sides, and we get: After sorting it out, 3x 2 + 2x − 1 < 0, that is, (x + 1)(3x − 1) < 0. Therefore, −1 < x < 13 . (2) As shown in Fig. 2.11, point P is a point in  ABC. Prove: AB + BC + CA > PA + PB + PC. Proof Since ∠ APC > ∠ PCA, then AC > PA. And since ∠ APB > ∠ PAB, then AB > PB. Since ∠ BPC > ∠ PBC, then BC > PC. Add these three formulas, then AB + BC + CA > PA + PB + PC.  (3) Solve the inequality lg x 2 − 1 < 1. 2 Solution Since√x 2 − 1 < 10, √ then x < 11. Namely, − 11 < x < 11.

Fig. 2.11 .

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(4) Solve the equation x 2 − (3 − 2i)x + 5 − 5i = 0.  2 Solution Turn the  original equation into x − 3x + 5 + (2x − 5)i = 0, x 2 − 3x + 5 = 0, then we get since the system of equations has no 2x − 5 = 0, solution, then the original equation has no solution. 18. Try to prove the following questions by using analysis method and synthesis method, respectively:   (1) If a, b, c, and d are all positive numbers, prove ab + dc ab + dc ≥ 4; (2) In  ABC, the opposite sides of angle A, B, and C are a, b, and c. If lg sinA, lg sinB, and lg sinC are arithmetic sequences and cx 2 + 2cx + a = 0 has two equal real roots, prove sinA = sinB = sinC 19. Prove the following questions by means of proof by contradiction: (x + y + 1)2 + (x − 2y + 4)2 = 0,  a+b   < 1; (1) If |x + y + 1| + |x − 2y + 4| = 0, then  1+ab √ √ x + y + 1 + x − 2y + 4 = 0, (2) Prove: x 2 + y 2 + z 2 is indecomposable; (3) In quadrilateral ABCD, E and F are the midpoints of AD and BC, respectively, and AB + CD = 2EF. Prove AB//CD. 20. Try to prove the following questions by means of the identity method: √ √ 3 3 (1) Prove 5 2 + 7 − 5 2 − 7 = 2; (2) If two sides of a triangle are cut by a straight line and the segments obtained are correspondingly proportional, then this line is parallel to the third side; (3) If an equilateral triangle is drawn inward with one side of a square as the base, then the third vertex of the triangle and the other two vertexes of the square form an isosceles triangle, and its base angle is equal to 15°.

Chapter 3

Secondary School Mathematics Thinking

Thinking is a complex process of mental activity. As stated by Qian Xuesen, “only the science of thinking, psychology and pedagogy are the basis of intellectual development”. According to Gao Shiqi, “the science of thinking is the science of cultivating talents”. Mathematical thinking plays an extremely special and important part in the science of thinking, and during the teaching of mathematics in secondary schools, students are guided to carry out thinking activities almost all the time and apply various methods and laws of thinking activities widely. To this end, we will elaborate on the methods and qualities of mathematical thinking and the development of mathematical thinking in secondary schools in this chapter.

3.1 Meaning of Mathematical Thinking In the process of cognition, in addition to sensation, perception, and memory, people must also find the answer to the questions in a roundabout and indirect way on the basis of experience, that is, to take pains to do the transformation work of “selecting the essential, sorting out, proceeding from one point to the other and from the surface to the center” based on rich perceptual knowledge materials in order to solve the problems. This kind of cognition process of solving problems in a roundabout and indirect way is the thinking activity. It is thus evident that thinking is the advanced stage of people’s cognitive activity, which is characterized by indirectness, generality, and being problem-based. The first important characteristic of thinking activity is to understand things indirectly through the results of their interaction or through other media. The understanding of things in general and the understanding of the connection and relations of general characteristics or regularities of things are the second important characteristics of thinking activity. Thinking is always closely related to problems. It is safe to say that without problems there would be no thinking, at least no positive and focused

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_3

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thinking. Therefore, being problem-based is the third important characteristic of thinking activity. In philosophy, thinking is discussed in terms of the relationship between the subject and object of thinking, i.e., the relationship between existence and consciousness. Logic mainly focuses on the laws and forms of correct construction of thinking, without involving the content of the object. Psychology is mainly concerned with the conditions and causes for the occurrence and development of normal human thinking, and the effect of various psychological phenomena on thinking. Mathematical thinking must be subject to the general laws of thinking, but necessarily has different characteristics and nature from general thinking. Mathematical thinking is the process of interaction between human brains and mathematical objects (numbers and shapes, etc.) and the understanding of mathematical laws (the essential characteristics of the objects) in accordance with general laws of thinking. Concepts, judgements, and inference are the basic forms of mathematical thinking. Common methods of mathematical thinking are observation, experimentation, analysis, synthesis, comparison, classification, abstraction, generalization, concretization, specialization, systematization, analogy, induction, deduction, imagination, and intuition. The types of mathematical thinking are divided into plane thinking (also known as one-plane thinking) and three-dimensional thinking (also known as multiplane thinking) by structure; into convergent thinking (also known as concentrated thinking) and divergent thinking (also known as difference-seeking thinking) by direction of search; into concrete thinking (also known as imaginal thinking), abstract thinking (also known as logical thinking), intuitive thinking (also known as creative thinking), and functional thinking (also known as dialectical thinking) by nature or level; into thinking-aloud and tangible thinking by reflex phenomena; into natural thinking, theoretical thinking, mathematical thinking, etc. by practical needs; and into reproducible thinking and creative thinking, etc. by quality of intelligence. As an important branch of educational science, mathematics pedagogy is closely connected with the science of thinking. A.A. Stolyar, a mathematics educator of the former Soviet Union, simply defined mathematics teaching as the teaching of mathematical (thinking) activity. According to him, the teaching of mathematics can be understood as the result of thinking activity and as the process of thinking activity as well. These two different understandings reflect the divergent perceptions of traditional educational theories on knowledge and ability, result and process. Starting from the need to cultivate talents, modern educational theories increasingly emphasize the process of teaching (i.e., the process of thinking) and the importance of cultivating students’ abilities, especially thinking abilities. Mathematics is the tool of thinking, the gymnastics of thinking, and the carrier of thinking training. It is obvious that secondary school mathematics education is closely related to the science of thinking.

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3.2 Methods of Secondary School Mathematics Thinking 3.2.1 Observation and Experimentation 1. Observation Observation is a method of understanding the essence and laws of things purposefully and selectively based on people’s perceptions. Mathematical observation is a method for people to examine the quantitative relationships and graphical properties of mathematical problems under objective conditions. In mathematics teaching in secondary schools, observation is often used to understand the essence of mathematics, reveal the laws of mathematics, and get the thinking methods of mathematics. For example, by observing, 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32 ... We can conjecture 1 + 3 + 5 + 7 + … + (2n − 1)  = n2 .  √ x  √ x 2+ 3 + 2 − 3 = 4. For another example, solve the equation   √ √ By observing, we find that 2 + 3 and 2 − 3 are reciprocals of each other. After supposing. √ x 2 − 3 = y, the original equation can be transformed into y 2 −4y +1 = 0, so as to facilitate the solution. 2. Experimentation Experimentation is a method of researching the essence and laws of things through the senses under artificially controlled conditions with the help of certain means (or tools) according to certain purposes. Observation and experimentation are closely related. Observation is the premise of experimentation, and experimentation is the confirmation and development of observation. Generally speaking, mathematics is not the science of experimentation, and observation and experimentation are not the main methods of mathematical research, but they have important applications in teaching and scientific research. For example, in mathematics teaching in secondary schools, for the theorem of the sum of interior angles of a triangle, Pythagorean Theorem, cone volume formula, and so on, we often resort to observation and experimental means to inspire, and then give theoretical proof. Euler’s formula F + V − E = 2 for the number of faces, vertices, and edges of a convex polyhedron is induced from the observation and experimentation on some special polyhedrons.

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3.2.2 Analysis and Synthesis 1. Analysis Analysis is a method of thinking in which the whole of a thing is broken down into parts or the individual features, factors, and levels of the whole are broken down and examined separately in your head. Analysis is a relatively independent process of thinking and consists of the following basic types: (1) Qualitative analysis: As an analysis to determine whether the research object has a certain property, it mainly solves the problem of “there is or there isn’t” and “is or isn’t”. (2) Quantitative analysis: It is analysis to determine the quantity of various components of the research subject, it mainly solves the question of “how many”. (3) Cause and effect analysis: As an analysis to determine the cause of a change in the research subject, it mainly solves the question of “why”. (4) Reversible analysis: It is an analysis to determine whether a phenomenon in the research subject can in turn be considered as a cause. (5) System analysis: It is an analysis to determine whether the research subject can be regarded as a developmental change system. In order to make a rational and effective analysis, the following principles should be observed: ➀ Analysis must reach the bottommost components (or the simplest factors). ➁ Analysis must be a new understanding of the subject under research. 2. Synthesis Synthesis is to connect the parts of a thing in your head, or to combine their features, characteristics or aspects, factors, levels, etc. to form a unified understanding of the research object as a whole mentally. A model is related to the overall understanding of the object, so the model establishment is the achievement of understanding of scientific synthesis, theoretically marking the completion of the scientific synthesis. The synthesis is mainly classified into the following types in the terms of models: (1) Synthesis of intuitive model: It refers to a kind of synthesis of the whole and structure of an object represented by familiar and observable charts from experience, that is, the synthesis of intuitive materials into a whole chart of the object. This kind of synthesis is characterized by being intuitive and visual, but relatively rough. (2) Synthesis of principle model: It refers to a kind of synthesis that systematically describes the overall structure of an object under abstract and theoretical conditions in order to reflect the characteristics and laws of the object. The establishment of a principle model reflects that the synthesis has reached a highly abstract stage of thinking.

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(3) Synthesis of mathematical model: It refers to a kind of synthesis that describes the characteristics, relations, and laws of an object as a whole in a mathematical way. The mathematical model also approximates simplified reflection and depiction of the object under pure conditions, and it reveals various properties of the object and the laws among them through more abstract quantitative relations. The following principles should be observed for synthesis: ➁ Synthesis must be combined with analysis; ➂ Synthesis must form an understanding of the object as a whole in a creative way. Analysis and synthesis are basic methods of thinking and the basic processes of thinking activities. Analysis and synthesis are closely related to each other in any kind of intellectual activity. Analysis is guided by the original theoretical synthesis, and the purpose of analysis is to transit to the next new synthesis. Synthesis is carried out on the basis of analysis, and the development process of scientific knowledge is a periodic movement of spiral rise from a low level to a high level (“synthesis - analysis - synthesis”). The development of contemporary natural sciences is distinctively characterized by, on one hand, high degree of differentiation, and on the one hand, high degree of synthesis, or synthesis of multiple disciplines, or synthesis within the discipline. The process of analysis and synthesis is endless as people’s understanding is inexhaustible. In secondary school mathematics, analysis and synthesis are often applied simultaneously. Analysis is often an approach to discovery, while synthesis is often an approach to demonstration. Try to answer the following questions using analysis and synthesis methods, respectively. √ √ √ √ ➀ Given a ≥ 3, prove a − a − 1 < a − 2 − a − 3; ➁ Two diameters AB and CD of are perpendicular to each other, and E is a point on AD . Prove: SABCE = 21 CE2 . 

3.2.3 Comparison and Classification 1. Comparison Comparison is a method of thinking that breaks down individual parts or features of a research object so as to determine their similarities and differences. Comparison can be carried out among objects of the same kind, among objects of different kinds, or among different aspects or parts of the same object. People break down a whole research object into parts and distinguish its characteristics, which is analysis. At the same time, people connect their corresponding parts to determine their similarities and differences, which is synthesis. Therefore, comparison is both a process of analysis and a process of synthesis.

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“Comparison is the key to discrimination”, and “everything is understood through comparison”. Comparison is an important method in the study and teaching of mathematics, and strengthening the teaching of comparison is conducive to students’ mastery of concepts and laws, enlightenment of thinking, discovery of laws, and breakthrough of difficult points in teaching. For example, when introducing the concept of arithmetic sequence, you can first ask students to compare the following sequences: 2,4,6,8,10,12, -3,-6,-9,-12,-15,-18,

5 3 7 9 1, , , ,2, , , 4 2 4 4 2,4,8,16,32,64, 1,2,5,-8,11,-13,

Through comparison, we find that sequences ➀, ➁, and ➂ have the same property: from the second term onward, each term in the sequence is equal to the preceding term of the sequence plus a constant, i.e., an = an−1 + d (d is a constant), and this sequence has the property: an =

an−1 + an+1 2

Using the method of comparison, the general term formula of the arithmetic sequence can also be found as follows: a1 = a1 , a2 = a1 + d, a1 = a1 + d = a1 + 2d. Comparing the above results again (which also contain abstraction and generalization), it is clear that an = a1 + (n − 1)d, In addition, comparison can simplify the study of similar problems by exploring similarities in different objects or exploring dissimilarities in identical objects to facilitate the study of the problem under discussion. The types of comparison include comparison of similarities, comparison of differences, comprehensive comparison, and so on. The following principles should be observed in the use of comparison: (1) The comparison can only be made if the objects have a definite connection with each other, i.e., the comparison should be meaningful;

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(2) Comparison must be made in the same relation, following certain steps; (3) Comparison of the same properties of mathematical objects should be complete and thorough. The comparison methods include the method of comparing difference and the method of comparing quotient. Try to answer the following questions using comparison method: ➀ Given a > b > 0, prove: a a bb > a b ba ; ➁ Prove: sin2 a + sin2 β > sin a · sin β + sin a + sin β − 1, 2. Classification Deriving from comparison, classification is a more complex method of thinking that classifies things with the same attributes into one category and things with different attributes into another category according to their commonalities and differences. Classification is inseparable from analysis and comparison. The only way to classify things is to analyze and compare them to find out their common properties. Classification is also a basic method commonly used in secondary school mathematics. For example, the classifications of logarithms, formulas, equations, inequalities, functions, triangles, quadrilaterals, polygons, conic curves, and position relationships between straight lines, straight line and plane, planes in the space, etc. are important elements of mathematics teaching in secondary schools. For example, the classification of elementary functions is as follows:

3.2.4 Abstraction and Generalization 1. Abstraction Abstraction is a kind of thinking activity which, on the basis of perceptual knowledge, leaves the concrete image of things and simply extracts a certain characteristic from it for understanding. The process of abstraction is roughly as follows: starting from the problem under discussion, and comparing and analyzing various empirical facts and eliminating the irrelevant factors, the important characteristics (universal laws and cause-and-effect relations) of the object under research are extracted to understand it, so as to provide some scientific basis or general principles for answering the problem. Now we take Euler’s study of “seven-bridge problem” as an example.

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In Königsberg, there were seven bridges connecting the islands to the river bank (Fig. 3.1). The question was raised whether it was possible to cross each bridge only once and return to the original place of departure. Euler first found that the size and shape of the islands and the river bank were irrelevant to the solution of the problem, so both islands and the land on the other side of the river could be reduced to one point, while the bridges and the roads between them could be drawn as curve segments, maintaining the essential feature of bridges connecting two places (two points). In this way, Fig. 3.1 was abstracted into Fig. 3.2, and the original problem was abstracted into a question of drawing by one line. There are roughly two types of abstraction: (1) Empirical abstraction is a kind of initial abstraction that takes the objective phenomenon of things as direct starting point. It is the abstraction of the characteristics of the object manifested. (2) Theoretical abstraction is a kind of deep abstraction formed on the basis of empirical abstraction. It grasps the causal relationship and regular relations of things, and the result of this abstraction is the law and theorem. For example, if the concrete image thinking is applied to the law of dividision of the same base powers, then 105 ÷ 102 = 103 ⇒ 105 ÷ 102 ÷ 105−2 , 610 ÷ 68 = 62 ⇒ 610 ÷ 68 = 610−8 , If empirical abstraction thinking is used, then. When Fig. 3.1 .

Fig. 3.2 .

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If theoretical abstraction thinking is used, then Since a m−n · a n = a m , and if a x · a n = a m . then m = x + n (the law of multiplication of the same base powers), since x = m − n, then a m−n · a n = a m , therefore a m−n ÷ a n = a m−n (definition of division). The following principles should be observed in the application of abstraction: ➀ Scientific abstraction must be universal; ➁ Higher levels of abstraction must be able to deduce lower levels of abstraction. The thinking process opposite to abstraction is concretization, which applies the laws obtained through abstraction and generalization into practice, so as to deepen people’s understanding of concepts and knowledge acquired. For example, the addition operations of rational numbers can be abstracted to obtain the equality a + b = b + a, and the simple calculation of 462 −362 is the concretization of calculation by using formula a 2 − b2 = (a + b)(a − b). 2. Generalization Generalization is a kind of thinking activity that mentally distinguishes real objects and real general things, and then unites them ideologically on this basis. Generalization is a way of understanding in which people pursue universality. In other words, generalization is a way of understanding from the individual to the general. The premise of generalization is the comparison of phenomena and objects with each other. Generalization is an important method of scientific discovery, the main types of which are: (1) Generalization of extrapolation type refers to the extension of a particular field to other fields. There are two forms: incomplete inductive generalization and parallel generalization. (2) Ascending generalization refers to the rise from knowledge of a single thing directly to a kind of generalization with understanding of universal laws, and this generalization is also known as generalization method. (3) Compound generalization refers to a generalization formed by combining and penetrating these two types of generalization. This kind of generalization is both

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a horizontal extrapolation, and a vertical ascending, and finally forms a kind of very universal basic principle. The following principles should be observed in the use of generalization: ➀ Generalization must be combined with comparison: ➁ Generalization must be combined with deduction.

3.2.5 Concretization, Specialization, and Systematization 1. Concretization Contrary to abstraction, concretization is a method of thinking that relates abstract mathematical facts (concepts, theorems, etc.) to the corresponding concrete materials to better understand the mathematical facts. Concretization can be used as a concrete verification of intuitive descriptions, abstract laws, or as an application of a property under specific conditions. For example, the abstract relationship “a(b + c) = ab + ac” is established for any number or formula, the law of “Pythagorean Theorem” in right triangles is established for any right triangle. They can also be verified with specific numbers, formulae, and specific right triangles, respectively. 2. Specialization Contrary to generalization, specialization is a method of thinking that “retreats” the mathematical facts under discussion to a special state (quantity or position relationship), so as to achieve the purpose of studying the general state. In secondary school mathematics teaching, the variables are often transformed into constants and any graphics are transformed into the graphics in special positions for research to get some kind of inspiration. This research method of “retreating for the sake of advancing” is actually the application of concretization and specialization in teaching. Example 1 In RtABC, ∠C = 90°, a, b, c are the lengths of three sides. Prove: a 2 + b2 < cn (n ≥ 3). Proof We can start with specific and special circumstances to solve the problem.

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When n = 3, a 3 + b3 − c3 = a 3 + b3 − c(a 2 + b2 ) = a 2 (a − c) + b2 (b − c) < 0, since a 3 + b3 < c3 ; When n = 4, a 4 + b4 − c4 = a 4 + b4 − c2 (a 2 + b2 ) = a 2 (a 2 − c2 ) + b2 (b2 − c2 ) < 0 since a 4 + b 4 < c4 ; Thus, we can obtain a n + bn − cn = a n + bn − cn−2 (a 2 + b2 ) = a 2 (a n−2 − cn−1 ) + b2 (bn−2 − cn−2 ) < 0,

Therefore, a n + bn < cn (n ≥ 3), Example 2 As shown in Fig. 3.3, AB is the diameter of , DE is the tangent passing through point C, and intersects tangents AD and BE at D, E respectively. Prove: AD · BE is a fixed value. Analysis From the conditions, point C can be any position, so we select a special position as shown in Fig. 3.4, and it is obvious that AD · BE = OC2 , which is a fixed value. And then we note that DA = DC, EB = EC, and it is not difficult to think of joining OD and OE, so as to find the way to prove it. Fig. 3.3 .

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Fig. 3.4 .

Proof (omitted). 3. Systematization Systematization is a method of thinking that arranges all kinds of relevant materials in sequence and incorporates them into a certain system for research. It is closely related to comparison, classification, abstraction, generalization, concretization, and other methods of thinking. The use of systematization methods is helpful to grasp the internal relations of things as a whole, to master the knowledge systematically and profoundly, to seize the core and to understand the ins and outs. In secondary school mathematics teaching, the knowledge is often systematized by making outlines, drawing charts, and so on. For example, after learning parabola, the following table can be used to systematize the related knowledge.

Property

Figure

Leftward x= e=1

Rightward x = − P2 e=1

Direction of opening

Directrix

Eccentricity

Focus P 2

O (0,0)   F − 2p , 0

O (0,0)   F 2p , 0

x-axis

x-axis

Vertex

x ≤ 0, −∞ < y < +∞

x ≥ 0,

( p > 0)

( p > 0)

−∞ < y < +∞

y 2 = −2 px

y 2 = 2 px

Axis of symmetry

Scope

Standard equation

e=1

y = − P2

Upward

O (0,0)   F 0, 2p

y-axis

y≥0

−∞ < x < +∞,

( p > 0)

x 2 = 2 py

P 2

e=1

y=

Downward

O (0,0)   F 0, − 2p

y-axis

y≤0

−∞ < x < +∞,

( p > 0)

x 2 = −2 py

3.2 Methods of Secondary School Mathematics Thinking 65

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3.2.6 Analogy, Induction, and Deduction 1. Analogy Analogy is a method of thinking that captures certain similar characteristics of things for comparison and contrast. Analogy has some connections with classification, and it is a form of thinking from the particular to the particular. The use of analogy can simplify the study of similar problems, and also helps to discover and promote certain properties, which is an important way to obtain discoveries or inventions. The application of analogy in mathematics teaching has an important role in enlightening the mind and developing ability. 2. Induction Induction is a method of thinking that divides a research object into parts, and synthesizes the research of each part to achieve the research of the whole. Induction has some connections with generalization and is a form of thinking from the particular to the general, or from the individual to the whole. 3. Deduction Deduction is a method of thinking that uses a universal or particular judgement to arrive at a special universal or particular judgement. Deduction has some connections with concretization, and it is a form of thinking from the general to the particular. Deduction is based on induction, induction is guided by deduction, deduction provides the theoretical basis for induction, and induction prepares the conditions for deduction. Interrelating and interpenetrating, they form a relationship of dialectical unity. In secondary school mathematics teaching, induction and deduction are often used in combination. Sometimes, the conjecture is made by induction first and then proved by deduction. Sometimes it is necessary to classify things first, then to make induction and conjecture, and then to deduce it. This reflects the comprehensive application of classification, comparison, analogy, induction, and deduction thinking. Example 3 Connect any two points on the perimeter of a unit square to form a curve, and divide the square into two parts of equal area. Prove: the curve length is not less than 1. Analysis Let the two points be P and Q, and the connecting curve length be l. First, the relative position of P and Q is classified, and there are three cases. Then these three cases are compared, and it is found that by studying one of the basic cases, the proof of other cases can be obtained by using analogy method. Proof . ➀ As shown in Fig. 3.5(1), the curve length at this time is ➁ As shown in Fig. 3.5(2), A and C are connected to intersect curve l at point S, and symmetric curve of curve with respect to AC is drawn, which is transformed into the case shown in Fig. 3.5(1), and thus the curve length is ;

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Fig. 3.5 .

➂ As shown in Fig. 3.5(3), the square axis of symmetry MN is drawn to intersect curve l at point S, and symmetric curve of curve with regard to MN is drawn, which is also transformed into the case shown in Fig. 3.5(1), and thus the curve . length is Summing up the above three cases, it is clear that the proposition has been proven.

3.2.7 Imagination and Intuition 1. Imagination Imagination, also known as scientific conjecture or scientific association, is a creative thinking activity that speculates on the causes and regularities of phenomena. Imagination should be based on a certain background of knowledge, and its purpose is to explore the answers to questions and put forward explanatory theories. There are following types of imagination: (1) Analogical imagination is a kind of imagination that imitates the trigger and envisions its similar creations. (2) Bound imagination is a kind of imagination that people, in order to solve a difficult problem, induced by the trigger, creatively deduce the general principle or law. (3) Compound imagination is a kind of imagination that combines analogical imagination and bound imagination. The following principles should be observed in the use of imagination: ➀ The theory proposed by scientific imagination must be capable of explaining the facts; ➂ The theory proposed by imagination must be testable. Example 4 Prove: Cn1 + 2Cn2 + 3Cn3 + · · · + nCnn = n · 2n−1 .

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Proof 1 Associate the formula n−1 0 1 2 + Cn−1 + Cn−1 + · · · + Cn−1 2n−1 = Cn−1

and the formula k−1 , (k = 1, 2, 3, . . .), kCkn = nCn−1

then Cn1 + 2Cn2 + 3Cn3 + · · · + nCnn n−1 0 1 2 = n(Cn−1 + Cn−1 + Cn−1 + · · · + Cn−1 ) n−1 =n·2 ;

Proof 2 Associate the formula Cnk = Cnn−k , (k = 0, 1, 2, 3, . . . , n).   and the formula n−1 = 21 · 2n = 21 Cn0 + Cn1 + Cn2 + · · · + Cnn n−1 = 21 · 2n =     1 Cn0 + Cn1 + Cn2 + · · · + Cnn n−1 = 21 · 2n = 21 Cn0 + Cn1 + Cn2 + · · · + Cnn 2 and suppose Sn = Cn1 + 2Cn2 + 3Cn1 + · · · + nCnn , then Sn = 0 · C0n + 1 · C1n + 2 · C2n + · · · + nCnn . and Sn = nCn0 + (n − 1)Cn1 + (n − 2)Cn2 + · · · + 0 · Cnn , add the above two formulas together, and we get 2Sn = n(C0n + C1n + C2n + · · · + Cnn ) = n2n , therefore Sn = n · 2n−1 . Proof 3 Associate the formula C0n + Cιn x + C2n x 2 + · · · + Cnn x n = (1 + x)n , Take the derivative of the above formula and let x = 1. Obviously, the above formula can be proved. 2. Intuition Intuition, also known as insight (inspiration), is also a creative thinking activity. In the history of science, many remarkable discoveries are often associated with it. Intuition is often manifested by reaching a true conclusion without analytical steps. Some people think it is a non-logical thinking activity; others think it as a compressed and simplified logical process that takes the form of “leap” and get the

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answer to a question by guess instantaneously, and it is apparently a “flash of insight” that pops into one’s mind. Intuition is sudden and accidental, but it is not arbitrary and does not come out of nowhere. Prolonged and intense logical thinking activity is often the prelude to and preparation for the generation of intuition, which is simply a change of thought, a reconsideration from a different perspective, and leads to scientific discovery under some kind of inspiration. As intuition is both creative and arbitrary, it is difficult for intuitive activity to have a strict and definite pattern, and it is obviously wrong to deny the role of intuition or to mystify it.

3.3 Quality of Secondary School Mathematics Thinking 3.3.1 Broadness of Thinking The broadness of thinking, also known as the divergence of thinking, is a kind of thinking quality that does not follow the conventions, seeks for variation, ponders problems, and seeks solutions from various perspectives and in many ways. The opposite is the narrow-mindedness of thinking, which is manifested as the closed state of thinking. In mathematics teaching, strengthening the teaching of basic knowledge and basic skills enables students to form a complete cognitive structure and skilled skills, which is the basis for developing the broadness of thinking. Therefore, we should guide students to develop the habit of thinking from multiple perspectives and in many ways, and strengthen the exercise and practical application of multiple solutions to and multiple variations of one question, which are undoubtedly beneficial to cultivate students’ broadness of thinking. For example, for the exercise of “writing out formulas equal to 1 as many as possible”, some students can only write out sin2 a + cos2 a = 1,

sin 90◦ = 1,

a 0 = 1(a = 0),

x2 sin x y2 = 1, 2 + 2 = 1, x→0 x a b

i4n = 1,

C◦n = 1,

1

loga b · logb a = 1, lim

2xdx = 1, 0

but some can write out twenty or thirty formulas, and others can write out hundreds of formulas. For example, when solving inequality |x| < |x + 1|, some students can find only one solution, but some can find many solutions applying the “zero method”, “elimination of absolute value sign”, “inequality properties”, “functional image”, etc.

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For example, for the exercise of “finding the Minimum value of x 2 + x42 ”, some students are only able to solve the question itself, while others are able to solve the following variant questions: If x > 0, prove: a − a1 (a < 0) = ; If x < 0, find the maximum value of x + x4 ; 16 If m > n > 0, prove: m + n + n 2 (m−n) ≥ 8, etc. The above distinction reflects the students’ difference in the broadness of thinking.

3.3.2 Profoundness of Thinking The so-called profoundness of thinking refers to a kind of quality of thinking that seeks the essence of the problems under study and the interconnections between problems in the process of analyzing and solving problems. Its opposite is superficiality of thinking, which is manifested as sluggishness of thinking. For example, students do not seek deeper understanding of the concepts they have learned, they follow suit while doing exercises, and they do not have a clear idea of how to solve the problems or grasp the method and essence of solving the problems. To overcome the students’ sluggishness of thinking and to cultivate the profoundness of students’ thinking, students should be guided to consciously think about the essential aspects of things, learn to understand the essence of things from the connections between things, and learn to understand things comprehensively. In this regard, a deeper understanding of the concepts involved can be gained through the teaching of discrimination and comparison. For example, distinguishing between: positive and non-negative numbers, negative and non-positive numbers; empty set ∅ and set{0}, acute angle and angles in the first quadrant; arcsin (sin x) and sin (arcsin x), etc. At the same time, the understanding of a proposition can be deepened by generalization and extension of the proposition. For example, prove that if a triangle is a regular triangle, then all three vertices of the triangle cannot be integers. Proof: Use the proof by contradiction. Let A, B, and C be the integers, that is, the coordinates of A(x1 , y1 ), B(x2 , y2 ), C(x3 , y3 ) are integers, and we may as well suppose that x1 = x2 = x3 , then kAB =

y2 − y1 y3 − y2 kAB − kBC , kBC = , tan B = | | x2 − x1 x3 − x2 1 + kAB · kBC

√ is a rational number, while tan B = tan 60◦ = 3 is an irrational number, resulting in mutual contradiction. Therefore the proposition is proved.

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However, in the above demonstration, only the conclusion that ∠B = 60° and tan 60° is an irrational number is used, which means that the demonstration has nothing to do with whether ∠A and ∠C are 60° or not. To this end, the following two new propositions can be extended: ➀ If a triangle has an angle equal to 60°, then all three vertices of the triangle cannot be integers. ➁ If the tangent value of one angle of a triangle is an irrational, then all three vertices of the triangle cannot be integers.

3.3.3 Criticalness of Thinking The so-called criticalness of thinking is a kind of thinking quality that is good at finding problems, raising questions, distinguishing right from wrong, and evaluating advantages and disadvantages. Critical thinking is a practical, thoughtful, and meticulous thinking. In order to cultivate students’ critical thinking, while teaching, we can intentionally raise some confusing concepts to guide students to analyze and identify; give some plausible judgements properly to inspire students to distinguish true and false; or deliberately give wrong answers to certain questions and organize discussions, and let students find out where and why they are wrong. Please be noted particularly that teachers should not poke fun at and reprimand students when they have onesided and superficial understanding in the process of independent thinking, and even resort to the so-called sophistry in and out of class. On the contrary, teachers should pay attention to timely encouragement, guidance, and inspiration, because this is the very reflection of developing critical thinking. For example, in a mathematics class of grade two of middle school, when the teacher wrote on the blackboard that “It is known that x1 +x2 are two roots of equation x 2 − 3x + 5 = 0. Find the value of x12 + x22 ”, since x12 + x22 = (x1 + x2 )2 − 2x1 x2 = (+3)2 −2×5 = −1, the quadratic sum of two numbers equals a negative value, which was beyond the students. A student interrupted “this question cannot be solved”, the teacher criticized the student on the spot. After obtaining x12 + x12 = −1, the teacher then proceeded to taunt the student. When the student argued, the teacher simply changed the equation to x 2 − 3x − 5 = 0. Obviously, this is wrong. For another example, in a mathematics class of grade two of high school, after teacher explained that “a 2 , b2 , c2 are known to be an arithmetic sequence. Prove: 1 1 · 1 · a+b are also an arithmetic sequence”, a student proposed “When |a| = b+c c+a |b| = |c|, the proposition may not be true”. The teacher affirmed the student’s opinion and praised the student for the attitude of positive thinking and being bold in distinguishing bet ween right and wrong, and guided the students to change the question to “When a 2 , b2 , c2 are known to be an arithmetic sequence, and the absolute 1 1 1 · c+a · a+b are also an arithmetic values of a, b, and c are not all equal, prove b+c sequence”. This would help students to develop their critical thinking.

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3.3.4 Flexibility of Thinking The so-called flexibility of thinking is a kind of thinking quality that can change the original plan in time with existing knowledge and experience according to the changes of things, not limited to outdated or inappropriate assumptions, and seek ways and methods to solve problems. The opposite is the rigidity of thinking and ossification of functions. In secondary school mathematics teaching, a one-sided emphasis on problem solving modelling can lead students to form stereotyped thinking, that is, following fixed ways of thinking and habits to think about and solve problems. For example, students have difficulty with factorization of a n+1 − 3a n + 2a n−1 because they are only good at turning a 3 · a 2 into a 5 , rather than turning a 5 into a 3 · a 2 , when using the formula for multiplying the same base powers. For another example, when simplifying sin(x − y) cos y + cos(x − y) sin y, some students are accustomed to expanding the sin(x − y) and cos(x − y) separately before calculating, rather than regarding (x − y) and y as two single angles. Stereotyped thinking has a positive inspiring effect on solving the problems of the same type, but has a negative effect on solving variant problems. In teaching we should give play to the positive role of thinking set, but also pay attention to the development of flexibility of thinking to overcome its negative effect. To cultivate flexibility of thinking, we should cultivate students to be good at analysis, analogy, association, and be good at self-regulation according to the specific situation, and have adaptability to changes. Therefore, it is necessary to pay attention to the cultivation of reverse thinking often while strengthening the positive thinking in teaching, and strengthen the training of reverse thinking through variant teaching, so that students can adapt to the rhythm of thinking changes. The following questions are left as exercises for the readers. 7 + lg 37 + lg 11 ; ➀ Find the value of lg 11 3 √ 2 2 ➁ Simplify tan a + cot a + 2; ➂ How is it that two sighted persons have a blind younger brother, but this blind younger brother does not have a sighted elder brother? ➃ Two people walked to the riverside at the same time, there was a boat on the shore that could only carry one person, but these two people still managed to cross the river by boat. How did they cross the river?

3.3.5 Sense of Organization of Thinking The so-called sense of organization of thinking is a kind of thinking quality that is good at summarizing and sorting out the knowledge acquired and making it organized, hierarchical and systematic. Its opposite is the messy state of thinking. Quite a few students are accustomed to thinking in a single way and cannot form a system of what they have learned, often relying more on the teacher’s review and

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summing up. They often feel that the knowledge they have acquired is disorganized, be at a loss for difficult exercises, have difficulty in expressing themselves clearly, which is a sign of a lack of ability to organize thoughts. The systematicness of mathematical sciences is very strong, and what is to be learned is closely related to the previous knowledge. In teaching, we should be good at guiding students to sort out and summarize the knowledge they have acquired, to connect the knowledge vertically and problem solutions horizontally, and to strengthen the understanding and consolidation of the knowledge they have learned. This is necessary to promote the methodization and systematization of students’ thinking and to improve their ability to solve problems, especially for higher grade students. The students should stick to doing this not only after learning each subject or each unit, but sometimes after learning each subunit, or even after each class, so as to develop the habit of self-summary and sorting. The only way to gradually improve your ability to organize your thoughts is to keep processing them.

3.3.6 Creativity of Thinking The so-called creativity of thinking refers to a kind of thinking quality of discovering new things, putting forward new ideas and solving new problems actively and creatively. Cao Chong weighed the elephant in ancient China, Sima Guang broke the vat to save the child, and Gauss, a German, calculated the sum of positive integers within 100 quickly at the age of 10, which showed this kind of valuable, unusual, and unconventional thinking quality. Chinese teenagers are bold in making innovations. However, due to the school’s undue emphasis on enrollment rate and people’s insufficient attention to science of mathematics education, the creative thinking of students is often restrained, leaving them in a state of conservative and closed thinking. Hence, in teaching, we should pay attention to carrying forward teaching democracy, advocating more thinking, and guiding students to think independently to analyze and solve problems. Meanwhile, we should use the discovery method, research method, and teaching method of learning and guiding pattern as much as possible to provide conditions for cultivating students’ creative thinking. Example Jack and John ride bicycles at constant speed in opposite direction at the same time from places A and B. After meeting on the way, Jack reaches place B after 4 h, and John reaches place A after 1 h. How many hours have Jack and John traveled during the whole journey? Solution 1 Suppose that Jack has traveled x hour(s), and John has traveled y hour(s), then after meeting, Jack travels x4 of the whole journey, and John travels 1y of the whole journey, so we have

x − 4 = y − 1, 4 + 1y = 1, x

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The solution is x = 6 (hours), y = 3 (hours). Solution 2 Suppose that Jack and John have travelled x hours respectively before meeting, then we have 4 1 + = 1, x +4 x +1 The solution is x = 2 (hours). Solution 3 Suppose that Jack and John have travelled x hours before meeting. Since the bicycle speeds remain the same, the time taken by Jack and John is proportional to each other during two legs of the journey, then we have 1 x = , 4 x The solution is x = 2 (hours). The third solution here is ingenious and unique, which is the result of creative thinking. The above qualities of thinking are a whole and they complement each other. The broadness of thinking is the basis and prerequisite to train students’ thinking. Only when they think more can they think more flexibly, more deeply and more skillfully. The profoundness and criticalness of thinking are the process of deepening the thinking training, and real knowledge can be acquired only by mastering the essential factors of things. The flexibility of thinking exists in the profoundness and criticalness of thinking, and only by grasping the essential factors of things, can one be agile and resourceful in the face of intricate and complex things. The sense of organization of thinking is the continuation of broadness, profoundness, criticalness, and flexibility of thinking, which is based on logical thinking and imaginal thinking to organize the structure of knowledge and summarize the laws of understanding. The creativity of thinking is the advanced state of thinking, which is the product of the mutual penetration, mutual influence, close coordination, and reasonable composition of the above qualities of thinking. Cultivating students’ thinking ability and improving their thinking quality is a long-term and arduous task, which needs to be started from an early age and requires multidisciplinary and multichannel coordination, and is particularly important in secondary school mathematics teaching.

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3.4 Cultivation of Mathematical Thinking Ability in Secondary Schools The development of mathematical thinking is an important task of mathematics teaching in secondary schools. But how to cultivate students’ thinking ability and what kind of thinking ability to cultivate are important subjects worth studying. Aiming at the actual situation of teaching of mathematical thinking in secondary schools, we only elaborate on the cultivation of divergent thinking, reverse thinking, and creative thinking here.

3.4.1 Divergent Thinking and Its Basic Approach to Development The so-called divergent thinking is a kind of thinking process that is off the beaten track, pursues variation, and seeks answers from multiple perspectives and in many ways. It has the characteristics of fluency, flexibility, originality, and so on. In secondary school mathematics teaching, paying attention to the training of divergent thinking can not only open up students’ ideas of solving problems, overcome the rigid and stereotyped thinking, and the defects of narrow ideas of solving problems, singular problem-solving method, and being at a loss in the case of a slight change in the questions, but also be of great significance for training students to be creative talents who are bold in exploring new methods and new theories. The training of divergent thinking can be strengthened from the following three aspects: 1. Enhancing the teaching of variants and transformations Fluency and flexibility are the qualities of divergent thinking. Students’ thinking is quick and unimpeded, that is, in a short period of time, they can bring together the concepts, theorems, formulas, methods, and techniques related to the problem under study and have them at their fingertips. This requires students to have the habit and ability to transform the propositional forms and research methods. For example, a common technique in the transformation of trigonometric expressions is to simplify to “1”. When “1” is evident or implied in an expression, the “1” formed by many square relations, reciprocal relations, multiple angle relations, special trigonometric values, etc., should immediately come to mind. For another example, when proving that two angles are equal in plane geometry, that these two angles are corresponding angles, two base angles of an isosceles triangle, corresponding angles in two congruent or similar triangles, and circumferential angles of the congruent arc (or equilateral arc) in the same circle (or equilateral circle) should immediately come to mind.

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At the same time, in the teaching of mathematics, in addition to variant language, the application of variant graphics, etc. should also be considered. 2. Advocating multiple solutions to and multiple variations of one question As stated by Albert Einstein, “it is often more important to raise a question than to solve it”. In teaching, in order to fully arouse students’ enthusiasm for learning and stimulate students to think positively, we should actively encourage students to ask questions. A good way to solve the problems is often inspired by the “strange question” raised. At the same time, we should strengthen the practice of multiple solutions to and multiple variations of one question, which plays a bridging role in improve the divergent thinking ability.   60 Example 1 It is known that sin θ · cos θ = 169 0 < θ < π2 . Find the value of tan θ. Solution 1 Apply the universal formula (omitted) and then find the values of sinθ and cosθ (omitted); Solution 2 First find the value of sinθ ± cosθ, Solution 3 Since sin θ cos θ =

60 169

⇒

sin2 θ+cos2 θ sin θ cos θ

=

169 , 60

then   Solution 4 Since θ ∈ 0, π2 sin θ cos θ =

12 5 60 = · , 169 13 13

and and it is known that θ is an interior angle of a right triangle, therefore (x + y + 1)2 + (x − 2y + 4)2 = 0, |x + y + 1| + |x − 2y + 4| = 0, √ √ x + y + 1 + x − 2y + 4 = 0, Example 2 After solving x 2 +y 2 = 0, students are asked to solve the following equations: This is transforming equation x 2 + y 2 = 0 into many other variations. 3. Strengthening the practice and examination of divergent thinking On the basis of students’ divergent thinking ability, it is undoubtedly important for students to improve the quality of their divergent thinking and to enhance their divergent thinking ability by strengthening the practice and examination of their divergent thinking skills.

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Fig. 3.6 .

Example 3 As shown in Fig. 3.6, in ABC, the bisectors of ∠A and its exterior angle intersect line BC and its extension line at E and F, and the tangent line of the circumcircle of ABC passing through A intersects CF at D. With no more additional lines, what conclusions can be deduced from this? And prove them. Students with less divergent thinking can conclude only that: ➀ AE⊥AF; ➁ ∠3 = ∠B; Students with slightly more divergent thinking can also conclude that: ➁ ∠DAE = ∠DEA, DA = DE ➂ ∠4 = ∠F, DA = DF; ➄ D is the midpoint of EF; Students with more divergent thinking, however, can continue to find and prove that: ➅ DA2 = DC · DB, DE2 = DC · DB, DF2 = DC · DB. ➆ EB · FC = EC · FB;

3.4.2 Reverse Thinking and Its Basic Approach to Development The so-called reverse thinking (also known as negative thinking) is a thinking process that is good at thinking from the opposite position, angle, level, and aspect, and can quickly transfer to another train of thought when there is an obstacle in one train of thinking, so as to solve the problem. As we know, the three major geometric problems were widely known in ancient Greece about 2400 years ago. For more than two thousand years, a substantial number of people have devoted themselves to researching them, but all of them ended in failure. Later, people felt that since the positive approach was fruitless, we may as well doubt the possibility of solving these three questions from the other side. This doubt inspired people’s minds. In 1837, Wantzel first proved that the problems of the duplication of a cube and the trisection of an angle couldn’t be solved with rulerand-compass construction, and in 1882, Lindemann proved the transcendence of π, and then the problem of quadrature of circle was also proved insolvable with rulerand-compass construction. This shows how deep-rooted the positive thinking is, and

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how important the reverse thinking is! In secondary school mathematics teaching, emphasis on the training of reverse thinking is instrumental in the development of flexibility, broadness, profoundness and other qualities of thinking, and conducive to overcoming the stereotype and rigidity of problem solving methods caused by directed thinking, as well as other shortcomings such as not being good at discovering new methods and new ideas. 1. Strengthening the reverse application practice of definitions, theorems, formulas, and laws In the teaching of concepts, the defined concept and the defining concept are identical in extension, i.e., the proposition and the converse proposition as definitions are equivalent. Therefore, when applying the definition, we can not only use the original proposition, but also pay attention to studying or applying its converse proposition. At the same time, since many theorems, formulas, and laws often have their own converse theorems, reversible formulas, and reversible laws, which provides a wealth of favorable conditions for the development of reverse thinking. Strengthening the reverse application practice of concepts, theorems, formulas, and laws can not only enable students to be familiar with the knowledge structure from multiple perspectives and to master their application in many aspects, but also be very beneficial to the development of students’ reverse thinking. For example, reverse application practice can be strengthened by answering the following questions. ➀ Fill in the blanks: n 3 ( )2 = n 9 ; If lg x = 2.3010, then x = _______. ➁ Move letters outside the root sign into it: A ∪ B = { x|x > −2}, A ∩ B = { x|1 ≤ x ≤ 3}______;  −y x 2 y(y ≥ 0)_____. ➂ Rewrite e|

dg ag ad |− f| | + k| | as a third-order determinant. ch bh bc

➃ . ➄ . ➅ Given that the distance between two points (a, 5) and (0, −10) is 17, find the value of a. ➆ It is known that sets A = {x| − 2 < x < −1 or x > 1}, B = {x|a ≤ x ≤ b} or, and if ➇ C0n a1 − C1n a2 + C2n a3 − C3n a4 + · · · + (−1)r −1 Crn−1 ar + · · · + (−1)n Cnn an−1 = 0, find the values of a and b.

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➈ . 2. Learning to apply backward inference and reverse solution If there is difficulty in solving a problem with a positive proof, it can be changed to a negative proof. If there is difficulty in applying synthesis method, it can be changed to the application of analysis method. If there is difficulty in proving the existence, it can also be changed to proving that it does not exist or is impossible, etc. This is backward inference and reverse solution. Example 4 A container has several liters of alcohol. One-third is poured out and then 20 L are poured in the first time, and more than half of the existing alcohol by 27 L is poured out the second time, and now there are 33 L left in the container, so how many liters of alcohol are there originally? Analysis The algebraic solution to this problem is positive thinking, while the idea of arithmetic solution is reverse thinking. From the last condition, we know 33 + 27 = 60 (liters) is half of the existing alcohol, so the existing alcohol is 2 × 60 = 120 (liters); then from the first condition, we get 120 − 20 = 100 (liters), which is two-thirds of the original alcohol, so the original alcohol is 100 ÷ 2/3 = 150 (liters). Example 5 Given that at least one of the following equations has a real root, try to find the range of real number a. x 2 + 4ax − 4a + 3 = 0, x 2 + (a − 1)x + a 2 = 0, x 2 + 2ax − 2a = 0. Analysis This problem is quite cumbersome if solved with positive thinking, but it is convenient if we use backward inference instead. Assuming that none of the three equations has a real root, then from.

we get − 23 < a < −1. This corresponds to three equations that have no real roots. Obviously, at least one of the equations has real roots when a ≥ -1 or a ≤ − 23 . 3. Developing awareness and ability of reverse thinking On the one hand, because students are used to positive thinking; on the other hand, because the reverse thinking is divergent, difference-seeking, and exploratory, it is

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difficult to master it, and most students lack the habit and ability of reverse thinking. Therefore, it is important to use different solutions of positive and reverse thinking in combination with more examples, so that students can experience these two different modes of thinking, stimulate the awareness of using reverse thinking, and gradually develop the ability to reverse thinking. Example 6 Given that six students line up in a row and that one of them does not stand at the head or the end of the row, so how many ways are there for them to line up? The positive solution is: If five other students stand at the head and the end of the row, there P52 ways. The student who does not stand at the head and the end can stand anywhere in the middle 4 positions, so there are P44 ways. These two ways both meet the conditions of the problem, so there are P52 · P44 = 480 ways. The reverse solution is: There are P66 ways without restrictions, and there are P55 ways for the head or end of the row respectively except for the person restricted, so there are P66 − 2P55 = 480 ways meeting the conditions. Example 7 160 people stand in a line, counting off from 1. Those who count odd numbers leave, those who remain count from 1 again, those who count odd numbers leave again, and so on, and finally one person remains. What is the number this person count for the first time? Solution If we follow the consequent process of the problem, it can be solved after a number of rounds by crossing out the eliminated ones, but it is too cumbersome. Now we think it over in reverse process, the person who remains finally must count 2 in the last round, 4 in the second last round, and 8 in the third last round…, so it is easy to know that the person counts 16, 32, 64, 128 through backward inference, that is, this person counts 128 in the first round. This example shows that if solved through positive thinking, some questions will be rather complicated. At this time we can guide students to use reverse thinking to seek solutions, enhancing their awareness of reverse thinking. The students will benefit from following the thinking strategy of “using reverse thinking if positive thinking is difficult” to arouse interest in reverse thinking.

3.4.3 Creative Thinking and Its Basic Approach to Development As a more advanced form of thinking, the so-called creative thinking is the ability to discover or solve problems that have not been discovered or solved by oneself or others by thinking extensively and deeply during the process of analyzing and

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solving problems. Creative thinking should be based on certain knowledge, experience, and skills, and make a conjecture to the problem under research through certain observation, association, analogy and induction. It is characterized by exploration, breakthrough, and innovation. Different from analytical thinking, it does not require sufficient reasons for every step forward. Instead, it is often a sudden realization. It is a form of cognition with insight and leap, which is a form of thinking completed in an instant and compressed in the line of thinking. There are narrow and broad definitions of creative thinking. Creative thinking, in a narrow sense, refers to the thinking activities coming into being for the first time in the history of cognition, which are unprecedented and have certain social significance and value. It includes the thinking process of discovering new things, revealing new laws, creating new methods, establishing new theories, inventing new technologies, developing new products, solving new problems, etc. Basically being not dependent or less dependent on the existing achievements, it can open up new fields. This kind of thinking ability is only possessed by a few people, and it is an advanced and sophisticated thinking activity, which plays an extremely important role in the progress of social history and the development of science and technology. Creative thinking, in a broad sense, refers to thinking activities that are new and original to specific thinking subjects. It also includes the above-mentioned thinking process of discovering new things, revealing new laws, creating new methods, and establishing new theories. However, it does not necessarily come into being for the first time, nor unprecedented, but it is the first discovery and off the track only for the specific thinking ability subject. This kind of thinking ability can be possessed by normal people. The creative thinking studied in the teaching of mathematics generally refers to creative thinking in a broad sense. As stated by Polya, the solving of unconventional mathematical problems is also truly creative work. To this end, we should free our minds and actively foster awareness of developing creative thinking in our teaching. 1. Sound knowledge, experience, and skills are the basis of creative thinking Example 8 There are two points C and D on the same side of line l. Find a point M in line l which has the largest field angle to points C and D (Fig. 3.7). Fig. 3.7 .

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The solution to this problem is not obvious. However, assuming that the moving point M moves gradually from left to right along line l, observe the change of ∠CMD at any time, and we can find: At the beginning, the field angle is very small, and as point M moves to the left, the field angle gradually increases. As it approaches point K, the field angle becomes smaller again (at point K, the field angle is equal to 0). So we initially conjecture that between these two extreme cases, there must be a point M0 which has the largest field angle to points C and D. If in combination with the knowledge that the exterior angle of a circle of an arc is less than the circumferential angle, we can further conjecture that as shown in Fig. 3.7, draw a circle tangent to line l and passing through points C and D, and the point of tangency M0 is what we’re looking for. However, whether there is only one circle that passes through points C and D and is tangent to line l remains to be further conjectured. It is thus evident that without certain knowledge, experience and skills (drawing skills), it is difficult to make conjectures here again and again. In finding solutions to problems, people often use the “empirical intuition method”, also known as the “fundamental quantity method”. For example, we know that the quantities related to triangles include side, angle, angular bisector, median, height, perimeter, area, circumcircle, inscribed circle radius, and so on. But each triangle has only three fundamental quantities, and when these three fundamental quantities are certain, other quantities can be uniquely determined. This is knowledge and experience, and sometimes contains certain skills. Thus, if the triangle is known, which quantities to be chosen as fundamental quantities (angles, sides, heights, etc.) depends on specific circumstances and gives the problem solver a certain amount of freedom, which often leads to “multiple solutions”. Example 9 As shown in Fig. 3.8, in ABC, AB = AC, ∠A = 100°, and the bisector of ∠B intersects AC at D. Prove: AD + BD = BC. Analysis The relevant quantities in this question have been restricted by two conditions AB = AC and ∠A = 100°, there is only one fundamental quantity, and which one shall be selected as the fundamental quantity? Experience suggests that BD should be chosen as the fundamental quantity. Suppose that BD = m, then m BC m AD = , = , sin 20◦ sin 100◦ sin 120◦ sin 40◦ the problem is then reduced to the proof of equation: Fig. 3.8 .

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m sin 20◦ m sin 120◦ sin 20◦ sin 120◦ + m = ⇒ + 1 = , sin 100◦ sin 40◦ sin 100◦ sin 40◦ and it’s a breeze. 2. Being good at observation, analogy, induction, and imagination is an important condition for creative thinking To think well, one must first observe well. Outstanding power of observation reflects the quality of mathematicians. To think well, we should be simultaneously good at analogy, induction and imagination. They are important methods of mathematical discovery, providing ideas for solving difficult problems and blueprints for invention and creation. Lenin said, “Imagination is the most valuable virtue”. Polya pointed out: “People always think of mathematics as just a systematic deductive science, but often ignore the characteristics of its formation process—it is also an experimental inductive science”. For example, we observe the arithmetic sequence {an }, where any three terms have the following equations:

Their coefficients “1, − 2, 1” are the same as the coefficients of the perfect square expansion of the binomial difference. Through ➀ minus ➁, we get a1 − 3a2 + 3a3 − a4 = 0, where the coefficients are “1, − 3, 3, − 1”. At this time, we conjure up whether there is an equation a1 − 4a2 + 6a3 − 4a4 + a5 = 0. Finally, induce the conjecture: for an arithmetic sequence {an }, the following equation is established Cn0 a1 − Cn1 a2 + Cn2 a3 − Cn3 a4 + · · · + (−1)r −1 Cnr −1 ar + · · · + (−1)n Cnn an−1 = 0, This is an interesting property between adjacent terms of an arithmetic sequence, which is not difficult to prove. 3. Strengthening the comprehensive application of knowledge is an important means to improve creative thinking Creativity is a kind of advanced ability to put forward new ideas, new theories and new methods independently in solving mathematical problems. To cultivate creativity, we should pay attention to cultivating students’ interest in mathematics and their creative consciousness, create a thinking situation to stimulate students’ desire for creativity, and train them through divergent thinking, intuitive thinking (inspiration) and the organic combination of all kinds of thinking. At the same time, we should pay attention to the combination of shapes and numbers, strengthen the mutual penetration and integrated application of knowledge, and provide a broad scenario for the cultivation of creativity.

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Example 10 It is known that the three sides of a triangle a, b, c are all integers, where a ≤ b ≤ c, and if b = n (positive integer), how many triangles like this are there?   Solution Suppose that the number of triangles is f n , find the expression for   f n , and then start with the discussion of n = 1, 2, 3,

(1) When n = 1, from b = l, we get a = c = l, f (l) = l; (2) When n = 2, from b = 2, we get f (2) = 1 + 2 = 3; a

b

c

1

2

2

1

2

2

2, 3

2

f (n)

(3) When n = 3, from b = 3, we get f (3) = 1 + 2 + 3 = 6; a

b

c

f (n)

1

3

3

1

2

3

3, 4

2

3

3

3, 4, 5

3

From this, we may conjecture that: f (n) = 1 + 2 + 3 + · · · + n =

n(n + 1) . 2

For further analysis, when b = 1, a has one value (a = l); when b = 2, a has two values (a = l, 2); when b = 3, a has three values (a = 1, 2, 3). This shows that when b = n, a has n values (a = l, 2,…, n). However, when a = l, c = n; when a = 2, c = n, n + l; when a = 3, c = n, n + 1, n + 2; when a = k, (l ≤ k ≤ n), from b ≤ c < a + b, we get n ≤ c < n + k. Hence, c has k values exactly, that is, c = n, n + l, n + 2,…, n + k − l. This also shows that the number of c value is exactly equal to the value of a, so we can draw a general conclusion:

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a

b

c

1

n

n (1 value)

1

2

n

n, n + l (2 values)

2

3

n

:

:

n, n + l, n + 2 (3 values) .. .

3 .. .

k

n :

(k values) .. .

k .. .

n

(n values)

n

n

f (n)

Hence, when b = n, the number of triangles meeting the conditions is f (n) = 1 + 2 + 3 + · · · + n =

n(n + 1) . 2 2

2

Example 11 Connect a point C on the ellipse ax 2 + by2 (a > b > 0) and focal points F1 and F2 to intersect the ellipse at A and B respectively, and the tangent lines at points A and B intersect at F. Prove: CF must be perpendicular to the tangent line passing through point C on the ellipse. Analysis It is rather cumbersome to prove this question by means of analysis method. If we use the property that the connecting lines of every vertex of elliptic circumscribed triangle and the tangent point of opposite side must be concurrent, it is more convenient to use plane geometry method. Proof As shown in Fig. 3.9, suppose that the tangents at A, B, and C form DEF, then AE, BD, and CF are concurrent. FA DC EB · CE · BF = 1, Based on Ceva’s theorem, we get AD Draw a parallel line to DE, which intersects extension lines of CA and CB at G and H respectively, therefore,

GF FA CE EB = , = , DC AD FH BF Multiply the above two equations, we get Fig. 3.9 .

GF FH

=

FA AD

·

DC CE

·

EB BF

= 1, GF = FH.

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And ∠1 = ∠2, hence ∠G = ∠H, CF⊥GH. and GH // DE, hence CF⊥DE. Review Questions and Exercises (II) 1. What are thinking and mathematical thinking, and what does it mean in teaching? 2. What are some common approaches to mathematical thinking? Try to give examples. 3. What are the basic qualities of mathematical thinking? Try to give examples. 4. What is reverse thinking and how to develop students’ ability of reverse thinking in secondary school mathematics teaching? 5. What is divergent thinking and how to develop students’ ability of divergent thinking in secondary school mathematics teaching? 6. What is creative thinking and how to develop students’ ability of creative thinking in secondary school mathematics teaching? 7. Try to draw up a set of exercises to illustrate the application of reverse thinking. 8. Try to draw up a set of exercises to illustrate the application of divergent thinking. 9. Try to illustrate the use of creative thinking with a specific exercise. 10. Try to illustrate a tentative idea for strengthening the thinking teaching with teaching content of a section of secondary school mathematics. 11. Observe the following equalities separately, sum up general laws, and prove your conclusions. c k + · · · + (a+b+···+h)(a+b+···+h+k) + (a+b)(a+b+c) l b+c+d+···+k+l = . (a+b+···+h+k)(a+b+···+h+k+l) a(a+b+c+···+k+l)

b a(a+b)

(3)

+

1=1 3+5=8 7 + 9 + 11 = 27 13 + 15 + 17 + 19 = 64 21 + 23 + 25 + 27 + 29 = 125 ........

(4)

1=1 1+2+4=7 1 + 2 + 4 + 5 + 7 = 19 1 + 2 + 4 + 5 + 7 + 8 + 10 = 37 1 + 2 + 4 + 5 + 7 + 8 + 10 + 11 + 13 = 61 ........

12. Try a variety of methods to solve the following exercises and explain the thinking methods used: (1) Prove: The two segments resulting from the division of an interior angle bisector of a triangle are proportional to two included sides of this angle correspondingly. (2) Suppose that x 2 + y 2 = z 2 , prove: for any positive real numbers x, y, z and m, n, there is  mx + ny ≤ z m 2 + n 2 . 13. Prove:

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c k + · · · + (a+b+···+h)(a+b+···+h+k) + (a+b)(a+b+c) l b+c+d+···+k+l = a(a+b+c+···+k+l) . (a+b+···+h+k)(a+b+···+h+k+l)

b a(a+b)

+

14. Using the method of conjecturing, prove that each term of the sequence 16, 1156, 111556, 11115556, … is a perfect square. 15. In ABC, prove: a = c cos B + b cos C, b = a cos C + c cos A, c = b cos A + a cos B, and try to write out a similar proposition using analogy association. 16. Using analogy association, answer the following questions: (1) If α + β + γ = kπ(k ∈ z), prove: tan a + tan β + tan γ = tan α · tan β · tan γ ; a−b b−c c−a a−b b−c c−a + 1+bc + 1+ca − 1+ab · 1+bc · 1+ca ; (2) Prove: 1+ab 2 (3) Given a + b + c = abc, prove: a(1 − b )(1 − c2 ) + b(1 − c2 )(1 − a 2 ) + c(1 − a 2 )(1 − b2 ) = 4abc..

Chapter 4

Secondary School Mathematics Methods

Method is the key to success, and scientific method is the soul of science. Since ancient times, people have attached great importance to the theoretical research of methods, trying to use the correct methods to understand and remold the world. In mathematics research, the correct research method is the premise of achievement. Now, some basic mathematical methods have become an important part of secondary school mathematics.

4.1 Meaning of Mathematical Methods 4.1.1 Meaning of Mathematical Methods Method is the way, approach, or means adopted by people to solve specific problems. Scientific methods refer to the general methods employed in scientific research. As a fundamental science for people to understand and remold the world, methodology is a general method theory for summarizing scientific discoveries or inventions. Any science has its own research methods, but the particularity contains the generality, and various specific methods contain the general methods and the general principles of methods of thinking. This kind of knowledge that studies the method problem from general methods, that is, the theory about general methods is methodology, also known as science of method. Different world outlook determines different methodology. According to the epistemology of dialectical materialism, people have gradually summed up the inherent laws and research methods, which is scientific methodology. It generally includes three levels. The first level: It refers to some specific research methods in various sciences, which are the research object of each science itself. For example, the spectral analysis method in physics, the quantitative analysis method in chemistry, the proof © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_4

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by contradiction, identity method, and method of undetermined coefficients in mathematics. The second level: It refers to methods that are generally applicable to many disciplines. That is, it is not unique to a discipline, but a method extracted from multiple disciplines, for example, observation, analysis, abstraction, experimentation, and so on. The third level: It refers to the methods applicable to all natural and social sciences research, namely philosophical methods or materialist dialectics, such as investigation, contradiction analysis, etc. Mathematical methodology is a theory that studies the laws and principles of mathematical discoveries and inventions. It is an emerging discipline to study and discuss the thinking methods of mathematics, the laws of mathematical development and the laws of mathematical discoveries, inventions, and innovations. It is the concrete embodiment of scientific methodology in mathematics, so it falls under the category of scientific methodology. For mathematical methods, people can make different classifications according to different standards. According to the degree of abstraction, mathematical methods can be classified as follows: Mathematical methods ⎧ ⎪ ⎨ Specific method (problem solving method) General method (logical method and test method) ⎪ ⎩ Mathematical thinking methods (generalization of specific methods and general methods

The specific methods are various specific methods to solve mathematical problems, such as proof by contradiction, method of undetermined coefficients, analytical method, etc. The specific methods are characterized by clear steps, clear procedures, specific operation, but the application scope is small. They are the lowest level of mathematical methods. According to their degree of pervasiveness, the specific methods, can also be divided into multiple levels. For example, the transformation methods include algebraic transformation method, geometric transformation method, and trigonometric transformation method. The geometrical transformation methods include method of congruent transformation, similarity transformation method, and inversion transformation method. The methods of congruent transformation include translation transformation method, symmetric transformation method, rotation transformation method and so on. The general methods mainly refer to various logical methods and test methods, with a higher level. The scope of research is not limited to a certain mathematical branch, but based on mathematics as a whole and is applicable for various mathematical branches. For example, the analysis method from a whole to its parts, the synthesis method from its parts to a whole, the deduction method from the general to the particular, the induction method from the particular to the general, and the analogy method from the particular to the particular, etc. are all logical methods. Screening method, non-standard problem solution method, etc., are all test methods. This chapter focuses on the introduction of general mathematical methods.

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In the process of the birth and development of mathematics, mathematical problems, mathematical knowledge, mathematical thought, and mathematical methods have always been developing harmoniously in a combined and interrelated way. To solve various mathematical problems in practice and theory, people are bound to create various kinds of mathematical thoughts and methods, and the corresponding mathematical knowledge comes one after another. For example, in seeking formula solutions to higher degree algebraic equations, Galois created the thought of “group theory”, thus promoting the development of algebra; Descartes created the thought of “number-shape combination”, thus establishing analytic geometry, and Newton proposed the thought of “fluxions”, thus creating differential and integral calculus, and so on. Mathematical thinking is a generalization of a specific class of mathematical methods or general methods and is the thinking strategy and adjustment principles throughout this class of mathematical methods. It restricts the direction of subjective consciousness in mathematical activities and plays a normalizing and regulating role in the choice and combination of methods. For example, the thought of number-shape combination is a generalization of a class of methods, such as coordinate method, trigonometry, complex number method, vector method, graphic method, etc., its thinking strategy is to connect the two basic objects of mathematical research (shape and number) for comprehensive research, giving full play to the respective advantages of algebraic and geometric theories to solve problems, and the thought of numbershape combination is formed by generalizing and escalating the basic spirit of this class of methods. The basic mathematical thinking in elementary mathematics is alphabetic algebra thought, logical reasoning thought, decomposition and combination thought, conversion and transformation thought, set and correspondence thought, number–shape combination thought, and so on. However, because mathematical thinking is inseparable from mathematical methods, the mathematical thinking and the specific methods or general methods from which the mathematical thinking is generalized are collectively referred to as the mathematical thinking method.

4.1.2 Approaches to Studying Mathematical Methods Mathematical methods and mathematics are produced at the same time and developed synchronously. They are an invaluable spiritual treasure accumulated in the development of mathematics and have a broad field of research and rich content. How to make a comprehensive conclusion and summary of the great legacy of research methods is an arduous work. To this end, it is necessary to attach importance to the following points. First, researchers should have a solid foundation of mathematical theory. Mathematical theory is the basic material to summarize mathematical methods. Only by possessing a large number of materials can research work be handy with facility. This requires us to be well read, or be proficient in a branch of mathematics, and

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improve our mathematical accomplishment, to lay a foundation for the research of mathematical methods. Second, researchers have to peruse a lot of mathematics history works. The history of mathematics not only records the dazzling research achievements of mathematicians, but also records the thinking activities when they are making scientific creations and the representations out of the activities. We can follow these representations to capture the laws of creative thinking, from which effective research methods can be extracted. Third, researchers should master the research methods of materialist dialectics. Mathematical methods are produced in production practice and research, develop with the development of mathematics, and guide the practice. The birth and development of mathematics is brimming with materialist dialectics, so it is evident that materialist dialectics must be employed to study mathematical methods. In short, to research mathematical methods, we must have sound mathematical accomplishment, solid mathematical theoretical foundation, and guided by materialist dialectics, closely associate them with the history of mathematical development, especially the history of the development of mathematical thinking, analyze major achievements of the mathematical findings, and summarize feasible research methods therefrom. This is the approach to studying mathematical methods on the whole.

4.1.3 Significance of Studying Mathematical Methods First, the study of mathematical methods is conducive to promoting the development of mathematics. As proved by research practice, the level of a mathematician’s research ability, in addition to having a strong mathematical theoretical foundation and strong thinking ability, is mainly determined by the mastery of the research methods and the flexibility to use these methods. Famous mathematicians such as Descartes, Euler, and Gauss are all models of consciously summarizing and controlling mathematical methods. Learning and studying mathematical methods can help us summarize practical experience systematically, escalate it to theory, make less or no detours in the research work; can improve our mathematical research ability, to make greater contributions to the development of mathematical theory and its application research. Secondly, the study of mathematical methods is conducive to giving play to the functions of mathematics: scientific function—mathematics not only provides simple and accurate formal language and reasoning basis for scientific research, but also provides quantitative analysis and calculation methods, which has become one of the important characteristics of modern scientific development; social function— mathematics thought and methods applied to social practice help to solve practical problems in various sciences and social practice; thinking function—mathematical thought and methods provide routes, procedures, and ways of thinking activities, serving as “thinking training gymnastics”.

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93

Thirdly, the study of mathematical methods is conducive to promoting the reform of mathematics teaching. The whole mathematics teaching content is, as it were, full of the content of mathematical method learning and application. Seen from the training objectives of normal colleges, systematic study and research of mathematical methods in secondary school mathematics pedagogy curriculum is beneficial for us to master the curriculum standards, deepen our understanding of secondary school mathematics textbooks, and improve our quality and ability, to create conditions to be competent for secondary school mathematics teaching actively. At the same time, in the macroscopic mathematical method, in addition to studying the general rules and methods of the development of mathematics itself, the growth law of mathematical talents is also probed into, and its research results will undoubtedly point out the direction of and provide theoretical guidance for cultivating mathematical talents. Fourthly, the study of mathematical methods is conducive to the enrichment and development of Marxist philosophy. As early as one hundred years ago, Engels pointed out that mathematics was full of dialectics. This will be further confirmed by the study of mathematical methods. In addition, concretization and abstraction, analysis and synthesis, induction and deduction, and other research methods are often used in mathematical exploration. An in-depth study of the dialectical relationship of unity of opposites of these methods can also greatly enrich the content of dialectical logic.

4.2 Reduction Method 4.2.1 Overview of Reduction Method Literally speaking the so-called reduction means transformation and reduction. The reduction method in mathematics is to normalize the mathematical problems, which refers to a means or method for people to summarize a new, outstanding, or unsolved problem through some transformation process into a kind of problems that have been solved or easily solved, so as to obtain the solution finally. This reduction method was regarded by Descartes, a French philosopher and mathematician in the seventeenth century, as a “universal solution” to mathematical problems. Polya, a famous mathematics educator, also thought highly of it and regarded it as the primary method to solve mathematical problems. Reduction is an important characteristic of mathematicians’ way of thinking, the first method to think about for people to solve problems, and also the most fundamental and typical method studied in mathematical methodology. Despite the numerous reduction methods, all the problem-solving processes by means of reduction method can all be simply summarized as shown in Fig. 4.1.

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Fig. 4.1 .

4.2.2 Direction of Reduction The direction of reduction can be briefly described as from the unknown to the known, from difficulty to easiness, from tediousness to conciseness. The direction of reduction, also known as the principle of reduction, includes: The principle of familiarity: transforming an unfamiliar question into a familiar one; The principle of simplicity: transforming a complex question into a simple question; The principle of intuitiveness: transforming an abstract question into a concrete question. The direction or principle of reduction may also be summarized as: Transforming a practical question into a mathematical question, mathematical question into an algebraic question, and algebraic question into a question of solving equations. Example 1 Suppose we know the sum of interior angles of a triangle is 180°, then how to calculate the sum of interior angles of a (convex) polygon? Analysis As shown in Fig. 4.2, obviously, the n-gon can be divided into (n − 2) triangles, then the sum of interior angles of the (convex) n-gon is (n − 2) 180°. That is to say, transforming the problem of the sum of interior angles of the unknown (convex) n-gon into the problem of the sum of interior angles of the known (n − 2) triangles. Example 2 Solve a system of equations.

Fig. 4.2 .

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95

Solution By ➀一5 × ➁, we get x 2 + y 2 + 2x y + 4x + 4y = 21, that is (x + y)2 + 4(x + y) − 21 = 0. Suppose x + y = u, then u 2 + 4u − 21 = 0, solve it and we have u 1 = 3, u 2 = −7. In this way, the solutions to the original system of equations are transformed into the solutions to the following two equations: 

x + y = 3, and xy = 2



x + y = −7 xy = 2

And this is not difficult to solve. In the above Examples 1 and 2, the unknown is transformed into the known, and the unfamiliar is transformed into the familiar, which reflects the application of the principle of familiarity in the reduction method. Example 3 Try to find the value of S = 1 + 2 + 3 + … + 100. Solution Obviously, it is quite troublesome to calculate it directly, but if we “double” it to get the problem * 2S, and then it is convenient to evaluate S from 2S. Since 2S = 1 + 2 + 3 + . . . + 99 + 100 + 100 + 99 + · · · + 3 + 2 + 1 = 101 × 100 = 10,100, 1 therefore, S = × 10,100 = 5050. 2 Example 4 As shown in Fig. 4.3, it is known that AC and BD are diagonals of cyclic quadrilateral ABCD. Prove: AB · CD + AD · BC = AC · BD = AC · BC(Ptolemy’s theorem).

Analysis Supposing that we have already known the method of proving that four segments are proportional through similar triangles, we have already known how to prove ad = bc. Obviously, simply find a point E in BD to make AB · CD = AC · BE (ABEACD), AD · BC = AC · ED(ABCAED), Fig. 4.3 .

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This only requires constructing ∠ BAE = ∠ DAC to prove the proposition. In the above Examples 3 and 4, the complex problems are transformed into simple problems, which reflects the application of the simplicity principle in the reduction method. Example 5 Try to calculate the minimum value of the function. z = f (u, ν) = (u − v)2 +



2 − u2 −

9 v

2 .

Analysis This is an extremum problem of a binary function, which is relatively abstract and complex, and it is rather difficult to calculate it directly. As shown in Fig. 4.4, it is√ intuitive if we regard f (u, v) as the square of the distance between two

points A (u, 2 − u 2 ) and B v, v9 , namely as z = f (u, v) = |AB|2 . This is because point A is on semicircle x2 + y2 = 2(y ≥ 0) and point B on hyperbola xy = 9. Thus, from Fig. 4.4, we get √ √ √ the minimum value AB = OB − OA = 3 2 − 2 = 2 2, √ that is, the minimum value of z = f (u,v) is AB^2 = (2 2)^2 = 8. In this way, we transform an abstract problem into a concrete problem through reduction, which reflects the application of the intuitiveness principle in the reduction method. Example 6 An iron ball floats on the mercury in the container, if we fill water into the container and cover the iron ball, will the iron ball go up, go down, or keep unchanged relative to the mercury surface? Solution As shown in Fig. 4.5, suppose the density of the liquid above is a, the density of the liquid below is b, the ball density is c, the ball volume is V, the volume of the part floating on the mercury is x, and the volume of the lower part is y. According to the Archimedes principle, namely the mass of floating body is equal to the mass of the discharged liquid, we can establish the equations: 

Fig. 4.4 .

ax + by = cV, x + y = V.

4.2 Reduction Method

97

Fig. 4.5 .

b−c c−a The answer is x = b−a V , y = b−a V Obviously, before filling water, a = 0, b = 13.60,c = 7.84, the answer obtained is x = 0. 424 V; after filling water, a = 1.00, the answer obtained is x = 0.457 V, Therefore, the proportion of the ball floating on the mercury in the overall volume of the ball increases after water filling, that is, the iron ball will go up. For further discussion, when b, c, and V remain unchanged, a increases from 0 b−c V to c, the denominator of x (b − a) gradually decrease, then x increases from b−a to V. This example further reflects the application of the reduction method. During the solution process, we first transform a practical problem into a mathematical problem, then into an algebraic problem, and finally into a problem of solving an equation or system of equations. In this way, the problem is solved, which also reflects the application of intuitiveness principle in the reduction method.

4.2.3 Methods of Reduction 1. Transformation method When we encounter a mathematical problem, we first consider whether it can be transformed into one or more simple problems. This kind of transformation has variant, transformation, condition transformation, conclusion transformation, and other cases, and it includes identical transformation, equivalent transformation, parameter transformation, coordinate transformation, geometric transformation, and so on. Sometimes only one-directional transformation is required, sometimes multidirectional transformation is required, which shows diversity, flexibility, and skill in specific applications.  √ 19 − 8 3, try to find the value of fraction Example 7 Suppose x = 4 3 2 x −6x −2x +18x+23 . x 2 −8x+15 Solution The direct substitution of the value of x into the original fraction will be too troublesome. If identical transformation is used, it can make hard things simple.  √ 2 √ √ 4 − 3 = 4 − 3, Since x = 19 − 8 3 =

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√ that is, x − 4 = − 3, square both sides, and we get x 2 − 8 x + 13 = 0, x 2 +2x+1)(x 2 −8x+13)+10 then transform the original fraction, and we get = ( = 5. (x 2 −8x+13)+2

Example 8 As shown in Fig. 4.6, given the length of three medians of  ABC, construct this triangle. Analysis According to Pappus Law of triangles, we get DO = 13 AD, BO = 23 , BE, CO = 23 CF, so extend AD to G to make OG = AO, then OG = 23 AD, so for BOG, OG = 23 AD, BO = 23 BE, BG = 23 CF and AD, BE and CF are known, so it is not difficult to construct BOG. After constructing BOG, it is not difficult to construct ABC. 2. Decomposition method As described by Descartes, the so-called decomposition method is “breaking down every problem under consideration into several parts, as needed and possible, to make them easier to solve”. But in many cases, the process has to be recombined to realize reduction completely. Polia said; “Decomposition and combination are important intellectual activities, making every part of it easier to solve”. This decomposition method, also known as superposition method, often appears in a variety of forms. In the calculation or proof of area and volume, the decomposition method is often used, which is specially known as shape segmentation method. The purpose of reduction is achieved through the segmentation of shape. As shown in Fig. 4.7, the segment area can be broken down into the difference between the areas of two graphs, and in Fig. 4.8, the tunnel sectional area can be broken down into the sum of the areas of two graphs, and so on. Example 9 As shown in Fig. 4.9, suppose ABCD is a quadrilateral, E and F trisect AB, and G and H trisect CD. Prove: SEFGH = 13 SABCD . Fig. 4.6 .

Fig. 4.7 .

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99

Fig. 4.8 .

Fig. 4.9 .

Proof (1) Join EG to divide EFGH into two triangles, and join DE and BG, then GH = 21 DG, EF = 21 EB, so S EHG = 21 S AEDG , S GEF = 21 S CEB , so SEFGH = SEHG + SGEF = 21 (SEDG + SGEB ) = 21 SEBCD ; (2) To seek the relationship between EBGD and ABCD, join DB again to divide EBGD into two triangles. Since SEBGD = SEBD + SGDB = 23 SABD + 23 SBCD = 23 SABCD , therefore, SEFGH = 13 SABCD . 3. Mapping method The mapping method, namely the relationship, mapping, and inversion method (the RMI method for short), is an important way to achieve reduction in modern mathematical studies. Compared with general reduction methods, this reduction method achieves a higher degree of abstraction, hence boasting of more important and extensive applications in mathematics. Professor Xu Lizhi, a famous mathematical educator in China, originally proposed this method from studying the series inversion theory, and first specialized in this method from the perspective of mathematical methodology, to make it approach elegance and refinements. √ √ Example 10 Prove: for any x ∈ R, x 2 − 8x + 20 + x 2 + 1 ≥ 5. Solution It is difficult to prove it by means of demonstrating inequalities in algebra. Therefore, we can consider mapping an algebra problem into a geometric problem with the help of the coordinate system, and obtaining an algebraic conclusion through a geometric conclusion. Establish a rectangular plane coordinate system, draw points A(0, l), B(4, 2), and M(x, 0), and symmetric point B of point B with regard to x-axis (4, −2) (Fig. 4.10). Since |MB| = |MB|,

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Fig. 4.10 .

therefore, |MA| + |MB| = |MA| + |MB| When |MA|+|MB | is minimum, points A, M, and B should  be in a straight line, 2 2  and its minimum is the length of segment √ AB , so we get √(0 − 4) + [1 − (−2)] 2 2 =  5. In this way, the taskof proving x − 8x + 20 + x + 1 ≥ 5 is proving (x − 4)2 + (0 − 2)2 + (x − 0)2 + (0 − 1)2 ≥ 5, which is obviously true. Example 11 In a planar convex quadrilateral with an area of 32 cm2 , the sum of the lengths of two opposite sides and one diagonal is 16 cm. Try to determine all possible lengths of the other diagonal. Solution This is a problem in plane geometry, which is more difficult to solve if we are limited to plane geometry methods. But it is easier if we use the coordinate method in mapping method to transform this geometric problem into an algebraic problem. For the sake of convenience, when selecting a coordinate system, make known points as many as possible on the coordinate axes. The coordinate system selected for this question is shown in Fig. 4.11.The four vertexes of the planar convex quadrilateral are sequentially. A(−a, 0), B(b, −b ),C(c, 0), D(0, d), According to the known conditions, we get Fig. 4.11 .

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1 1 (c + a)d + (c + a)b = 32. 2 2  √ And |AB| + |CD| + |AC| = (a + b)2 + b2 + c2 + d 2 + (a + c) = 16, that is S  BCD =

And it is obvious that b + d ≤

 √ (a + b)2 + b 2 + c2 + d 2

Substitute it into ➀, we get (c + a) [16 − (a + c)] ≥ 64, that is, [(c + a)—8]2 ≤ 0 → a + c = 8, and only the equal sign can make the above inequality established, so we get b + d = 8, c = 0, a + b = 0, so the answer is a = 8, b = −8, therefore, the other diagonal is |BD| =

 √ b2 + (b + d)2 = 8 2 (cm).

4. Specialization and generalization method The so-called specialization is to “withdraw” the mathematical fact in question to its special state (number or position relationship) so as to achieve the purpose of studying the general state. In mathematics, variables are often transformed into constants and arbitrary figures into special figures or special positions to gain some enlightenment, which is the concrete embodiment of specialization. The so-called generalization is to put the specific problem in question in the general state for thinking so as to achieve the idea of solving the specific problem. The train of thought is from the particular to the general, and from the general to the particular, namely from the concrete to the abstract, and from the abstract to the concrete, and they check and complement each other. It is another common method in the reduction method. For the specialization method, we have illustrated it in Sect. 2.2, Chap. 2. The generalization method is illustrated as follows. Example 12 What is the value of x so that the two roots of the equation of x 7x 2 − (k + 13)x + k 2 − k − 2 = 0 are within (0, 1) and (1, 2) respectively? Solution Now we perform a generalization and put the equation in a function for consideration. We may as well suppose f (x) = 7x 2 − (k + 13)x + k 2 − k − 2. As shown in Fig. 4.12, this function graph can only cross the x-axis within (0, 1) and (1, 2), namely, f (x) must have a sign reversal within this interval, so it is necessary and sufficient conditions are: ⎧ ⎧ ⎧ 2 ⎨ (k − 2)(k + 1) > 0 ⎨ f (0) > 0 ⎨ k −k−2>0 f (1) < 0 ⇒ k 2 − 2k − 8 < 0 ⇒ (k + 2)(k − 4) < 0 ⎩ ⎩ ⎩ k 2 − 3k > 0 k(k − 3) > 0 f (2) > 0

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Fig. 4.12 .

Fig. 4.13 .

The solution of this system of inequalities may as well be expressed by the number axis (Fig. 4.13). Thus, their common solution are −2 < k < −1 or 3 < k < 4.

4.2.4 Understanding Reduction Dialectically 1. Core thought of reduction The core thought of reduction lies in a word “change”, and this kind of “change” is often a thinking process of solving problems. Solving mathematical problems, especially more complex mathematical problems, often requires some pondering, which has its complex thinking process. It is necessary to analyze the process and characteristics of this thinking activity to solve problems by means of reduction purposefully and effectively. First, the process of reduction is a process of thinking activity to identify and classify problems. In the problem solving, the identification and classification of the problem is basically the identification of mathematical models and graphics. Of course, to form this correct and rapid identification capacity, it is necessary to improve the ability to observe and generalize the problems, sometimes reasoning is required to transform the original problem appropriately, enabling it to become an identifiable mode.

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Second, the process of reduction is a thinking process of continuous analysis and synthesis of problems. In solving a problem, we always analyze the problem first, make clear what the known conditions are, what the conclusion is, and then compare the conditions with the conclusion to find out the internal relations between them, which is synthesis, and this analysis and comprehensive process needs to be carried out repeatedly. Third, the process of reduction is a process of dialectical unity of stereotyped thinking and variation. Reduction is a transformation from the unknown to the known, the complex to the simple, the difficult to the easy, and is induced under a certain stereotyped thinking. However, due to incomplete analysis, unclear identification, we may often apply mechanically the existing knowledge, methods, and experience mistakenly, thus misleading the reduction and hindering the solution of the problem. 2. Practice of reduction (1) Reduction is a purposeful practical activity. During reduction, we always “keep our eyes on the goal”, namely always consider this question: how can we achieve the purpose of solving the original problem? How can we achieve the purpose of solving the original problem in the best way and form? For example, how can we best find the unknown quantity in the problem, and how can we better prove the conclusions in the problem? (2) Reduction is a repeated exploratory practical practice. During reduction, how do we determine the direction of reduction and application of the reduction method? It often requires repeated exploration. As pointed out by Polya in Mathematical Discovery, if there are several possible approaches and none of them is a sure thing, you’d better explore each road slightly before going too far along a certain road, because any road may lead you to a cul-de-sac. Therefore, you should not be limited to a certain approach too early. (3) Reduction is a selective and flexible practical activity. For an unfamiliar and complex problem, there are often diverse ways to achieve reduction. We should consider both the possibility of realizing reduction, but also the best way to realize reduction, so as to solve the problem more quickly and better. How to choose in the selective and flexible ways of reduction? In Mathematical Discovery, Polya also pointed out the “principle of selecting the best qualified”, which means giving priority to the less difficult over the more difficult, the more familiar over the less familiar, and the simpler over the more complex. 3. Limitations of reduction Finally, it should be pointed out that although the reduction method plays an important role in mathematical research, it also has certain limitations. Specifically speaking first, not all problems can be solved through reduction. For example, it is obvious that the aforesaid reduction “from difficulty to easiness, and from tediousness to conciseness” cannot be continued forever and indefinitely, that is, reduction can never be endless.

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Secondly, it can be seen from the previous analysis that the key to applying the reduction method to solve the problems is whether we can find the correct reduction direction and method. Therefore, although reduction is mainly or ultimately manifested as a method to solve problems, its successful application is based on “mathematical discovery” and reduction is based on “foresight”. Third, placing too much emphasis on reduction is not conducive to the development of mathematics. If we emphasize reduction heavily and too much, it is not conducive to the innovation of thinking, and consequently it will be difficult for mathematics to develop. Therefore, as far as the research of mathematical methodology is concerned, we cannot settle for the analysis of the reduction method, but must engage in new research. For example, we should first discuss the problem about mathematical discovery methods.

4.3 Discovery Methods 4.3.1 Analogy Method 1. Meaning of analogy Analogy is a form of thinking that, based on the identity or similarity in certain aspects of two different objects (such as characteristics, attributes, relationships, etc.), they may be the same or similar in other aspects. As an inference from the particular to the particular in the thinking process, it is a basic and important means to look for truth and find truth, and one of the most important and fundamental methods of mathematical methodology. In mathematical research, number and formula, equation and inequality, one variable and multivariate, lower degree and high degree, plane and space, number and shape, finity and infinity, and so on, are common analogies. Through analogy, we find a certain connection between their concepts, properties, and laws, so as to lay a solid foundation for systematic learning. Here we’ll make a detailed discussion on three important and common analogies. 2. Analogy between plane and space While learning solid geometry, people often compare the problems to be resolved with related problems in plane geometry; while learning space analytic geometry, it is often compared with related problems in plane analytic geometry to give us some useful enlightenment. For example, as we know, two lines in the plane l1 : A1 x + B1 y + C1 = 0 and l2 : A2 x + B2 y + C2 = 0

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105

l1 //l2 ⇔

A1 B1 C1 =

= , A2 B2 C2

In two planes in the space π1 : A1 x + B1 y + C1 z + D1 = 0 and π2 : A2 x + B2 y + C2 z + D2 = 0, A1 B1 C1 D1 = =

= π1 //π2 ⇔ A2 B2 C2 D2 Their equations and properties have some similarities. Example 1 Find the length of radius of inscribed sphere in a regular tetrahedron. Solution For this question, we might as well use analogy to find that in the plane, the ratio of the radius to the height of the inscribed circle of a regular triangle is 1:3; therefore, we can conjecture that in the space, the ratio of the radius of the required inscribe sphere to the height of the regular tetrahedron may also be a constant (Fig. 4.14). Based on this idea, the above question is not difficult to solve. As shown in Fig. 4.15, without loss of generality, we suppose that the edge length of the regular tetrahedron is 1. Based on relevant knowledge, it is easy to calculate the height of the √ 6 regular tetrahedron AH = 3 . Now assume that the ratio of the radius to the height of the inscribed sphere is x. Based on the fact that OH = OP, we can get the value of x. (Since A-BCD is a regular tetrahedron, therefore H and P is the center of regular triangles BCD and ACD respectively). Since OH = OP = x · AH = therefore, BO = BP − OP = While BH = Fig. 4.14 .

√

3 , 3

√

6 x, and 3 √ √ 6 6 − x 3 3

BP = AH = =

√

6 (1 3

√

6 , 3

− x).

substitute the above formulas into OB2 = OH2 + BH2 , and we get

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Fig. 4.15 .

 √ 2  √ 2  √ 2 6 6 3 2 x + (1 − x) = , 3 3 3 that is, 2(1 − x)2 = 2x z + 1. x= √

1 . 4

√

The answer is 41 × 36 a = 126 a. Thus the ratio of the radius to the height of the inscribed sphere of the required regular tetrahedron is 1:4. In other words, if the edge length of a regular tetrahedron is a, the radius of the inscribed sphere is. 3. Analogy of shape and number In mathematical studies, analogies of number and shape are often applied in two opposite directions, that is, we can understand the relevant properties of the “numbers” through the comparison with “shapes”, and we can also understand the relevant properties of the “shapes” through the comparison with the “numbers”. In addition, we can research “number” problems through graphical method and image method, and research “shape” problems through algebraic method, trigonometric method and coordinate method. See specific examples below. Example 2 Solve the inequality: |x + a| + |x| 0, a = b} etc. Based on this analogy, we can further induce the following general inequality:     a 2 − b2 , a 2 + b2 (x1 , x2 , . . . , xn are all non-negative real numbers). Moreover, suppose x1n = A1 , x2n = A2 , · · · , xnn = An , and the following basic inequality can also be obtained from the above formula:  A1 + A2 + · · · + An ≥ n A1 A2 · · · An . n In other words, the arithmetic mean of n non-negative real numbers is not less than their geometric mean. For another example, under the conditions 0 < a < l, 0 < b < 1 and 0 < c < 1, it is easy to prove: (l − a)(l − b) = l − a − b + ab > l − a − b. Then, we can further prove that: If 0 < a < l, 0 < b < 1 and 0 < c < 1, we can get (1 − a)(l − b)(1 − c) > l − a − b − c. On the basis of this analogy, we can induce the following general inequality:

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If 0 < a1 < 1, 0 < a2 < 1, . . . , 0 < an < 1, then (1 − a1 )(1 − a1 ) · · · (1 − an ) > 1 − a1 − a1 − · · · − an . 3. Induction based on an abstract analysis A general induction is not as simple as the above that are obtained through direct analogy, but requires a process of abstract analysis. Euler said: “The science of mathematics requires observation and experiment as well”. To this end, induction method is also known as “experiment induction method”. Many conjectures in mathematics are based on observation and experiments. Here, we present a typical example of the successful use of induction method to make a scientific discovery. The Euler’s theorem of convex polyhedrons refers to that the Euler characteristic of any convex polyhedron p is 2, that is, X ( p) = F + V − E = 2, where the F, V, and E represent the number of faces, vertexes, and edges of a convex polyhedron respectively (corresponding to the first letter of Face, Vertex, and Edge respectively). Observe these special polyhedrons (Fig. 4.19), such as ➀ cube, ➁ triangular prism, ➂ pentagonal prism, ➃ rectangular pyramid, ➄ triangular pyramid, ➅ pentagonal pyramid, ➆ octahedron, ➇ tower top, ➈ truncated cube, etc., which are all listed in the table below (Table 4.1). A closer look at these data and comparing the relations therein may suggest the following two conjectures: Does the number of vertexes V increase with the number of faces F? But comparing a cube with an octahedron, we find that V > F for one, and V < F for the other. Does E increase with F or V ? But comparing a truncated cube with a tower top, we find that E increases from 15 to 16, while V decreases from 10 to 9; and comparing an octahedron and a pentagonal prism, we find that E increases from 12 to 15, while F decreases from 8 to 7. So neither F or V consistently increases with E. The above shows that our attempt to establish a consistent regularity has failed. But we are not reconciled to admit that our original idea was completely wrong. Through further observations, we find that while neither F nor V consistently increases with E, it seems that the “general trend” is to increase. When we examine these arranged data, it is obvious that F + V continuously increases. Just find the F + V of each polyhedron and compare it with E value, and we are surprised at the results because all the nine cases in the table satisfy the relational expression F +V = E +2

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Fig. 4.19 .

Table 4.1 Quantitative relation of faces, vertexes, and edges of polyhedrons No

Polyhedron

Faces (F)

➀

Cube

6

Vertexes (V )

➁

Triangular prism

5

6

9

➂

Pentagonal prism

7

10

15

➃

Rectangular pyramid

5

5

8

➄

Triangular pyramid

4

4

6

➅

Pentagonal pyramid

6

6

10

➆

Octahedron

8

6

12

➇

Tower top

9

9

16

➈

Truncated cube

7

10

15

8

Edges (E) 12

So, we derive a conjecture that for any polyhedron, the number of faces plus the number of vertexes is equal to the number of edges plus 2. To support this conjecture, we may as well examine some new examples to enhance confidence. For example, examine an icosahedron and a dodecahedron. It is not hard to see that they all satisfy the law of F + V = E + 2. 4. Analogy and induction are important methods of mathematical discovery

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In scientific research, people tend to, first through the observation and analysis of things, put forward a reasonable speculation using analogy and induction. This speculation may be a specific conclusion, or a specific way to solve the problems. Therefore, analogy and induction have become important methods of mathematical discovery. Polya thought highly of this approach, calling it plausible reasoning specifically, which is a quasi-logical form. However, speculation may not be reliable and correct. Only through demonstration can the possible results and possible solutions be tested and improved until the final reliable solution is obtained. Since the mathematical research activity consists of two processes of discovery and demonstration, it is clear that the general mode of mathematical research activities is shown in Fig. 4.20. However, as mentioned in the previous analysis, the process of demonstration and discovery is often interdependent and interpenetrative in practical mathematical research; and more often than not what is finally obtained is not the original speculation, but improved conclusion. Therefore, the research activity of mathematics is a process that contains repeated discoveries, speculations, and repeated demonstrations, which are shown in Fig. 4.21. 5. Reunderstanding of analogy and induction (1) Relationship between analogy, induction, and reduction As mentioned earlier, analogy and induction are a kind of methods to draw new conjectures from known facts through comparison, namely they are generalization and development from the known to the unknown. Therefore, in this sense, analogy method, induction method, and reduction method are directly opposite, because the reduction method emphasizes the transformation from the unknown to the known, and as far as its final representation is concerned, it clearly falls under the category of demonstration. But in practical mathematical research, analogy method, induction method, and reduction method are often interdependent and interpenetrative. For example, analogy and induction often indicate possible directions for the realization of reduction, while the reduction provides the necessary proof for the conjecture

Fig. 4.20 .

Fig. 4.21 .

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obtained through analogy and induction. Therefore, reduction and conjecture actually constitute two aspects of a contradiction. From the perspective of connection, analogy method, induction method and reduction method have more obvious points in common: what they emphasize is the connection of things, that is, we are required to look at problems with an interconnected point of view, not only being good at exploring the unknown from the known, but also being good at obtaining the known from the unknown. Therefore, from a methodological perspective, we should also pay attention to the following: First, when solving the problems, attention should be paid to the analysis of relevant problems. As stated by Descartes, “when a problem arises, we should be able to think about whether to test some other problems first would be beneficial, and think about what other problems should be tested and in what order”. Second, through the analogy and inductive analysis of the relevant problems, the direction of solving the problem can be clear. For example, the reduction from the unknown to the known, from difficulty to easiness, from tediousness to conciseness; analogy of plane and space, number and shape, finity and infinity, and so on, they can actually be regarded as guidance on indicating the direction of solving the problem. Thirdly, after a certain problem is solved, attention should be paid to analyze whether it can lead to new results, that is, whether the results obtained can be further developed and promoted. (2) Striving to improve the level of speculation and demonstration For mathematical research, we should learn how to speculate, and how to demonstrate as well, and constantly strive to improve the level of speculation and demonstration. Therefore, in the mathematics textbooks and teaching in secondary schools and colleges, the principle of induction–deduction interaction should be adopted. We should not only teach students to learn how to use scientific induction to guess conclusions, conditions, theorems, and demonstration methods, but also enable them to learn how to transit from exploratory deduction to a pure form of deduction, and to build all the proof of the foreseeable reasonable propositions or theorems on the basis of logical deduction. Secondly, we should also have a dialectical understanding of the rigor of mathematics. As is known, mathematics is based on rigorous demonstrations. However, mathematical research is not always so rigorous, because only through bold exploration and imagination can we find the truth; and only through repeated experiments and improvement can we finally find the right way to solve the problems. Thus, rigorous logical analysis must be supplemented by bold exploration and attempt. In other words, in mathematical research, we should strive to be rigorous, but not to be bound by the requirement of rigor. Finally, as stated by Polia, we should adopt “cautious optimism” in mathematical research, that is, we should have confidence in our ability, make bold speculations and demonstrate them with optimism; but we should not be credulous, and should be ready to correct any of our beliefs. In short, “sticking to truth and always being optimistic” is the attitude an excellent mathematician should have.

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4.3.3 Association Method 1. Meaning of mathematical association The so-called association refers to a kind of conscious and purposeful thinking activity, and it is the psychological activity of thinking of another thing from the things currently perceived or thought about, or then imagining other things therefrom. Mathematical association is a thinking method connecting with existing knowledge, skills, and experience for imagination based on observation and according to the characteristics of the objects or problems under research. It is a reproducible imagination, and the basis of analogy, simulation, induction, conjecture, and plausible reasonings. In imaginal thinking, people think of the representation of another thing from the representation of one thing, and then form a new image and even an image system. All these should be realized through association. In abstract thinking, the concept is formed through the thinking of several things of the same type “from one point to another, and from the outside to the inside” (and analysis and comparison, which can also be broken down into several specific association steps), and the remolding of “refining and sorting out”. Judgement is formed by the relationship between concepts, and inference is formed by the relationship between judgements, and all these should be realized through association. Association is a regular thinking activity. No matter how complex the thinking process and how large the span in between each step is, it is composed of specific and different types of associations, which is independent of whether people’s thinking is consciously aware of this. The purpose of researching association is to enable people to consciously strengthen the awareness of association, follow the rules of association for thinking, so as to improve the efficiency and quality of thinking. The problem-solving strategies formulated by Polya, a famous mathematics educator, in the book How to Solve It are high generalization of the application of mathematical association and has become a model of the world. 2. Classification of mathematical associations (1) Meaning association The so-called meaning association refers to the association of their specific meanings aroused by the concepts reflected by the conditions and conclusions in the questions. All concepts comprehended are related to a certain meaning in thinking, and this thinking relation forms a thinking channel, which is conducive to the emergence of association. If this meaning is also related to the specific method of problem solving, then the problem can be readily solved. Example 4 Judge the relationship between set A and set B: A = {x||a − b| < x < a + b, a > 0, b > 0, a = b}

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  B = {x| |a 2 − b2 | < a 2 + b2 , a > 0, b > 0, a = b} Solution Based on the geometric meaning of inequalities, we naturally associate the intervals on the axis with the set of values of x determined by the inequalities, so we think that the problem can be transformed into the relationship  between open

judging   √ 2 2 2 2   a − b , a + b , the concept of interval (|a − b|, a + b) and open interval this “relationship” is manifested as (is further associated with) inclusion, overlapping, or disjoint. From the specific meaning of this relationship, we further associate that the value size of the interval endpoints should be compared, which further leads to the association of the differencing method, so that the following series of problemsolving processes occur. √ First compare the values of a + b and a 2 + b2 : Since a > 0, b > 0, (a + b)2 − (a2 + b2 ) = 2ab > 0, √ therefore 0 < a2 + b2 < (a + b)2 , 0 < a 2 + b2 < a + b √ Then compare the values of |a − b| and a 2 + b2 : Since a > 0, b > 0, |a − b|2 − |a2 − b2 | = |a − b| (|a − b| − |a + b|) < |a − b| (a − b| − |a − b|) = 0,

√ therefore 0 < |a − b|2 < |a2 − b2 |, 0 < |a − b| < a 2 + b2  √ then, (|a − b|, a + b) ⊃ ( |a 2 − b2 |, a 2 + b2 ) Therefore, A ⊃ B. (2) Goal association The so-called goal association is that goals become a thinking orientation that stimulates thinking to pursue it when people research problems. Therefore, goals can spur thinking to search extensively in the mind, thus leading in the related associations needed to achieve the goal. The so-called related associations here are those in subjective consciousness caused by parts (or whole) of the problem. Whether they really meet the objective needs of achieving the goal should also be subject to the practical test. Of course, the objects involved in the goal can only arouse the goal awareness in the thinking under the premise of being understood, so the goal association is often intertwined with the meaning association. Example 5 It is known that the quadratic equation with real coefficients x 2 + ax + b = 0 has two real roots α and β. If |α| < 2, |β| < 2, prove: 2|a| < 4 + b and |b| < 4. Proof From it has real roots α and β, we associate its meaning, conditions, and the relationship between two roots.

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From |α| < 2, |β| < 2 and equal conditions in formula ➃, and inspired by the goal awareness of proving |b| < 4, we conduct an extensive thinking search in the mind (often known as repeated thinking), and the useful associations constitute the following train of thought (i.e., something obtained from organizing the thinking process). Apparently, |b| = |αβ| = || · |β| < 2 · 2 = 4. To prove the conclusion of 2|a| < 4 + b (goal consciousness), we use the conditions |α| < 2 and |β| < 2 (also a kind of goal consciousness) to associate that ➁ should give specific content to α and β, so we associate (meaning association) that we might as well let √ √ −a +  −a −  a= , β= . 2 2 The following series of transformation are gradually caused further under the guidance of goal consciousness:  √  |a| < 2 −2 < a < 2 −2 < −a+2√ < 2 ⇒ ⇒ |β| < 2 −2 < β < 2 −2 < −a−2  < 2 √   0 ≤ √ < 4 + a 8a > −4(4 + b) ⇒ ⇒ ⇒ 2|a| < 4 + b. 0≤  1 − x. Analysis From a question of numbers, we associate the mathematical thought and method of number–shape combination, and further associate it with using functional expressions and graphs. So let f 1 (x) =



2 − x 2 , f 1 (x) = 1 − x,

The original inequality is transformed into f 1 (x) > f 2 (x), During this process of association, in general, it is a stylized association, but in terms of its content, it is inseparable from meaning association, goal association, and resemblance association. With these associations, the thinking content is gradually concretized. Based on Fig. 4.23, then we get the solution set 

 √ √ 1− 3 0), which is the locus equation of point P.

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4.3.5 Aesthetic Method 1. Mathematical beauty As once asserted by Laplace, an ancient philosopher and mathematician, “where there is numbers, there is beauty”. As a science of beauty, mathematics not only has a strong aesthetic charm, but also has its definite content. The basic characteristics of mathematical beauty are simplicity, symmetry, unity, and singularity. ➀ Beauty of simplicity, also known as beauty of concision, is an important symbol of mathematical beauty. The forte of mathematical theory is that it can reveal the regularities of the quantities and their relationship in the real world in the simplest way. Poincare once said, “simplicity is a beauty”. We frequently say that “this solution is very beautiful”, and what is called “beauty” here often includes the meaning of simplicity. In this regard, Diderot once pointed out: “the so-called beautiful answer in mathematics refers to a simple answer to a difficult and complex question”. As pointed out by von Neumann, “people require a mathematical theorem or mathematical theory can not only use simple and beautiful method to describe and classify a large number of individual cases without connections with each other congenitally, but also expect it to be beautiful in architectural structure”. For example, for mathematical expression ρ = 1−eepcos θ , it contains fewer symbols and simple structure, being striking and intuitive, but it summarizes five curve equations: straight line, circle, ellipse, hyperbola, and parabola, which reflects the simplicity of mathematical relationships. For another example, the relativistic formula E = mc2 is so clear and concise, it has guided people to do experiments for years, and opened up new source of energy—nuclear energy. The simplicity of mathematics is fully manifested in the application of mathematical symbols, abstract form of mathematics and abstract methods, methods and skills of solving problems, so it is safe to say that the whole mathematics is full of simplicity. ➁ Beauty of symmetry is also called well-proportioned beauty. Symmetry is one of the commonalities of all things in nature. Symmetric patterns and symmetrical buildings are seen everywhere. Symmetry is used in paintings, it is also used in literary works, and the beauty of symmetry in mathematics is one of its remarkable features. In geometric figures, there are so-called point symmetry, line symmetry, and plane symmetry. A sphere is point-symmetric, line-symmetric, and plane-symmetric. Pythagoras believed: “The most beautiful in all solid figures is the sphere, and the most beautiful in all plane figures is the circle”. The reasons for this acclaim may be mainly based on the symmetry and uniformity of the sphere and the circle. The binomial expansion also shows a beauty of symmetry:

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125

(a + b)n = Cn0 a n + Cn1 a n−1 b + Cn1 a n−2 b2 + · · · + Cnn−2 a 2 bn−1 + Cnn−1 abn−1 + Cn0 bn . There is a symmetry between many reciprocal operations. As an example, we give two formulas of logarithmic and exponential operations, where exp x represents ex . log exp

 i  i

xi = xi =

 i 

log xi , exp xi .

i

➂ Beauty of unity is also known as beauty of harmony. In mathematics, different parts, and parts and whole often pursue harmonious unity. i is the simplest virtual number, and it is a virtual unit, just like 1 is a real unit; 1 and i are two simplest numbers. On the other hand, there were two numbers (π and e) that have been elusive to people for a long time, and people didn’t realize that they were transcendental numbers until the second half of the nineteenth century, so they are, of course, the most complex numbers. These two simplest numbers 1 and i, and two most complex numbers π and e, seem to be unrelated somehow or other, but these four numbers can be unified in one formula: e xi + 1 = 0. People are amazed at the unity of mathematics. Unity is indeed one of the important features of mathematical beauty, and it is also one of the goals to be pursued by mathematics. People always try to link beauty of simplicity to beauty of unity. For example, the simplest addition is a case in point. After the introduction of negative numbers, subtraction can be unified into addition. After the introduction of reciprocals, division can be unified into multiplication. After the introduction of logarithms, multiplication and division can be unified into addition. So, with logarithms, power, extraction of root, etc. can be reduced to multiplication, and finally to addition. Even the calculus approximation calculation, approximate solution of differential equations, etc., can be reduced to addition. It turns out that there is also unity among so many operations above, which are unified into the simplest addition under certain conditions. ➃ Beauty of singularity is also known as beauty of peculiarity. Symmetry, uniformity, and unity all reflect the coordination and harmony of mathematics. But if there is nothing more, mathematics is bound to be monotonous. Mathematics can develop only when singularity appears, people understand singularity, and seek new harmony and unity. So the complete beauty of this picture of mathematics

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undoubtedly includes its singularity. Bacon once said, “None of magnificent things has not some singularities in harmony!” As we all know, when irrational numbers and imaginary numbers emerged initially, people were all amazed and felt incomprehensible. Of course, they are beyond reproach today. In the seventeenth century, Fermat guessed that all numbers n like 22 +1 were prime numbers, and by the eighteenth century, Euler pointed out that n when n = 5, 22 + 1 was not a prime number. This proved that Fermat’s conjecture n was wrong. So far, people seemed to completely forget numbers like 22 + 1. But a miracle occurred in 1796, when Gauss, aged 19, proved that the regular heptadecagon n could be constructed with ruler and compass, but seventeen was 22 + 1! Before the epoch, people could construct regular triangles and regular pentagons, but the attempt to construct regular heptagons and regular hendecagons with ruler and compass failed. In 1796, Gauss constructed a regular heptadecagon. What a singular and shocking event! Greatly inspired by this singular discovery, Gauss chose mathematics as his lifelong career. With simplicity, symmetry, unity, combined with novelty, this beautiful picture will be more radiant. 2. Mathematical beauty is an important method of mathematical discovery We have studied imagination and intuition earlier. Intuition is direct perception, a way of thinking and a sudden vision in the mind, also known as inspiration. Archimedes figured out the crown mystery, Gauss found the sum of 1 + 2 + 3 + · · · + 100 very quickly, and Newton discovered universal gravitation through the fall of an apple, these are three famous examples of intuition. Adama said: “Invention is choice, but choice is only dominated by aesthetic sense of science”, Poincaré said: “Without a highly developed intuition of beauty, one cannot be a great mathematical inventor”. Here, we illustrate the application of mathematical beauty through several examples. Example 11 The sum of the coefficients of various terms in the expansion of (2x − y)100 is ( ). (A) 1024; (B) 512; (C) 8; (D)1 Solution Let (2x − y)100 =

100

i=1 ai j x i yi . 100 Actually we only need to find i=1 ai j . 100 Just let x = y = 1, then i=1 ai j = 1.

So the answer is (D). This is the application of beauty of simplicity in mathematics. Example 12 In an arithmetic sequence, a6 + a9 + a12 + a15 = 20, then S 20 = ( ).

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127

(A) 200; (B) 75; (C) 30; (D) 100 Solution Since a6 + a15 = a9 + a12 = 10, then S20 = (a1 + a10 ) + (a2 + a19 ) + · · · + (a10 + a11 ) = 100, therefore, the answer is (D). We discover the relations through observation here. It is the application of the beauty of symmetry in mathematics. Example 13 It is known that f (x) is a rational function and f (2x) + f (3x + l) = 13x 2 + 6x + l, and then f [ f (x)] = ( ). (A) x 2 − 2x + l; (B) x 2 − l; (C) x 4 ; (D) x 2 − 2x Solution Since f (x) is not less than quadratic, then f [ f (x)] is not less than biquadratic. Therefore, the answer is (C). We discover the relations through analysis here. It is the application of the beauty of unity in mathematics. Example 14 In the known  ABC, prove: a z + bz + cz cos A cos B cos C = + + . 2abc a b c Analysis The relationship between the left side and the right side is not coordinated as the left side is about the relationship between sides while the right side is about the relationship between angles. To this end, we can harmonize the right side with the left side, or vice versa, which is not difficult to do. From cos A cos B cos C 1 b2 + c2 − a 2 1 a 2 + c2 − b2 + + = · + · a b c a 2bc b 2ac 1 a 2 + b2 − c2 + · c 2ab a 2 + b2 + c2 = 2abc The proposition is thus proved. Finally, it should be noted that when mathematical beauty is employed to solve problems, people often resort to a kind of “consummation”. Through consummation, the incomplete is made to be complete, ununity to unity, asymmetry to symmetry, which is conducive to the solution of problems. For example, find the regularity of series 34 , 23 , 58 , 35 , . . . Through consummation, it is transformed into:

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Fig. 4.25 .

3 4 5 6 , , , ,... 4 6 8 10 Thus the form becomes harmonious, and of course its regularity is found. sin x For Another example, when finding the value of lim x2 , first transform into

1 2

·

sin x 2 x

x→0

sin x 2 x

which becomes harmonious, and it is easy to get sin x sin x 1 1 2 = lim x 2 = x→0 x 2 x2 →0 2 2 lim

Example 15 As shown in Fig. 4.25, there are four points P, A, B, and C on the sphere with radius of R, and PA, PB and PC are perpendicular to each other, PA = PB = PC = x. Try to find the value of x. Analysis: At the first glance, this question is difficult to get under way, but through observation, we find the line segments in the figure is incomplete, and the figure feels incomplete. If we employ “consummation” method to form a cube with PA, PB, and PC as the edges, thus forming a complete figure. We can prove that this cube√is an inscribed cube, its diagonal length is 2R, and it is not difficult to obtain x = 23 3R.

4.4 Demonstration Methods The demonstration method in mathematics is an important part of mathematical methodology. As a matter of fact, after a practical problem forms a mathematical problem after analysis, abstraction and generalization, to seek for its laws, people make conjectures through association. Only after demonstration can its laws be determined. At this time the demonstration is of great significance. As for proof methods, the common methods include analysis or synthesis methods; direct proof or indirect proof methods; and calculation proof method as well. The first two types of methods have been introduced in Chap. 2. Therefore, this section mainly focuses on calculation proof method. For some propositions, especially those about geometric proof questions, if after observation and analysis, you still have no problem-solving idea, then you may as well

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129

achieve the purpose of proof through calculation. This method of using calculation for proving is called calculation proof method. The calculation proof method includes algebraic method, trigonometric method, coordinate method, complex number method, vector method, etc. For these proof methods, the train of thought is simple, and easy to get under way, saving too many auxiliary lines, or even requiring no auxiliary line. 1. Algebraic method When studying some geometric problems of metric relations, represent the relevant line segments, angles, and areas with unknown numbers, establish corresponding relational expressions according to known conditions, and then find the answer through identical transformation or solution of equations in algebra. This method that does not rely on the coordinate system and uses algebraic knowledge to research geometric problems is called algebraic method. Example 1 In ▱ABCD, ∠A is an acute angle, and AC2 · BD2 = AB4 + AD4 . Prove: ∠A = 45°. Proof As shown in Fig. 4.26, let AB = a, AD = b, AC = m, BD = n, then m 2 + n 2 = 2(a 2 + b2 ) and m 2 · n 2 = a 4 + b4 , Based on the relationship between the coefficients and roots of quadratic equations with one unknown, m2 and n2 are two 2 2 2 4 4 roots of equation √x − 2(a + b )x + a + b = 0. Solve this equation, and we get 2 2 x = a + b ± 2ab. While ∠ A is an acute angle, x 16 + x −16 = (x 8 + x −8 )2 − 2 = 22072 − 2 = 4870847;. √ Then, based on the cosine theorem, n 2 = a 2 + b2 − 2ab cos A, so cos A = 22 , ∠ A = 45°. 2. Trigonometric method When studying some algebraic or geometric problems, trigonometric knowledge can be employed to obtain answers, which is called trigonometric method. The trigonometric method used to solve algebraic problems is mainly trigonometric substitution method; for geometric problems, the relationship between line segments and angles are mainly transformed into trigonometric function relationship, and the geometric problems are solved through trigonometric identical transformation, solving trigonometric equations or proving trigonometric inequalities. The solution of Example 9 in Chap. 2 is an example of trigonometric method. Now see another example. Fig. 4.26 .

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Fig. 4.27 .

Example 2 In ABC, it is known that ∠A = 60°, a = 1 (Fig. 4.27). Prove: b + c ≤ 2. Proof This is a geometric inequality problem. We can apply sine theorem or cosine theorem to establish the expression of b + c. By the sine theorem, we get

a sin A

=

b sin B

=

c sin C

=

b+c sin B+sin C

1 (sin B + sin C) ◦ sin √60 4 3 B +C B −C = sin · cos 3 2 2 √ √ 3 4 3 B −C = · · cos 3 2 2 B −C ≤2 = 2 cos 2

b+c =

The equality is established only when ∠ B = ∠ C, that is, b = c. Or by the cosine theorem, we get b2 + c2 − 2bc cos 60◦ = a 2 = 1 1 + bc = b2 + c2 ≥ 2bc, so bc ≤ 1. Thus, (b + c)2 = b2 + c2 + 2bc = 1 + 3bc ≤ 4, that is, b + c ≤ 2. The equality is established only when b = c. 3. Coordinate method After establishing the coordinate system and setting the coordinates of the points on the figure given and curvilinear equations, convert geometric problems into algebraic problems, then apply algebraic knowledge for solution, and then give geometric meaning, so as to get the answer to geometric problems. This method is called coordinate method. There are three main steps to solve geometric problems using the coordinate method: The first step, select an appropriate coordinate system to determine the coordinates of key points and the equations of key curves on the principle of facilitating

4.4 Demonstration Methods

131

calculation. For example, you can select the symmetry axis of the figure, intersecting straight lines and other special lines as the coordinate axis, and the endpoint, center of fixed segment, symmetry center of symmetric figure or other special points as the origin. The second step, set the coordinates of the points and equation of the curve. Generally, try your best to reduce the parameters as far as possible, and make the structure of the curvilinear equation as simple as possible, to facilitate calculation and reasoning. The third step, computation and reasoning. For this purpose, in addition to making clear related concepts and being familiar with related formulas and equations, attention should be paid to using the geometric properties of the figures given. The solution of Example 3 in Sect. 3.3 is the coordinate method. Here is another example. To facilitate calculation, a line segment in the figure is taken as unit length. Example 3 Four squares are constructed outward with each side of quadrilateral ABCD as the side length, and M, N, P, and L are the centers of these squares. Prove: PM⊥NL. Proof We adopt the coordinate method. As shown in Fig. 4.28, select rectangular coordinate

a+b a−b  system, let A (a, 0), B (b, 0), C (c, d) and D (0, l), then AB = b − a, M , 2 . 2 By I∼ =I , we get BF =EC = d, FG = EB = b − c, hence G(b + d, b − c), and ); , b−c+d we have N b+c+d 2 2 

; , b−c+d Similarly, by II∼ =II , we get H (c − d + l, c + d), P c−d+1 2 2 1−a By III∼ , , =III , we get R (a − 1, −a), L a−1 2 2 · b−c+d−1+a = −1. Since kPM · kNL = c+d+1−a+b c−d+1−a−b b+c+d−a+1 Therefore, PM ⊥ NL. 4. Complex number method The way to apply complex number knowledge to solve mathematical problems is usually called the complex number method. This is a special algebraic method. The basic idea of using the complex number method to solve geometric problems is to draw the complex plane based on the characteristics of the problem, select the corresponding complex number representation, convert the geometric problem into Fig. 4.28 .

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Fig. 4.29 .

complex number problem according to the known conditions, and solve the problem through calculation and reasoning. Example 4 In rectangular ABCD, AB = 1/3 BC, E and F are trisection points of segment AD. Prove: ∠AEB + ∠AFB + ∠ADB = 90°. Proof As shown in Fig. 4.29, establish the complex plane and let AB = 1, then z 1 = 1 + i, z 2 = 2 + i, z 3 = 3 + i; z 1 · z 2 · z 3 = (1 + i)(2 + i)(3 + i) = 10i, so ∠AEB + ∠AFB + ∠ADB = arg Z 1 + arg Z 2 + arg Z 3 = arg(Z 1 Z 2 Z 3 ) = arg(10i) = 90◦ 5. Vector method The so-called vector method is a method to transform the geometric problem into a corresponding vector algebra problem, and then apply the knowledge of vector algebra to solve the problem. This method is often simple and effective for parallel, vertical, concurrent, collinear, cross-ratio, and other problems in geometry. Example 5 As shown in Fig. 4.30, in tetrahedron ABCD, AB ⊥ CD, AD ⊥ BC. Prove: AC ⊥ BD. Fig. 4.30 .

4.5 Test Method

133

−→ −→ − → −→ Proof To simplify representations, let AB = b, AC = c, AD = d, then BC = −→ −→ c − d, C D = d − c, D B = b − d. −→ −→ Since AB ⊥ CD, AB · CD = 0., that is, b·(d − c) = 0. Therefore, b · d = b · c ➀ Similarly, by AD ⊥ BC, we get d · c = d · b ➁ Compare formulas ➀ and ➁, we have c · d = c · b, c·(d − b) = 0, − → −→ that is, AC · BD = 0. Therefore, AC ⊥ BD.

4.5 Test Method As stated by G. Polya, a famous mathematics educator, “Mathematics is not only a science of demonstration, but also a science of experimentation”. The observation, analysis, analogy, induction, specialization, and other methods that we have grasped are actually test methods, though not made clear at that time. In learning and studying mathematics, people often first carry out mathematical tests purposefully and systematically to obtain useful information and provide necessary basis for analyzing and solving problems before they enter the stage of exploration, discussion, and research. This section starts with the basic idea of mathematical tests and focuses on some applications of mathematical test methods in mathematical conjecture and mathematical problem solving.

4.5.1 Basic Idea of Mathematical Test Method Mathematical test method originates from the elementary number theory. Its basic idea is to determine the test plan according to the known conditions, eliminate the false and retain the true through item-by-item test, and find the answer to the original question. The thinking process of researching problems by means of mathematical test method is as shown in Fig. 4.31.

Fig. 4.31 .

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Any test is closely linked to observation. Observation is the premise of test, test is the confirmation and development of observation, observation is based on test, and test cannot be separated from observation. Therefore, to be good at using test methods, one must be diligent in observation, good at observation, and carry out purposeful and planned observations carefully. Example 1 Try to find the prime numbers within 1–100. Analysis To find the prime numbers within 1–100, we need to test each natural number between 2 and 100. To facilitate testing, the definitions of prime number and composite number can be used to eliminate the composite numbers from 2 regularly, so as to find the prime numbers as soon as possible. Solution Write out all the natural numbers within 2–100. The first number 2 is a prime number, keep it; cross-out all multiples of 2 after 2; the first remaining number after 2 is 3, which is a prime number, keep it; then cross-out all multiples of 3 after 3; the first remaining number after 3 is 5, which is not a multiple of any number before it, so it is also a prime number, keep it; then cross-out all multiples of 5 after 5; in this way, after limited tests, the remaining numbers constitute the table of prime number within 100, as shown in Fig. 4.32. Example 1 is a typical example of researching problems with test method. The method used is first proposed by ancient Greek mathematician Eratosthenes, so this method is often known as sieve of Eratosthenes. As a practice, readers can continue to make table of prime numbers within 200, 500, 1000, etc. using sieve of Eratosthenes. Fig. 4.32 .

4.5 Test Method

135

4.5.2 Test and Conjecture While studying problems with tests, each test can provide people with certain information. So, for some structurally complex mathematical problems, if we cannot find an appropriate solution for the time being, we might as well use the basic idea of test method, arrange several expedient tests first, then examine the test results with incomplete induction, explore the internal relations between conditions and problem or conditions and conclusion, and guess the direction of problem-solving. Example 2 Let series {an } be an = 3m(3m + 1). Prove: Each item of {an } can be expressed as a product of two adjacent integers. Analysis This is a proof question about the properties of general term of series. The key is to understand the specific structure of two adjacent integers. Therefore, we can use the basic idea of test method, first examine several terms at the beginning of the series, to provide useful conjectures for further solution. We have a1 = 12 = 3 × 4, a2 = 1122 = 33 × 34, a3 = 111222 = 333 × 334, a4 = 11112222 = 3333 × 3334, ... Accordingly, with incomplete induction, we guess that for an, there may exist · · · 1 22 · · · 2 = 33 · · · 3 × 33 · · 34 an = 11        · n

n

(n−1)

n

In this way, with this as the goal, the proposition can be proved. Proof Examine the general term of the series, we get an = 11 · · · 1 22 · · · 2     n

n

= 11 · · · 1 ×10n + 11 · · · 1 ×2     (n−1)

n

= 11 · · · 1 ×(10 + 2)   n

n

But · · · 9 +3 10n + 2 = 99   n

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= 3 × (11 · · · 1 ×3 + 1)    (n−1)

= 11 · · · 1 ×(10n + 2)   n

Let m = 11 · · · 1 ∈ N+ , then an = 3m(3m + 1).   n

Here, 3m and 3m + 1 are two adjacent integers. Thus the proposition is proved. Example 3 Suppose x ∈ R and x 2 − 3x + 1 = 0. Try to determine the single-digit n n number of x 2 + x −2 (n ∈ N+ ). Analysis This question is actually to determine the single-digit number of x 2 + x −2 for a different positive integer n. For easy examination, tests can be arranged for n = 1, 2, 3, 4. According to the known conditions, x 2 − 3x + 1 = 0, apparently x = 0. After both sides are divided by x, we get x + x −1 = 3. n

n

When n = 1, x 2 + x −2 = (x + x −1 )2 − 2 = 32 − 2 = 7; When n = 2, x 4 + x −4 = (x 2 + x −2 )2 − 2 = 72 − 2 = 47; When n = 3, x 8 + x −8 = (x 4 + x −4 )2 − 2 = 472 − 2 = 2227; When n = 4, x 16 + x −16 = (x 8 + x −8 )2 − 2 = 22072 − 2 = 4,870,847; n n From the above tests, we can conjecture that the single-digit number of x z + x −z may be 7 for any positive integer n, so we may consider proving the conjecture obtained by means of mathematical induction. Proof The single-digit number of x z + x −z is constantly 7 for any positive integer n. n

n

It is to be proved by means of mathematical induction as follows. (1) Laying a foundation. When n = 1, as calculated in the above analysis, the conclusion is clearly tenable. (2) Induction. Assume the conclusion holds when n = k, that is, the single-digit k k number of x z + x −z is 7, then when n = k + 1, we have x2

k+1

+ x −2

k+1

= (x 2 + x −2 )2 − 2 k

k

Based on the induction, assume the single-digit number of x 2 + x −2 is 7, k k so the single-digit number of (x 2 + x −2 )2 is 9, thus the single-digit number of k k k+1 k+1 (x 2 + x −2 )2 − 2 that is, x 2 + x −2 must be 7. In other words, when n = k + 1, the conclusion is also true. Therefore, the conclusion is true for an arbitrary natural number n. k

k

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137

4.5.3 Non-standard Problems The so-called non-standard problems usually refer to the mathematical problems that lacks conventional logical connection between the conditions and the problem, also known as unconventional problems. Generally, it is difficult to use conventional thinking methods directly to solve such kind of problems, while it is often likely to achieve success to explore problem-solving ideas by means of test method. Example 4 Restore the natural numbers of the dividend, divisor, and quotient (taking × as the natural number from 0 to 9, and the first digit is not zero): 

M M2 → M Analysis The dividend in question is a six-digit number, with 9 × 105 possibilities; the divisor is a three-digit number, with 9 × 102 possibilities. If combined, we need to conduct 81 × 107 tests, which is like looking for a needle in the ocean. However, if we notice the product of one-digit and k-digit number is a k-digit number or k + 1-digit number, the number of tests can be reduced, and the answer can be found as soon as possible. Solution For the sake of discussion, we number various lines in the expression. According to the known conditions, the divisor is a three-digit number, and the tens’ digit of the quotient is 8; if the product of a three-digit number and 8 is a threedigit number, we can try multiplying 1×× and 2×× by 8, the product of the former may have three digits, while the product of the latter has four digits, so the divisor takes the form of 1 × ×. For easy calculation, the divisor and quotient are denoted by: 1ab, c8d. Since the numbers in lines 2 and 6 are four-digit numbers, and the number in line 4 is a three-digit number, then c > 8, d > 8. Therefore c = 9, d = 9, that is, the quotient is 989. Note that l ab × 8 is a three-digit number and 13b × 8 must have four digits, so a must be less than 3, in other words, a may be 0, 1 or 2. If a = 0, that is, the divisor is 10b, then 10b × 9 should be a four-digit number, which is obviously impossible. If a = 2, that is, the divisor is 12b, then 12b × 8 is a three-digit number with the first digit being 9; while the first digit in line 3 is up to 9; thus, the first digit in line 5 should be 0, which is contradictory with the known conditions. Thus, a cannot be 0 and 2, so only 1 can be taken, that is, the divisor is 11b. Note that 11b × 9 has four digits and 11b × 8 is a three-digit number with the first digit less than 9; try multiplying 111 × 9 = 999, which is a three-digit number,

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and 113 × 8 = 904 is a three-digit number with the first digit being 9. So the b must be 2, that is, the divisor is 112. Based on the above analysis, the divisor in question is 112, the quotient is 989, and the dividend is 112 × 989 = 110,768, meaning the expression is (1) (2)

110768 112 1008 989

(3)

996

(4)

8 96

(5)

1008

(6)

1008 0

Example 4 above is a reduction problem for numbers. When solving such problems by means of test method, make full use of related properties of positive integers, e.g., the sum (difference) of even numbers is an even number; the sum (difference) of odd numbers is an even number; the sum (difference) of an even number and an odd number is an odd number; the product of even numbers is an even number; the product of odd numbers is an odd number; the product of an even number and an odd number is an even number; the conditions for the product of k-digit number and single digit number is a k-digit number or k + 1-digit number; characteristics of positive integers, etc., to reduce the number of trials and improve test efficiency. Example 5 Suppose a, b, c, d ∈ N+ , a < b < c < d, and 1 1 1 1 + + + = 1. a b c d Find the value of a, b, c, d. Analysis This is a quaternary indefinite equation. The key of solving the problem by means of test method is to narrow down the test range, reduce the number of tests, and determine the test plan reasonably. To this end, we need to make full use of the known conditions to determine the value range of a, b, and c successively. Solution Based on the known conditions, we get 1 < a < b < c < d, that is, 1 1 1 > > > a b c

4.5 Test Method

139

then cos 48◦ + cos 24◦ − cos 12◦ − cos 84◦ Apparently, a4 > 1, so 1 < a < 4. If a = 3, then the minimum of b, c, d should be 4, 5, 6 respectively, then the 19 < 1, maximum of a1 < a1 + b1 + 1c + d1 is 13 + 41 + 15 + 16 = 20 This indicates that a isn’t 3, so only 2 can be taken; If a = 2, then b1 + 1c + d1 = 1 − 21 = 21 , Obviously, b3 > 21 , so 2 < b < 6. Examine various possibilities of b, we have 3 , (1) When b = 5, we have 1c + d1 = 21 − 15 = 10 So

2 c

>

3 , 10

5 < c < 7,

2 hence, c can only be 6. At this time, d1 = 1 − 21 − 15 − 16 = 15 , We can see that d is not a positive integer, so there is no solution in this case.

(2) When b = 4, we have 1 1 1 1 1 + = − = , c d 2 4 4 so 2 1 > , c 4

4 < c < 8,

Through simple calculation, we get: When c = 7, d has no positive integer solution; when c = 6, d = 12; when c = 5, d = 20; (3) When b = 3, we have

1 c

+

1 d

=

1 2

−

1 3

= 16 ,

so (4) 2 1 > , c 6

3 < c < 12.

Similarly: When c = ll, 6, 5, 4, there is no positive integer solution; when c = 10, d = 15; when c = 9, d = 18; when c = 8, d = 24; when c = 7, d = 42. To sum up, there are six sets of positive integer solutions for a, b, c, d that meet the set conditions: (2, 4, 6, 12), (2, 4, 5, 20), (2, 3, 10, 15),

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(2, 3, 9, 18), (2, 3, 8, 24), (2, 3, 7, 42). The above three examples show that the integer solution problem of indefinite equations is a dynamic topic. When solving it with the test method, the method is flexible and requires high skill. In usual practice, we should seek the optimal way to solve the problem through multiple comparisons.

4.6 Teaching of Secondary School Mathematics Thoughts and Methods 4.6.1 Secondary School Mathematics Thoughts There are mainly the following mathematical thoughts in secondary school mathematics. 1. Letter algebraic thought Replacing numbers with letters is the first mathematical thought that secondary school students have come to contact with, and it is also the most important and fundamental mathematical thought of elementary algebra and mathematics as a whole. Since the nineteenth century, algebra has developed into a general theory of formal operations. Algebra has been developing continuously with the extension of the letter meaning from number → vector → matrix → tensor → spinor → hypercomplex number and other quantities of various forms. In mathematics, numbers are replaced with letters, and the relationships among quantities, changes in quantities, and reasoning and calculations among quantities are expressed in symbolic forms (including numbers, letters, graphs and charts, and various specific symbols), namely a complete set of formal mathematical language. For example, using a vector to indicate the size and direction of a force, using point P (x, y, z to indicate the spatial position of an object, using S = f (t) to indicate the relationship between displacement and time, using “⊥” to indicate perpendicularity, “//” to indicate parallelism, “∈” to indicate belonging to, “∃” to indicate existence, b “∀” to indicate arbitrariness, “ a ” to indicate definite integral from a to b, etc. For mathematics itself, it is the extensive use of symbols (a universal international language) that facilitates the statement of problems, and expression of reasoning and quantitative calculations, greatly simplifies and accelerates the process of thinking, to make mathematics a dynamic running system. For example, the birth of computer language brought the epoch-making computing technology revolution. The “fourcolor problem” that needed the efforts of several generations to complete could be completed in more than a thousand hours of calculation; on the contrary, without this formal mathematical language, it would be difficult to describe even a simple natural law, let alone the research of inherent regularities of complex systems, qualitative and quantitative analysis.

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In short, it is the existence of mathematical language that makes mathematics a tool to describe the world and an important means to store and exchange information. Therefore, Engels once spoke highly of mathematics: “Mathematics: dialectical aid and expression”. People “symbolize” it and think of it as a golden bridge to the palace of science. 2. Decomposition and combination thought When the mathematical problems cannot be solved in a unified form, we can decompose the range involved in the known conditions into several subsets, study the local solutions to the problem in each subset, and then obtain the solution to the original problem by combining various local solutions. This is the decomposition and combination thought, whose method is called the classified discussion method. Decomposition and combination is one of the important mathematical thoughts. For complex calculation problems, construction problems, and demonstration problems, etc., the use of decomposition and combination thought to solve them can help people carry out comprehensive and rigorous thinking and analysis to obtain reasonable and effective ways to solve problems. In secondary school mathematics, decomposition and combination is an effective method of thought for equation problems, proof and solution of inequalities, judgement and proof of function monotonicity and various problems with parameters, etc. When using the decomposition and combination method to solve the problems, it is very important to divide the set of known conditions given in a scientific way. The rules of division should be followed to prevent repetition or omission in the decomposition. 3. Thought of reduction and transformation Reduction refers to transformation and reduction, which is a kind of thought to reduce outstanding or unsolved problems, through transformation process, to familiar normative problems or problems that have been solved to obtain the solution finally. We have discussed it in detail in Sect. 4.2 of this chapter. To achieve “reduction”, “substitution” (also known as transformation) is often resorted to in mathematics. In algebra, there are identical transformation of analytical formulas, equivalent transformation of equations and inequalities; in geometry, there are congruent transformation, similarity transformation, projective transformation, and isometric transformation; in analytic geometry, there are coordinate transformation, graphic transformation, etc. Transformation is a means, whose purpose is to reveal what remains unchanged therein. To explore the means of transformation for the unchanged purpose constitutes the idea and skill of solving problems. Generally speaking, when solving problems by means of transformation, the conditions of the problem can be transformed, its conclusion can also be transformed; the internal organizational structure of the problem can be transformed, its external representation can also be transformed; you can start with the local quantity, and you can also start with the overall quality. Thus, in a sense, transformation is closely related to reduction, but transformation is more flexible than reduction.

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4. Thought of set and correspondence After the introduction of letters, the research objects of mathematics expand continuously. Polynomials, determinants, equations, inequalities, linear transformations, and even events in probability theory, strategies in game theory, signals on computers and other forms of non-pure mathematical research objects are becoming more and more, forcing people to seek unified views and powerful means to deal with them, which is the ideological basis of the birth of set and correspondence theory. Set theory as a mathematical language is particularly simple, and it has only one basic verb “belong to”, denoted by “ ∈ ”, from which other concepts can be defined: “not belong to” (∈), / “include” (⊂), “be equal” (=), etc. With these concepts, coupled with logical languages such as “or” and “and”, we can define union operation (∪), intersection operation (∩) and difference operation, complement operation among sets. Thus, the basic operations of set theory are established, forming an algebraic structure. After the establishment of the set concept, the concepts that can only be expressed in daily language appear simple and clear, and can be denoted by unified symbols, which is more conducive to understanding and research. For example, the set of all even numbers can be written as: M = { x ∈ z|x = 2n, n = 0, ±1, ±2, . . .}(Z represents the set of integers) In general, M = {x ∈ z| p(x) } is used to represent set M comprised of those x with property p in Z. After the set theory is determined, and then through the relations and operations of ⊂ , = , ∪ , ∩ and so on, we can formally express many mathematical formulas and content with symbols. Correspondence is the most powerful research tool of set theory. Without correspondence, it would be difficult to “start” set theory, and even the simple “counting” number cannot be carried out, let alone modern mathematics. Set and correspondence is a very important basic idea of modern mathematics. At present, scattered in secondary school mathematics, it is relatively briefly introduced. But examples of its application can be seen everywhere, and with the deepening of reform of mathematics, its teaching status will be increasingly strengthened. The establishment of the concept of function mainly depends on correspondence, and transformation, a special case of correspondence, is more widely used in secondary school mathematics. Factorization is a kind of identical transformation, solving (system of) equations and (system of) inequalities is using equivalent transformation, and if we, from the point of view of set and correspondence, explain and understand permutation, combination, parameter equations, necessary and sufficient conditions and many other contents, they will be much clearer and deeper, conducive to students’ grasp of these knowledge, but also conducive to the flexible use of knowledge. 5. Equation and function thought

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“What is an equation” is a controversial question for a long time in the mathematics teaching literature. But the teaching practice has proved that even if the students do not know what an equation is, they know how to solve equations. Therefore, starting from the teaching reform, we can play down the concept of equation and do not have to spend too much energy and time on it. As argued by Stolyal, a Soviet mathematical educator: “These difficulties will be eliminated if we jump out of the box of classical understanding of mathematical objects and explain the general concept of equation with the concepts of modern logic, especially logic function (predicate)”. The thoughts of equation and function are important thoughts for dealing with mathematics of constants and variable mathematics and have important methodological significance in solving general mathematical problems. Equations and functions are very important part of secondary school mathematics, and a systematic research has been made for various equations and basic elementary functions. A rather complex problem can often be solved satisfactorily by simply looking for equivalent relations, formulating one or several equations (systems of equations) or functional relations. For example, the complex four operations application problems in arithmetic become much easier if solved by means of (systems of) equations; it is often easier to use extreme values of functions to find the distance between straight lines in different planes in geometry. In addition, inequalities and equalities also have many similar qualities. From the perspective of functions, numbers, formulas, and equations can be unified, which is the main basis of modern mathematics. Because the functions fully embodies the basic mathematical thoughts, such as set, correspondence and mapping, etc., enabling secondary school mathematics to approach the modern level of science of mathematics, and enabling the students to obtain basic, profound and useful thoughts and methods of advanced mathematics. 6. Shape–number combination thought Mathematics takes the quantitative relations and spatial forms in the real world as its object of research, while the shapes and numbers are interrelated and interconvertible. Converting the quantitative relation of a question into the property question of a figure, or the other way round, is a very important thinking strategy in mathematical activities. This idea of solving the problems is the basic method of thought of shape– number combination. Throughout the history of mathematics, the shape–number combination not only makes geometric problems obtain powerful algebraic tools, but also makes many algebraic problems distinctly intuitive, thus opening up new research directions. For example, through shape–number combination, Descartes converted geometric problems into algebraic problems, combined algebra and geometry that have been split up for a long time, and opened up a new era of mathematical development. Not only has the resulting analytical geometry become an immortal milestone in the history of mathematics, but his research is a glorious model of applying the thought of shape–number combination. For another example, in modern mathematics people regard a function as a “point” and the entirety of first-class functions as a “space”,

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thus leading to the concept of infinite dimensional space, which is also the result of successful application of shape–number combination thought. The solution of a system of differential equations can thus be reduced to a fixed point problem of geometric transformation in the corresponding function space, making the abstract analysis problem obtain intuitive geometric meaning. The shape–number combination is a very important thought in elementary mathematics and advanced mathematics, which has a unique strategic guidance and regulation role in solving mathematical problems. As specific methods of shape–number combination, the common methods to convert geometric problems into problems of quantitative relations are generally algebraic method, coordinate method, trigonometric method, complex number method, vector method; the common methods to convert problems of quantitative relations into problems of graphic properties are graphic method, graph method, and geometric method and so on.

4.6.2 Teaching of Mathematical Thoughts and Methods in Secondary Schools 1. Teaching principles of mathematical thoughts and methods in secondary schools Mathematical thoughts and methods have become an important part of mathematics teaching content in secondary schools. In this regard, we should follow the following principles. Objective-oriented principle. The teaching of mathematical thoughts and methods is one of the important teaching objectives in the new curriculum standard. On the basis of studying the curriculum standard and textbooks carefully, teachers should understand the importance of the teaching of mathematical thoughts and methods, make a comprehensive plan for the teaching of mathematical thoughts and methods, clarify the specific objectives of each stage, and put forward the specific plan for implementation. Principle of permeability. The presentation of knowledge and skills in high school mathematics textbooks is usually a clear line, while the introduction of mathematical thoughts and methods is a dark line, which permeates in the corresponding mathematical knowledge and skills. It requires teachers to combine the teaching practice, be good at digging, revealing, and summarizing the mathematical thoughts and methods therein, to enable students to understand the significance of mathematical thoughts and methods. Principle of generalization. As mathematical thoughts and methods are contained and integrated in the mathematical knowledge system, to enable students to understand and master mathematical thoughts and methods in order to develop the ability to solve problems, teachers must, aiming at the teaching practice, actively guide students to participate in the process of refining and generalizing mathematical thoughts and

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methods purposefully and systematically in order to reveal the essence of mathematical thoughts and methods, and make clear the connection between mathematical thoughts and methods and mathematical knowledge. Principle of practicality. The development level of students’ mathematical thoughts and methods ultimately depends on the process of students’ participation in mathematical activities themselves. In teaching, teachers should strive to create a teaching atmosphere, provide materials and opportunities for students’ thinking activities, guide students to actively participate in the practice of mathematics activities to be edified and experience the fun of application in practice, and develop a good habit of pondering problems. Only in this way can the learning of mathematical thoughts and methods be truly effective. 2. Approaches to the teaching of mathematical thoughts and methods in secondary schools (1) Permeate and experience mathematical thoughts and methods during the process of knowledge acquisition The occurrence process of mathematical knowledge is actually the occurrence process of mathematical thoughts and methods. Therefore, in the process of concept formation, derivation of conclusions, thinking about methods, revealing the rules and discovery of problems, etc., they are all excellent opportunities to penetrate mathematical thoughts and methods into students and carry on thinking training. To this end, in the teaching of concepts, it should be conducted based on the way of concept formation or concept assimilation to enable students to undergo and experience the vivid process of concept generation, guide students to reveal the thinking processes and thoughts and methods hidden in the process of concept generation, instead of simply providing the definitions; in the teaching of theorems, properties, laws, formulas, rules and other conclusions, we should guide students to actively participate in the process of exploration, discovery, and derivation of these conclusions, find the causality of each conclusion, instead of making premature conclusions; in the process of reasoning, we should make the existing judgement linked up and interrelated, try to generate as many as tentacles of thinking as possible from existing judgements, facilitate the thinking chain, achieve the efficient operation of the thinking network, constantly launch new judgements and new thinking results under the guidance of mathematical thoughts and methods, instead of looking for correlations mechanically. (2) Refine and summarize mathematical thoughts and methods through review, summary and lectures Mathematical thoughts and methods are not only reflected in the solution of a specific problem, but also reflected in the treatment of many problems, so it has the guiding significance of general rules. Because the same content can show different mathematical thoughts and methods, and the same mathematical thought and method is often distributed in many different knowledge points, therefore, during afterclass summary, unit summary, review and general review, we should pay attention

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to sorting out, refining and summarizing mathematical thoughts and methods and their systematic teaching horizontally and vertically. At the same time, offer special lectures to explain the ins and outs, connotation and extension, and function of mathematical thoughts and methods, which can help students understand and master mathematical thoughts and methods better and systematically. (3) Master and deepen mathematical thoughts and methods through problem solving As we know, the mathematical problem is generated in the process of constantly exploring and discovering the internal relations among mathematical knowledge hidden in the problem situation, and it is the carrier of mathematical activities. The mathematical problem solving refers to the process of, with clear goals but not knowing the way or method to achieve the goals, applying a series of directed cognitive operations to make it a new advanced rule and apply it to the mathematical problem situation. In short, mathematical problem solving “is essentially the process of constant transformation of propositions and repeated application of mathematical thought and methods”. Mathematical thoughts and methods exist in the solution of mathematical problems, and the step-by-step transformation of mathematical problems all follows the direction indicated by the mathematical thoughts and methods. Therefore, through mathematical problem solving, cultivate students’ problem consciousness, induce students’ creative motivation, and embed problems in living thinking activities, so that we can guide students to develop and master mathematics thoughts and methods in the process of learning and applying mathematics, and promote the development of their thinking ability. Review Questions and Exercises (III) 1. What are the methods and mathematical methods, and what are the ways and significance of studying mathematical methods? 2. Define the meaning of mathematical knowledge, mathematical problem, mathematical thought, mathematical method, and mathematical thinking method, and illustrate them with examples. 3. What is reduction, what is the direction and method of reduction, and why is reduction an important feature of mathematicians’ thinking? 4. What is analogy, and what are the common forms of analogy in secondary school mathematics? Illustrate them with examples. 5. What is induction, what types of induction are there, and why is induction not only a method of mathematical discovery, but also a method of mathematical demonstration? 6. What is association and intuition, and how are association and intuition applied in mathematical research and problem solving? Illustrate them with examples. 7. What is aesthetic method, and what does it mean in mathematical research and study? Illustrate it with examples. 8. What is mathematical discovery, and what are the main methods of mathematical discovery? Illustrate them with examples.

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9. What is the significance of demonstration in mathematics, and what specific methods are commonly used? Illustrate them with examples 10. What is test method in mathematics, what is its basic idea, and what’s the significance of the mathematical test method in mathematical research and problem solving? 11. What is the non-standard problems in mathematics, and how are they generally solved? 12. What are the basic thoughts in secondary school mathematics, and what is its specific significance? Try to illustrate it together with examples. 13. What are the teaching principles of mathematical thoughts and methods in secondary schools, and what are the basic teaching approaches? 14. Why has the teaching of mathematical thoughts and methods become an important part of secondary school mathematics teaching. How are you going to teach it? 15. Factorization: (1) (a − x)y 3 + (y − a)x 3 + (x − y)a 3 ; (2) (x + y + z)5 − x 5 − y 5 − z 5 . 16. Evaluation: (1) cos 48◦ + cos 24◦ − cos 12◦ − cos 84◦ ; (2) tan 9◦ − tan 27◦ − tan 63◦ + tan 81◦ . 17. Solve the following equations: (1) (2) (3) (4)

6x 4 − 25x 3 + 12x 2 + 25x + 6 = 0; 6X + 4X = 9X ; √ √ x 2 + 4x +√6 − √x 2 − 4x + √ 6 = 2; 2(x + 1) = x − x + 8 + 2 x(x + 8).

18. Solve the following systems of equations: ⎧ ⎨

x + y + z = 0, (1) x 2 + y 2 − z 2 = 20, ⎩ 4 x + y 4 − z 4 = 560,

⎧ ⎨ x + y + z = 1, (2) x 2 + y 2 + z 2 = 13 , ⎩ 3 x + y 3 + z 3 = 19 ,

19. f (x) = x 4 + Ax 3 + Bx 2 − 8x + 4 is known to be the perfect square expression of quadratic trinomial g(x). Find f (x) and g(x). 20. It is known that the graph of a quadratic function passes through point (2, − 12), and the intercepts on both axes are −8. Find this quadratic function. 21. If x and y satisfy (x − 1)2 + (y + 1)2 ≤ 9, find the extreme value of s = 4x − 3y. ⎧ 2 2 ⎪ ⎨ x + x y + y = 19, 22. It is known that positive numbers x, y and z satisfy to find the value of x + y + z.

⎪ ⎩

y 2 + yz + z 2 = 37, try

z 2 + zx + x 2 = 28,

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23. It is known that a, b ∈ R + and a 2 + b2 = 1, c2 + d 2 = 1, and a, b, c, d are positive. Prove: ac + bd ≤ l. √ √ 24. It is known that a, b ∈ R + , and a 1 − b2 + b 1 − a 2 = 1. Prove: a2 + b2 = l. √ √ 25. If a ∈ R, try to find the minimum value of f (a) = a 2 + a + 1+ a 2 − a + 1. 26. In isosceles triangle ABC, height AD on base BC and height BE on side AC intersect at H, K is the midpoint of AH, EF is perpendicular to BC at F. Extend AD to G to make DG = EF. Prove: BG⊥BK. 27. In ABC, AD is the height on BC, and AD = BC. H is the orthocenter, and M 1 is the midpoint of BC. Prove: HM + DM = 2 BC. 28. In ABC, AD ⊥BC, P is a point in AD, straight lines BP and CP intersect AC and AB at points F and E respectively. Prove: ∠ADE = ∠ADF. 29. In ABC, construct squares ABDE and ACFG outward with AB and AC as a side respectively, let M be the midpoint of BC, the centers of two square are P and Q respectively. Prove: PM = QM, and PM⊥QM. 30. In quadrilateral ABCD, the areas ABD: BCD: ABC = 3:4:1, M and N = CC ND , and B, M, and N are collinear. are on AC and CD respectively and AM AC Prove: M and N are the midpoints of AC and CD respectively. 31. Let ABCD be a convex quadrilateral with opposite sides not parallel to both sides, the extension lines of AB and CD, and DA and CB intersect at E and F respectively, M and N are midpoints of AC and BD respectively. Prove: The extension line of MN bisects EF. 1 1 = 3, Find the value of x 1989 + (x−1) 32. It is known that x + x−1 1989 . 2 33. It is known that real numbers a, b, and c satisfy b = 6 − a, c = ab − 9, Try to find the value of a and b. 34. It is known that the hyperbola passes through points A(−2, 4) and B(4, 4), one of its focus happens to be the focus of parabola. Find the locus of the other focus.y 2 = 4x 35. Given n arbitrary squares (n > 1, n ∈ N+ ), prove: After cut off, they can form a new square. 36. Suppose x + y + z = 0, x yz = 0. Find the value of 

1 1 + x y z





1 1 +y + x z





1 1 +z + x y



37. As shown in Fig. 4.33, A1 B1 C1 D1 is an oblique section of a cuboid, where AB = 4, BC = 3, AA1 = 5, BB1 = 8, CC1 = 12. Find the volume of this geometric solid. 38. Try to determine the single-digit number of 31001 · 71002 · 131003 . 39. Find the positive integer solutions to the following equations:

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Fig. 4.33 .

Fig. 4.34 .

(1) x y = x + y (2) x1 + 1y = 15 (3) x y + 3x − 5y = 3 (4) 53x y + 5x − 7y = 1 40. Below are the expressions of two eight-digit numbers divided by a three-digit number. Try to find their divisors and quotients respectively, and explain the reasons (where x is denoted as a natural number from 0 to 9, and the first digit cannot be 0) (Fig. 4.34).

Chapter 5

Mathematical Ability in Secondary Schools

Mathematics teaching in secondary schools aims not only at imparting knowledge, but also at cultivating ability, and the cultivation of ability is increasingly important. This chapter only expounds the significance of cultivating ability, the basic requirements and cultivation problems of operational ability, logical thinking ability, spatial imagination ability, and the crucial problem-solving ability in secondary school mathematics curriculum standard.

5.1 Significance of Mathematical Ability 5.1.1 Relationship Between Knowledge and Ability Knowledge is the sum of people’s understanding of objective things. At present, the mental state where the brain’s mechanism fits in with a certain activity and can complete a certain activity successfully is called ability (or special ability). In mathematics, ability generally refers to operational ability, logical thinking ability, spatial imagination ability, and the resulting ability to analyze and solve problems. The mental characteristics where the brain’s mechanism understands and transforms objective things in social activities are called intelligence (or general ability). In mathematics, intelligence generally refers to attention, power of observation, memory, imagination, and thinking ability, etc. Knowledge and ability have different connotations. Knowledge is the reflection of the phenomenon and essence of objective things, while ability is the capability of people to complete certain activities successfully, which belongs to individual’s mental state or mental characteristics; knowledge is acquired by man, while ability is related not only to man’s innate factors, but also to man’s acquired environment, education, and other factors; the acquisition of knowledge is endless, and its development is relatively fast, while the development of ability is limited, and its development is relatively slow. In teaching practice, some students have high scores but © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_5

151

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low abilities, and other students have low scores and high abilities, which shows that knowledge and ability cannot be equated, and the ability and its development degree cannot be measured mechanically by the knowledge they have mastered. Secondly, knowledge and ability are interrelated and interacted. People need both knowledge and ability to engage in any activity. Usually, ability gradually forms and develops in the process of mastering knowledge, while the formed ability, in turn, affects the speed, depth, and breadth of mastering knowledge. In other words, mastering knowledge is the condition and foundation of developing ability, and ability is the premise and result of mastering knowledge.

5.1.2 Significance of Cultivating Mathematical Ability in Secondary Schools 1. Developing ability is the trend of teaching reform In the past 50 years, the international modernization movement of secondary school mathematics education has been characterized by improving teaching principles and methods and placing ability training in a more important position than learning and memorizing existing knowledge. The United States has clearly declared that it was the top priority of education in the 1980s. Therefore, ability training not only has been the proposition of many Chinese and foreign educators, but has become a trend of development of international mathematics teaching reform at present. 2. Training ability is the need of building the Four Modernizations The mathematization of contemporary sciences and the mutual infiltration of basic disciplines of mathematics have promoted the emergence of many new disciplines. How can mathematics education adapt to the challenge of this new situation? Although some educators advocate early education and life-long education, they still cannot solve the contradiction between the limited life and the infinite knowledge. This requires people to select the quintessential part from the confluence of mathematical knowledge and learn in the best way and method. On the other hand, pay attention to the development of intelligence, cultivate out ability to acquire new knowledge, and use the key of intelligence to open the door of knowledge. Secondary school education is a basic education. While acquiring the necessary basic knowledge, the students must also acquire the ability of application and further learning. Therefore, only by paying attention to the cultivation of ability, can students become knowledgeable and change their limited knowledge into powerful productivity to contribute to the construction of Four Modernizations. 3. Training ability is an important task of secondary school mathematics teaching Our educational policy is to make the educated develop vividly and vigorously in moral, intellectual, physical, aesthetic, and labor aspects. “Intellectual education” is

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not “knowledge education”, not imparting knowledge simply, but including knowledge imparting, ability training, and many other contents. Since the founding of the People’s Republic of China in 1949, the Secondary school Mathematics Teaching Syllabus has been formulated 8 times, the specific requirements for cultivating ability have been clearly put forward in the last seven versions, which fully shows that cultivating ability is an important task of secondary school mathematics teaching. At present, in the secondary school mathematics teaching, since the “excessive assignments tactic” and “more lecturing, more practice, more test” phenomena still exist, students are overburdened, which not only affects the improvement of secondary school mathematics teaching quality, but also affects students’ further study in the future. It has become a task of top priority to turn things around, follow the law of education, implement the new curriculum standard conscientiously, especially pay attention to the ability training to improve the quality of secondary school mathematics teaching.

5.1.3 Basic Approaches to Cultivating Mathematical Ability in Secondary Schools How to cultivate ability in secondary school mathematics teaching is a very complex and wide-ranging subject. Among them, the reform of curriculum, textbooks, and teaching methods is the main subject. In a sense, the reform of teaching methods is particularly important. In terms of the characteristics of secondary school mathematics, the cultivation of mathematical ability should still focus on “cultivating operational ability, logical thinking ability and spatial imagination ability” mentioned in the Secondary school Mathematics Curriculum Standard. Here is some analysis on the common problems of how to cultivate ability. 1. Improving the consciousness and enthusiasm of learning is the premise of cultivating ability. The external cause is the condition, and the internal cause is the basis. Only by improving students’ consciousness and enthusiasm of learning, making them goaloriented, determined, maintaining strong learning enthusiasm and generating strong internal motivation, can they not only learn basic knowledge well, but also cultivate their ability. 2. Mastering the basic knowledge of mathematics is the basis of cultivating ability. Knowledge and ability complement each other. Without knowledge, cultivating ability becomes water without a source and a tree without roots. In the teaching of basic knowledge, we should pay attention to teaching students regular knowledge and the regularity of knowledge, so as to make them master knowledge in a clear and systematic manner, and “be ready for students’ own purposes”. For example, if students have a vague understanding of absolute value, it is impossible for them

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to make a correct judgement in the calculation or proof related to absolute value; if they don’t master the steps and methods of solving quadratic equations with one unknown, it’s difficult for them to have the ability to solve quadratic equations with one unknown, etc. 3. Improving teaching methods and organizational forms is the key to cultivating ability. In order to improve teaching methods and organizational forms, people have carried out discussions on the optimal teaching model. It is considered that the teaching process of “lead-in, development and ending” is a better teaching model at present. The so-called lean-in is to raise students’ awareness, desire, and interest in acquiring new knowledge, mobilize necessary experience, understand and master existing learning methods, induce ideal learning mental function to acquire new knowledge, and this is the key stage of learning success; the so-called development is to enable students to actively participate in thinking activities, preliminarily understand and master the knowledge through appropriate teaching methods on the basis of lead-in, which is the core stage of teaching; the so-called ending means the turn from the learning of mastering the essence of teaching to the active realistic mastering stage on the basis of the first two stages, and when the inward direction is adopted here, it is the further mastering and consolidation; when the outward direction is adopted, it is transfer and flexible use. Better migration is obtained through consolidation, and better consolidation is obtained through transfer. 4. Paying attention to the penetration and integration of knowledge of various disciplines is an important measure to cultivate ability. At present, there is a one-sided view that the cultivation of operational ability is the business of algebra teaching and the cultivation of logical thinking ability is the business of geometry teaching. In fact, operations include algebraic operation, geometric operation, and analytical operation. The translation, symmetry, rotation, stretching, inversion, affine, and other transformations in geometry are geometric operations. At the same time, logical thinking is not only “geometric”, but there is operational type, which is a kind of rather advanced logical inference form. As for spatial imagination, it is not only limited to three-dimensional space, but it can be two-dimensional (plane) or one-dimensional (straight line) space, and even can be developed to n-dimensional space. The cultivation of certain ability cannot be carried out in isolation, and all disciplines of mathematics should strengthen the penetration of knowledge and the comprehensive cultivation of ability. 5. Improving teachers’ knowledge and professional level is an important condition for cultivating students’ ability. The cultivation of ability is carried out under the correct guidance and strict demonstration of teachers. This requires teachers to have a wide range of knowledge, high mathematical accomplishment, and strong ability, and not to be satisfied with existing experience. Only by strengthening professional development, possessing

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a vast amount of information and striving to improve our knowledge level and professional ability can we meet the requirements of cultivating ability. At the same time, due to the change in teaching form, students’ thinking is more open, their imagination is richer, and they often raise many “strange” problems. This requires teachers to prepare lessons carefully, design teaching plans meticulously, and predict all possible situations in teaching adequately, so as to be able to harness the teaching process accurately and complete the teaching tasks step by step, and constantly push the cultivation of ability to a new level.

5.2 Cultivation of Operational Ability 5.2.1 Operational Ability of Secondary School Mathematics Operation is a deductive process of transforming concrete objects according to operational rules and formulas. The operations of secondary school mathematics should not only be understood as numerical calculation, but also include identical transformation of formulas, equivalent transformation of equations and inequalities, elementary operation, transcendental operation, differential and integral operation of functions, measurement and calculation of various geometric quantities, preliminary calculation of probability and statistics, relevant correspondence and transformations, etc. Although all kinds of operations have their own meanings, rules, and formulas, they all require the existence, uniqueness, and simplicity of their results. Generally, there are two explanations for operation: One is to interpret operation as “associative law”. If an operation “*” is introduced into set M = {a, b, c …}, that is to say, “*” operation can be implemented for any two elements a and b in M, that is, there is a unique element c ∈ M, which is the result of this operation implemented on elements a and b (associative law), usually expressed as a * b = c. The other is to regard operation as a “function”. In essence, the associative law makes the elements in M exactly correspond to one element of this set, that is, 2 implementing the of all possible ordered pairs set M of the elements in  mapping 2 set M to set M M → M is called operation “*” in set M (specifically, binary operation). Operational ability refers to not only the ability to execute operations correctly according to rules and formulas, but also the ability to understand operational principles and seek reasonable and simple operational ways according to the conditions of problems. The operational ability has the following characteristics. First, from the perspective of the development of mathematics, the operational ability is hierarchical.

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The operations of different types develop gradually from the simple to the complex, from a low level to a high level, and from the concrete to the abstract. For example, if you don’t know real arithmetic, you can’t know complex operation, if you don’t master formula operation and solution of equations and inequalities, you can’t master transcendental operation, if you don’t master finite operation, you can’t implement infinite operation, etc. This shows that operational ability develops gradually with constant expanding of mathematics knowledge and increasingly improved level of abstraction. In addition, the process from mastering knowledge to forming operational ability is also hierarchical. For the same problem, to carry out operation reasonably and flexibly, we must raise skills to the level of ability, that is, to combine skills with thinking activities to form operational ability. In other words, for the same problem, different people apply different knowledge and methods, and the operation process is different. Second, from the perspective of operation process, the operational ability is comprehensive. Not existing in isolation, operational ability is interrelated and interpenetrated to memory ability, observation ability, comprehension ability, reasoning ability, imagination ability, and expression ability. The memory ability is the “assistant” of operational ability, and it is beneficial to complex operations to memorize the most fundamental and important formulas, their deformations and common data firmly for a long time. The observation ability and imaginative ability are the starting point of operational ability. Many errors in operations are usually inseparable from students’ lack of observation ability and imagination ability. Understanding the questions and being good at “equivalent deformation” of formulas help to improve operational flexibility. Furthermore, the operation process is substantially to derive a result based on operation definition and rules, so it is also a reasoning process. Therefore, the operational ability is the embodiment of comprehensive ability, and the cultivation of operational ability must be carried out in a variety of ways.

5.2.2 Basic Approaches to Cultivating Operational Ability 1. Mastering the concepts and properties related to operations is the premise of cultivating operational ability. In order to cultivate students’ operational ability, first of all, students should understand and master the relevant concepts, properties, formulas, and rules of various operations, which is the premise of ensuring correct and reasonable operations. For example, if students don’t understand the meaning of they’ll  quadratic radicals,  have difficulty in simplifying algebraic expressions like (x − 5)2 − (1 − x)2 .

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Similarly, if we don’t understand the concepts and properties of exponents, logarithms and tangent functions, we’ll have difficulty in calculating (4 tan 15◦ )25 ×  √ (0.25 tan 75◦ )24 , log√2−1 2 + 2 2 , etc. 2. Memorizing the relevant data and formulas is the basis of cultivating operational ability. In order to cultivate students’ operational ability, students should improve their memory ability, memorize some commonly used data and formulas; master some oral arithmetic and quick calculation methods. In terms of frequently used data, we should memorize: Squares of 1–20, and cubes of 1–10; Values of 2√n (n √ = 1,√. . . , 10), 3n (n = 1, . . . , 5), 5n (n = 1, . . . , 5); Values of 2, 3, 5 and lg 2, lg 3, lg 7 (accurate to 0.0001); Pythagorean triples: 3, 4, 5; 5, 12, 13; 7, 24, 25; 8, 15, 17, etc.; Trigonometric function values of special angles, i.e. trigonometric function values of 0°, 30°, 45°, 60°, and 90°, etc.; Properties of positive integers divisible by 3, 6, 9, 2, 4, 8, 5, 11, 25, etc.; Values of π, e (accurate to 0.0001), etc. In terms of quick calculation, for example, we should master the method of calculating the following: 27 × 33 = (30 − 3)(30 + 3) = 900 − 9 = 891; √ √ 21 × 14 × 6 = 3 × 7 × 2 × 7 × 2 × 3 = 3 × 7 × 2 = 42 3. Mastering general rules and methods of operations is the key to cultivating operational ability. To cultivate students’ operational ability, we should enable students to master general methods and rules of operations. For example, in secondary school mathematics, there is a rule of order of operations of exponentiation, extraction of a root, multiplication, division before addition and subtraction; there is a rule of removing parentheses in sequence of innermost parentheses before outermost parentheses (or outermost parentheses before innermost parentheses); there is a calculation rule of the part before the whole, and a calculation rule of simplification before substitution for algebraic expressions. At the same time, there is elimination method for transforming multivariate into one unknown, degree decreasing method for transforming high degree into low degree, a series of general methods for transforming high level into low level and transforming the cumbersome into the simple, etc. In terms of relevant knowledge, there is a rule of approximate calculation, rule of exponent and logarithm operations, rule of trigonometric and inverse trigonometric operations, rule of set operations, rule of differential, and integral operations, etc.

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Only by mastering the general rules and methods of operation, can we have a foundation for relevant calculation, otherwise improving the operational ability is nothing but empty talk. 4. To be good at observation, analysis and reasoning is a necessary condition to improve operational ability. In order to cultivate students’ operational ability, it is necessary to improve students’ reasoning ability in operation, because each operation step must be well grounded, which is inseparable from reasoning.   √ For example, Simplify sin 50◦ 1 + 3 tan 10◦ . Original formula   √ sin 10◦ = sin 50 1 + 3 · cos 10◦

√ 2 sin 50◦ 1 3 2 sin 50◦ sin 40◦ ◦ ◦ = cos 10 + sin 10 = ◦ cos 10 2 2 cos 10◦ ◦

=

2 sin 50◦ cos 50◦ sin 100◦ cos 10◦ = = = 1. cos 10◦ cos 10◦ cos 10◦

This requires flexible use of addition theorem and multiple-angle formula to carry out identical transformation and reasoning of trigonometric expressions. Here, it is particularly important to improve students’ reasoning ability, number (or formula) deformation ability, and geometric transformation ability.  1 the key is to master the deformation For example, during calculation of n(n+1) 1 1 = n1 − n(n+1) . For another example, as shown in Fig. 5.1, given the straight n(n+1) line l1 ||l2 , A and B are two fixed points outside l1 and l2 respectively. Find two points C and D on l1 and l2 to make AC + CD + DB the shortest. Obviously, we need to master the deformation method of the figure. Translate the common perpendicular D C  to BE position, then when A, C, and E are in a straight line, AC + CD + DB is the shortest. 5. Accumulating operation skills is a magic weapon to improve operational ability. There is no doubt that there are skills in operations, especially there are skills in assumption, transformation, substitution, use of parameters, use of concepts, use of Fig. 5.1 .

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theorems, and so on in operations. In order to yield twice the result with half the effort, we must strengthen the training of operation skills and pay attention to the accumulation of operation skills. Example 1 It is known that the lengths of three sides of a triangle are 24, 32, and 40 respectively. Find the maximum angle. Solution If the cosine theorem is used, the calculation is obviously cumbersome. However, through observation, we find 24:32:40 = 3:4:5, which constitute a Pythagorean triple, so the maximum angle of 90° can be obtained easily. Example 2 Solve the equation (16x 2 − 9)2 + (16x 2 − 9)(9x 2 − 16) + (9x 2 − 16)2 = 625(x 2 − 1)2 . Solution It is obviously not advisable to expand the known equation into a quartic equation with one unknown, but if the original equation is transformed into [(16x 2 − 9) + (9x 2 − 16)]2 − (16x 2 − 9)(9x 2 − 16) = 625(x 2 − 1)2 and we find that (16x 2 − 9) + (9x 2 − 16) = 25(x 2 − 1) that is, [(16x 2 − 9) + (9x 2 − 16)]2 − (16x 2 − 9)(9x 2 − 16) [(16x 2 − 9) + (9x 2 − 16)]2 and we get (16x 2 − 9)(9x 2 − 16) = 0, x1,2 = ± 43 , x3,4 = ± 43 . It would be rather ingenious. 6. Forming the habit of step-test is an important aspect of improving operational ability. The operation process is often interfered by non-intellectual factors. It is common for students to be perfunctory and careless. Therefore, it is very important to develop the test habit, master test skills, and improve test efficiency. Here, we should not only have the consciousness of test, but also develop the ability to ask ourselves in the operation process: is each step grounded? Is the calculation correct? Is it written correctly? Is this the end result? Only by focusing on every little thing and going ahead steadily and surely can we ensure correct operation.

5.3 Cultivation of Logical Thinking Ability 5.3.1 Logical Thinking Ability of Secondary School Mathematics The logical thinking ability of secondary school mathematics is to think correctly and reasonably according to the correct laws and forms of thinking, that is, the ability to analyze, synthesize, abstract, generalize, infer and demonstrate the mathematical

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objects. It plays a core role in the ability training, and is an indispensable basic ability for learning mathematical theory and applying mathematical knowledge. From the mental characteristics of secondary school students, the thinking ability of secondary school students is in a rapid development period. The first grade students in middle schools mainly focus on imaginal thinking, the second and third grade students tend to focus on empirical type logical thinking, while the high school students’ thinking shifts from empirical type to theoretical type. It is of great significance to train students to develop into logical thinking from imaginal thinking, and shift from empirical-type logical thinking to theoretical type thinking smoothly in middle schools. The basic requirements for cultivating logical thinking ability include two interrelated aspects: in the aspect of formal thinking, we can use concepts correctly, make judgements properly, make analysis and synthesis, comparison and classification, abstraction and generalization, inference and demonstration logically when thinking about and solving problems. It is required that we should have a clear thinking, clear cause and effect, be well-grounded, meticulous, and rigorous; in the aspect of dialectical thinking, we can understand, master, and apply mathematical knowledge from the viewpoint of interrelation and interaction, the viewpoint of movement, change and development, the law of the unity of opposites, the law of mutual change of quality and quantity, and the law of negation of negation when thinking about and solving problems.

5.3.2 Basic Approaches to Cultivating Logical Thinking Ability 1. Combine basic knowledge teaching to cultivate logical thinking ability Knowledge and ability always complement each other. In the process of imparting mathematical knowledge to students, logical thinking ability can be cultivated. As long as we take knowledge instruction as the carrier of cultivating ability, and infiltrate or introduce the rules and methods of logical thinking during imparting knowledge, we may achieve good results. Logical thinking is rational knowledge. In order to cultivate logical thinking ability, first of all, students should be enabled to feel the distinct feeling, perception and representation, and form concrete, vivid, and visual perceptual knowledge. Then, through analysis and synthesis, abstraction and generalization and other thinking activities, the perceptual materials are processed, sorted out, and transformed to form concepts and judgements. Finally, the objects of thinking are expressed in normative mathematical language. The specific cultivation steps can be carried out according to the following procedures: (1) First, make students understand by insight, and make them attain hazy perception. For example, when introducing the concept of angle, the textbook first

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gives examples: “The hour hand and minute hand of a clock and two open feet of a pair of compasses all provide people with image of angle.” Then it analyzes and synthesizes: “They are all composed of two rays, and the two rays have a common endpoint”. Finally, it abstracts and generalizes: “A figure composed of two rays with a common endpoint is called an angle”. This process of forming the concept is from perceptual cognition to rational knowledge. In the perceptual stage, we make students understand “angle” by insight and make them have a hazy perception of angle. (2) Then, explain in words to make students have a clear understanding. Students’ understanding of logical thinking methods starts with a hazy perception. After understanding by insight for a period of time, we can clearly tell them the characteristics, structure and thinking rules of concept, judgement, reasoning, and other thinking forms at an appropriate time. (3) Teach students by our own example. In teaching, the teacher should set a good example for students to follow. The teacher’s language and writing on the blackboard should be accurate, rigorous, well-organized, well-grounded, and logical; the narration of students’ answers to questions (including homework) should be logical, and the logical mistakes made by students should be corrected meticulously, carefully, and timely. 2. Strengthen basic thinking training to cultivate logical thinking ability Learning how to swim by swimming is a figurative expression of cultivating students’ ability. To cultivate logical thinking ability, it is necessary to make students learn how to think by thinking, we must train students’ basic skills of logical thinking in a purposeful and planned way, which can be carried out around the basic forms of formal thinking and the basic points of dialectics. (1) Thinking training of concepts In general, concrete things are abstracted through the presentation of concrete reality prototypes. For example, the triangles of various forms are abstracted into triangles, and the equations with concrete coefficients are abstracted into the equations with literal coefficients. Students are required to express mathematical concepts in accurate language. (2) Thinking training of judgements We should pay attention to cultivate students’ habit of serious examining and teach them general methods of examining. After explaining four basic forms of propositions, we can carry out stage training on the transformation of these four forms, to enable students to deeply understand the truth or falsehood, equivalence, and substitution property in reasoning of these four forms of propositions. (3) Thinking training of reasoning Every time a new proof method is introduced, the reasoning format of this proof method must be explained to the students through examples, and then the students

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are required to demonstrate and solve problems according to the reasoning format. The key is to require students to master the following four most fundamental and commonly used reasoning formats: forward inference format of synthesis method, backward inference format of analysis method; three-step format of proof by contradiction; and discussion format of enumeration method. At the same time, in the process of reasoning, we should actively guide students to associate, to achieve “insight” through association, and find problem-solving ideas. To guide students to associate, we should assign some problems for vertical and horizontal association, to enable students to learn how to associate through association practice. (4) Training of basic viewpoints of dialectics In the problem-solving thinking, we should consciously use the thinking method of dialectics. For example, we can follow the philosophy of “the universality of contradiction residing in its particularity” to think about the problems with general conclusions. For example, for such problems as fixed value, fixed point, fixed shape, and fixed position, we can start with particular cases to find out the conclusion of general cases, transforming “the general” into “the particular”. 3. Correct logical errors through the analysis of counterexamples To carry out logical thinking training on students, we can set positive examples, and then correct their logical errors through the analysis of counterexamples, in order to deepen students’ understanding of logical thinking. The following common practice of counterexamples can be adopted. ➀ Point out and correct the students’ logical mistakes in answering questions or homework. ➁ Analyze typical logical errors. Select typical logical errors and mathematical sophistry questions, explain and analyze them in the classroom or seek answers through the mathematical blackboard newspaper and mathematical world, to enable students to distinguish between right and wrong, arouse students’ attention, so as to avoid mistakes. The following are two logical errors that students are likely to make in solving problems. Example 1 Prove the base changing formula for logarithms: logb N = are both positive numbers, and a = 1, b = 1). Proof 1

1 logb N =

loga N loga b

(a, b

loga N loga b loga N lg N = ÷ = lg b loga 10 loga 10 loga b

loga N

Proof 2 Since b loga b = N , we take the logarithm of both sides to base b, log N therefore, loga b × logb b = logb N a

log N

that is, logb N = loga b . a As a matter of fact, both proof 1 and proof 2 have made a mistake of circular reasoning.

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Fig. 5.2 .

Example 2 In ABC, AB > AC, AD is the median of side BC, and AE is the bisector of ∠A. Prove: AE < AD. Proof Since AB > AC, Therefore ∠C > ∠B ➀ It can be seen from Fig. 5.2 that ∠EAB > DAB, that is, 21 ∠A > ∠D AB ➁ ∠AD E = ∠D AB + ∠B. ➂ ∠AE D = ∠E AC + ∠C = 21 ∠A + ∠C ➃ From ➀, ➁, ➂, and ➃, we get ∠D F B > ∠D AB, Therefore, AE < AD. The proof here seems very rigorous, without any flaw. In fact, “it can be seen from Fig. 5.2 that ∠E AB > ∠D AB” is an intuition instead of a logical proof, resulting in loss of rigor of demonstration. To make a rigorous demonstration, it is necessary to prove that AE is between AD and AC. For this reason, we can make auxiliary lines (Fig. 5.3). Extend AD to F to make DF = AD, then ADC ∼ = F D B, ∠D AC = ∠D F B, while BF = AC < AB, ∠D F B > ∠AB. And AE is the bisector of ∠A. Therefore, AE is between AD and AC. Then, repeat the above demonstration process to obtain rigorous proof. Fig. 5.3 .

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5.4 Cultivation of Spatial Imagination Ability 5.4.1 Spatial Imagination Ability of Secondary School Mathematics Imagination is a kind of special thinking activity, that is, to imagine something that has never been perceived in the mind, or to create an image of something or phenomenon that has never been perceived. Spatial imagination ability in mathematics refers to the ability to imagine the shapes, structures, sizes and positions of objects (objective spatial forms). In order to cultivate the spatial imagination ability, the first thing is to make students acquire the mathematical knowledge about spatial forms, which not only involve solid geometry, but also include the contents of plane geometry, analytic geometry, and shape–number combination. The imagination ability of plane figures is the basis of developing the spatial imagination ability, and the teaching of plane geometry is also the content of cultivating the students’ spatial imagination ability. Therefore, it is a basic task of secondary school mathematics teaching to cultivate the spatial imagination ability. Generally, there are four requirements for cultivating the spatial imagination ability: first, being able to imagine geometric figures from objects with simple shape, and imagine the shape of objects from geometric figures; second, being able to decompose complex plane figures into simple and basic figures; third, being able to find out basic elements and their relations in the basic figures; fourth, being able to construct or draw the figures according to the conditions.

5.4.2 Basic Approaches to Cultivating the Spatial Imagination Ability 1. Strengthen the teaching of basic knowledge Both reproducible imagination and creative imagination are based on certain knowledge and experience. The process of acquiring basic knowledge is also a process of gradually forming the concept of space and developing the spatial imagination ability. Understanding and mastering relevant mathematical concepts, mathematical propositions and mathematical methods will help to reproduce the relevant spatial form clearly in the mind, and help to express this spatial form in geometric language (i.e., geometric figures) correctly. For example, through intuitive figures, students begin to get a preliminary image of the positional relationship of two circles. Only after students understand and master the theoretical knowledge that “the positional relationship of two circles is determined by the metric relations between the distance

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of circle centers and two radii” and understand the essence of the positional relationship between two circles, can they really develop the imagination ability of positional relationship between two circles. 2. Carry out intuitive teaching with the help of physical models Spatial forms and concepts can be visualized through physical models. Guiding students to observe and analyze physical models, making models, measuring the real objects on the spot and other practical activities can help students gradually form the spatial concepts, and make the spatial forms concretized and visualized in students’ mind. In this way, they can gradually manage to think about spatial forms without the help of physical objects, models, and graphics. The deeper the image is, the richer the imagination is. Therefore, it is an indispensable way to cultivate students’ spatial concepts and spatial imagination ability to use visual aids such as physical models for teaching. 3. Strengthen the training of graph recognizing and drawing The concept of space is a thinking process of alternate action of imaginal thinking and logical thinking. The best language to express this kind of thinking is geometric figure, which can express the spatial forms most simply and intuitively. Therefore, strengthening the training of graph recognizing and drawing is the best way to cultivate the concept of space. In mathematics teaching, we must pay attention to graph drawing teaching, and teachers should play an exemplary role in graph drawing, especially in the initial stage of geometry teaching, we should pay more attention to the training of graph recognizing and drawing. For example, we should make students observe and recognize the same geometric figure from different angles, and make students describe the figure according to text description. 4. Cultivate the concept of space through shape–number combination Shape (graphics, images, charts) has the characteristics of concretization and visualization; number (quantitative relations) has the characteristic of generalization and abstraction. Shape–number combination is the combination of intuition and abstraction, perception, and thinking. The training of shape-number combination in a planned way can connect geometry, algebra with triangle, to cultivate the spatial imagination ability effectively. The meaning of solving problems by means of shape–number combination can be summarized into two aspects. First, according to the structural characteristics of numbers, we can construct corresponding geometric figures by arousing representation or creating imagination, and use the characteristics and laws of figures to help solve the problem when we want to prove the inequality  of numbers. For example, √ √ 2 2 2 2 a + b + (a − 1) + (b − 2) ≥ 5, if we can imagine a triangle with three points (a, b), (1, 2), (0, 0) as the vertexes, we can prove the proposition by using the theorem that the sum of two sides of a triangle is greater than the third side. In this way, the abstract is changed into the visualized, making hard things easy. Second, the graphic information is partially or completely transformed into numerical information to weaken or eliminate the reasoning components of the shape, so that the

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problem of shape to be solved is reduced to the study of quantitative relationship. For example, by using the coordinate method to prove geometric problems, the elusive geometric problems can be incorporated into the stereotyped ways of quantitative calculation. 5. Strengthen the training of spatial imagination Ability is always produced and improved with people’s activities. An important way to cultivate the spatial imagination ability is to strengthen the training of spatial imagination. In teaching, we should often raise some questions that can cause students to think about spatial imagination to train them. For example, when teaching a geometric concept, require students to imagine various physical prototypes of this concept. For another example, when discussing the solution of system of linear equations with two unknowns, we can ask students: When the coefficients of the corresponding terms of two equations are not proportional, why can there be only one solution? The students are required to give geometric explanation. We can also, in combination with the teaching content, ask and guide students to do some imagination exercises. For example, jigsaw puzzle exercises, graph decomposition exercises, or finding several triangles and quadrangles in a complex geometric figure, etc. are effective exercises to improve the spatial imagination ability. This means that through teaching and exercises, students can imagine real objects from the models, and draw figures from real objects; they can imagine real objects from the figures, until they have the ability to imagine solid figures from plane figures. Example 1 It is known that cube ABCD—A1 B1 C 1 D1 , where K and L are the midpoints of edges AB and BC respectively. Try to find the area of the cross section passing through three points D1 , K, and L, and the dihedral angle between this section and the bottom side. Solution As shown in Fig. 5.4, suppose the edge length of the cube is a, it is not difficult to imagine that this section intersects edges DC and DA at G and H. respectively, and intersects edges AA1 and CC 1 at E and F respectively, and E and F are trisection points of their edges, and the section is a pentagon, where EF =

√

2a,

√ 2 1 a K L = AC = 2 2 DS = DS =

7 24



3 3√ DB = 2a 4 4 √

D D12 + DS 2 =

34 a 4

The area of the pentagonal section can be obtained by further calculation, S = √ 17a 2 .

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Fig. 5.4 .

And θ = arctan

 √  2 2 . 3

5.5 Cultivation of Problem Solving Ability The ultimate purpose of cultivating students’ operational ability, logical thinking ability, and spatial imagination ability is to cultivate students’ ability of analyzing and solving problems. Therefore, the cultivation of problem solving ability is not only the comprehensive embodiment of the above three abilities, but also the main indicator of improving the quality of mathematics teaching. To this end, we are going to introduce the basic knowledge of problem solving and the approaches to cultivating problem solving ability.

5.5.1 Basic Knowledge of Problem Solving 1. Classification of secondary school mathematics exercises There are a variety of classification methods of secondary school mathematics exercises. The common methods include: By the different requirements of the questions, they can be divided into calculation questions, proof questions, construction questions, application questions and so on; by the different forms of problem solving, they can be divided into examples, oral answers, exercises, review questions, reflection questions, game questions, and so on; by the different solutions, they can be divided into free solution questions (such as free-response questions, essay questions, etc.) and fixed solution questions (such as yes–no questions, multiple choices, etc.). 2. Selection and arrangement of secondary school mathematics exercises During secondary school mathematics teaching, the mathematics exercises must be selected and arranged carefully according to the teaching purpose. Here, careful selection is to choose those exercises that are typical, representative, inspiring, in

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diverse forms, can play the role of drawing inferences about other cases from one instance, but also has a certain degree of difficulty, and likely to develop students’ thinking ability; arrangement is to be from the shallow to the deep, from the simple to the complex, from the easier to the more advanced, not only to proceed gradually in proper sequence, with a certain gradient; but also to be coherent, in a certain cycle. 3. Basic requirement for secondary school mathematics problem solving Secondary school mathematics problem solving must meet the basic requirements of being correct, reasonable, concise, clear, and complete. In other words, in the process of problem solving, expression and operation, reasoning, construction, and the results must be well-grounded. We should make every effort to use a simple and rapid problem solving method with a certain skill and can solve all problems in question or find all results completely. The problem solving process must be well organized, clear and meet certain specifications. In secondary school mathematics problem solving, students pay more attention to the requirements for being correct, reasonable, and concise, but somewhat ignore the requirements for being complete. Let’s look at the following example. Example 1 It is known that the sides of triangle ABC are an arithmetic sequence, a > b > c, and A (− 1, 0), C (1, 0). Find the locus of point B. Solution Through analysis, it is not difficult to obtain the locus is an elliptic arc.  Since 2a  = √a + c = 2b = 4, c = 1, then, b = 3. 2 2 Therefore, the locus is x4 + y3 = 1. In fact, the solution is complete only after the conditions a > b > c and that ABC is a triangle are added. The locus equation is 2 x2 + y3 = 1 (− 2 < x < 0), namely two elliptic arcs to the left of y-axis (Fig. 5.5). 4 Obviously, failure to discuss the solutions to the equations with literal coefficients; failure to test the solutions to fractional, radical, and transcendental equations; failure to consider the practical meaning of the solutions to application problems; and failure to pay attention to the value range of parameters when general equations and parametric equations are reciprocally transformed, are all manifestations of the incompleteness of solution. In order to meet the above problem solving requirements, a review should be conducted after the problem solving, that is, continuing the analysis and research on the completed problem solving activities. The review of problem solving includes testing the solution, discussing solving method, and generalizing the results. If we Fig. 5.5 .

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can persist in doing so and develop the habit of reviewing problem solving, it will undoubtedly play an important role in improving our problem solving ability.

5.5.2 Thinking Process of Problem Solving Solving a mathematical problem, especially solving a rather complex mathematical problem, often requires some pondering, which has its own complex thinking process. It is necessary to analyze the process and characteristics of this thinking activity to solve the problems with a purpose and highly effectively. 1. The process of mathematical problem solving is a thinking activity process of problem identification and classification. To solve mathematical problems, we must first identify the problems, analyze their characteristics, and classify them accurately, so as to find the solutions with the corresponding thinking method. Example 2 As shown in Fig. 5.6, regular triangle ABE and regular triangle ACD are constructed outward with sides AB and AC of ABC as the side. Prove: BD = CE. This is a plane geometry problem. If we can identify that as long as we can prove AB D ∼ = AEC, we can prove BD = CE. Thus the problem is reduced to the problem of proving these two triangles are congruent, and then the problem will be solved. In solving problems, the basic point of identification and classification of problems is the identification of mathematical models and figures. Of course, in order to form this correct and rapid identification ability, it is necessary to improve the ability to observe and generalize the problems; sometimes reasoning is needed to transform and deform the original problem into an identifiable model. 2. The process of mathematical problem solving is a process of thinking activity process of continuously analyzing and synthesizing the problems. In problem solving, analysis and synthesis are always closely linked. Whenever we solve a problem, prove a theorem, or complete a construction, etc., we always analyze the problem first, make clear what the known conditions are, what the desired conclusion is, and then compare the conditions and conclusions to find out the internal connection between them, which is synthesis. And this process of analysis and synthesis needs to be carried out continuously and repeatedly. Fig. 5.6 .

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Example 3 Given that a, b, and c are all positive integers, 1 < a < b < c, and (ab − 1) (bc − 1) (ca − 1) is divisible by abc. Try to find the values of a, b, and c. Solution (Analysis) Since (ab − 1) is not divisible by a and b, (bc − 1) is not divisible by b and c, (ca − 1) is not divisible by c and a, while (ab − 1) (bc − 1) (ca − 1) is divisible by abc. (Synthesis) Therefore, (ab − 1) is divisible by c, (ca − 1) is divisible by b, (bc − 1) is divisible by a, (Analysis) Since (ab − 1)(bc − 1)(ca − 1) = a 2 b2 c2 − ab2 c − a 2 bc − abc2 + ab + bc + ca − 1 = abc(abc − a − b − c) + ab + bc + ca − 1, therefore, ab + bc + ca − 1 is divisible by abc, (Reanalysis) Let ab − 1 = kc, then ab > kc, we have a > k · bc , while bc > 1, Therefore, a > k. And, ab + bc + ca − 1 = c (a + b + k), Since a + b + k can be divisible by b, then a + k is divisible by b, Therefore, a + k = pb. ➀ And from k < a < b, we get a + k < 2b. ➁ (Re-synthesis) From ➀ and ➁, we have P < 2b, so we get P < 2, p = l. So a + k = b, then a + b + k = 2b. Then from a + b + k is divisible by ab, and ab is divisible by 2b, we get: k = 1, a = 2, b = 3, c = 5. This is a continuous process of “analysis and synthesis”. Synthesis is made on the basis of analysis and analysis is made on the basis of synthesis. At the same time, when putting forward various ideas or choosing the best plan to solve the problem, it is also realized through “analysis–synthesis”. Therefore, the process of problem solving is a process of continuous analysis and synthesis, and the strength of this analysis and synthesis ability will directly determine the depth and breadth of problem solving thinking activities. 3. The process of mathematical problem solving is a thinking activity process of dialectical unity of stereotypes and variations of thinking. To solve a mathematical problem, after analyzing the meaning of the problem, or changing or deforming the original problem properly, it is often transformed into a similar problem that can be solved, or into a simpler problem by using the existing knowledge, methods, and experience. Obviously, this kind of thinking process is induced by certain thinking stereotype. Without this thinking stereotype, there is no certain ability to solve problems. However, due to the lack of clear analysis and identification of the problem, the existing knowledge, methods, and experience are often wrongly applied indiscriminately, which will mislead the thinking and hinder the correct solution of the problem.

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Example 4 Find real number a so that equation x 2 + (a + 2i)x + 2 + ai = 0 has real roots. Some students solve the problem like this: Since the original equation has real roots, then  ≥ 0, that is, (a + 2i)2 − 4(2 + ai)√= a 2 − 12 ≥√0 The solution is a ≥ 2 3 or a ≤ −2 3. In fact, when a = 4 (4 > 2), the two roots of the equation – 1 − i and – 3 − i are imaginary numbers, so obviously the above solution is wrong. The reason for this error is the influence from the habit of solving the equations with real coefficients, so the discriminant which can only be used for the roots of equations with real coefficients is mechanically applied to the discrimination of the roots of equations with complex coefficients. This is the negative effect of stereotypes of thinking. The correct solution is: The original equation is transformed into (x 2 + ax + 2) + (2x + a)i = 0.

x 2 + ax + 2 = 0 2x + a = 0,

√ we get a = ±2 2

From √ √ √ √ When a = 2 2, x = − 2; when a = − 2 2, x = 2. In the problem solving, on the one hand, students should be guided to pay attention to the use of existing knowledge and experience, give full play to the positive role of stereotype of thinking; on the other hand, they should be guided to pay attention to overcoming the negative effect of stereotypes of thinking to prevent rigidity of thinking. Only by advocating variation thinking, cultivating the quality of flexibility and vastness of thinking, and dialectically unifying the stereotypes and variation of thinking, can we really improve our ability to solve problems.

5.5.3 Cultivation of Problem Solving Ability 1. Fall into a habit of examining the problems carefully Problem examining is the basis of problem solving. Students’ mistakes in solving the problems or difficulty in solving the problems are often caused by the failure to examine the problems carefully or not being good at examining. (1) Make clear the questions Examining aims at making clear the question and the grammatical structure of the proposition. For example, try to find the number of positive integer solutions to inequality x 2 − 5x − 3 < 0. In this case, we’re required to find the number of solutions, not the positive integer solutions themselves.

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When examining the problem, it is necessary to make clear the meaning of key words such as “include”, “be included in”, “divide”, “divided by”, “greater than”, “not greater than”, “less than”, “not less than”, “positive”, “non-positive”, “negative”, “non-negative”, “increase”, “increase to”, and make clear the logical relations of common narrative ways, such as “if…then…”, “If…, …”, “Given…, prove…”, “…is…the condition…”, “On condition that…”, etc. (2) Dig implied conditions The so-called implied conditions refer to the conditions that are given but not obvious in the question, or the conditions that are not given but implied in the meaning of the question. For the former, it is necessary to transform the non-obvious conditions into obvious conditions. For the latter, it is necessary to dig the conditions implied in the question according to the set conditions. In a sense, to fall into a habit of and improve the ability of examining problems, it is important to improve students’ ability to dig implied conditions and transform the unknown into the known. For example, solve the equation 

√ x   √ x 3+2 2 + 3 − 2 2 = 6.

  √ √ There’s an implied condition here: 3 + 2 2 ∗ 3 − 2 2 = 1. After finding this implied condition, the original equation is transformed into 

√

3+2 2



x

+ 

1

√ 3+2 2

x = 6.

It’s not hard to figure out x1 = 2, x2 = − 2. For another example, it is known that the two real roots of equation x 2 −(k − 2)x + k 2 + 3k + 5 = 0 are x1 and x2 . Find the maximum value of x12 + x22 . Quite a few students find the maximum value by Vieta theorem   x12 + x22 = (x1 + x1 )3 − 2x1 x2 = (k − 2)2 − 2 k 2 + 3k + 5 = −(k + 5)2 + 19. So they draw a conclusion that the maximum value is 19. However, it is wrong, because the implied condition, the value range of k, is ignored in problem solving. In fact, since x1 and x2 , are two real roots, then we have   4  = (k − 2)2 − 4 k 2 + 3k + 5 ≥ 0, that is − 4 ≤ k ≤ − 3

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Therefore, we can only take k = − 4, so the maximum value of x12 + x22 should be 18. 2. Pay attention to summarize problem solving methods and key points After learning some contents, pay attention to summarize the methods and key points of solving some problems, which is beneficial to improve the ability of solving problems. For example, in algebra, the following methods and points can be summarized: √ (1) When a is any real number, the value of a 2 must be differentiated in three conditions: √ a > 0, a < 0, a = 0, or it is transformed into the absolute value by virtue of a 2 = |a| for discussion. (2) When the radicals are added or subtracted, they must first be converted into similar radicals; when multiplying or dividing them, they must first be converted into radicals with same degree. (3) For factorization, we have the method of extracting the common factor, formula method, group multiplication method (including the method of completing the square, cross-multiplication for quadratic trinomials), synthetic division method, method of undetermined coefficients, rotation method, etc. (4) The steps of using equations to solve application problems are: examining the problem, setting the variable, formulating (formulating the equivalent relations), solving the equation, checking, and answering. Among them, the auxiliary methods to analyze equivalent relationships include translation method, image method, simulation experimentation method, etc. (5) The methods of solving quadratic equations with one variable include direct radication method, method of completing the square, factorization method, formula method, image method, and so on. (6) The idea of solving fractional equations is to transform them into integral equation, the idea of solving irrational equations is to transform them into rational equations, and the idea of solving transcendental equations is to transform them into algebraic equations. However, the possibility of extraneous and losing roots must be considered when solving fractional equations, irrational equations, and transcendental equations. (7) The idea of solving equations of higher degree is order reduction and transformation, and the idea of solving system of equations is elimination, order reduction and transformation. (8) The definition domain of functions is often considered from the following aspects: ➀ The denominator is not zero; ➁ The radicands beneath radical sign with even numbers is greater than or equal to zero; ➂ In logarithmic expressions, the antilogarithm is greater than zero and the base number is greater than zero and not equal to 1; ➃ The base number of negative exponents is greater than zero; ➄ In tan x, x = kπ ± π2 (k ∈ Z );

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➅ In cot x, x = kπ(k ∈ Z ), etc. (9) The steps of constructing a function graph include setting values, fixing points and connecting lines. According to the definition, the steps of finding derivatives are to calculate difference, difference quotient, differential quotient. According to the definition, the steps of finding definite integral include division, approximation, summation, taking the limit. (10) To solve permutation and combination problems, we should, first of all, distinguish whether they are related to the order or not. The related ones are permutation problems, and the unrelated ones are combination problems. Secondly, for complex problems, we should consider the form of event completion, the addition principle is applicable to the events completed separately by each part, and the multiplication principle is applicable to the events completed continuously by segments. Of course, we can still list some overall or local methods and key points. Due to limited space, we hope readers can sum them up by yourselves. 3. Pay attention to the accumulation of problem-solving skills The usual solutions to quite a few mathematical problems are often cumbersome and tedious, but some solutions are very concise, clear and enlightening. This problem solving method is a skill that yields twice the result with half the effort. In the process of problem solving, we should not only pay attention to checking step by step to prevent errors, but also pay attention to problem solving skills step by step to prevent fussiness. Example 5 Given a =

√4 , 5−1

find the value of 21 a 3 − a 2 − 2a + 1.

√ √ 5 + 1, a − 1 = 5,   then, original formula = 21 a(a − 1)2 − 5(a − 1) − 3

Solution Since a =

√4 5−1

=

  √ 1  √ 5 5+1 −5 5−3 =1 2 Example 6 It is known that (z − x)2 = 4(x − y)(y − z). Prove: x + z = 2y. Proof 1 From the given conditions, we get (z − y + y − x)2 = 4(x − y)(y − z)   (x − y)2 + 2(x − y)(y − x) + (y − x)2 − 4(x − y)(y − z) = 0, that is, [(x − y) − (y − x)]2 = 0, (x + z − 2y)2 = 0, therefore, x + z = 2y. Proof 2 Let (x − y)t 2 + (z − x)t + (y − z) = 0. From the given conditions, it is obvious that  = (z − x)2 −4(x − y)(y − z) = 0 in the equation, so this equation has a repeated root t = 1,

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175

Then, according to the relations between the roots and the coefficients, we get y−z z−x = 2, or x−y = 1, − x−y After simplification, we get x + z = 2y. Example 7 Given that the parabola passes through three points A (− 1, 1), B (3, 0), C (− 2, 0), and the symmetry axis is parallel to the y-axis. Find the equation of this parabola. Solution Based on the given conditions, we can let the parabolic equation be y = a(x − 3) (x + 2), and then based on the condition that it passes through point A (− 1, 1), we get a = − 14 . So the desired parabolic equation is y = − 41 (x − 3)(x + 2). Obviously, the above three problem solving methods are very brief, with high skills. 4. Pay attention to multiple solutions to and multiple variations of one problem The so-called multiple solutions to a problem is to consider as many different solutions to the same problem as possible. Emphasizing multiple solutions to a problem is conducive to cultivating students’ ability to apply mathematical knowledge comprehensively. For example, some application problems of linear equations with one variable can be solved by means of system of linear equations with two variables; some algebraic problems can be solved by means of trigonometric and geometric methods; some geometric problems can be solved by means of algebraic, trigonometric, and coordinate methods, etc. Example 8 It is known that 0 < α, β < 2 sin 2β = 0. Prove: a + 2β = π2 .

π , 2

and 3 sin 2α + 2 sin 2β = 1, 3 sin 2α −

Example 9 Prove that in a right triangle, the sum of the hypotenuse and the height on the hypotenuse is greater than the sum of two legs. For Example 8, 10 solutions have been summarized; For Example 9, 22 solutions have been summarized. Can you write out more than five different solutions to each example? The so-called multiple variations of a problem refer to a problem that is appropriately changed into several problems that are different from the original problem in content, but with the same or similar solution method. The multiple variations of a problem is conducive to broadening students’ vision, deepening their knowledge, drawing inferences about other cases from one instance, comprehending by analogy, and improving their ability to solve problems. Example 10 After solving equation x 2 + y 2 = 0, can you find the real number solutions to the following equations respectively? ➀ (x + y + 1)2 + (x − 2y + 4)2 = 0; ➁ |x + y + 1| + |x − 2y + 4| = 0;

176

➂

5 Mathematical Ability in Secondary Schools

√

x +y+1+

√

x − 2y + 4 = 0.

  (z − y)2 + 2(z − y)(y − x) + (y − x)2 − 4(x − y)(y − z) = 0, [(z − y) − (y − x)]2 = 0, (x + z − 2y)2 = 0. Example 11 After proving that the sum of the sum of four consecutive integers and 1 gives a perfect square, we can change it into the following different questions, that is, we can apply the conclusion of x(x + 1)(x + 2)(x + 3) + 1 = (x 2 + 3x)(x 2 + 3x + 2) + 1 = (x 2 + 3x)2 + 2(x 2 + 3x) + 1 = (x 2 + 3x + 1)2 to answer the following questions conveniently: √ ➀ Evaluate 25 · 28 · 27 · 28 + 1. ➁ Factorize: x(x − 1)(x − 2)(x − 1) + 1, (3x − 8)(3x − 7)(3 − 6)x(3x − 5) + 1. ➂ ➃ ➄ ➅

Solve equation x(x + l)(x + 2)(x + 3) = − 1. Solve inequality (x + 2)(x + 3)(x + 4)(x + 5) > 143. 2x−7 . Find the domain of function; y = x(x+1)(x+2)(x+3)+1 Find the extremum of the function and graph it; y = √ 1 + x(x + 1)(x + 2)(x + 3). ➆ What is the curve represented by equation (y + 1)(y − 1) − x(x + 1)(x + 2)(x + 3) = 0? ➇ From how many elements do we take four elements at a time so that the number of all permutations is 93,024? The above exercises are quite difficult for all grades, but if students are trained on multiple variations of a problem at the appropriate time, they can be readily solved, which is conducive to improving the ability to solve the problems. 5. Pay attention to shape-number combination Numbers are characterized by abstraction and generalization, and shapes are characterized by concretization and visualization. In solving some mathematical problems, if we can pay attention to applying the shape–number combination and complementing each other, we will often get twice the result with half the effort.

5.5 Cultivation of Problem Solving Ability

177

Fig. 5.7 .

Example 12 What is the value of m, when the system of equations

√ y = a2 − x 2 y = x +m

(a > 0)

has one solution, two solutions, and no solution? Solution As shown in Fig. 5.7. Obviously, solving the system of equations becomes the problem of finding the intersection point of line family y = x + m and semicircle √ y = a 2 − x 2 , and it is not difficult to find when −a ≤ m < √a, the system has one solution; when a ≤ m ≤√ 2a, the system has two solutions; and when m > 2a or m < −a, the system has no solution. Example 13 Let |z| = 1, find the extreme values of |z + 2 + i|. Solution As shown in Fig. 5.8, the problem of finding the extreme values becomes the problem of finding the distance between point√P (− 2, − 1) and O. Obviously, √ PO = 5, the maximum value of |z + 2 + 1| is 5 + 1, and the minimum value is √ 5 − 1. 6. Pay attention to the generalization and association of propositions The generalization of a proposition is to generalize the conditions of the proposition so as to infer a more general conclusion. The so-called association of a proposition is to appropriately change the conditions and conclusions of a proposition after solving a problems, and to extend and expand them vertically and horizontally, so as to obtain new conclusions. Through the generalization and association of propositions, we often grasp the solution of not only a problem, but also a group of problems and a class of problems. If we can stick to doing so, we can cultivate students’ habit of delving into the exercises, stimulate our creative spirit in mathematics, which is Fig. 5.8 .

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undoubtedly very beneficial to improve our problem solving ability and innovation ability.  √ For example, after finding the value of 5 5 5 . . ., we can generalize it to find the value of   √ √ n n n a a a · · · (a > 0) ⇒ a a a · · · (a > 0, n is a natural number greater than 1)  n√ n1 ⇒ a n2 a 3 a · · · (a > 0 m i , n i are natural numbers greater than 1) and it with finding the value of associate   √ √ 5 + 5 + 5 · · · ⇒ a + a + a · · · ⇒ · · · and so on. For another example, after demonstrating that the sum of the distances from any point inside a regular triangle to three sides is a constant value, we can generalize this point from the inside to the outside; generalize regular triangle to isosceles triangle and any triangle in turn; generalize regular triangle to regular n-gon, etc. The linear equations can be generalized to circle system equations, elliptic equations, quadratic curve equations, any plane curve equations, etc. successively. Analogy association can also be carried out from many aspects. For example, the analogy association can be carried between fractional numbers and algebraic fractions, factors and factorization, the relationship between roots and coefficients of quadratic equations with one variable and equations of higher degree with one variable, congruence and similarity, equations and inequalities, rectangular coordinates and polar coordinates, exponents and logarithms, plane and space, etc. Example 14 Solve system of equations ⎧ ⎨ x + y + z = 3, x 2 + y 2 + z 2 = 3, ⎩ 3 x + y 3 + z 3 = 3,

1  2  3 

We can make an analogy between it and the system of equations of

x+y=a x 2 + y2 = a

Example 15 As shown in Fig. 5.9, let H be the orthocenter of acute triangle ABC, abc AH = m, BH = n, CH = p, BC = a, CA = b, AB = c. Prove: ma + nb + cp = mnp . Proof 1 It is rather cumbersome to apply proportion property. Proof 2 In ABC, since tan A · tan B · tan C = tan A + tan B + tan C, after comparing with it, we can get tan A =

BE a CF b AD c = , tan B = = , tan C = = AE m BF n CD p

5.5 Cultivation of Problem Solving Ability

179

Fig. 5.9 .

Fig. 5.10 .

Therefore, it is not difficult to prove the proposition. Example 16 Take points D, E, and F on each side of acute triangle ABC to minimize the sum of DE + EF + FD. Solution For this problem, it can be associated with the problems “find a point in a known line to minimize the sum of the distances between it and two known points outside the line” and “find a point P inside the known angle AOB to make the sum of distances between P and OA and OB be PC + PD and minimize the sum of PC + CD + PD”, thus solving this problem. As shown in Fig. 5.10, let point D be found, and draw symmetric points D1 and D2 of D with regard to AB and AC, then DE + EF + FD = D1 D2 . ∠D1 AD2 = 2∠BAC, AD1 = AD2 = AD. By the cosine theorem, in AD1 D2 , D1 D22 = AD12 + AD22 − 2 AD1 · AD2 cos ∠D1 AD2   = 2 AD 2 − 2 AD 2 1 − 2 sin2 ∠B AC = 4 AD 2 sin2 ∠B AC. So D1 D2 = 2 AD sin ∠B AC, which is the minimum ⇒ AD is the minimum ⇒ AD is the altitude.

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5.6 Problem Solving and Its Teaching 5.6.1 Meaning of Mathematical Problems Generally speaking, mathematical problems are usually divided into pure mathematical problems and mathematical application problems. The pure mathematical problems can be divided into two categories, one is conventional problems, usually the application and combination of some conventional algorithms or mathematical methods, mainly referring to the traditional practice questions and exercises in secondary school textbooks or teaching references; the other is unconventional problems. Such problems generally cannot be solved by imitating or applying relevant mathematical algorithms and rules mechanically, and can be solved only through exploration and flexible use of various mathematical knowledge, mathematical thoughts and methods. They mainly include the problems from real life and production and from other disciplines that need to be solved by means of mathematical knowledge.

5.6.2 Problem Solving After Hilbert delivered a speech Mathematical Problems at the International Congress of Mathematicians in 1900, solving mathematical problems has become a motive force to motivate mathematicians to advance the development of mathematics. Since the 1980s, “problem solving” has been one of the centers of international mathematics education reform, and become an important measure to implement innovative education and cultivate students’ innovative ability. However, the “problem” in problem solving does not include conventional mathematical problems with established methods, but refers to unconventional mathematical problems (also known as non-standard problems) and the application of mathematics. People have different understandings of “problem solving” and thus give many meanings. First, American Professor Berg said, “The real reason to teach mathematics is that mathematics has a wide range of applications, and teaching mathematics should be conducive to solving various problems”. “Learning how to solve problems is the purpose of learning mathematics”. This is a kind of view taking problem solving as a learning purpose. Second, the National Advisory Committee on Mathematics Education (NACOME) in the United States believes that problem solving is a basic mathematical skill, and they have done a lot of explorations and researches on how to define and evaluate this skill. This is a kind of view taking problem solving as a skill.

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181

Third, Kirkcroft and others in the UK believe that “discussion, research, problem solving and exploration should be added to the forms of teaching”, and he pointed out: “In the UK, teachers are far from adopting problem-solving as a form of mathematics”. This is a kind of view taking problem-solving as a form of teaching. Fourthly, according to Lebranse of the United States, “the cognitive structure about the process that the individual has formed is used to deal with the problems facing the individual”; the National Council of Supervisors of Mathematics (NCSM) further defines problem solving as “the application of previously acquired knowledge to a new unfamiliar process”. This is a kind of view taking problem solving as a process.

5.6.3 Teaching of Mathematical Problem Solving In view of the important role of mathematical problem solving in cultivating students’ ability (especially creative thinking ability) and strengthening the application consciousness of mathematics, mathematical problem solving and its mathematical problems have attracted extensive attention from the education circles of mathematics in China. The following is a brief discussion on its teaching. 1. Create problem situations Generally speaking, the mathematical problem solving starts from a new mathematical problem situation, it is an exploration process to use the known knowledge to seek the way to solve the problem and to achieve the purpose of solving the problem. Therefore, in the secondary school mathematics teaching, the introduction of new teaching content from realistic problems that students are familiar with is not only conducive to creating vivid problem situations to stimulate students’ thirst for knowledge, but also conducive to making students naturally acquire mathematical knowledge and skills, and helps to train their exploring spirit and creative thinking ability as well. Example 1 A couple deposits a sum of money in the bank on each birthday since their only child is born to pay the tuition at college for the child in the future. The current college tuition is about 4000 yuan a year, which will increase at a rate of 5% a year in consideration of rising prices. The current annual bank interest is 3.60%, which is assumed to remain unchanged in the following 18 years, calculated by compounding. If when the child goes to college at 18, the parents have saved the tuition for 4 years, so how much money should they deposit on each birthday? Solution The total tuition for 4 years is 16,000 yuan, which, at the rate of increase of 5%, should be the following sum after 18 years. 16,000(1 + 5%)18 ≈ 16,000 × 2.4066 = 38,506

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Let the couple deposit x yuan every year. Calculated by compounding, the capital and interest after r years is x(1 + 3.60%)r . The couple have deposited 18 sums of money, and their capital and interest is 18 

x(1 + 3.60%)r ,

r =1

 then from r18=1 x(1 + 3.60%)r = 38,506, we get x = 1504 yuan. That is so say that the couple need to deposit about 1504 yuan in the bank every year to pay their child’s intuition at college. 2. Design mathematics problems meticulously In order to cultivate students’ ability to solve problems, in addition to organizing students to practice the basic exercises and skill training problems in the textbooks, special attention should be paid to collecting, sorting, selecting, and designing unconventional mathematical problems and practical application problems, which is very necessary for the reform of mathematics teaching and the promotion of students’ mathematics learning. Example 2 Eight persons go to a party. If they shake hands once every two persons, how many times will they shake hands? This is an unconventional problem. We can guide students to explore a variety of solutions. Solution 1 Tabulation method We use 1, 2, 3 …, and 8 to represent eight persons respectively, use symbol 1–2 to indicate that persons 1 and 2 shake hands once, and so forth. It can be seen from the table below that they shake hands 28 times in total. Person

1

2

3

4

5

6

7

8

Times of handshakes

1

0

1–2

1–3

1–4

1–5

1–6

1–7

1–8

7

2

0

0

2–3

2–4

2–5

2–6

2–7

2–8

6

3

0

0

0

3–4

3–5

3–6

3–7

3–8

5

4

0

0

0

0

4–5

4–6

4–7

4–8

4

5

0

0

0

0

0

5–6

5–7

5–8

3

6

0

0

0

0

0

0

6–7

6–8

2

7

0

0

0

0

0

0

0

7–8

1

8

0

0

0

0

0

0

0

0

The times of handshakes are 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

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Solution 2 Graphic method We can regard eight persons as eight vertices of a convex octagon, that is, to establish a mathematical model of octagon, then the times of handshakes of eight persons are the sum of the number of sides and diagonals of the octagon, that is, 8+

8(8 − 3) = 28 2

Solution 3 Induction method We can simplify the problem and draw a general conclusion by analogy. Obviously, when the number of persons is 2, 3, 4, 5 …, the times of handshakes are 1, 3, 6, 10 … respectively. This is a first-order arithmetic sequence, and the seventh term is 28, which means that 8 persons shake hands 28 times. Similarly, it can be further inferred that n persons shake hands for a total of 1/2n(n + 1) times. 3. Enable students to master problem solving strategies In teaching, attention should be paid to helping students to sum up the strategies and methods of problem solving, and make them form their independent consciousness. For example, ➀ ➁ ➂ ➃ ➄

Tabulation and graphic analysis method; Method of constructing mathematical models; Try to solve related simple problem; Generalize or specialize the problem; Transform the problem to simplify it, and so on.

4. Organize teaching support activities To strengthen the teaching of “problem solving”, first of all, it should be incorporated into the teaching plan. Because more teaching time is often needed, it should be carried out selectively and emphatically in order to inspire students. At the same time, according to the actual situation, lectures can be offered or interest groups can be set up to carry out activities under the guidance of teachers, and students can study and solve the problems collectively, and write the problem solving report, respectively. In this regard, we should advocate active exploration in order to sum up the teaching experience in line with China’s national conditions. Finally, it should be pointed out that the inquiry-based learning topic has become a new content of the current secondary school mathematics teaching. In terms of its learning style, it is discovery and inquiry learning; as far as the learning content is concerned, it is “problem solving” learning. In this regard, it should be implemented

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in accordance with the teaching requirements of the curriculum standards. Due to limited space, it will not be detailed here. Review Questions and Exercises 1. What is knowledge and ability, and what is the significance of developing ability in secondary school mathematics? What do you think is the fundamental approach to developing secondary school mathematics ability? 2. What is operational ability? Try to illustrate how to develop it with examples. 3. What is logical thinking ability? Try to illustrate how to cultivate students’ logical thinking ability with examples. 4. What is the spatial imagination ability? Try to illustrate how to cultivate students’ spatial imagination ability with examples. 5. What types of secondary school mathematics exercises are there, what are the basic requirements for solving secondary school mathematics exercise and the thinking process of solving problems? Illustrate them with examples. 6. What are the basic ways and methods to improve students’ ability to solve problems? Illustrate them one by one with examples. 7. What are the principles for selecting and arranging mathematical exercises? Try to arrange a set of exercises for the teaching content of a certain section of secondary school algebra and geometry. 8. What is the significance of paying attention to summarizing the methods and essentials of problem solving? Try to summarize the methods and essentials of solving (proving) application problems and proving that triangles are congruent with linear equations with one unknown. 9. What is problem examining, and what requirements should be met for the problem examining? Illustrate them with examples. 10. What is problem solving and how should the teaching of mathematical problem solving be conducted? 11. What are multiple solutions to and multiple variations of one problem? Solve the following problems from this viewpoint: (1) It is known that a + b + c = l. Prove: a 2 + b2 + c2 ≥ 13 ; (2) Solve the inequality |2x − 1| < |x − 1|; (3) It is knowing CD is the median of side AB of ABC, CE is the median of side AD of ADC, and AD = AC. Prove: BC = 2CE; 

(4) It is known that P is a point on BC of the circumcircle of regular triangle ABC. Prove: PA = PB + PC; (5) As shown in Fig. 5.11, it is known that in rectangle ABCD, A A√1 = A1 A2 = A2 A3 = A3 A4 = A4 D = a, AB1 = B1 B2 = B2 B = 3a, and A1 E||AB. Prove: ∠B1 A1 E + ∠B2 A1 E + ∠B DC = π2 . 12. What is the generalization and association of propositions? Try to solve the following problems from this point of view:   (1) It is known that a > 0, b > 0. Prove: (a + b) a1 + b1 ≥ 4;

5.6 Problem Solving and Its Teaching

185

Fig. 5.11 .

(2) Prove: 12 + 22 + 32 + · · · + n 2 = 16 n(n + 1)(2n + 1); (3) It is known that AC⊥AB, BD⊥AB, AD intersects BC at E, EF⊥AB at F, and AC = p, BD = q, EF = r, AF = m, FB = n. ➀ Represent r/p, r/q with m and n respectively; ➁ Prove: r/p + r/q = 1. (4) The sum of the distances from any point on the base of an isosceles triangle to two waists is a fixed value; (5) It is known that the two ends of line segment AB which is 2a long slide on two mutually perpendicular straight lines. Find the locus of the midpoint of line segment AB; (6) In ABC, D1 and D2 are the trisection points of side AB, and E 1 and E 2 are the trisection points of side AC. Prove: S D1 D2 E2 E1 = 13 SABC . 13. Select the review questions of a certain chapter of middle or high school mathematics, find multiple solutions to and multiple variations of one problem, and generalize and extend the propositions. 14. It is known that l + a + b = 0. Prove: (1 − a)2 + (1 − b)2 ≥ 29 . 2 2 15.  Given that a, b,  x, and y are all positive numbers, and x + y = 1. Prove: 2 2 2 2 2 2 2 2 a x + b y a y + b x ≥ a + b. 16. It is known that in triangular pyramid P-ABC, PA⊥BC, PA = BC = l, and the common perpendicular ED of PA and BC ED = h. Prove: V P−ABC = 16 l 2 h. 17. As shown in Fig. 5.12, in trapezoid ABCD, given that |AB| = 2|C D|, point E −→ divide the directed line segment AC into constant ratio λ, the hyperbola passes through points C, D, and E, with A and B as the focus. Find the value range of the hyperbolic eccentricity e when 23 ≤ λ ≤ 43 . 18. In order to promote the development of freshwater fish farming and control the price within an appropriate range, a local government decides to provide government subsidies for freshwater fish farming. Suppose the market price of freshwater fish is x yuan/kg, and the government subsidy is t yuan/kg. According to market research, when 8 ≤ x ≤ 14, the daily market supply of freshwater fish P kg and the daily demand Q kg approximately satisfy the relations: P =

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Fig. 5.12 .

 1000(x + t − 8)(x ≥ 8, t ≥ 0), Q = 500 40 − (x − 8)2 (8 ≤ x ≤ 14). The market price when P = Q is called the market equilibrium price. (1) Express the market equilibrium price as a function of government subsidies and find the domain of definition of this function; (2) In order to keep the market equilibrium price not higher than 10 yuan/kg, how much is the government subsidy per kilogram at least? 19. The distance between places A and B is s (km), and a car travels at a constant speed from place A to place B, with the speed not exceeding c (km/h). It is known that the hourly transportation cost of the car (in yuan) consists of a variable part and a fixed part: the variable part is proportional to the square of speed (km/h), with the proportionality coefficient is b; and the fixed part is a yuan. (1) Express the total transportation cost y (yuan) as a function of speed v (km/ h), and point out the domain of definition of this function; (2) How fast should the car travel in order to minimize the total transportation cost? 20. Shanghai is located at 30° N and 120° E and Los Angeles is at 30° N and 120° W. Try to find the arc length (small arc) between Shanghai and Los Angeles on the 30° N circle, and find out the arc length (small arc) on the big circle passing through Shanghai and Los Angeles, and explain why China Eastern Airlines usually fly through the Aleutian Islands in Alaska, USA when flying from Shanghai to Los Angeles.

Chapter 6

Secondary School Mathematics Learning

Learning mathematics is an extremely complex process of psychological activity. Some laws of psychology are of great significance to guide secondary school mathematics learning. Starting from the actual needs of secondary school mathematics teaching, this chapter makes a preliminary discussion on some basic issues of secondary school mathematics learning theory, learning psychology, and learning methods.

6.1 Characteristics of Secondary School Mathematics Learning What is learning? This is a familiar but difficult question to answer. At present, people tend to believe that learning refers to the process of knowledge and experience acquisition and behavioral changes. In other words, not all behavioral changes are learning, only the behavioral changes based on accumulated knowledge and experience are learning, and learning is a process of continuous gradual improvement. The research on learning theory in ancient China was earlier. In modern times, foreign learning theories have developed rapidly, with many academic schools and different viewpoints. In summary, there are two representative basic viewpoints.

6.1.1 Associationistic View of Learning This is the view of stimulus–response association represented by Thorndike, Pavlov, Skinner, Bandura, etc.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_6

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Thorndike believes that “learning is association”. It is a process of continuous “trial and error” until success. Through experiments in which the salivation reaction of dogs was induced by the combination of certain sound and meat several times, Pavlov believed that learning was the formation of a temporary neural connection, and a classic conditioned reflex. Through the experiment in which a white mouse stepped on the joystick occasionally to get food, and then continued to press the joystick until it was full, Skinner believed that learning was a conditioned reflex of operating a certain tool under reward. Bandura believed that learning was a kind of observation and imitation and a process of continuous self-reinforcement.

6.1.2 Cognitive View of Learning This is a cognitive view represented by Gestalt, Tolman, Bruner, Ausubel, etc. Through observations that the chimpanzee could produce similar behavior after picking up a short stick and hitting a banana from a high place, Gestalt believed that learning was a process of comprehension. Through experiments in which rats walked out of the maze, Tolman believed that learning was a potential cognitive structure, lying in neither reward nor in the process of self-reinforcement. Bruner and Ausubel believed that learning is the organization and reorganization of cognitive structure. They emphasized not only the function of original experience, but also the internal logical structure of learning material itself. In summary, it is thus clear that learning is a certain change that occurs after learners have undergone certain training. This change is complex, in the aspect of movement, emotion, and cognition. The psychological function of this change is also diverse, including conditioned reflex, trial and error, imitation, and comprehension. The reasons for these changes include the factors of learning situation and learners themselves, etc. The author agrees with the viewpoint of cognitive theory and studies the learning of secondary school mathematics on this basis. Secondary school mathematics learning is carried out in accordance with the requirements of secondary school mathematics teaching plan and purpose. It is a process of long-lasting behavioral changes caused by gaining mathematical knowledge and experience. It has the following striking features: 1. Secondary school mathematics learning is a rediscovery based on human discovery. In the process of secondary school mathematics learning, on the one hand, students mainly master indirect experience, and know the truth obtained by predecessors through discovery; on the other hand, the rediscovery of students has been processed by didactics, making their discovery process be a “shortcut” with moderate difficulty,

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which can be achieved through efforts. For example, the concept of number came into being thousands of years ago. It was not relatively complete until the nineteenth century, and people’s understanding of it has gone through a long historical process. However, when secondary school students learn the concept of number, it is a process of rediscovery, which is different from the long and primitive process of the ancients’ discovery of the concept of number. It has been processed by certain didactics, and adjusted both in order and difficulty, so that students can fully grasp it in just a few years. 2. Secondary school mathematics learning is carried out in a purposeful and planned way. Generally speaking, human learning is carried out through trial and error, and has gone through a long process. However, as far as the learning of secondary school mathematics is concerned, its content has been processed by didactics, and the learning process is carried out in a purposeful and planned way under the guidance of teachers. This kind of learning is based on curriculum standards, certain courses and textbooks, under the guidance of teachers, and has certain methods for reference. This avoids the tortuous road of repeated exploration in learning and can ensure a better learning effect in a short time. 3. Secondary school mathematics learning focuses on knowledge acquisition and ability training. Since most of the current secondary school textbooks adopt deductive methods to form mathematical facts into a unity, they thus conceal its vivid and lively discovery and invention historical process. Therefore, in learning secondary school mathematics, we should not only master formal mathematical conclusions truly, but also grasp the rich facts behind the formal conclusions, learn to observe and analyze, improve the ability of abstraction and generalization, the ability of logical reasoning and mathematical thinking. Only when the basic ability is improved can we truly learn secondary school mathematics well and have the ability to further study and research mathematics. As for the types of secondary school mathematics learning, by learning method, they can be divided into reception learning and discovery learning. By psychology of learning, they can be divided into rote learning and meaningful learning. We should attach importance to meaningful reception learning and meaningful discovery learning. For example, look at the following set of symbols (Fig. 6.1).

Fig. 6.1 .

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Fig. 6.2 .

The set of symbols has been used to represent containers, sports, water, writing, knowledge, electricity, people, control, devices respectively, and form more than 1000 different patterns to represent vehicles, ships, communications, letters, sailors, drivers, etc. for experiments (Fig. 6.2). In the first two weeks, the capacitance of comprehensive memory was twice as much as that of rote memorization, and it became seven times after two weeks. At present, a learning method of “SQ3R” is popular abroad. “SQ3R” is the acronym of five English words: Survey, Question, Read, Recite, and Revise. “SQ3R” is also called five-stage learning method. This method enables the comprehensive application of some psychological principles related to learning and memory, which can be applied to all disciplines in principle. In terms of secondary school mathematics learning, we should do a good job of thinking, self-study, taking lessons, problem solving, memorizing, reviewing, and application, etc., and master the learning methods in each step. Among them, the thinking and problem solving have been explained above. In addition, attention should be paid to mastering the methods of self-study methods, methods of taking lessons, review methods, memorizing methods, and application methods, etc.

6.2 Process of Secondary School Mathematics Learning 6.2.1 Cognitive Structure As we know, the knowledge structure of mathematics, in a broad sense, refers to the knowledge system, classifications, procedures and internal relations among knowledge of a certain subject of mathematics; in a narrow sense, it refers to the basic concepts, basic theorems, formulas, rules and basic relations among them of a certain unit, etc. For example, the research object of unit “straight line and plane” in solid geometry is: two basic elements that make up simple spatial figures—straight line and plane. The knowledge structure of this unit is to study three sets of relations: the relative position of straight lines; the relative position of straight line and plane; the relative position of planes. The internal relations of these three sets of relations are as follows (a and b represent two different straight lines, and α and β represent two different planes):

6.2 Process of Secondary School Mathematics Learning

a//b b∈a

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 ⇒ a//α or a ∈ α

⎫ a//α ⎬ ⇒ a · b a and b determine the only plane a that makes β//a b//α ⎭ a ∩ b = ∅  a⊥α ⇒ α ⊥ β etc. a∈β Their research methods are as follows: Qualitative—Intersecting, parallel, perpendicular, different planes, etc.; Quantitative—Distance and angle. This unit contains 4 axioms and 12 basic theorems. Among them, there are the decision theorem and property theorem of two parallel lines, the decision theorem and property theorem of a straight line perpendicular to a plane, the decision theorem and property theorem of two parallel planes, the decision theorem and property theorem of two perpendicular planes, the three perpendicular lines theorem and its converse theorem, the theorem of two equal angles and the projective theorem, etc. Cognition is the whole process in which the perceived information is converted, simplified, stored, restored, and applied in human brains. In cognitive activities, the input information is processed by organizing or reorganizing to form a generalized general cognitive model, namely cognitive structure. The so-called cognitive structure of mathematics is the structure of mathematical knowledge in students’ minds. The mathematical cognitive structure of each secondary school student has its own characteristics, and the characteristics of individual cognitive structure in content and organization are called cognitive structural variables. They can be divided into: ➀ General (long-term) cognitive structure variables—The content and organizational characteristics of all knowledge structure in mathematics subjects of secondary school students, and these characteristics affect their future achievements in mathematics subjects. ➁ Special (temporary) cognitive structure variables—The content and organizational characteristics of the concepts and propositions in their cognitive structure that affect this new knowledge unit and have a direct relation when secondary school students learn a relatively small knowledge unit. For example, when students learn the concept of fractional exponent, there may be the following four types of short-term cognitive structure variables: The first type, one can master the operations of integer exponents, but is vague about the concept and theories. The second type, one can be more skilled at the operations of integer exponents and can explain relevant theories, but is not clear about why the concept and operations of exponents should be generalized.

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The third type, one is clear about the concepts and more proficient in operations, but only rests on understanding the meaning of defined new exponents. The fourth type, one can further realize that only after introducing zero exponent and fractional exponents can fractional and integral, rational and irrational expressions be made harmonious and unified, so as to facilitate operations. In fact, the students with the fourth cognitive structure can truly understand and be in a state of “determined” and “puzzled”. With only a few directions, they may raise a question themselves “Can it be generalized to irrational exponents and complex exponents?” The knowledge structure of mathematics is the research object of mathematicians, and the cognitive structure of mathematics is the research object of psychologists. The differences are as follows: ➀ The knowledge structure of mathematics is a summary of experience accumulated by predecessors in studying mathematics in practice. It is objective and an external thing to students. The cognitive structure of mathematics is a cognitive model that students gradually form in their own minds when they study mathematics. It is subjective and an internal and psychological thing for students. ➁ The knowledge structure of mathematics is organized in a certain order in the textbooks, which students can master through learning. The cognitive structure of mathematics is the intelligent activity mode for students to recognize these mathematical contents. It can be divided into right or wrong, good or bad. To a certain extent, it reflects the ability of learning mathematics to adapt to the learning of similar mathematics knowledge. ➂ The content of the same mathematical knowledge structure can be mastered through different mathematical cognitive structures. The accumulation of pure mathematical knowledge does not mean the formation of mathematical cognitive structure. The cognitive structure of mathematics has a development process from the simple to the complex, from a low level to a high level. However, there is a close connection between knowledge structure and cognitive structure of mathematics. This is because the cognitive structure of students in learning cannot be separated from the knowledge structure of mathematics. When a certain pattern of cognitive structure is formed, the relevant knowledge structure is mastered correspondingly. At the same time, in the process of learning mathematics, if people discover new cognitive models through creative thinking, they can enrich the teaching content and thus develop or reorganize the knowledge structure of mathematics in turn. In fact, the process of learning mathematics can, so to speak, be a process in which human knowledge structure of mathematics is transformed into the learners’ knowledge structure of mathematics. It is also the result of unique mathematical cognitive structure formed by predecessors in solving mathematical problems, which is transformed into the common knowledge wealth of human.

6.2 Process of Secondary School Mathematics Learning

Situation

New

Input stage

Learning content Generating a new cognitive structure of mathematics

Original cognitive structure of mathematics

Operation stage

Forming a new

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Interaction stage Expected goal

cognitive structure of mathematics

Fig. 6.3 .

In mathematics teaching, we should strive to master the knowledge structure of teaching content and be good at analyzing the cognitive structure of students’ learning. Only when the two are combined organically and reach the optimum point, can the best teaching effect be achieved. 1. General process of secondary school mathematics learning According to the cognitive theory of learning, the learning process of secondary school mathematics is a cognitive process of mathematics. This process includes input stage, interaction stage of new and old knowledge, and operation stage. The general model is shown in Fig. 6.3. First, learning originates from learning situation. The input stage is actually to provide students with new learning contents and create learning situation. In this learning situation, the student’s original mathematical cognitive structure conflicts with the new learning content, resulting in a psychological need for learning new knowledge, which is the key to the input stage. Therefore, at this stage, the new contents provided by teachers should be suitable for students’ abilities and interests, to stimulate their internal learning motivation. After the learning need come into being, the students’ original mathematical cognitive structure and new learning content take effect, and enter the interaction stage in two basic forms: assimilation and adaptation. Among them, the process in which the new knowledge is incorporated into the original mathematical cognitive structure and further expands the original knowledge is called assimilation. If the new knowledge does not have proper knowledge related to it in the original mathematical cognitive structure, then the original mathematical cognitive structure must be reorganized or partially reorganized to form a new mathematical cognitive structure and integrate the new knowledge, and this process is called adaptation (regulation). For example, in the case of students learning negative rational numbers, when the concept of negative rational numbers is input, students screen out the mathematical cognitive structure in their minds that can incorporate negative rational numbers—the cognitive structure of positive rational numbers. According to this cognitive structure, students reform negative rational numbers and establish a connection with positive rational numbers: the properties of negative rational numbers are opposite to positive rational numbers, and the addition and subtraction of negative rational numbers can

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be defined by positive rational numbers, etc. In this way, negative rational numbers are assimilated into the cognitive structure of positive rational numbers, and the original cognitive structure of positive rational numbers is expanded to the cognitive structure of rational numbers. For another example, when students learn analytic geometry, due to the difference between elementary geometry and analytic geometry research methods, students cannot simply rely on assimilation method to learn analytic geometry based on the original cognitive structure of elementary geometry. It is necessary to reform the original cognitive structure of elementary geometry, and realize the correspondence between shapes and numbers through the establishment of a coordinate system. Only after learning the part of the curves and equations, they can gradually adapt to the learning of analytic geometry. The last stage of learning process is operation stage, which is essentially the process of forming a new mathematical cognitive structure through exercises and other practical activities based on the new mathematical cognitive structure generated at the second stage. Its characteristic is to further develop mathematical thinking activities and guide students to solve mathematical problems, and the purpose is to make the cognitive structure of mathematics generated by students better-constructed and achieve the expected teaching goals.

6.3 Intelligence and Non-intelligence Factors 6.3.1 Intelligence Factors What is intelligence? At present, there is a multiplicity of views on the definition of intelligence in psychological circles. Chinese scholars tend to believe that intelligence is the synthesis of various stable psychological characteristics that guarantee people to carry out cognitive activities successfully. It is composed of five basic factors: power of observation, memory, imagination, thinking ability and attention, etc. Among them, power of observation is the foundation and thinking ability is the core. In the process of mathematics learning, thinking ability and imagination begin to differentiate, gradually forming logical thinking and spatial imagination ability, which become an important part of mathematical ability. For this reason, here we only give a brief introduction to power of observation, memory, and attention. 1. Power of observation Power of observation refers to the ability to observe the characteristics of things comprehensively, correctly, and deeply. The power of observation in mathematics learning is mainly manifested in the ability to observe the characteristics of things in such two aspects as “number” and “shape” quickly, and discover the internal connection from the form and structure of the problem. For example, observe some

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special figure or relations from a geometric figure, and observe its characteristics and mutual relations from a mathematical formula. To improve power of observation, we must learn how to observe, be good at observation, observe in a purposeful and step-by-step way, from different levels and different aspects, and observe through the appearance by grasping the main levels, main aspects and main characteristics. √ √ For example, for equation X − 3+ 5X − 1 = 0, people with weak observation ability can only conclude that this is an irrational equation, while those with strong observation ability can also find that this is an equation with no solution. For another example, for the proposition “It is known that 0 < a, b < 1”, prove:    a 2 + b2 + (1 − a)2 + b2 + a 2 + (1 − b)2  √ + (1 − a)2 + (1 − b)2 ≥ 2 2. Observing the form of the inequality to be proved, we can find that as long as the inequality of 

x2

+

y2

√ 2 (x + y) (x, y ∈ R), ≥ 2

is applied, the proposition can be proved, that is,    a 2 + b2 + (1 − a)2 + b2 + a 2 + (1 − b)2 √  2 2 2 + (1 − a) + (1 − b) ≥ 2 √ [a + b + (1 − a) + b + a + (1 − b) + (1 − a) + (1 − b)] = 2 2. If you associate the modulus of complex number when observing this inequality, you can find the complex number proof. If you associate the relevant line segments in the square, you can find the graphic method. The two proof methods discussed here are left as an exercise for the readers. 2. Memory Memory is the ability to maintain and reproduce the impression left in the mind of things that people have known or done. It can be divided into three basic stages: memorization, retention, and reproduction (or recognition). Mathematical memory is the ability to remember the laws, structures, problem-solving methods and exploration principles of mathematics. To improve the memory ability of mathematics, firstly, it is necessary to have a deep understanding of the content learnt and master

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its essence and internal connections. Secondly, we should pay attention to the use of appropriate memory methods. The common methods are as follows: ➀ Imaginal memory method This is a memory method that endows abstract mathematical knowledge with a vivid visual image, that is, it combines the meaning and image of mathematical objects. For example, through graphs of exponential and logarithmic functions, it is easy to remember their related properties. ➁ Analogical memory method This is a memory method by mastering the similarities and differences of things of different quality with same forms and things of same quality with different forms. For example, for arithmetic sequence and geometric sequence, analogical memory can be used in the definition, general term formula, mean term, formula of the sum of first n-terms, etc. ➂ Systematic memory method This is a memory method that organizes and systemizes the characteristics of related things and their internal relations through comparison and classification to form a knowledge network. For example, the knowledge of equations, functions, inequalities, and the knowledge about ellipse, hyperbola, and parabola in secondary school mathematics can be systematized in the form of tables to help memorize. 3. Attention Attention refers to the ability of man’s mental activity to direct to and focus on a certain object. It has the qualities of scope of attention, stability of attention, distribution of attention and shifting of attention, etc. The scope of attention, that is, the span of attention, can clearly grasp the number of objects at the same time. Generally speaking, the more regular the perceptual objects are arranged, the stronger the overall quality or the richer the personal knowledge and experience is, the wider the scope of attention is. The stability of attention, that is, the persistence of attention, refers to the fact that attention stays on a certain thing or activity for a long time, which is the characteristic of attention in time. Generally speaking, if the activity is rich in content, changeable and can arouse interest, and the attention can be stable and lasting. The distribution of attention refers to the fact that attention is directed to more than two different objects or activities at the same time. Generally speaking, the distribution of attention is conditional. In the same activities, the attention is often focused on unfamiliar activity according to the level of proficiency. The shifting of attention refers to the rapid and proactive shifting of attention from one object to another according to new needs. For example, when sketching the graph of a function, attention is shifted from the research of function properties to finding the value of function, and then to sketching the graph to achieve the intended goal.

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6.3.2 Non-intelligence Factors In a broad sense, the so-called non-intelligence factors are a general term for all psychological factors other than the intelligence factors that are beneficial to people in various activities (including learning activities). In a narrow sense, nonintelligence factors are composed of five factors: motivation, interest, emotion, will, and personality. From the analysis of their own factors that affect students’ meaningful learning of mathematics, it is clear that in addition to their own cognitive structure, cognitive development level (thinking development level), intelligence and other cognitive factors, students are also strongly affected by their own non-intelligence factors. Only when intelligence factors and non-intelligence factors develop in a coordination, can good learning benefits be produced. The development from acquiring basic knowledge to the combination of acquiring basic knowledge and cultivating ability, and from cultivating intelligence to the combination of cultivating intelligence and nonintelligence factors, is the in-depth development of mathematics teaching reform. Here is a brief description of learning motivation, learning interest, and learning will in non-intelligence factors. 1. Learning motivation Learning motivation is a driving force to promote students’ learning directly. Motivation comes from needs. When people have a certain need, they have a desire to satisfy their needs. When there are conditions for them to satisfy their desire, the motivation and enthusiasm for action arise. Learning is the need of human society and everyone. It is not only the need of the country and society, but also the need of personal development for secondary school students to learn mathematics. Starting from the need, we can effectively cultivate and stimulate students’ learning motivation. Due to the influence of society and the complexity of educational situation, different students have different learning motivations, and even the same student sometimes interweave several different learning motivations. Some psychologists divide the learning motivations of students into two types: the motivation of pursuing success and the motivation of avoiding failure. The motivation of pursuing success refers to the learning motivation of learners who try to use their talents to overcome learning obstacles, complete learning tasks, and achieve excellent results. When Chen Jingrun, a mathematician, was in secondary school, he developed a strong desire to learn mathematics after listening to the teacher’s vivid explanation of the story of Goldbach Conjecture. This is a motivation of pursuing success. The motivation of pursuing success is a positive motivation. Relatively speaking, the motivation of avoiding failure is a negative one. At the same time, various experiments show that when the motivation of pursuing success is greater than that of avoiding failure, people are likely to achieve good results. Otherwise, people are unlikely to achieve good results.

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The learning motivation of primary and secondary school students goes through a development stage from the near to the distant (i.e., development from direct and close motivation to indirect, long-term motivation), from incomplete maturity and incomplete correctness to maturity and correctness gradually, and from low to high. Therefore, motivation is constantly changing with age and knowledge. 2. Learning interest Interest is the tendency of people who are fond of certain activities or strive to know certain things, and is connected with certain emotions. Interest is also generated on the basis of needs, formed and developed in the course of life practice. The interest of learning mathematics is a tendency of students to approach or recognize mathematics objects and learning mathematics activities. The ancients said: “They who love it are better than those who know it”. Keen interest is an important factor in learning mathematics well. In teaching, experienced teachers always try their best to arouse students’ interest in learning pointedly through a variety of ways. Interests are divided into direct interest and indirect interest. Direct interest is the interest aroused by the need for things themselves. In the secondary school mathematics curriculum, special attention should be paid to strengthening the teaching of stimulating students’ direct interest. As far as indirect interest is concerned, one only thinks the future results of the things or activities are important, but takes no interest in such things themselves. For example, some students are not fond of mathematics, but in view of the need for the construction of the Four Modernizations or a sense of historical responsibility, they study mathematics hard. This is an indirect interest. These two types of interests are interconvertible. For example, students who realize the importance of mathematics to the construction of the Four Modernizations and study hard have cultivated an interest in learning mathematics, which becomes a hobby, so indirect interest is converted into direct interest. 3. Learning will Will is a psychological activity process in which people make conscious efforts to achieve a predetermined purpose. With correct learning motivation and keen learning interest, various difficulties will be encountered in the process of learning mathematics. How to strengthen our confidence, treat difficulties seriously and then overcome them, so as to obtain knowledge, skills, and abilities, we will have experienced a process of will. In mathematics learning, quite a few poor students are not full of self-confidence, self-esteem, and self-respect, but full of self-doubt, self-abasement, and selfabandonment. Instead of going upstream, they flinch from a difficulty. This is a manifestation of lack of strong will. Therefore, only by cultivating students’ tenacious will and great perseverance can they improve their mathematical achievement.

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Like engaging in scientific research, learning mathematics doesn’t have a favorable wind all the way. It is full of difficulties and obstacles. Only those who are persistent and dauntless to overcome all kinds of difficulties can achieve good results and even make inventions and innovations, as stated by Marx, “There is no royal road to science, and only those who do not dread the fatiguing climb of its steep paths have a chance of gaining its luminous summits”. A good will quality is proactive, independent, persistent and decisive, which is a necessary condition for learning mathematics well. In order to cultivate students’ will to learn mathematics well, it is necessary to, in combination with teaching, carry out education on learning purpose and ideals regularly, cultivate patriotism and collectivism, stimulate learning responsibility, and help them develop strong faith. At the same time, a strong will is formed in difficulties, so some difficult situations must be consciously created in teaching to allow students to temper their will. Be strict with them in learning, and guide and encourage them in difficult situations, so as to gradually improve their learning will and develop a good will quality.

6.4 Teaching in Conformity with the Laws of Psychological Activities Mathematics teaching must be in line with the laws of students’ psychological activities. To this end, we should pay attention to the following aspects. 1. Make clear the purpose of learning and stimulate the interest in learning. As we all know, at the beginning of middle school, the introduction of negative numbers is new to students, so they have a strong thirst for knowledge and are very enthusiastic about learning. Specifically, they listen attentively, complete the homework earnestly, and are able to cite some applications of rational numbers in connection with reality. An experienced teacher will cherish the student’s enthusiasm for learning, in order to cultivate their interest in learning algebra and develop good learning habits. It is thus clear that psychological motivation is the driving force for learning mathematics, and active psychological behavior plays a positive role in learning. To this end, in addition to the necessary ideological and political work, we should also cultivate learning interests and hobbies and arouse the enthusiasm for learning from the intellectual charm of mathematics itself. Its charm lies in the fact that mathematics is a tool of all sciences, a gymnastics to train thinking, a scientific language, and an important method of exploration and discovery. 2. Cultivate attentiveness and maintain strong enthusiasm for learning. Attention can be divided into incidental and spontaneous attention. Due to the age characteristics of secondary school students, sustained attentiveness is generally limited. It is the case in classroom teaching, as well as in a certain stage of learning.

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For example, some students feel bored when learning the operations of rational numbers, think they are similar to the operational rules learnt in primary schools, and slack off in their studies. As a result, they learn quickly, do exercises quickly, but make many mistakes; and they make more mistakes in multiplication and division of rational numbers than in addition and subtraction of rational numbers, and even draw an absurd conclusion that “negative number + negative number = positive number”. For this purpose, targeted teaching should be conducted according to the psychological characteristics of students. First of all, we should attract the attention of students. The methods are as follows: ➀ Organize classroom teaching well and create a sound teaching environment; ➁ Organize the teaching content well to ensure appropriate classroom capacity and difficulty level; ➂ Encourage students to participate, explain profound theories in simple language, be focused and well-organized; ➃ Pay attention to revealing the internal relations between thinking process and knowledge; ➄ Pay attention to integrating theory with practice and strengthening application; ➅ Pay attention to the flexibility and pertinence of teaching methods; ➆ Pay attention to analyzing the students’ errors, etc. 3. Eliminate difficulties in a timely manner and build up confidence in learning. Mathematics is strictly logical and must be studied step by step. Students often encounter obstacles in learning new concepts, new theories, and new methods. If not eliminated in time, they will often present difficulties to subsequent learning and even result in loss of confidence. This is a negative effect of reverse psychology in learning. For example, most students feel accustomed to learning algebra and their grades are relatively normal. But after starting to learn plane geometry, they have difficulty in learning the rudiments, are unable to think, and write the demonstrations. As time goes on, the sense of fear and serious differentiation will be produced. If it is not reversed in time, the fear of difficulties arises, resulting in serious differentiation. If not reversed in time, it will have a direct effect on the success or failure of this subject teaching. At this time, it is necessary to slow down the teaching progress appropriately so that students can gradually reach the standards for concept, language, figures, and demonstration, and then build up confidence in learning to achieve better teaching effect. 4. Adjust the cognitive structure at any time according to the characteristics of new knowledge. As revealed by the principle of cognitive learning, new learning must be suitable for learners’ cognitive development level at that time. When new knowledge is inconsistent with the original cognitive structure, it is necessary to adjust the original cognitive structure to adapt to the new learning needs, and establish a new cognitive structure as the basis for further learning.

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For example, students have difficulty in learning the concept of functions, because in the past when they learn the identical transformation of algebraic formulas, solve equations and inequalities, they obtain the results through operations, and the focus is on “operations”. Now, the dependence relations between quantities are examined from the perspective of changing, and the focus of research is on “relationship”, and the means of expression is to elicit analytical formulas, tables, or sketch graphs. In this way, the original cognitive structure cannot adapt to the new cognitive structure. Together with the shortcomings of textbooks, if we do not improve the teaching methods, it will inevitably cause confusion among students. For example, “for a car traveling at 40 km/h, try to analyze the relationship between travel time and distance”. If we are not eager to ask students to elicit functional relations, but let students analyze the variables and their relations in the changing process with a free hand. And then from many similar examples, through analysis, comparison, synthesis, abstraction, and generalization, enable students’ understanding of the function concept to change from generalization to differentiation to synthesis, forming a new cognitive structure, which facilitates the understanding of the function concept. 5. Give play to positive transfer and overcome the interference of negative transfer. Transfer is a kind of psychological phenomenon, which is the impact of prior learning on subsequent learning. Those having a positive effect are called positive transfer. The most commonly used associations, analogies, and transformations in mathematics are all positive transfers. Those having a negative effect are called negative transfer. For example, when students start to learn algebra, they are likely to confuse positive numbers with integers, and take both positive numbers and integers as positive integers one-sidedly. This is a negative transfer caused by the concept of natural numbers. The following errors often occur among students: (a ± b)2 = a 2 ± b2 ,

 a 2 ± b2 = a ± b, |a ± b| = |a| ± |b|,

lg(a ± b) = lg a ± lg b, sin(a ± b) = sin a ± sin b These are negative transfers caused by the distributive law of multiplication. Students forget to test the roots when solving fractional and irrational equations; this is a negative transfer caused by solving integral equations. When learning geometry, students have difficulty in adapting to the demonstration of plane figures and are likely to confuse the properties and demonstration of spatial figures with plane figures; students are not accustomed to shifting from research on constants to variables, from finity to infinity, etc. In addition to the students’ weak logical thinking ability and weak imagination, one of the important factors is the influence of negative transfer. Moreover, this kind of negative transfer is somewhat stubborn, which is

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caused by many factors. It shows that negative transfer in mathematics learning can lead to conceptual confusion, operational errors, and affect intellectual development and ability improvement. In mathematics teaching, we must pay attention to the positive transfer of learning and overcome the negative influence of negative transfer. In this regard, people have accumulated some good experience. For example, pay attention to concept teaching to enable students to clarify the essential attributes of concepts; through discrimination and comparison, overcome the negative influence of stereotyped thinking and cultivate the ability to seek different thinking; consolidate and summarize the knowledge acquired in time to make it systematized and organized; instruct students to learn how to adopt the learning method of “exploration, research, and self-summary” for active learning, promote positive transfer, and improve ability, and so on. Review Questions and Exercises 1. What is learning? What are the two representative views on learning in contemporary times? 2. What are the characteristics of secondary school mathematics learning? Try to illustrate the difference between secondary school mathematics learning and mathematical discovery or invention. 3. What are the types of mathematics learning in secondary schools? Which type of learning methods should be adopted and why? 4. What links should be done well in learning secondary school mathematics, and what should be paid attention to in each link? 5. What are mathematical knowledge structure and cognitive structure? What is the relationship between them? 6. Give an example to illustrate the general process of secondary school mathematics learning, and what should be paid attention to at each stage of the learning process? 7. What are the intelligence and non-intelligence factors in mathematics learning? What are their effects on learning? How can we combine intelligence and nonintelligence factors in mathematics learning? 8. How can we do a good job of teaching according to the laws of psychological activities of secondary school students? 9. Take some concepts in the current geometry and algebra textbooks in secondary schools as examples respectively to illustrate how secondary school students can learn mathematical concepts well. 10. Take some theorems and formulas in the current algebra and geometry textbooks in secondary schools as examples respectively to illustrate how secondary school students can learn mathematical propositions well.

Chapter 7

Secondary School Mathematics Curriculum Standard

The core of the new round of basic education reform in our country at the turn of the century is the development of national curriculum standards, which not only reflect the basic ideas of new curriculums directly, but also provide a basis for the compilation of textbooks and teachers’ classroom teaching, and provide a guide for the management and evaluation of new curriculums. This chapter intends to discuss the connotation, characteristics, and significance of mathematics curriculum standards, analyze the difference between curriculum standard and syllabus, and introduce mathematics curriculum standards for the stages of compulsory education and general high school.

7.1 Curriculum Standards and Their Significance 7.1.1 Curriculum Standards 1. Connotation of curriculum standards Many developed countries in the west organized their forces and invested a lot of money to develop or revise various curriculum standards at the end of the twentieth century. In our country, all national curriculum standards (experimental draft) for compulsory education stage were initiated in early 2000 and were promulgated in 2001. At the same time, all national curriculum standards (experimental draft) for general high schools were also developed and officially published in April 2003.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_7

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The curriculum standard is a programmatic document that stipulates the curriculum level and curriculum structure of a certain stage, and is a national basic norm and quality requirement for basic education curriculums. As further pointed out in the Basic Education Curriculum Reform Outline (for Trial Implementation) (hereinafter referred to as the Outline) promulgated by the Ministry of Education of our country in 2001, the national curriculum standards are the basis for textbook compilation, teaching, evaluation, and test question setting, and are the foundation for national management and evaluation of curriculums. They reflect the country’s basic requirements for students at different stages in terms of knowledge and skills, processes and methods, emotional attitudes and value orientations, etc., stipulate the nature, objectives, and content framework of each curriculum, and put forward teaching and evaluation suggestions. Thus, it can be seen that the curriculum standards include the following connotations: The curriculum standards are formulated by subjects or curriculums; The curriculum standard specifies the nature, objectives, and content framework of the curriculum; The curriculum standard provides guiding teaching principles and evaluation suggestions; The curriculum standard stipulates the basic requirements for knowledge and skills, processes and methods, emotional attitudes and values, etc. that students at different stages should meet; However, the curriculum standards do not include specific content such as key teaching points, difficult points and time allocation, etc. 2. Characteristics of curriculum standards As the soul of the entire basic education, curriculum standards are the core of the systematic project of basic education reform. Curriculum standards have the characteristics of being programmatic, instructive, fundamental, and flexible. Being programmatic means that the national curriculum standards represent a country’s basic will and policy to implement basic education reform, where the basic education at all levels and stages must take them as a guide to action. Being instructive means that the curriculum standards will have important guiding significance for textbook compilation, teaching, and evaluation, and are the starting point and destination of textbooks, teaching and evaluation. Being fundamental means that the basic requirements for knowledge and skills, processes and methods, emotional attitudes and values, etc. stipulated in the curriculum standards are the basic quality for every citizen and the level that a sound student can achieve through hard work. Being flexible means that instead of stipulating the key teaching points and difficult points and giving any suggestions on teaching time, the curriculum standards leave the teaching content to the mercy of schools and teachers, resulting in teachers’ greater initiative and flexibility in handling the textbooks.

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7.1.2 Syllabus 1. Connotation and characteristics of syllabus In fact, the Nanking National Government promulgated the Interim Curriculum Standards for General Education as early as 1912, which clearly put forward the curriculum standards as guidance document for school education, and it has been followed in our country for nearly 40 years. During the early days of the New China, curriculum standards (draft) for all subjects in primary schools and individual subjects in secondary schools were also promulgated. After 1952, the Soviet Union’s educational model was fully transplanted in our field of education, and the concept of syllabus was used to replace the curriculum standard according to the Soviet Union’s practice. For fifty years that followed, the educational circles of our country have always regarded the syllabuses as programmatic documents to guide school education although the syllabuses have been adjusted and revised many times. The syllabus is a national programmatic document for each curriculum in basic education stage, and a guiding document for teachers to study textbooks and conduct teaching work. It stipulates the curriculum objectives and teaching aims and puts forward the outline of teaching contents and the basic requirements for teaching. The syllabus has the following characteristics: (1) The starting point and focus of the syllabus is to guide the teaching work. The syllabus should not only stipulate the teaching objectives and teaching contents clearly and precisely, but also make arrangements for the teaching sequence of teaching contents. These teaching requirements and stipulations prompt teachers to pay attention to knowledge points and learning outcomes. (2) The curriculum objectives of the syllabus are achieved through teaching aims. For many years, the mathematics syllabus of our country has taken the training of students’ acquisition of knowledge, skills, and abilities as the primary aim, and taken the development of thinking ability as the core of ability training, i.e., it has taken the teaching of “double basics” (basic knowledge and basic skills) as the core and focused on three basic abilities (operational ability, logical thinking ability and spatial imagination ability). It is safe to say that the mathematics teaching in our country for decades has been characterized by our focus on “double basics” and three basic abilities. (3) The syllabus highlights the teaching requirements for various knowledge points and basic skills. The mathematics syllabus of our country stipulates the status and focus of “double basics” clearly through the four levels of “understanding”, “comprehension”, “mastery” and “flexible application”, and it is also the highest requirement for textbooks, teaching and evaluation, neither textbooks, teaching nor evaluation (especially the high school entrance examination and the college entrance examination) can break through this upper limit. This is to say that the syllabus has direct influence on textbook compilation, teaching work and evaluation, with strict and rigid control.

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(4) In terms of the selection and arrangement of contents, the syllabus emphasizes the systematic, logical, and complete knowledge structure. For example, textbook compilation and teaching are conducted by algebra, solid geometry and analytic geometry in accordance with the high school mathematics syllabus in order to highlight the systematicness and logicality of the knowledge of various branches; as for trigonometric functions, we need to learn not only the basic concepts, graphs, and properties, but also a lot of trigonometric identical transformation formulas, to emphasize the completeness of knowledge structure. In this way, while teaching, teachers often start with knowledge, and adopt deduction to deduce new knowledge from what has been acquired. Therefore, teaching under the guidance of the syllabus is more conducive to students’ mastery of systematic knowledge. The above characteristics of the syllabus enable it to play an active role in the imparting of scientific and cultural knowledge and the training of basic skills. In the past decades, within the framework of syllabus, teachers have devoted a lot of time and energy to the exploration and research into the teaching of “double basics”, accumulated extremely rich experience, and summed up many effective teaching methods, and these experience and methods have helped to cultivate batches of talents with solid basic theories for higher-level schools, and cultivate builders and successors with high cultural literacy for all walks of life in the society as well. However, with the development of society, especially the rapid development of science and technology, the process of global economic integration has accelerated sharply, and the international competition has become increasingly fierce. International competition is fundamentally the competition between technology and talents, while the key to the competition between technology and talents lies in education. In order to meet the needs of cultivating talents in the new century, people have begun to reflect on the traditional outlook on talents, traditional teaching models and methods. Obviously, people’s ingrained educational thought, curriculum concept, teaching methods, and teaching ideas within the framework of syllabus cannot be cast off overnight. However, if we do not break through this framework, it is impossible to address the deep-rooted problems of syllabus completely and to achieve the goals of educational reform truly. So what’s the problem with the traditional syllabus? ➀ The teaching objectives are too sweeping and general, which not only makes it difficult to implement the teaching of “double basics” and the cultivation of practical ability, but also makes it difficult to put forward specific suggestions and requirements for the cultivation of innovative consciousness and personality traits. ➁ The teaching contents are complex, difficult, and outdated, the teaching requirements are inflexible, the students are overburdened, and it is difficult to meet the teaching requirements of the syllabus. ➂ The teaching contents are presented linearly, paying excessive attention to completeness, systematicness, and logicality but ignoring students’ cognitive rules and existing life experience.

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➃ The curriculum management is too rigid and uniform, which is not conducive to teachers giving play to their creativity and students’ individualized development, etc. 2. Differences between curriculum standard and syllabus Though both are programmatic and guiding documents for teaching work, curriculum standard and syllabus are different mainly in the following six aspects. (1) The syllabus fits in with elite education while the curriculum standard is based on mass education. The syllabus is too demanding and rigid, without flexibility and selectivity, causing most students to be overburdened, which can only be conducive to elite education. As a “minimum standard”, the curriculum standard is a standard that the vast majority of students can achieve, and is oriented to every teenager and focuses on the development of every student. The compulsory education should not be elite education, but high school education starts from the construction of a new curriculum system, with different content series designed, implemented hierarchically, and taught in a differentiated way, which not only embodies the idea of mass education, but also meets the different development needs of high school students, and is in line with the development trend of today’s secondary education. (2) The syllabus focuses on the requirements for students’ knowledge and skills, while the curriculum standard is aimed at students’ quality improvement in an all-round way. The syllabus focuses on the requirements for students in terms of knowledge and skills, while the curriculum standard is aimed at the requirements of the future society for the national quality. The goal of basic education is to train future builders. With the rapid development of science and technology and economic globalization in the twenty-first century, the future society has presented new requirements for people’s quality. As a programmatic document of the country’s basic requirements for future national quality, the requirements for students’ quality in various disciplines or fields should become the core part of curriculum standards. Therefore, this curriculum reform aims to promote students’ all-round development and establishes the three-in-one curriculum objectives of knowledge and skills, process and methods, emotional attitude, and values. (3) The syllabus places emphasis on teaching work, while the curriculum standard focuses on the overall implementation process of the curriculum. The syllabus emphasizes the teaching work of teachers and ignores the implementation process of the curriculum. The curriculum standard focuses on various factors of curriculum implementation and puts forward a series of guidance and suggestions on teaching methods, teaching means, learning methods and teaching evaluation, etc. For example, it is suggested that mathematics teaching should start from students’ life experience and existing knowledge background and provide them with sufficient opportunities to engage in mathematics practices and exchanges; teachers should be the organizers, facilitators, and collaborators of students’ learning; students should carry out independent exploration,

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cooperation and exchange, and active thinking and operating practice; schools should attach importance to the use of modern teaching aids, and use computers (calculators) as a powerful tool for research and problem-solving. Regarding the evaluation method, the curriculum standard proposes to weaken the discrimination function of evaluation and give full play to its incentive function; have both quantitative and qualitative evaluation; place emphasis on both formative and process evaluation; and evaluate students’ acquisition of knowledge and their emotional attitudes displayed in learning activities. (4) The syllabus usually presents the teaching contents linearly, while the curriculum standard highlights the spiral mode of presentation. The teaching contents are generally presented linearly in the syllabus, while in the curriculum standards, especially the curriculum standards for compulsory education, the contents are presented spirally, to enable the arrangement of teaching contents to be more in line with the cognitive rules of students. For example, the mathematics curriculum standard divides the contents into four fields: numbers and algebra, space and graphics, statistics and probability, practice and comprehensive application, and covers all three stages. Starting from the second stage, the learning in each learning field will be an extension and expansion of the previous stage, thus forming a step-by-step, spiraling and deepening process. (5) The curriculum management in the syllabus is rigid, while the curriculum management in the curriculum standard is flexible. Within the framework of the syllabus, the curriculum management is basically the national first-level management, and some regions have the second-level right of management, both schools and teachers are faithful implementers of the national curriculum plan and syllabus, and they have no autonomy in curriculum implementation. However, the curriculum standard is the basic requirement put forward by the state for students’ quality in a certain aspect or field, and it is a standard for all students. It provides some guidance and suggestions on teaching objectives, teaching content, implementation, and evaluation of teaching and textbook compilation, but compared with the syllabus, this impact is indirect, instructive and flexible, leaving a lot of choice and flexibility to teaching and evaluation. For example, the curriculum standard encourages teachers to “teach with textbooks” instead of “teaching textbooks”, and teachers should become users, creators and developers of teaching resources. At the same time, this curriculum reform also takes the implementation of three-level management policy as an important goal, providing policy guarantee for localities and schools to implement the national curriculum creatively. (6) The curriculum standard has more reasonable framework than the syllabus, which is conducive to the teaching work. The framework of the curriculum standard refers to the main components of specific content and structure of curriculum standard. The 18 curriculum standards for compulsory education promulgated for the first time in our country, although each with its own characteristics, are basically the same in structure, roughly including preface, curriculum objectives, content standards, implementation suggestions, appendices, and other parts. The objectives are stated in three aspects: knowledge

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and skills, process and methods, and emotional attitudes and values. This is obviously different from the syllabus in the past. This framework is characterized by more comprehensiveness, richer content and clearer targeting, which is conducive to teaching work.

7.1.3 Significance of Curriculum Standard As an important document for the implementation of the new curriculum reform, the national curriculum standard is also an extremely important milestone in the history of the development of our country’s basic education curriculum, and its promulgation and implementation will lead to profound changes in primary and secondary education in our country, with far-reaching significance. 1. The curriculum standard marks a substantial breakthrough in the mathematics teaching reform in our country. The change from the syllabus to the curriculum standard is not only a change in the name, but also a replacement of educational concept, a switching of educational perspective, and a repositioning and design of mathematics curriculum and teaching content in a deeper sense. The curriculum standard endowed with new connotations, along with new concepts, new curriculum structure and content system brings a new feeling to teachers, it will exert an immeasurable influence and motivation on the entire basic education. 2. The curriculum standards are conducive to the comprehensive promotion and specific implementation of quality-oriented education. Quality-oriented education is the ideal basic education in our country, but over the years, the educational status quo of primary and secondary schools has not been fundamentally changed. The new curriculum holds aloft the banner of qualityoriented education, and the curriculum standards are the quality standards of qualityoriented education at the corresponding stage. The national curriculum standards for compulsory education is common and unified basic requirements formulated by the state for a certain learning stage, rather than the highest requirements. Multilevel curriculum standards have been formulated according to the characteristics of high schools, which greatly meet the different needs of high school students. Therefore, in this sense, the national curriculum standards fully reflect the basic requirements of civic literacy education, making the comprehensive promotion and concrete implementation of quality-oriented education become a reality. 3. Curriculum standards are conducive to improving the quality of mathematics teaching. Over the past 50 years, our country has been using the syllabus and textbooks, which fit in with the education model under the planned economy. With the development of society, people pay more and more attention to the quality of education, and their

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needs are more and more diversified. The curriculum standards have changed from the syllabus that mainly focuses on the teaching content to focusing on the teaching process, experience, and results, and this change marks a clear and scientific quality standard for compulsory education stage in our country. This standard includes not only students’ learning quality, but also life and development quality, as well as the quality of textbook compilation, teachers’ teaching, curriculum evaluation, and management. 4. Curriculum standards are conducive to the implementation of textbook diversification. Curriculum standards define uniform quality standards, not just content standards. At the same time, replacing the previous syllabus prepared by semester or academic year with the academic stage reserves a certain space for curriculum development for textbook compilers, which offers opportunity for the truly diversified development of textbooks, and also makes it possible to compile distinctive textbooks. 5. Curriculum standards are conducive to the innovation of teaching and learning methods. The reform of teaching and learning methods is one of the goals pursued by new curriculums. Combining the characteristics of the subject and according to the requirements of outcome, experiential and expressive goals, the curriculum standards strongly advocate teachers’ innovating their own teaching behaviors, making bold attempts within the framework stipulated in the standards, encouraging students to participate and practice actively, think independently and inquire cooperatively, so as to realize the reform of students’ learning method, in order to develop students’ ability to collect and process information, acquire new knowledge, analyze and solve problems, and communicate and cooperate. 6. Curriculum standards are conducive to giving play to teaching autonomy. An important symbol of teachers’ professionalization is teachers’ professional autonomy. The basic requirements are designed by stages in curriculum standards, but the means to achieve such requirements are not stipulated, which reserves space for the textbook compilers and provides the possibility for teachers’ innovative teaching. Teachers can re-develop the textbooks completely according to the on-site resources, students’ experience and their own advantages, design adaptive programs, and carry out creative teaching. For example, in order to achieve the goal of allowing students to “experience the exploration process of the Pythagorean theorem” in the Standard, in addition to traditional methods such as measurement and counting the number of squares, teachers can also make full use of computer and network resources for exploration, such as using the Geometer’s Sketchpad to explore the trilateral relations of right triangles from multiple perspectives and in multiple ways; make use of network resources to collect various proof methods and related historical materials of

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the Pythagorean theorem, and understand the research on the Pythagorean theorem in ancient China, etc. 7. Curriculum standards are conducive to reducing the actual burden on students. Because the national curriculum standard is a unified basic requirement for all students and a requirement where everyone can achieve and experience success, it is an important measure to reduce the burden on students. As long as a student meets the unified basic requirements, he is successful. Undoubtedly, this will reduce the psychological burden and learning pressure on students. The curriculum standard proposes a variety of evaluation methods, and also advocates the establishment of a harmonious, democratic and equal teacher-student relationship, to improve students’ psychological environment, and create a relaxed, friendly and happy atmosphere for learning. In this way, the psychological burden on students will be eased.

7.2 Introduction to Mathematics Curriculum Standard for Compulsory Education 7.2.1 Nature and Basic Concepts of Mathematics Curriculum 1. Nature of mathematics curriculum As maintained by the Standard, “mathematics learning is a process in which people qualitatively grasp and quantitatively describe the objective world, form methods and theories through gradual abstraction and generalization, and apply them widely”. “Mathematics can help people better explore the laws of the objective world, make appropriate choices and judgements for a large amount of complicated information in modern society, and provide an effective and simple means for people to exchange information. As a universally applicable technology, mathematics helps people to collect, organize and describe information, establish mathematical models, then solve problems, and create value for the society directly”. These functions of mathematics reflect that mathematics cannot be replaced by any other subject. Mathematics curriculum is one of the most important and fundamental curriculums in the stage of compulsory education. It should not only consider the characteristics and value of mathematics, but also follow the psychological laws of students learning mathematics. The mathematics curriculum should provide students with the connection between mathematics and the real world, emphasize starting from the students’ existing life experience, and enable students to experience the process of abstracting practical problems into mathematical models, solving and applying them personally; mathematic curriculum should, through mathematization process, enhance students’ collective awareness of cooperative learning, develop their ability to use mathematics to explain relevant facts and phenomena, and use mathematics to

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communicate; mathematics curriculum should emphasize students’ all-round development in thinking ability, emotional attitude and values, etc. while experiencing mathematical activities to understand mathematical knowledge, ideas, and methods. 2. Basic concepts of mathematics curriculum Advanced concepts are an important basis for constructing the mathematics curriculum standard for compulsory education. The study and understanding of the basic concepts in the Standard help us to establish “people-oriented” education concept, the outlook on talents of “diversity of talents, and everyone amounting to something”, the view of educational quality of “all-round development of morality, intelligence, physique and aesthetics”, and the concept of educational value of “laying a foundation for students’ lifelong development and happiness”. Six basic concepts are proposed in the Standard. (1) The mathematics curriculum should highlight the fundamentality, popularity, and expansibility and be oriented to all students. As pointed out by the Standard, “the mathematics curriculum in the compulsory education stage should highlight the fundamentality, popularity and developmental and expansibility, making mathematics education accessible to all students, to achieve: Everyone learns useful mathematics; Everyone has access to essential mathematics; Different people gain different development in mathematics”. This basic concept is the cornerstone of constructing the entire Standard, which represents a new mathematics curriculum concept and practice system. ➀ Everyone learns useful mathematics. Useful mathematics means that mathematics as educational content should meet the needs of students’ future social life, adapt to the requirements of students’ personality development, and be beneficial to inspire thinking and develop intelligence. Useful mathematics should be closely related to students’ real life and previous knowledge and experience, and should be attractive and interesting content to them. Useful mathematics should be suitable for students to understand and master in limited study time. ➁ Everyone has access to essential mathematics. This refers to that useful mathematics should and can be mastered by every student. It means that the contents and teaching requirements stipulated in the Standard are the most fundamental ones, and a successful experience that every child with normal intelligence can obtain under the guidance of teachers and with his/her own efforts. ➂ Different people gain different development in mathematics. Every student boasts rich life experience and knowledge accumulation, and every student has his/ her own way of thinking and strategies to solve problems. “Different people gain different development in mathematics”, which means that mathematics curriculum should be oriented to every distinctive individual, and adapt to the different development needs of each student. Therefore, mathematics curriculum

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should cover a wide range of areas, which include not only themes for students to think, explore, and operate, but also some implied original growing points of modern mathematics, to enable every student to have an opportunity to touch, understand and dig into the interesting mathematical problems, meet the mathematical needs of each student to the greatest extent, and unlock the intellectual potential of each student to the utmost extent. (2) Mathematics is the basic tool for human practices and an important part of modern civilization. Mathematics is an indispensable tool for people’s life, labor, and learning. It can help people to process data, perform calculations, reasoning, and proof. It can provide mathematical models of natural and social phenomena, provide language, ideas and methods for other sciences, and is the foundation of all major technological developments; it plays a unique role in improving people’s reasoning ability, abstracting ability, imagination and creativity; mathematics is also a human culture, and its content, ideas, methods and language have become an important part of modern civilization. The above-mentioned understanding of mathematics in the Standard centers on the close relationship between mathematics and human development and real life. Therefore, it advocates that mathematics curriculum must fully reflect the completely harmonious connection between human life and mathematics. This is to say that the content of mathematics curriculum should fully consider the trajectory of human activities in the process of mathematics development, and be close to the real life that students are familiar with. Constantly communicating mathematics in life and mathematics in textbooks to integrate life with mathematics, such mathematics curriculum can be beneficial for students to understand and have a passion for mathematics, and make mathematics become the main driving source for students’ development. (3) The mathematics curriculum provides a variety of learning methods, resulting in students’ lively and invigorating learning methods. The Standard points out that the content of students’ mathematics learning should be realistic, meaningful and challenging, and the content should be presented in different ways to meet diverse learning needs. This concept of the Standard emphasizes the process in terms of content, which is not only closely related to the cultivation of innovative consciousness and practical ability, but also makes students’ exploration experience and the experience of acquiring new discoveries become an important way to learn mathematics. First of all, the content of mathematics curriculum should not only include some readymade results of mathematics, but also include the forming process of these results. Through the forming process of mathematical results, students can experience the joy of mathematical discovery, enhance their confidence in learning mathematics well, form awareness of application and innovation, and achieve substantial development and improvement of human intellect and emotion. Secondly, mathematics learning activity should be a lively, active and personalized process, because the above-mentioned changes in mathematics curriculum content necessarily require

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meaningful and matching learning methods. The learning method of mathematics can no longer be a single and boring way dominated by passive listening and practice, it should be a process full of vitality, and a process where students’ subjectivity, initiative, and independence are constantly generated, displayed, developed, and elevated. Of course, listening carefully, classroom practice and homework, etc. are still important learning methods of mathematics. The new curriculum focuses more on learning methods that were short of and easily overlooked in the past. (4) Students are the masters of mathematics learning, and teachers are the organizers, facilitators, and collaborators of mathematics learning. The Standard points out that mathematics teaching activities must be based on students’ cognitive development level and existing knowledge and experience. In other words, mathematics teaching should be based on the development of students, and take students’ personal knowledge, direct experience and real world as important resources of mathematics teaching. Students have rich life experience and knowledge accumulation, and have a certain level of cognition, including a lot of experience in mathematics activities, especially the strategies of applying mathematics to solve problems. The mathematics teaching activities should provide them with situations and opportunities to explore mathematical knowledge and master basic skills independently, and leave more time for students to think, operate and communicate, to make students become the real masters of learning activities. However, to this end, the teacher’s role must be changed. Teachers should change from a knowledge imparter to a facilitator of students’ development, from the authoritative position of classroom space dominator to the role of organizer, facilitator, and collaborator of mathematics learning activities. Among them, the role of organizer includes organizing students to discover, search, collect, and utilize learning resources, and organizing students to create and maintain a positive psychological atmosphere in the classroom and during the learning process, etc.; the role of facilitator includes guiding students to design appropriate learning activities, guiding students to activate the prior experience required for further exploration, and guiding students to conduct in-depth exploration and brainstorming around the core problem; the role of collaborator includes establishing a humane, harmonious, democratic and equal teacher–student relationship, to enable students to be motivated, inspired, guided, and advised in an atmosphere of equality, respect, trust, understanding, and tolerance. (5) Strengthening the incentive function of evaluation, and establishing a various and diversified system of evaluation. The Standard points out that the main purpose of evaluation is to understand students’ mathematics learning process comprehensively, stimulate students’ learning and improve teachers’ teaching; an evaluation system with multiple evaluation objectives and various evaluation methods should be established. The traditional evaluation is characterized by quantification. In practice, quantification is often reduced to “queuing” tests characterized by finding faults and “deductions”. Although “queuing” also has certain incentive effect, it is likely to

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make those students who are full of enthusiasm for learning begin to doubt their own abilities and become less and less confident, which is in complete violation of the original purpose of evaluation. Judging from the process of teaching and learning activities, evaluation is always to help students understand themselves, build selfconfidence, encourage independent learning, and improve and perfect teaching and learning process at the same time. Therefore, in terms of evaluation, it is important to try to find the strengths of students, pay attention to encouraging students more, and enable them to find their strengths and progress more. Of course, encouragement should also be moderate, and more importantly, it helps students to recognize the interesting, useful, and genial aspect of mathematics, and at the same time, come to realize some truths that are helpful to personhood and life. (6) Making full use of modern information technology to make it a powerful tool for learning mathematics. Modern information technology provides feasible plans, technologies, methods, and tools for the ideal of mathematics curriculum reform and is an important guarantee for creating a new mathematics learning environment and realizing the concept of mathematics curriculum reform. Therefore, during the implementation of the new mathematics curriculum, we must vigorously develop and make full use of the functions of modern information technology in educational activities, making it a powerful auxiliary means of mathematics teaching and a powerful tool for students to learn mathematics and solve problems. The Standard also points out that modern information technology should “be committed to changing students’ learning styles, so that students are willing and have more energy to throw themselves into realistic and exploratory mathematical activities”. This is to say that computers, networks, and other modern information technologies will become an important means to develop students’ interests and will help students to free themselves from some tedious, boring, and repetitive work, so that they have more time and opportunities to think and explore, which will not only stimulate students’ interest, but also cultivate students’ creative ability effectively.

7.2.2 Objectives of Mathematics Curriculum As the core content of the Standard, mathematics curriculum objectives reflect the requirements of the Standard for future citizens in terms of basic qualities related to mathematics, and also reflect the educational value of mathematics curriculum for the sustainable development of students. It fundamentally clarifies the basic questions about mathematics curriculum such as “why students learn”, “what mathematics students should learn” and “what mathematics learning will bring to students”. The Standard divides mathematics curriculum objectives into overall objectives and stage objectives. We only introduce and analyze the overall objectives here. The overall objectives are divided into general objectives and specific objectives. The general

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objectives are the core of overall objectives, so they are also called the overall objectives; the specific objectives are the specific elaboration of the overall objectives in four objective fields of “knowledge and skills, mathematical thinking, problem solving, emotion and attitude”. 1. Overall objectives of mathematics curriculum (1) To enable students to acquire important mathematical knowledge (including mathematical facts, mathematical activity experience) necessary for adapting to future social life and further development, as well as basic mathematical thoughts and methods and necessary application skills. In the elaboration of this objective, the understanding of mathematical knowledge has changed. Mathematical knowledge not only includes “objective knowledge” (such as various operational rules, area and volume formulas, and established mathematical thoughts and methods, etc.), but also includes students’ own “subjective knowledge”, i.e., personal knowledge and mathematical activity experience with distinct individual cognitive characteristics. They only belong to a specific learner himself/ herself, reflect the understanding of corresponding mathematical objects at a certain learning stage, and are empirical. (2) To enable students to preliminarily learn to use the mathematical way of thinking to observe and analyze the real world, to solve problems in daily life and in the study of other disciplines, and to enhance the awareness of applying mathematics. This objective reflects that the Standard positions mathematics learning in the compulsory education stage to promote the overall development of students, that is, to cultivate students to understand the environment and society in which they live with a mathematical perspective, learn to think mathematically, and use mathematical knowledge and methods to analyze things and think about problems. This is to say that the new mathematics curriculum will no longer emphasize providing students with systematic mathematical knowledge first, but focus more on providing students with mathematics with a realistic background, including the mathematics in their lives, the mathematics they are interested in and the mathematics conducive to their learning and growth. The important result of students’ mathematics learning is no longer just how many standardized mathematical problems they can solve, but whether they can find mathematics in the realistic background and whether they can apply mathematics to think about and solve problems. (3) To enable students to experience the close connection between mathematics and nature and human society, understand the value of mathematics, and enhance their understanding of mathematics and confidence in learning mathematics well. This objective shows that a good mathematics curriculum should make students realize that “mathematics is a civilization of human society, and it plays a huge role

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in the development of human beings in the past, at present and in the future”. Mathematics as a teaching content should not be simply regarded as abstract symbolic operations, graphical decomposition and proof, it reflects various relationships, forms, and laws existing in realistic situations. In addition, this objective also shows that learning mathematics well is not the privilege of a few, but the right of every student. The mathematics curriculum is set for every student, and every student with normal physical and mental development can learn mathematics well, achieve the objectives set forth in the Standard, and enhance his/her confidence in learning mathematics well. (4) To enable students to have a preliminary innovative spirit and practical ability, and to be fully developed in emotional attitude and general abilities. This objective shows that starting from realistic situations to acquire mathematical knowledge through the process of exploration, thinking and cooperation, what we gain is self-confidence, sense of responsibility, matter-of-fact attitude, scientific spirit, consciousness of innovation and practical ability, which are far more important citizen qualities than entering schools of a higher grade. To sum up, compared with previous mathematics teaching objectives, the curriculum objectives set by the Standard have richer connotations and more reasonable structure, and are more closely related to the development of students’ basic quality. 2. Specific objectives of mathematics curriculum The new curriculum concepts of “being people-oriented” and “cultivating wellrounded people” require that the objectives of mathematics curriculum should not only enable students to acquire necessary mathematical knowledge and skills, but also include the development in inspiring thinking, problem solving, emotion and attitude, and other aspects. Therefore, the Standard lists “mathematical thinking, problem solving, emotion and attitude” together with “knowledge and skills” clearly as the specific objective fields of the curriculum, and gives a more detailed description of them, which is one of the features of the Standard. (1) About the “knowledge and skills” objective The Standard believes that basic knowledge and basic skills are still the focus of students’ mathematics learning, but what needs to be reflected on is what the basic knowledge and basic skills that students should spend time and energy firmly mastering are nowadays. The Standard believes that with the progress of society, especially the rapid development of science, technology and mathematics, the understanding of basic knowledge and basic skills should keep pace with the times, and some basic knowledge and basic skills that were valued many years ago are no longer the focus of mathematics learning today or in the future, such as some complex operational techniques and proof techniques that are far beyond the students’ cognitive level and comprehension ability, and those fabricated question types that are only related to competitions, etc. On the contrary, some knowledge, skills, or mathematical thoughts and methods that have not been valued in the past have become the

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basic knowledge and basic skills that students must master. For example, the ability to select an appropriate algorithm based on the actual background, the ability to use a calculator to process data, the ability to read data, the ability to process data and make inferences based on the results obtained, the consciousness to grasp and apply the law of changes between variables in the process of change, etc., are the basic mathematical literacy that a citizen should possess, and the basic knowledge and basic skills that must be mastered. It is worth noting that process objective appears in knowledge and skills objectives for the first time, such as “experiencing the process of abstracting some practical problems into number-shape problems”, “experiencing the process of exploring the shape, size, positional relationship and transformation of objects and figures”, “experiencing the process of raising questions, collecting and processing data, making decisions and predictions”, etc. This is to say that the process of acquiring knowledge and skills itself is the experience process of mathematics learning activities. (2) About the “mathematical thinking” objective The mathematical education in the compulsory education stage is a kind of civic education, which brings students more than just solving more mathematical problems. Students will face different challenges in the future—some need to learn or study more mathematics, so it is very important for them to be able to “think about mathematics”; others will basically not need to solve pure mathematics problems after employment, so for them, “thinking about mathematics” is a need, but what is more important is perhaps being able to “think mathematically”, that is, when faced with various problem situations (especially non-mathematical problems), they can think about problems from mathematical point of view, discover the mathematical phenomena therein, and apply mathematical knowledge and methods to solve problems. Therefore, there are two aspects that constitute this objective— being able to think about mathematics and to think mathematically, which should become important objectives for students to learn mathematics. (3) About the “problem solving” objective Our students solve a lot of problems almost every day. But the “problem-solving” emphasized in the Standard is not equivalent to these problem-solving activities. First of all, in terms of content, the problems mentioned in the Standard are not limited to pure mathematical problems, especially different from those problems that can be solved only through non-thinking activities such as “identifying problem types, recalling solutions, and imitating examples”. The problems in question can be pure mathematical problems or various problems presented in the form of nonmathematical problems. But regardless of problem types, the core is that they can be solved by students through observation, pondering, speculation, communication, reasoning, and other thinking activities. Secondly, in terms of specific connotations, the requirements in the Standard are multifaceted, including preliminarily learning

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to raise and understand problems from mathematical point of view, and being able to apply the acquired knowledge and skills to solve problems comprehensively. (4) About the “emotion and attitude” objective This objective is related to the understanding of quality-oriented education in mathematics classroom. The Standard believes that the mathematics classroom is a qualityoriented education classroom, and many basic qualities of a qualified citizen, such as curiosity about natural and social phenomena, thirst for knowledge, down-toearth attitude, rational spirit, independent thinking, cooperation and communication ability, self-confidence to overcome difficulties, willpower, creative spirit and practical ability, can be cultivated through mathematics teaching activities. Regarding the objectives in four aspects above, the Standard further points out that the objectives in these four fields are a closely related organic whole and play a very important role in human development. But in fact, from the elaboration of the specific objectives for each stage, it is not difficult to find that the Standard has an obvious positioning for the understanding of the curriculum objectives in four aspects—students’ development in mathematical thinking, problem solving, emotion and attitude, etc. is more important than the development in knowledge and skills, because the former is the foundation of every student’s lifelong sustainable development, no matter what occupation he/she will take up in the future.

7.2.3 Content of Mathematics Curriculum The Standard divides the content of mathematics curriculum into four fields: “numbers and algebra”, “space and graphics”, “statistics and probability”, “practice and comprehensive application”, and makes specific provisions for the learning content of each stage, as shown in Table 7.1. The table shows that although the mathematics curriculum of compulsory education is presented in three different stages, most of the contents of the subsequent stage and the previous stage are repeated, but this is by no means a simple reproduction, it reflects that the knowledge and some important concepts, thoughts and methods learned in the subsequent stage are the extension and expansion of the previous stage, which is a gradual, spiraling, and deepening process. Therefore, the teaching content of the entire compulsory education stage is an inseparable organic whole, and a curriculum content system that has been designed meticulously and planned systematically. The content, objectives, and teaching requirements of mathematics curriculum of grades 7–9 (middle school) are introduced in brief as follows. 1. Numbers and algebra In this field, students will learn rational numbers, real numbers, equations and systems of equations, inequalities and systems of inequalities, functions, etc., explore numbers, shapes, and the relations and laws contained in practical problems, master

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Table 7.1 Content structure of mathematics curriculum of compulsory education Stage

Stage 1 (Grades 1–3)

Stage 2 (Grades 4–6) Stage 3 (Grades 7–9)

Numbers and algebra

Understanding of numbers Operations of numbers Common quantities Exploration of laws

Understanding of numbers Operations of numbers Common quantities Exploration of laws

Numbers and formulas Equations and inequalities Functions

Space and graphics Understanding of graphics Measurement Graphics and transformations Graphics and positions

Understanding of graphics Measurement Graphics and transformations Graphics and positions

Understanding of graphics Graphics and transformations Graphics and coordinates Graphics and proof

Statistics and probability

Preliminary statistical activities Indeterminacy phenomena

Simple statistical process Probability

Statistical probability

Practice and comprehensive application

Practical activities

Comprehensive application

Task-based learning

some tools to effectively represent, process and communicate quantitative relationships and change laws preliminarily, develop sense of symbols, experience the close connection between mathematics and real life, enhance awareness of application, and improve the ability to apply algebraic knowledge and methods to solve problems. In teaching, we should pay attention to enable students to understand the basic quantitative relationship and change laws in actual background, and enable students to experience the process of establishing mathematical models, estimating, solving, and verifying the rationality of solutions from practical problems; strengthen the connection between equations, inequalities, functions and other contents; introduce the geometric background of related algebraic content; and avoid tedious operations. 2. Space and graphics In this field, students will explore the basic properties of basic graphics (linear type, circles) and their interrelationships, further enrich their understanding and feeling of spatial graphics, learn the basic properties of translation, rotation and symmetry, appreciate and experience widespread application of transformation in real life, learn the method of locating objects using the coordinate system, and develop the concept of space. The reasoning and demonstration is learnt from the following aspects: During the process of exploring the properties of graphics, cooperating and communicating with others and other activities, students develop plausible reasoning, and further learn to think and express in an orderly manner; on the basis of accumulating certain activity experience and mastering certain graphic properties, students prove some

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basic properties of triangles and quadrilaterals starting from several basic facts, so as to realize the necessity of proof, understand the basic process of proof, master the format of proof by means of synthesis method, and get a preliminary feeling of axiomatic thought. In teaching, we should pay attention to the connection between the content of learning and real life, enabling students to go through the exploration process of observation, operation, reasoning, imagination, etc.; pay attention to the understanding of the proof itself, rather than the quantity and technique of proof; focus on the basic process and basic methods of proof, and take proof as a natural continuation and necessary development of exploratory activities. 3. Statistics and probability In this field, students will experience the necessity of sampling and the idea of using samples to estimate the population, further learn the methods of describing data, further understand the meaning of probability, and be able to calculate the probability of simple events. In teaching, we should pay attention to the connection between the content of learning and daily life, production, nature, society and science and technology fields, to enable students to experience the important effect of statistics and probability on decision making; enabling students to understand the whole process of data processing, and make a reasonable judgement based on the statistical results; enabling students to understand the meaning of probability in specific situations; we should strengthen the connection between statistics and probability; and we should avoid turning the study of this part of the content into the practice of number operations, but the strict representation of relevant terms is not required, etc. 4. Practice and comprehensive application In this field, students will explore some challenging research tasks and develop the awareness and ability to apply mathematical knowledge to solve problems. At the same time, students further deepen their understanding of relevant mathematical knowledge and understand the relationship between mathematical knowledges. Compared with the practical activities and comprehensive application of the first two stages, the task-based learning will focus more on the exploratory and research nature of the activities and will connect mathematics with social and other subject knowledge more, to enable students to further experience different mathematics knowledges and the connection between mathematics and the outside world, and understand the methods of research-based learning preliminarily to improve students’ practical ability and innovative consciousness. In teaching, students should be guided to propose tasks based on their life experience, think about the tasks they face actively, express their views clearly and be able to solve some problems. The content of task-based learning is not necessarily completed in class, and students are encouraged to make use of their extracurricular time to collect data, conduct investigations, and other activities.

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7.2.4 Evaluation of Mathematics Learning Paying attention to the evaluation of mathematics learning is a feature of the Standard that is different from the syllabus. The Standard puts forward specific evaluation suggestions for each stage in the section of “curriculum implementation suggestions”. Several evaluation suggestions for the third stage (grades 7–9) are as follows: Pay attention to the evaluation of students’ mathematics learning process; Evaluate students’ basic knowledge and basic skills appropriately; Pay attention to the evaluation of students’ ability to discover and solve problems; Evaluation subjects and methods should be diversified; The evaluation results should be presented in a combination of qualitative and quantitative methods. The Standard gives a detailed description for each of the above suggestions. Some suggestions are also equipped with examples and samples to enhance their operability. For example, regarding the suggestion that “evaluation subjects and methods should be diversified”, the Standard points out that self-evaluation, peer evaluation, teachers’ evaluation, parents’ evaluation should be combined with relevant professionals’ evaluation, and the evaluation methods should be diversified, including written examination, oral examination, homework analysis, etc., as well as classroom observation, after-school interview, practical work, establishment of growth record books, writing essays and activity reports, etc. These diversified and various evaluation methods are the results of experiments of evaluation reform carried out by some primary and secondary schools in our country for decades, and practice has proved that they are effective. The Standard collects them for promotion and use, which is not only an affirmation of these experimental results, but also an indispensable and important measure to construct a new curriculum evaluation system.

7.3 Introduction to the General High School Mathematics Curriculum Standard Since the high school mathematics curriculum needs to meet several types of high school students with different requirements for mathematics, the selection of curriculum content and the establishment of curriculum flexibility mechanism are the focus of the development of high school mathematics curriculum standard. In this regard, our development team has carried out extensive and in-depth international comparative research, analyzed the current situation of domestic high school mathematics curriculum, combined with the needs of social development in our country in the future and the characteristics of general high schools, and reached the following basic consensus first:

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Compared with European countries, the total knowledge quantity of high school students in our country is relatively small, which is not conducive to the growth of “top talents”. More than half of high school students in our country enter college, so it is necessary for different people learn different mathematics, that is, to increase students’ selectivity to learn mathematics. At present, several series of courses are generally offered after public mathematics curriculum in European countries, and the credit system is implemented. To this end, the “mathematics as a public basic course” in our country should also be divided into several levels to meet the needs of different students. It is preliminarily considered that the minimum level may be the mathematical common sense that average citizens should have. Functions and calculus, probability and statistics, and vector geometry are the core contents of high school mathematics in countries around the world, and these parts are just the weakness of mathematics curriculum in our country. It is a general trend to increase the connection between mathematics and other subjects and daily life. The teaching of computing technology and mathematical modeling will become increasingly important, and the connection between mathematics, literature and art, and social sciences are becoming increasingly close. The combined education of liberal arts and sciences for outstanding talents to improve mathematical accomplishment will continue to be supported, and the connection between mathematics and human culture should be included in the content of secondary schools. The above consensus has laid the ideological foundation of the high school mathematics curriculum standard. After in-depth investigation and research, a preliminary idea has been formed. It included the objectives, basic concepts, basic framework, and main contents of the curriculum. The General High School Mathematics Curriculum Standard (Draft for Comment) was published on the Internet at the end of 2002, revised on the basis of soliciting opinions from all walks of life widely, and finally the High School Mathematics Curriculum Standard was officially published by the People’s Education Press in April 2003. At present, high school textbooks to match the new curriculum standard are in preparation, and the substantive experimental stage of the new high school curriculums will begin in the fall of 2004. We introduce some of the contents of the High School Mathematics Curriculum Standard in brief below.

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7.3.1 Nature and Basic Concepts of Mathematics Curriculum 1. Nature of mathematics curriculum The new high school mathematics curriculum should have the following properties: (1) It is a main course of general high schools after compulsory education, which contains the most fundamental content in mathematics, and is a basic course for cultivating citizen quality. (2) It plays a fundamental role in understanding the relationship between mathematics and nature, mathematics and human society, understanding the curriculum value and cultural value of mathematics, improving the ability to raise, analyze and solve problems, forming rational thinking, and developing intelligence and innovative consciousness. (3) It helps students to understand the application value of mathematics, enhance application awareness, and develop the ability to solve simple practical problems. (4) It is the basis for learning physics, chemistry, technology and other courses and further study. At the same time, it lays the foundation for students’ lifelong development and formation of a scientific world outlook and values and is of great significance to improve the quality of the whole nation. 2. Basic concepts of mathematics curriculum “The new full-time high school mathematics curriculum standard developed at the turn of the century should have the characteristics of the times to adapt to the changing social environment and personality development of young people”. To this end, the following basic concepts are proposed in the High School Mathematics Curriculum Standard in combination with the nature of the high school curriculum. Building a common foundation and providing a development platform; Offering a variety of courses to adapt to personalized selection; Advocating a proactive and exploratory learning style; Focusing on improving students’ mathematical thinking ability; Developing students’ mathematical application consciousness; Keeping pace with the times to understand “double basics”; Emphasizing the essence and paying attention to moderate formalization; Embodying the cultural value of mathematics; Focusing on the integration of information technology and mathematics curriculum; Establishing a reasonable and scientific evaluation system.

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7.3.2 Basic Framework of Mathematics Curriculum 1. Curriculum framework Designing the curriculum in a modular way is an important feature of the new high school curriculum structure. Its basic structural framework is shown in Fig. 7.1. Note: In the figure, the small rectangular boxes represent modules (36 class hours each), and the diamond boxes represent special topics (18 class hours each). 2. Notes on the framework High school mathematics curriculum is divided into two parts: compulsory and elective. (1) The compulsory course is set for the first graders in high schools. The compulsory course consists of 5 modules, namely Mathematics 1 to Mathematics 5, with 2 credits for each module, totaling 10 credits, and they are the contents that every student must learn. (2) Elective courses are set for the second and third graders in high schools. There are 4 series of elective courses, of which series 1 is set for students who wish to develop in humanities, social sciences, etc., consisting of 2 modules, with a total of 4 credits. Series 2 is set for students who wish to develop in science and engineering, economics, etc., consisting of 3 modules, with a total of 6 credits. Series 3 and 4 consist of several special topics, with 1 credit for each special topic. Series 3 and 4 boast rather rich materials. With the development of the curriculum, these contents will be further expanded, enriched and perfected. The contents they cover are all basic mathematical contents. Not only those who wish to develop in science and engineering, economics, etc., but also those

Fig. 7.1 .

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who wish to develop in humanities and social sciences should be encouraged to take these elective courses actively. 3. Suggestions on students’ course selection Students’ aspirations and their own conditions are different, and different schools and majors have different requirements for students in mathematics. Students can select different course combinations independently and can also make appropriate adjustments according to their own situations and conditions. The following are the course selection suggestions provided in the High School Mathematics Curriculum Standard. (1) Students complete the compulsory course of 10 credits and meet the requirements in mathematics for graduation from high schools. (2) On the basis of completing compulsory course of 10 credits, students who wish to develop in humanities, social sciences, etc., can have two options: One is to study elective 1–1 and elective 1–2 in series 1 and earn 4 credits, then select any 2 special topics in series 3 and earn 2 credits, totaling 16 credits; The other is that if students are interested in mathematics and wish to obtain higher mathematical literacy, they can earn 4 credits in series 4 at the same time in addition to the 16 credits earned as per the requirements above, totaling 20 credits. (3) On the basis of completing compulsory course of 10 credits, students who wish to develop in science and engineering (including some economics) can also have two options: One is to study all three modules in Series 2 and earn 6 credits, then select any 2 special topics in Series 3 and earn 2 credits, and select any 2 special topics in Series 4 and earn 2 credits, totaling 20 credits; The other is that if students are interested in mathematics and want to obtain higher mathematical literacy, in addition to the 20 credits earned as per the above requirements, they can select any 2 special topics in series 4 and earn 4 credits at the same time, totaling 24 credits.

7.3.3 Objectives of Mathematics Curriculum The high school mathematics curriculum in the new century should, on the basis of mathematics curriculum of the nine-year compulsory education, enable students to further improve the mathematical literacy necessary for future citizens, so as to meet the needs of personal development and social progress. Specifically, it should: (1) Enable students to acquire necessary basic mathematical knowledge and skills, comprehend the essence of basic mathematical concepts and mathematical

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(2)

(3)

(4)

(5)

(6)

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conclusions, understand the background behind concepts and conclusions, etc. and their application, realize the mathematical thoughts and methods contained therein, as well as their role in subsequent learning, and experience the process of mathematical discovery and creation through different forms of autonomous learning and inquiry activities. Improve students’ basic abilities such as spatial imagination, abstraction and generalization, reasoning and demonstration, operation, solution, and data processing, etc. Improve students’ ability to raise, analyze, and solve problems mathematically (including simple practical problems), to express and communicate mathematically, and to acquire mathematical knowledge independently. Develop students’ mathematical application and innovation consciousness, and strive to think and make judgements about some mathematical models contained in the real world. Improve students’ interest in learning mathematics, build confidence in learning mathematics well, and develop a perseverant inquiring spirit and scientific attitude. Enable students to develop a certain mathematical vision, understand the scientific value, thinking value, application value and cultural value of mathematics gradually, develop a habit of critical thinking, advocate the rational spirit of mathematics, and realize the aesthetic significance of mathematics, so as to further establish a world outlook of dialectical materialism and historical materialism.

7.3.4 Content of Mathematics Curriculum 1. Mathematics as a compulsory course The principle of determining the content of compulsory courses in high schools is to meet the basic mathematical needs of future citizens and to provide necessary mathematical preparation for students’ further study. The contents of 5 compulsory modules are: Mathematics 1: Set, concept of functions and basic elementary functions (exponential, logarithmic, power functions); Mathematics 2: Preliminary solid geometry, preliminary plane analytic geometry; Mathematics 3: Preliminary algorithms, statistics, probability; Mathematics 4: Basic elementary functions II (trigonometric functions), vectors in the plane, trigonometric identical transformations;

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Mathematics 5: Solving triangles, sequence, inequalities. 2. Mathematics as an elective course The principle of determining the content of elective courses in high schools is to satisfy students’ interests and needs for future development and lay a foundation for students to study further and acquire higher mathematical literacy. The elective courses consist of 4 series. The contents of series 1 are: Elective 1–1: Common logical terms, conic curves and equations, derivatives and their application; Elective l–2: Statistical cases, reasoning and proof, extension of number systems and introduction of complex numbers, program charts. The contents of series 2 are: Elective 2–1: Common logical terms, conic curves and equations, vectors in the space and solid geometry; Elective 2–2: Derivatives and their applications, reasoning and proof, extension of number systems and introduction of complex numbers; Elective 2–3: Principles of counting, statistical cases, probability. Series 3 includes several special topics. The current special topics recommended in the High School Mathematics Curriculum Standard are: Selected lectures on history of mathematics, information security and cryptography, geometry on the spherical surface, symmetry and group, Euler’s formula and classification of closed surfaces, trisection of an angle and number field expansion, totaling 6 special topics. Series 4 also includes several special topics. The current special topics recommended in the High School Mathematics Curriculum Standard are: Selected lectures on geometric proof, matrices and transformations, sequences and differences, coordinate systems and parametric equations, selected lectures on inequalities, preliminary elementary number theory, preliminary optimization and experimental design, overall planning method and preliminary graph theory, risk and decision making, switching circuit and Boolean algebra, totaling 10 special topics. 3. Mathematical inquiry, mathematical modeling, and mathematical culture Mathematical inquiry, mathematical modeling, and mathematical culture are important contents throughout the entire high school mathematics curriculum. These contents are not set separately, but infiltrate in each module or special topic. At least one complete mathematical inquiry and mathematical modeling activity should be arranged respectively in high school stage. (1) Mathematical inquiry, namely mathematical inquiring task-based learning, refers to the process of students conducting specific autonomous inquiry and learning around a mathematical problem. This process includes observing

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and analyzing mathematical facts, raising meaningful mathematical problems, guessing and seeking appropriate mathematical conclusions or laws, and providing explanation or proof. Mathematical inquiry is a new learning method introduced in high school mathematics curriculum. It helps students to have a preliminary understanding of the process of generating mathematical concepts and conclusions, understand the relationship between intuition and rigorousness preliminarily, try the process of mathematical research preliminarily, experience the passion of creation, and develop a rigorous scientific attitude and an undaunted spirit of scientific exploration; it helps to cultivate students’ habit of questioning and reflection, cultivate students’ ability to discover, analyze, raise, and solve mathematical problems; it helps to develop students’ innovative awareness and practical ability. (2) Mathematical modeling is the process of applying mathematical thoughts, methods and knowledge to solve practical problems, and has become an important and basic content of mathematics teaching at different levels. The basic structure of mathematical modeling process is shown in Fig. 7.2.

Fig. 7.2 .

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Mathematical modeling is a new way of mathematics learning, which provides students with a space for independent learning, helps students to experience the value and role of mathematics in solving practical problems, the connection between mathematics and daily life and other subjects, the process of comprehensively applying knowledge and methods to solve practical problems to enhance application awareness; it helps to stimulate students’ interest in learning mathematics and develop students’ innovative awareness and practical ability. (3) Mathematics is an important part of human culture, the product of the progress of human society, and the driving force for social development. Through the study of mathematical culture in high schools, students will have a preliminary understanding of the interaction between mathematical science and the development of human society, experience the scientific value, thinking value, application value and humanistic value of mathematics, broaden their horizons, seek the historical trajectory of mathematical progress, stimulate the understanding of the driving force of mathematical innovation, be exposed to excellent culture, appreciate the aesthetic value of mathematics, so as to improve their own cultural literacy and innovation consciousness. The following topics are provided for the study of mathematical culture for reference in the High School Mathematics Curriculum Standard: Birth and development of numbers; Euclid’s Elements and axiomatization thought; Birth of plane analytic geometry and number–shape combination thought; Calculus and limit thought; Non-Euclidean geometry and relativity problem; Birth of topology; Binary system and computers; Computational complexity; Data in advertising and reliability; Trademark design and geometric figures; Mathematical problems arising from golden section; Mathematics in arts; Infinity and paradox; Television and image compression; Mathematics in CT scanning; Military and mathematics; Mathematics in finance; Coastlines and fractals; System reliability.

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7.3.5 Suggestions for Teaching of Mathematics Curriculum (1) The curriculum content should reflect the requirements of the information age for mathematics education. ➀ With the rapid development of modern information technology, algorithms play an increasingly important role in science and technology and social development and are increasingly integrated into many aspects of social life. Algorithmic thinking has become a kind of mathematical literacy that modern people should have. Therefore, in the High School Mathematics Curriculum Standard, algorithms are directly included in the content of compulsory courses of high schools as an independent part, to enable students to, on the basis of the preliminary experience of algorithm thinking in the compulsory education stage, learn to use the computer symbol system to represent mathematical content and mathematical propositions, learn to design program charts, express mathematical processes with programs and algorithms, experience the role of program charts in solving problems, and realize the close connection between computer theory and technology and mathematical science. ➁ The most important mathematical knowledge that every citizen needs to know in the twenty-first century is data processing. In the content of probability and statistics in mathematics as a compulsory course of high schools, statistics is arranged before probability, in the order of “raising problems, collecting data, organizing data, interpreting data, studying data characteristics, and making statistical judgements”. Through analyzing and solving practical problems, students experience the thought of estimating the population and its characteristics with samples, go through the whole process of data collection and processing systematically, and realize the difference between statistical thinking and deterministic thinking. ➂ Three modules of “mathematical inquiry”, “mathematical modeling”, and “mathematical culture” are added to the compulsory and elective content in High School Mathematics Curriculum Standard, providing students with a broader space for development, and providing materials and opportunities to change the ways students learn. This is a continuation and development of the current research-based learning and will make the content of mathematics curriculum richer and more stereoscopic. (2) Mathematics teaching should focus on the infiltration of modern mathematical thoughts and methods and the use of mathematical tools. ➀ About sets The purpose of learning symbols such as “belonging to, union and intersection” in high school mathematics curriculum is to facilitate expression, but there is no need to introduce many formulas of set operations. It is not a study of set theory here, and sets only serve as a language. Logic training should be carried out in combination with actual mathematical language, but excessively formalized symbolic calculus

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and expression methods such as truth tables actually make no contribution to the improvement of students’ logical thinking ability. ➁ About functions and equations The teaching of functions is the core content of high school mathematics. The functions in high school stage involve simple irrational equations, trigonometric equations and exponential equations, which are not elaborated. It mainly enables students to understand the relationship between the roots of equations and function graphs, and pay attention to obtaining approximate solutions to various equations by means of computers and graphic calculators. ➂ About calculus Calculus is listed as an elective course in the Secondary School Mathematics Curriculum Standard, and only “derivatives and their applications” are learnt, without involving limits and integrals. This is a big difference from traditional syllabus. Regarding “calculus”, the starting point of Secondary School Mathematics Curriculum Standard is that for calculus in secondary schools, it is inadvisable to pursue completeness or start from the general concept of limit, and derivatives are directly introduced instead (i.e., the thought of rate of change). When limits are involved, only intuitive understanding is required. Learning derivatives in secondary schools not only paves the way for colleges, but also provides an opportunity for students who will not learn calculus in the future to understand the thought of rate of change. In the terms of secondary school mathematics curriculum itself, its study also helps students to further understand the properties of functions, and using derivative to find the monotony interval of functions, extreme values and prove inequalities can embody its application value in secondary school mathematics. ➃ About probability and statistics The teaching of probability and statistics in secondary schools is a weakness in mathematics teaching in China, which needs to be made up vigorously. In the traditional processing method, the sample and the population are always abstractly defined first for students to remember and understand. In fact, the concepts of probability, population and sample are very complex, and it is difficult for secondary school students to explain them in a strict way clearly, so only description is required. The teaching of probability is mainly to cultivate students’ concept of randomness, enable students to understand that the value rules of random variables is described by probability distribution, be able to process random phenomena from the viewpoint of randomness, and know that statistical results are presented probabilistically and there may be errors. ➄ About matrix Matrix has never been formally included in the secondary school mathematics curriculum in our country previously, but as a mapping from vector set to vector set, it constitutes the algebraic representation of geometric transformation, and it

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has become the basic representation tool of modern mathematics. Therefore, matrix methods can be included in elective modules or special topics. ➅ About rational thinking High school is a key stage for cultivating students’ high-level rational thinking ability. Therefore, students who are interested in rational thinking or have greater development potentials in this area should be provided with corresponding learning opportunities, and selected lectures on geometric proof is a special topic set up for this purpose. Since there is no complete geometric content about the “relationship between circles and straight lines” in the compulsory education curriculum, circles and straight lines will be taken as materials in this special topic, to enable students to experience the rational way of thinking of synthesis method “from the cause to the effect” and the analysis method of “backward inference” through appropriate geometric proof. This special topic also includes the triangular projective theorem and theorems related to conic section. ➆ About solid geometry The teaching content of solid geometry was processed by means of synthesis method in the past, but this time, drawing on the processing methods of many foreign countries, the standard adopts both synthesis method and vector method, dominated by the vector method. Review Questions and Exercises 1. What are the connotations and characteristics of a curriculum standard, and what is its significance? 2. What are the connotations and characteristics of a syllabus, and what are the main differences between the curriculum standard and the syllabus? 3. What are the nature and basic concepts of mathematics curriculum standards of compulsory education in our country? 4. What are the general objectives and specific objectives of mathematics curriculum of compulsory education in our country? 5. What are the content and teaching requirements of mathematics curriculum of compulsory education in our country? 6. What are the nature and basic concepts of the mathematics curriculum standard for general high schools in our country? 7. What is the basic framework of the mathematics curriculum of general high schools in our country? 8. What are the teaching contents and teaching suggestions of mathematics curriculum of general secondary schools in our country? 9. Compare and analyze the teaching content of a chapter or section in the Standard and the traditional content in terms of teaching concepts, teaching methods and teaching requirements. 10. Evaluate the teaching of the new curriculum based on a certain lesson.

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11. Select a topic and write an essay about your experience of new curriculum teaching. 12. Analyze self-learning and teaching using new curriculum standards, and find what measures should be taken to make rapid progress.

Chapter 8

Teaching Form and Means of Secondary School Mathematics

Teaching form is the study of how to organize teaching activities and how to control and use teaching time and space effectively. Teaching means is to study what media or material conditions can be used to impart knowledge and cultivate students’ ability effectively. Teaching form and means are two very important and closely related aspects in the implementation of teaching work.

8.1 Teaching Form of Secondary School Mathematics 8.1.1 Overview of Teaching Form Teaching activities are usually carried out by teachers and students making use of certain textbooks and material conditions in a specific time and space. In order to achieve the expected teaching objectives, teachers, students, textbooks, material conditions for teaching, teaching time and space need to be organized and arranged reasonably, to utilize various factors of teaching effectively and bring the effectiveness of teaching into full play. This is the problem to be studied and solved by the organizational form of teaching. As there are many factors restricting the teaching form, the teaching form also shows diversity and hierarchy accordingly.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_8

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First of all, any teaching activity is carried out in a specific space. In different space environment, there are different organizational forms of teaching. There are two organizational forms of teaching: internal and external environment. The organizational form of teaching in internal environment, that is, the organizational form of current class teaching system, includes students’ preview, class attending, homework, exams and extracurricular learning activities, etc. The organizational form of teaching in external environment includes extracurricular visit, social survey, internship, practical activities, and so on. Secondly, any teaching activity is carried out in a specific time and in a certain time sequence according to the need and possibility. For general regular schools, teaching plans are implemented and teaching activities are carried out according to the development stages of primary school, middle school, high school, and university. Of course, there are also different teaching forms and sequences within a semester or within a teaching unit. At present, class teaching is mainly adopted in regular primary and secondary schools in China, while correspondence teaching, broadcast teaching or television teaching are available in workers’ spare-time schools. Some technical schools also adopt the teaching form of the combination of class teaching and on-site teaching. The organizational form classroom teaching of class teaching system first came into being in the sixteenth and seventeenth centuries in Europe. In the seventeenth century, Comenius, a Czech educator, demonstrated and elaborated it theoretically on the basis of summing up the experience of predecessors. Since the beginning of the twentieth century, class teaching system has become the basic teaching form widely adopted in the world. Compared with the old-style private schools and the system of academy of classical learning, this system has the following obvious advantages: ➀ Mass teaching is carried out to the class, with an average of three to four teachers and about 50 students in a class, thus making full use of human, material, and financial resources and greatly improving the efficiency of school running. ➁ The unified program and textbooks are used, with unified teaching plan and requirements, which is helpful for the teachers to improve their professional skills, guarantees the student to learn systematic and complete scientific knowledge, and enhances the teaching quality. ➂ The teaching activities are carried out in a unified way according to a strict work and rest regime, which is beneficial to the leading role of teachers, to the cultivation of students’ collective spirit, and to their physical and mental health development. Of course, classroom teaching also has its disadvantages. In particular, this kind of teaching form is carried out under the mode of unified content, unified time, unified requirements, and basically unified activity, so it ignores the individual differences in students’ basic conditions and cognition, which is not conducive to the development of students’ initiative spirit and individuality. Therefore, individual instruction should be strengthened in teaching, and attention should be paid to teaching students in accordance with their aptitude to give full play to students’ initiative of learning.

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8.1.2 Main Work of Mathematics Lesson Mathematics classroom teaching is the basic form of mathematics teaching in schools. The classroom teaching of secondary school mathematics, namely mathematics lesson, should mainly include the following work. 1. Acquiring new knowledge One of the main purposes of secondary school teaching is to enable students to master the necessary basic knowledge and develop the basic ability under the guidance and with the help of teachers. Therefore, the primary and most fundamental work of secondary school mathematics lesson is to acquire new knowledge of mathematics and develop related abilities. This kind of work is usually called teaching new lesson, whose content mainly includes learning new mathematical concepts, mathematical theorems, and mathematical thinking methods, and cultivating and developing related abilities and intelligence. 2. Reviewing and consolidating the acquired knowledge In the process of acquiring new knowledge, in view of secondary school students’ receptivity, it must be supplemented by necessary review and consolidation work. Among them, some of the reviews are arranged before teaching new knowledge to check the learning situation of previous lesson, some is arranged for digesting the knowledge in class, and some is arranged after learning a chapter and a unit, so as to summarize, sort out and systematize the knowledge acquired, and transform the knowledge into ability. 3. Assigning, checking and guiding the students’ homework In order to learn secondary school mathematics well, students must complete appropriate homework in and out of class independently. Some of the exercises are arranged in class, so that teachers can make an inspection tour, give individual guidance, or make collective corrections after finding common problems. Sometimes for some difficult exercises, teachers can give appropriate tips and let students discuss it, which reduces the level of difficulty for students to finish homework independently after class, and also facilitates homework correcting.

8.1.3 Type and Structure of Mathematics Lesson According to the teaching objectives and tasks of each lesson, the main types of mathematics lesson in secondary schools include new lesson, exercise lesson, inquiry lesson, review lesson, evaluation lesson, and introductory lesson, seminar, field

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survey lesson, examination lesson, etc. The structure and characteristics of the first five lessons are introduced as follows: 1. New lesson The main task of a new lesson is to teach new mathematics knowledge. It is a common and most important type in mathematics lessons, which can be divided into brand new lesson, secondary new lesson and comprehensive new lesson, etc. Since acquiring new knowledge is closely related to what has been learned, and there is a digestion and consolidation process to accept new knowledge, the basic structure of this type of lesson generally includes such five steps as review, lecturing, exercise, summary, and homework. As new lessons focus on explanation traditionally and can be diversified in form, this type of lesson is a new lesson in terms of content, a comprehensive lesson in terms of form, and a lecture or lecture-exercise lesson in terms of didactics. The above “five-step” originates from Kairov’s didactics of the Soviet Union. In fact, with the change in students’ grades and the deepening of educational reform, it is not difficult to see different meanings given to this type of lesson. Chinese scholars have changed the original “five-step” to the new “five-step” of review and thinking, innovation situation, inquiry into new lessons, consolidation and reflection, summary and exercises, or problem situation, student activities, constructivist teaching, application of mathematics, and review and summary. 2. Exercise lesson The main task of exercise lesson is to learn or consolidate what students have learned and develop skills and technique by doing exercises under the guidance of the teachers. Because before exercises, students must read the textbooks, and review the relevant knowledge, or the teacher makes the necessary prompts and induction, therefore the basic structure of this type of lesson generally has four steps: review, exercise, summary, and assignment of homework. 3. Inquiry lesson Inquiry lesson is a new type of lesson coming into being under the new situation of educational reform. The main task of inquiry lesson is to cultivate students’ preliminary inquiry ability by teachers and students participating in inquiry practice aiming at the research topics stipulated in the syllabus and textbooks. The inquiry lesson needs to combine in-class and out-of-class activities, and its basic structure generally includes four steps: topic introduction, extracurricular investigation and inquiry, in-class communication, and joint summary of teachers and students. 4. Review lesson The main task of review lesson is to deepen the understanding and memory of the acquired knowledge through induction and reorganization under the guidance of the teachers, and to systematize it, as well as to mend the faults and solve difficult problems. Review lessons are in the form of stage review, semester review, graduation review, etc. Therefore, the basic structure of this type of lesson generally has four

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steps: review (providing outline), key point explanation, summary, and assignment of homework. 5. Evaluation lesson The main task of evaluation lesson is to analyze the extracurricular homework at a certain stage or the results of a certain exam in order to correct mistakes and promote further learning. Evaluation can enable students to make clear the correctness of their own solutions, help poor students find out the reasons for the mistakes, and discover the direction of their efforts. Explanation enables everyone to be improved, so the basic structure of this type of lesson usually has four steps: brief introduction, key point explanation, summary, and assignment of homework. Introductory lesson, seminar, field survey lesson, examination lesson, etc. also have certain basic structure and common teaching steps. Readers are expected to summarize them on their own.

8.2 Common Teaching Means of Secondary School Mathematics The teaching means commonly used in secondary schools can be roughly divided into two categories: language-mediated teaching means and teaching aids-mediated means. The language-mediated teaching means can be divided into two categories: the use of oral language and the use of written language. The teaching aids-mediated means can be divided into two categories: the use of ordinary teaching aids and the use of modern teaching aids, namely, ⎧ Language: Oral language ⎪ ⎪ ⎪ ⎪ Written language: Blackboard-writing, textbooks ⎪ ⎪ ⎪ ⎪ Homework and correction ⎨ Teaching means Teaching aids: Ordinary teaching aids: Wall charts, records ⎪ ⎪ Models, physical objects ⎪ ⎪ ⎪ ⎪ Modern teaching aids: Slide, film, television, recording ⎪ ⎪ ⎩ Video, computer, teaching machine

At present, teachers’ language and blackboard-writing are the frequently and widely used teaching means, followed by ordinary teaching aids and modern teaching aids.

8.2.1 Mathematical Language Language is a tool for people to communicate their thoughts and feelings. In teaching activities, teachers’ language is not only a general tool to exchange ideas, but also a tool to impart systematic scientific knowledge and influence students’ thoughts, feelings, and moral qualities, which has special significance.

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Mathematics is a natural science with a high degree of scientificity and strict logic and has higher requirements for language. For example, “extend a line segment by 2 times” and “extend a line segment to twice its original length”. Though the difference lies in a single word, they have entirely different meanings. In the former case, the extended line segment is 3 times the original length, while in the latter case, the extended segment is 2 times the original length. For another example, “none of a, b and c is zero” and “a, b and c are all not zero”, which are quite different in meaning. The former has only one case, while the latter contains as many as seven cases. Mathematical language, also known as symbolic language, is an improved natural language that uses words, symbols, and graphs, and has the characteristics of conciseness, accuracy, clarity, changing “variable”, trinity of words, word meanings and symbols, and mutual interpretation between intuitive language (graphs and symbols) and abstract language (word meanings), etc., which cannot be matched by other sciences. The mathematical language is well known for its conciseness, accuracy, and clarity. Its changing “variable” means that the elements and laws revealed in mathematical language have a “position” nature, for example. (a + b)2 = a 2 + 2ab + b2 Here a and b can represent any element or thing. Its words, meanings, and symbols are trinity. For example, the word “parallel” refers to a specific position relationship between straight lines in a plane, between straight lines, planes, and line and plane in the space, it is indicated by a special symbol “//”, and can be represented by specific graphs. The graphs here should also include the schematic diagrams, tables, model diagrams, thought analysis diagram, Venn diagram in set theory and some structural diagrams of logical relations that are commonly used in mathematics. A textbook of analytic geometry is essentially a book of mutual interpretation between intuitive language and abstract language. For example, for plane analytic geometry: Intuitive language

Abstract language

Point

Real number pair (x, y)

Line

Ax + By + C = 0

Quadratic curve

Ax 2 + Bx y + C y 2 + Dx + E y + F = 0

Point on a circle

(x0 , y0 ) meets x 2 + y 2 = R 2 , i.e. x02 + y02 = R 2

The application of mathematics teaching language in secondary schools must achieve the following objectives. 1. Language should be purposeful. Mathematical language should have a clear purpose. For example, when a teacher introduces a new lesson “binomial theorem”, if the teacher just says “What did we learn in the last lesson? What are we going to learn today?” Students must feel at a

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loss and confused. If the teacher writes the topic (e.g., “binomial theorem”) on the blackboard at the very beginning, there is nothing special in it, and students do not understand why they’re going to learn this content. However, if through reviewing formulas (a ± b)2 , (a ± b)3 , asking them how to evaluate (a ± b)4 , (a ± b)5 , …, (a ± b)n and then bringing forward the binomial theorem, students will have a clear purpose, and the teaching will be very natural, and attract students’ attention. Therefore, the application of mathematical language must be clearly purposeful. It is not allowed to follow one’s inclinations, stray from the topic, or abuse symbols or figures. 2. Language should be popular and easy to understand. Being “popular” means not quoting unfamiliar or difficult terms based on the students’ age, existing knowledge level and life experience. For example, it is not appropriate to quote terms such as “correspondence”, “mutual exclusion” and “necessary and sufficient to grade one students in middle schools, and symbols conditions” n lim an ” to grade one students in high schools. such as “n!, ∞, i=1 n→∞

3. Language should be succinct and accurate. Mathematics is one of the accurate sciences. Mathematical language is required to be extremely succinct, accurate, and rigorously logical. The description of concepts, theorems and rules must be strict, complete, and accurate, which cannot be fabricated or simplified at will. √ For example, x 2 , |a − 2| and x − 1 are nonnegative numbers, which cannot be said as positive numbers. The sum of any two sides of a triangle is greater than that of the third side, where the word “any” cannot be omitted. The accuracy of mathematical language is also reflected in the correct reading and accurate graphing. For example, (a + b)2 should be read as the square of the sum of (pause) two numbers a and b, rather than the square of a and b. sin2 α should be read as the square (pause) of sin α, rather than sin square α. A straight line should be expressed as shown in Fig. 8.1(1), not as shown in Fig. 8.1(2). The perpendicular line of a triangle should be graphed as shown in Fig. 8.1(3), not as shown in Fig. 8.1(4).

Fig. 8.1 .

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In addition, the correct use of language is also reflected in the application of conjunction words, such as “if”, “only if”, “if and only if”, “and”, “or”, “both…and”, “all”, “not all”, “none”. The usage of “if”, “only if” and “if and only if” is the same as “as long as”, “must” and “as long as and must”. Among them, “if” (as long as) indicates a sufficient condition, “only if” (must) indicates a necessary condition, and “if and only if” (as long as and must) indicates a necessary and sufficient condition, for example “if A is established, B is established”, “only if A is established, B is established”, “if and only if A is established, B is established”. “And” is the abbreviation of “both…and”, which means that both of them are satisfied at the same time and are indispensable. “Or” is the abbreviation of “either…or”, which means that the two are independent and optional. For example, a quadrilateral with “a pair of opposite sides parallel and equal” is a parallelogram; the solution(s) to (x − 2)(x − 3) = 0 is x = 2 or x = 3 (“and” is not permissible here); while the solutions to equation (x − 2)(x − 3) = 0 are x = 2 and x = 3 (“or” is not permissible here). From the perspective of set, “and” and “intersection”, “or “and “union” have corresponding relations. “Not all” is the negation of “all”, which generally has many cases, while there is only one case of “none”. “a and b are not both even numbers” is the negation of “a and b are both even numbers”, and at this time there are three cases: “a is odd and b is even”, “a is even and b is odd” and “a is odd and b is odd”. But if “neither a nor b is even number”, there is one case: both a and b are odd numbers. It can be seen that “not all” includes “none”, that is, “none” is a special case of “not all”. 4. Language should be inspiring and interesting. “A bad teacher imparts the truth”, while “a good teacher makes students discover the truth”. In the process of teaching, teachers should use inspiring language to inspire students to observe carefully and think actively, so as to discover the laws and explore the truth. We should also strive to make our mathematical language vivid and interesting, so that students are in high spirit, relaxed and lighthearted. For example, when explaining the sequence, teachers can introduce the story of rice accumulated on the chessboard in ancient India. When explaining the limit, teachers can introduce the philosophy of “cutting off a half of a foot long stick every day, it will never be exhausted”. When explaining the ellipse, teachers can introduce the progress and achievements of China’s man-made earth satellites and space technology. When explaining the extreme value, teachers can introduce the design of some containers, etc. Of course, the so-called vivid and interesting mathematical language here must not become a mere formality, it is different from vulgar talk. In order to make the language inspirating, vivid and interesting, sometimes it can be supplemented by “postures”. Correct posture, proper movement, gesture, eye shift and facial expressions can all play a role in supporting language, sometimes even more powerful and more effective than language. In order to make the language inspirating, vivid and interesting, teachers should also pay attention to the use of “variant” language in

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mathematics teaching. For example, when explaining the concept of absolute value, the following exercises can be arranged: ➀ ➁ ➂ ➃

What number has an absolute value that is itself? What number has a greater absolute value than itself? What number has an absolute value that is the opposite number of itself? What number has the absolute value equal to 3?

When judging the relations between rectangles and rhombus, the following exercises can be arranged: ➀ Is any rectangle a rhombus?

(No)

➁ Is any rhombus a rectangle?

(No)

➂ Is there a rectangle that is a rhombus?

(Yes, square)

➃ Is there a rhombus that is a rectangle?

(Yes, square)

➄ Is there a rectangle that is not a rhombus?

(Yes, rectangles with unequal adjacent sides)

➅ Is there a rhombus that is not a rectangle?

(Yes, non-square rhombus)

8.2.2 Mathematics Blackboard-Writing Blackboard-writing is of great significance in mathematics teaching because a lot of knowledge in mathematics teaching is passed on through blackboard-writing. What’s more, problem solving, demonstration, graphing, and so on are often demonstrated by teachers through blackboard-writing. Mathematics teaching is usually a process in which the teacher keeps explaining while writing on the blackboard. Therefore, the quality of blackboard-writing has a great influence on the effect of mathematics teaching. Blackboard-writing is a necessary basic skill for mathematics teachers. In order to improve the quality and effect of blackboard-writing, we should strive to meet the following requirements. 1. Blackboard-writing should be planned. In order to make full use of blackboard-writing, first of all, teachers should have a thoughtful plan. A new teacher, in particular, must consider the blackboard-writing plan when preparing lessons. According to the requirements of teaching objectives, teachers should have a clear idea about which should be written on the left, which should be written on the right, which can be erased first, and which can be erased later or even retained until the next class. Teachers must not write freely without too much hesitation, resulting in writings in patches, badly organized, inappropriately detailed,

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without focus, and not of uniform size, and even writes and erases repeatedly, which is a waste of time and affects the teaching effect. 2. Blackboard-writing should be clear. Blackboard-writing is for the whole class to see, so the characters should be upright and neat, so that students can see clearly. For lower grade students in secondary schools, teachers should follow the requirements for ruler-and-compass construction strictly for demonstration. For higher grade students in secondary schools, teachers who have some basic skills in graphing can adopt freehand drawing, but should be based on the principle of not affecting the accuracy and intuition of graphs. In order to facilitate students’ observation, the important parts should be highlighted with colored chalk, which should not be abused. In order to emphasize the planned way of blackboard-writing, some young teachers try to keep the completeness of blackboard-writing as much as possible, making the blackboard densely filled with writings, but if students can’t read it clearly, it will also affect the teaching effect. 3. Blackboard-writing should be complete and standardized. Blackboard-writing cannot be as detailed as textbooks. Generally, it should concentrate on the outline and the main points. However, attention should be paid to the completeness and normativeness of the blackboard-writing of concepts, theorems and rules. For example, when writing exponential function y = a x , it is necessary to note that (a > 0, a = l), and a general triangle cannot be graphed into a special triangle. Among them, as for the problem-solving format and the graphing method of geometric figures, teachers’ blackboard-writing plays an important demonstration role. Even for a word or a sentence, a punctuation mark, and the use of a dotted line, teachers should strive to be complete and standardized. If teachers don’t pay due attention, it will not only bring trouble to homework correcting, but also make students develop bad habits and even result in incomplete knowledge or mistakes. 4. Blackboard-writing should be conducive to thinking and memorizing. Blackboard-writing should follow the laws of cognition from the shallow to the deep, from the easier to the more advanced, from the surface to the center, from the particular to the general, and from the known to the unknown, which is conducive to inspiring students to think, understand, and memorize. It is important to compare confusing concepts and connect related knowledge together. For example, when learning the definition of trigonometric functions, the blackboard-writing can be as follows: [], [].

π π   π ; (3) θ ∈ , (1) θ ∈ R; (2) θ ∈ 0, 4 4 2 r x cos a = , sec a = , r x

8.2 Common Teaching Means of Secondary School Mathematics

tan α =

245

x y , cot α = . x y

When learning the cosine theorem, the blackboard-writing can be as follows: a 2 = b2 + c2 − 2bc cos A, b2 = c2 + a 2 − 2ca cos B, c2 = a 2 + b2 − 2ab cos C. For another example, in order to obtain the properties of the first and last digits of common logarithms, the blackboard-writing can be as follows: lg 3.408 = 0.5325, lg 34.08 = lg(10 × 3.408) = lg 10 + lg 3.408 = 1.5325, lg 340.8 = lg(100 × 3.408) = lg 100 + lg 3.408 = 2.5325, lg 0.3408 = lg(0.1 × 3.408) = lg 0.1 + lg 3.408 = 1.5325, ............ In this way, it is convenient for students to find the regularity therein.

8.2.3 Mathematical Teaching Aids The common teaching aids in the mathematics teaching in secondary schools include wall charts, pictures, models, physical objects, etc., such as wall charts of functions in algebra, related models in geometry, and compasses, rulers, set squares, protractors, ellipsographs, etc. The proper use of teaching aids can help students understand and master some complex and abstract spatial forms and quantitative relations. In teaching, teachers should proceed from the reality, make and use some teaching aids with whatever is available, create conditions to acquire some practical modern teaching aids, and give full play to the role of teaching aids in teaching. When using and demonstrating the teaching aids, the following requirements should be met: ➀ The timing of demonstration should be appropriate. If they are demonstrated too early, students will not understand the teacher’s intention, which will distract students’ attention. If they are not tidied away in time after demonstration, it will also distract students’ attention.

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➁ The position of demonstration should be appropriate and suitable. When the teaching aids are small, teachers should carry them and walk between rows of seats so that each student can observe. ➂ The purpose of demonstration should be clear. During the demonstration, teachers should guide the students to observe, analyze, and summarize it, and rise from perceptual knowledge to rational knowledge, to form the concepts, master the laws, and improve the ability of abstraction, generalization, and inquiry.

8.3 Modern Teaching Means of Secondary School Mathematics Modern teaching means is to apply the modern scientific and technological achievements as a means in the field of teaching. It is known as audio-visual teaching abroad and electrical audio-visual instruction in China. Specifically, it is the use of slides, films, televisions, audio and video recordings, broadcast, electronic computers, teaching machines and other modern tools to assist teaching activities. The use of modern teaching means is an important aspect of the modernization of secondary school mathematics education. It has the advantages of expanding the educational coverage, reducing the teaching difficulty, facilitating timely consolidation, improving teaching efficiency, etc., which is of great significance for accelerating and developing education on a large scale and improving the quality of teaching. The following is a brief introduction to the most common types of modern teaching means in the mathematics teaching in secondary schools.

8.3.1 Teaching with Slides and Projection A slide projector is an optical amplifier using the principle of convex lens imaging to provide still frames. There are two types: automatic slide projector and simple slide projector, and their operating principle is the same, as shown in Fig. 8.2. A projector is an optical amplifier for the ease of writing and based on the slide projector, and it is the most widely used in teaching with slides today. The key to teaching with slide projectors and projectors is to master how to make slides and screen sheets. The vast majority of slides and screen sheets at present are 135 Fig. 8.2 .

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photographic films, which are produced using photographic method. Readers can refer to professional books on this subject.

8.3.2 Teaching Means of Videoplaying and Video Shooting 1. Videoplaying After connecting the VIDEO OUT of a videoplayer (or VCR) to the VIDEO IN of a TV set, and connecting the AUDIO OUT of the videoplayer (or VCR) to the AUDIO IN of the TV set, turn on the power switch on the videoplayer (or VCR), set the TV to AV mode, insert the videotape into the videoplayer (or VCR), and press the Play button to play. 2. Camera shooting Camera shooting is to use the video camera to capture real objects and scenes on a video tape in the form of magnetic signals. Its main operations are: first make preparations such as turning on the power, putting in the video tape and opening the lens cap, then carry the camera on shoulder (or placing it on a tripod), gaze at the image displayed on the camera viewfinder, press the Record button when needed, and make dynamic composition. When shooting, pay attention to adjusting the white balance, use pushing, pulling, shaking, moving, following and other camera shooting skills as needed, and achieve manual focusing and exposure compensation.

8.3.3 Multimedia Teaching Means Multimedia is a technology that can acquire, process, edit, store, and display two or more different signal media at the same time. It has the characteristics of integration, controllability and interactivity, etc. 1. Preparation of materials in multimedia courseware The materials in multimedia courseware include sound, graphics, images, animations, video images, and so on. Different materials should be prepared by different means. Sound: The sound in multimedia courseware includes language, music, effect sound and so on. In general, for the preparatory work, the language is recorded by professionals, the music may be composed exclusively or ready-made music can be borrowed; the effect sound can be recorded voluntarily or found from the sound effect library. Graphics: Graphics can be created using graphics authoring tools or graphics drawing tools provided by multimedia authoring tools. Simple graphics can also be created with the drawing tools in Word, WPS (WPS2000) and other software.

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Images: Images can be made using “drawing” accessory in Windows or image making tools, scanned by a scanner, shot by a digital camera, or made using drawing program provided by multimedia authoring tools, etc. Animations: Animations can be designed using the programming language, designed and produced using the special animation authoring tools, or designed and produced using the animation production functions provided by the multimedia production tools. Video images: The videos in multimedia are generally captured by the dynamic video capture card from the camera, VCR or DVD player, and formatted by the video editor such as editing, compressing, synchronous format conversion of audio and video, etc., to form the video files in AVI format or other formats to be saved for use. Texts: The texts can be input using Word, WPS and other word processors, which can set the character pattern, font, type size, color, style etc. of the text, and characters can also be deformed, shaded, networked, three-dimensional, etc. 2. Application of multimedia in classroom teaching First of all, use appropriate software to combine the prepared multimedia materials organically to complete the integration of multimedia materials so as to be effectively used in teaching. There are often two types of multimedia courseware: classroom demonstration type and independent learning type. When making classroom demonstration type courseware, WPS2000, Powerpoint97, Powerpoint2000, and other software can be used to demonstrate all materials according to teaching needs. If making multimedia courseware for students’ independent learning, it is necessary to integrate all kinds of materials with special multimedia authoring tools. There are various forms of multimedia teaching. Among then, multimedia classroom teaching is usually carried out in the multifunctional classroom equipped with multimedia computers, projectors, video presenters (or digital cameras). For this reason, it is necessary to first understand the usage of the projector’s corresponding display equipment. The projector is generally fixed in the classroom. It is a device that can be connected to a variety of input devices (such as computers, presenters, video recorders, VCDs, DCDs, etc.) and can convert the electrical signals sent to it into optical signals to be projected on the screen. The operation of the projector mainly include connecting the signal source to the projector, power on, power off, image adjustment and other function selection, etc. When using it, we should first pay attention to the normative use, and then understand the meaning of various functions, and select them correctly in the teaching. Video presenter is a device that uses a lens to image the physical object (or text) onto an OCD chip, which transfers it into electrical signal, and then transfers it into video signal to be transmitted through the video output interface to the television for presenting, or transfers the analog electrical signal into digital electrical

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signal and then transmits to the projector through the S-232C and other digital interface for presenting. The usage of physical object presenter mainly include making connections, power on and off, focus, zoom, white balance adjustment, and other operations. If a digital camera is used instead of a video presenter, the content to be presented is usually captured and stored in the digital camera before class, and then the digital camera is connected to a TV or projector for presenting in class.

8.3.4 Network Teaching Means Network teaching refers to the use of computer networks for teaching. Computer networks are widely used in all walks of life and play an increasingly prominent role in teaching. China has formed four major computer networks, namely China Public Computer Interconnection Network (CHINANET), China Education and Research Network (CERNET), China Science and Technology Network (CSTNET) and China Golden Bridge Network (CHINAGBN), among which CERNET is a network sponsored by the Ministry of Education and specially designed for educational purpose. Network teaching is characterized by extensive teaching resources, learning sociality, time flexibility, famous teachers, and knowledge buildability. The main functions of the network and its role in education are: 1. Network communication Network communication mainly takes the form of E-mail. Electronic mail (E-mail) is a tool for people to exchange information in the form of mails on the Internet through computers. E-mails are not only simple, flexible, and intuitive to use, but also multifunctional. The same mail can be sent to many people at once. In education, many different teaching modes such as teacher–student, student–student, teacher–teacher, teacher or student-experts can be realized by means of E-mails. 2. Remote login to computer system Remote login is the access to and use of remote computer systems via the Internet. The remote login enables students to connect their own computers with the remote teacher’s computer and make their own computers as the terminal of the remote computer so as to use the system resources of the teacher’s computer. In this way, the student’s computer actually becomes an emulational terminal of the teacher’s host computer, and the teacher specially opens up an area for students to log in, and arranges various courseware, to facilitate students to carry on the individualized learning. 3. Network information service Network service function is the unique and attractive function of the Internet, including information inquiry service and information resource establishment

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service. There are many information inquiry tools on the Internet, such as WWW, Archie, WAIS, etc. The Internet is not only a place where people ask for information, but also a place where information is released and stored. The Internet information is widely distributed in various information servers. The function of network information service is particularly important in network teaching. Teachers can release learning information through the webpage, and students can learn through the web browsing function. At the same time, teachers can set a password to limit the browsing of non-students, and students can also save some of the contents to the hard disk for further study after class if necessary. Usually, teachers can teach simply by setting up corresponding teaching materials and some simple interactive questions on the home page. 4. Discussion function The discussion function makes use of the Internet to allow many people to participate in a discussion at the same time. The teacher opens up a special discussion board where the teacher and students can discuss a topic through text input for heuristic instruction. This method requires teachers and students to be able to input Chinese character information skillfully. In addition, teachers are required to be able to control the whole discussion process. When necessary, they can interrupt students’ discussion and shift to unilateral information transmission from the teachers. On the Internet, the news discussion function is widely used in scientific research, which is helpful to cultivate students’ scientific research ability. In a word, network teaching is a new form of teaching. In the future, any teacher and student will be inseparable from it. Therefore, teachers must be familiar with network functions, be able to use network tools with ease, be able to design and make their own teaching web pages, and be able to make full use of this teaching form in their own teaching and scientific research as soon as possible. Review Questions and Exercises 1. What is the teaching form, and why is it diverse and hierarchical? 2. How did the class teaching system come into being and develop, and what are its advantages and disadvantages? 3. What are the main tasks of mathematics lesson and why? 4. What are the main types of mathematics lessons, and what is the structure of these main types? 5. What are teaching means and what are the most commonly used teaching means? 6. What is mathematical language and what are its characteristics? 7. What requirements should mathematics teaching language meet? Try to give examples. 8. What is the significance of mathematics blackboard-writing, and what requirements should blackboard-writing meet? Try to give examples. 9. What are some common teaching aids in secondary school mathematics teaching, and what requirements should be met when applying them?

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10. What are modern teaching means and what is the significance of using modern teaching means in the mathematics teaching in secondary schools? 11. Investigate and understand the use and production of teaching aids in mathematics teaching in current secondary schools through educational probation or practice. 12. What is teaching with slides and projection, and what contents in current mathematics teaching in secondary schools are suitable for teaching with slides and projection? 13. What are the teaching means of videoplaying and video shooting? How to operate the video playing and camera shooting, and what requirements should be met? 14. What are the multimedia teaching means? What are the characteristics of multimedia teaching? How are the materials in the multimedia courseware generated and produced? 15. What are the network teaching means? What are the characteristics of network teaching means? What are its main functions? 16. Select a section from the textbook, and try to design the concrete plan of using modern teaching means.

Chapter 9

Teaching Principles and Methods of Secondary School Mathematics

Teaching principles and teaching methods are the product of a certain social ideological system, while the social ideological system is a reflection of a certain social structure. The teaching principles and methods of secondary school mathematics have always been two important and closely related topics in mathematics teaching. The research on them is theoretically and practically significant.

9.1 Teaching Principles of Secondary School Mathematics The teaching principles are the reflection of teaching laws, the crystallization of teaching experience, the basic requirements to guide teaching work, and also the basic norms that teachers must abide by in teaching work. They have the requirements of directionality, scientificity, generality, and completeness. Secondary school mathematics teaching is a special educational science, with its own characteristics and special regularities. How to propose a set of teaching principles system that meets the requirements of modernization and is suitable for national conditions is an important issue to be solved urgently in the current mathematics education circle. Subject to the constraints, we can only introduce a few principles of mathematics teaching that people have reached a consensus on.

9.1.1 Principle of Combining the Concrete with the Abstract 1. Concreteness and abstractness Mathematics takes the spatial forms and quantitative relations in the real world as the research object. It is very vivid and concrete, with realistic prototypes and concreteness. At the same time, it is abstract, and this abstractness is also manifested as a high degree of generality. Generally speaking, the higher the level of abstraction of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_9

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mathematics, the stronger its generality; the abstractness of mathematics also has the characteristics of reabstraction. For example, the development from numbers to formulas and from functions to mappings has experienced many abstractions. Mathematical abstractness is also manifested in the extensive and systematic use of mathematical symbols, which is unrivaled by other sciences. People’s cognitive process always starts from practice. Only by taking objective things as the foundation can we obtain abstract concepts and abstract propositions in mathematics. On the contrary, some abstract mathematical content can be described with concrete examples, and even some abstract mathematics thoughts and mathematical methods often have very realistic and concrete backgrounds, even a higher degree of abstractness is no exception. For example, the trigonometric function of acute angle is a relatively concrete abstract concept, its generality is relatively weak, and the specific content it relates to is relatively less. However, when it advances to trigonometric function of any angle, it is manifested as a circular function, and the specific content involved includes relative periodic motion. When it is expanded to a numerical function, the specific content involved includes relative periodic motion. 2. Limitations of secondary school students’ abstract thinking ability The current secondary school students, especially those in lower grades, have weak abstracting ability, so they are dependent on concrete materials during learning, separate the concrete from the abstract, and have difficulty in grasping the relationship between abstract conclusions. For example, they are not apt to accept the “quantities with opposite meanings” if not through a certain number of examples; have difficulty in understanding the “concept of locus”; and are often unable to draw a figure and confound the known with proving for the proposition “two line segments obtained by the extension line of the bisector of the exterior angle of a parallelogram cutting off two nonadjacent sides are equal”, etc. For another example, they have difficulty in finding the law and prove it from the following known conditions 1 1 + 1×2 2×3 1 1 + 1×3 3×5 1 1 + 1×4 4×7 1 1 + 1×5 5×9 .........

1 1 n + ··· + = 3×4 n(n + 1) n+1 1 1 n + + ··· + = 5×7 2n + 1 (2n − 1)(2n + 1) 1 1 n + + ··· + = 7 × 10 3n + 1 (3n − 2)(3n + 1) 1 1 n + + ··· + = 9 × 13 4n + 1 (4n − 3)(4n + 1) +

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It is thus evident that now it is necessary to gradually raise reasonable requirements for abstractness in teaching to improve students’ abstracting ability in a step-by-step and planned manner. 3. Combining the concrete with the abstract during teaching The principle of combining the concrete with the abstract is a reflection of the law of unity of opposites of abstract thinking and the vivid and concrete in mathematics and is determined by the laws of commonality and particularity of people’s process of cognition in the teaching process. In other words, teaching should start from the students’ perception, from the concrete to the abstract, and then from the abstract to the concrete, to enable the students to form correct concepts and make judgements and inferences. The specific methods can be as follows: (1) Pay attention to introducing and expounding mathematical concepts through examples. Form visual images and provide perceptual materials through physical intuition, simulacrum intuition, or language intuition. For example, introduce quantities with opposite meanings through rise and fall of temperature, getting in and out of goods and other examples; introduce the concept of similarity through image changes under certain conditions, etc. (2) Pay attention to introducing special cases to explain general laws. For example, for solving quadratic equations with one unknown, we generally start with x 2 = a, then learn (x + a)2 = b, and then learn ax 2 + bx + c = 0, which is easier for students to accept. It must be pointed out that from the perspective of mathematics teaching, cultivating students’ abstract thinking ability is the goal, and the concrete and intuitive is only a means. If we do not pay attention to cultivating students’ abstract thinking ability, they will not be able to learn mathematics well; on the contrary, if we do not rely on the concrete and intuitive, it is also difficult to cultivate abstract thinking ability. However, if we only stay in the perceptual stage, it will surely dampen students’ enthusiasm for learning and affect the further development of thinking ability. Only by combining the concrete with the abstract constantly and repeating the cycle of concreteness–abstractness–concreteness in teaching can we continue to learn in depth and breadth.

9.1.2 Principle of Combining Theory with Practice 1. Improving the theoretical level of secondary school mathematics appropriately For a long time, the theoretical level of mathematics teaching in secondary schools is not high: it lacks theoretical guidance, emphasizes memory and imitation, and neglects understanding and systematization; lacks teaching of mathematical thinking and methods; and lacks introduction to ideas and methods of solving some difficult and tricky exercises, etc.

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Raising the theoretical level is by no means profound content, hidden relationships and ingenious methods, the supplementation of extracting root formulas of cubic and quartic equations in one unknown, or the introduction of “nine-point circle”, “Euler line”, and other contents. It lies in giving better play to the universal guiding significance of the theory and in strengthening the teaching of general principles and methods. For example, the synthetic division and remainder theorem in algebra have higher theoretical and practical value than some specific methods and techniques of factorization; using differential method to solve problems related to tangent and extreme value, and using integral method to solve problems related to area, volume, etc., can improve secondary school students’ theoretical level in mathematics. To raise the theoretical level, the key is to attach importance to the teaching of mathematics thinking methods in secondary schools, strengthen students’ thorough understanding and mastery of relevant principles and methods, and correspondingly improve their ability to solve problems. For example, for the understanding of the contracted multiplication formulas, the key is not only to understand their derivation, but also to truly understand the arbitrariness of the relevant letters and the plasticity of the formulas, and the basic deformation and generalization of these formulas, i.e., to derive the square formula of difference of two terms from the square formula of sum of two terms and then derive the square formula of sum and difference of three or multiple terms, etc. 2. Strengthening the integration with practice vigorously For a long time, secondary school mathematics teaching has not been adequately integrated with practice, its content is too outdated, and it does not coordinate well with the current construction of Four Modernizations and the teaching of other subjects in secondary schools. Strengthening the integration with practice does not lie in borrowing a large number of examples to illustrate the application of mathematics, but in enhancing the concept of quantities and cultivating students’ ability to apply mathematical knowledge through examples. Therefore, when introducing examples, we should not only pay attention to the typicality and conciseness of the examples, but also pay attention to the updated content and processing methods in connection with practice. This requires strengthening integration with the knowledge acquired by secondary school students, especially physics, chemistry, biology, and other disciplines; teachers are required to keep a close eye on the current new situation in industrial and agricultural production and scientific research and pay attention to abstracting mathematical content from practice as a supplement to textbooks. For example, when explaining the application of the scale and the coordinate system, the following similar examples can be supplemented. Example 1 The statistical table of the quantity of products of various groups in a factory is as follows:

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Group

Quantity A

B

C

D

E

1

2

3

4

0

2

2

4

0

1

3

5

3 .. .

3

4

5

0

3

Where A, B, C, D, and E represent the quantity of containers filled with goods, the quantity of large cases, the quantity of small cases, the quantity of boxes and the quantity of remaining pieces respectively, and they are all in senary system (i.e., 6 pieces make a box, 6 boxes make a case, etc.). Find the production volume of each group and the total production volume of the factory. Example 2 Two tests A and B were performed on ten patients suffering from diseases α and β, and the measured data is as follows: Disease α Disease β

A

9

8

7

4

7

6

9

9

8

B

8

7

6

1

4

2

9

5

3

8 6

A

2

3

7

4

4

9

1

8

2

4

B

4

4

1

4

3

0

5

0

5

2

If the measured data A of a patient is 8 and data B is 4, try to infer the specific disease the patient may suffer from. If taking the A value of the disease as the abscissa and B value as the ordinate, we draw the distribution diagram of diseases α and β on the coordinate plane and then can infer that the patient may suffer from disease α based on the measured data of the patient. 3. Combining theory with practice during teaching The combination of theory with practice is the basic tenets and excellent style of Marxist-Leninist epistemology and methodology and the embodiment of the law of cognition in teaching. It should be noted that the theory of secondary school mathematics comes from practice. Only by focusing on integrating theory with practice can we acquire more complete knowledge and develop students’ ability to analyze and solve problems better, and at the same time, the criterion for testing mathematical theory can only be practice. For this reason, when learning new concepts, on the one hand, we should try our best to introduce them through the examples that students are familiar with and pay attention to obtaining mathematical concepts and laws through scientific abstraction, generalization, and necessary logical reasoning from practical problems, to enable students to realize that this mathematical knowledge is a reflection of the real world; on the other hand, after learning new concepts and new theorems, we must

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not only complete certain exercises, but also pay attention to applying the acquired knowledge into practice as much as possible, to gradually cultivate students’ ability to analyze and solve problems, and lay a foundation for applying the knowledge acquired into practice. Of course, while strengthening theoretical teaching, we should pay special attention to the revelation of mathematical thinking and methods, and the teaching of general principles and methods, to enable students to thoroughly understand and firmly grasp these thoughts, principles, and methods; while strengthening the integration with practice, we must update our concept, material selection and processing methods to enable students to truly understand the dialectical relationship between theory and practice and to prevent unhealthy tendencies of pragmatism and dogmatism.

9.1.3 Principle of Combining Rigorousness with Being Realistic 1. Rigorousness Rigorousness is one of the basic characteristics of mathematics. It requires that the description of mathematical content must be concise and accurate, the derivation, demonstration, and system arrangement of conclusions must be rigorous and elaborate, with logical rigorousness and certainty of conclusions. As the rigorousness of scientific mathematics, a complete system of axioms is generally proposed first, from which a series of theorems are logically derived, without any intuitive element. However, the rigorousness of mathematics has the relativity of development, and it has been continuously enriched and improved with the development of history. For example, the concept of function has gone through seven stages of development before it becomes rigorous gradually, and Euclidean geometry did not become truly rigorous until the Hilbert’s system of axioms was established at the end of the nineteenth century. Generally speaking, pure mathematics focusing on theoretical research has higher requirement for rigorousness, while applied mathematics focusing on applied research has relatively low requirement for rigorousness. The rigorousness of mathematics also has a process of gradual improvement with the development of people’s cognitive ability. When we start to learn mathematics, it is often not rigorous, and our understanding depends on intuition. For example, a point is understood as a very small ball, similarity as resemblance, ∞as a very large number, limit as proximity, etc. Only after these concepts are systematically studied,

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their true meanings are clarified, the stage of rational cognition is entered through in-depth discussion can the requirements for rigorousness be reached. 2. Being realistic Being realistic means doing what you are capable of, i.e., teaching what is acceptable to students. This is because the physical and psychological development of adolescents are in stages, and there are obvious differences in the level of thinking development, understanding, and receptivity at various age stages. Therefore, in mathematics teaching, the age characteristics of adolescents must be considered in arranging courses, processing textbooks, designing teaching methods, etc., so as to have a gradual adaptation and improvement process to the rigorousness of teaching. The teaching requirements are neither too high to surmount nor too low. Obviously, the requirement for being realistic can neither be ignored nor accommodated. For example, when students begin to learn equations with literal coefficients, there are often certain restriction on the literal coefficients. Only after an appropriate stage can this restriction be given up and students be allowed to discuss the solutions. This is the concrete embodiment of implementing the principle of being realistic. 3. Combining rigorousness with being realistic while teaching The combination of rigorousness and being realistic is determined by the essence of science of mathematics and the characteristics of mathematics teaching and is the reflection of the law of unity of opposites of the rigorousness of mathematics and being realistic of students’ cognitive ability in teaching. Combining rigorousness and being realistic is an important principle that must be followed in formulating curriculum standards, compiling textbooks, and the entire teaching process and should be implemented in the aspects of students’ mastering basic knowledge, developing basic abilities, using mathematical language, etc. In the actual teaching work, the teaching requirements must be appropriate and clear; the language must be rigorous and concise, with meticulous thinking, correct, and well-arranged ideas; teach from the shallow to the deep, from the easier to the more advanced, from the near to the distant, from the simple to the complex, from the known to the unknown, and from the specific to the abstract, from the particular to the general. While emphasizing the rigorousness, the acceptability of students cannot be ignored; while emphasizing being realistic, the scientificity of the content cannot be ignored. Only by combining the two organically can the quality of teaching be promoted.

9.1.4 Principle of Combining Imparting Knowledge with Cultivating Ability 1. Relationship between knowledge and ability As pointed out in Chapter Four, knowledge is the sum of people’s understanding of objective things, and ability is people’s capability to complete certain activities

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successfully. Knowledge and ability are different and interrelated and interacted as well. In other words, mastering knowledge is the condition and basis for developing ability, and the ability is the premise and result of mastering knowledge. It should be pointed out that skill is a mode of activity of completing certain tasks, and it is individual and part-based, while ability is integrated and assembled. The skill focuses on technical mechanical operation, while ability focuses on active processing of thinking. It can be seen that skills are the basis for developing ability, and ability is the comprehensive development and improvement of skills. Mathematics teaching from “double basics” to “imparting knowledge and cultivating ability” is a further development and improvement. The current secondary school mathematics curriculum standard has made specific provisions on the content and requirements for related skills: to be skilled in calculations (can calculate), be skilled in table lookup (can look up), be able to use drawing tools correctly (can draw), be able to carry out preliminary reasoning (can demonstrate), etc. Therefore, we must pay attention to skill training as the current secondary school mathematics, especially middle school mathematics, is mostly technical knowledge. 2. Significance of cultivating secondary school mathematics ability One of the notable features of the international modernization movement of secondary school mathematics education over the past fifty years is putting the cultivation of ability in a more important position than the mastery of knowledge. Therefore, the cultivation of ability has always been the proposal of many Chinese and foreign educators and has become a trend in the current international mathematics teaching reform as well. The mathematization of contemporary sciences and the interpenetration of various basic disciplines of mathematics have promoted the emergence of many new disciplines. This requires people, on the one hand, to learn the best content in the best way, mode and method; on the other hand, to attach importance to the development of intelligence, cultivate ability, and use the key of intelligence to open the door of knowledge one by one to meet the challenges of this new situation. Our educational policy is to enable the educated to get a lively development in moral, intellectual, physical, esthetics, and labor education. Instead of being “knowledge education”, intellectual education is not simply imparting knowledge, but includes imparting knowledge, cultivating ability, and developing personality. Among the Secondary School Mathematics Syllabus formulated 8 times since the founding of New China, specific requirements for cultivating ability have been clearly put forward for the last 7 times. Following the laws of education, especially paying attention to ability cultivation, has become a top priority for improving the quality of secondary school mathematics teaching. 3. Combining imparting knowledge with cultivating ability while teaching The combination of imparting knowledge and cultivating ability is a teaching principle of dialectical materialism. The combination of imparting knowledge and cultivating ability is conducive to acquiring knowledge and wisdom. In essence, it is a

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reflection of the law of unity of opposites of teaching and development and knowledge and intelligence. It is restricted by the laws of the intellectual development of adolescents and is also determined by the rapid development of contemporary science and technology and the urgent need of mathematics teaching reform. In the reform of secondary school mathematics teaching, how to combine imparting knowledge with cultivating ability is a relatively complex issue involving a wide area. First, we should first make clear the essence of knowledge and ability, the difference and connection between knowledge and ability, and be clear about the importance of imparting knowledge and cultivating ability and the specific requirements and tasks in the teaching of each discipline and unit of mathematics. This is the general prerequisite and basis for implementing the teaching principle of combining imparting knowledge with cultivating ability. Second, in terms of the commonalities between imparting knowledge and cultivating ability, we should attach importance to educating students about learning objectives, stimulate their interest in learning, strive to acquire basic knowledge, master basic methods and basic skills; improve teaching methods and organizational form of teaching, which is the key to imparting knowledge and cultivating ability; pay attention to the interpenetration and comprehensive application of knowledge in various disciplines of mathematics, which is an important measure to impart knowledge and cultivate ability; and improve the cultural and professional level of teachers, which is an important condition to impart knowledge and cultivate ability. Third, in teaching activities, there is a sequential order between knowledge and ability, and each has its own emphasis, but they are also interactive, interrelated and reinforcing each other. Therefore, in secondary school mathematics teaching, we should integrate knowledge impartment and ability development and strive to achieve simultaneous development. For example, in the teaching of multiplication formulas: (a − b)(a + b) = a 2 − b2 , (a ± b)2 (a 2 ± 2ab + b2 ), (a ± b)(a 2 ∓ ab + b2 ) = a 3 ± b3 , (a ± b)3 = a 3 ± 3a 2 b + 3ab2 ± b2 In terms of knowledge, it is to enable students to understand the meaning of formulas, master their derivation process, and use them to perform multiplication. When applying the above formulas for factorization, it is to enable students to master the steps of reverse use of the above formulas and develop factorization ability by means of formula method. However, at a certain stage in the teaching process, the requirements for knowledge and ability have their own emphasis, but this process is not entirely separate and individually closed, because the elements of factorization ability are included in the teaching of multiplication formulas, and while cultivating the ability of factorization, it is necessary to continuously deepen and consolidate the

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knowledge of multiplication formulas, lay the foundation for acquiring knowledge of fractions and equations, etc.

9.1.5 Principle of Combining Shape with Number 1. Shape and number are two basic contents of mathematics As a science with a long history, mathematics has now developed into a huge treasure house of three to four hundred branch disciplines. However, the entire subject of mathematics has always been extracted, evolved, and developed around two basic concepts of shape and number. “The objects of pure mathematics are the spatial forms and quantitative relations in the real world”. The concept of number originated from counting. Later, for the sake of calculation, the rules and methods of calculation were generated, natural numbers, integers, fractions, irrational numbers and real numbers came into being, and after further research and abstraction, imaginary numbers, n-dimensional numbers, ideals, etc. came into being again. The same is true for the origin of the concept of shape. Objects in nature always exist in various shapes. Human’s practice starts from the study of straight lines, triangles, circles, and other fundamental figures, to the study of some complex figures, and from the study of shapes of concrete things to the study of abstract pure shapes in mathematics. Although the treasure house of mathematics is becoming increasingly abundant and the realm of mathematics is gorgeous, their research objects are still “very realistic materials”, and shape and number are still two basic contents of mathematics. 2. The combination of shape and number is the idea and method of mathematics The concepts of shape and number have their own definite meanings respectively, but there is an essential connection between them. People have discovered this connection a long time ago. For example, in the Song and Yuan dynasties in China, people used the geometric algebraization method, where some geometric features were represented by algebraic expressions. Especially after Descartes, a French mathematician, established the connection between shape and number through the coordinate system and laid the foundation for a new analytic geometry in the seventeenth century, this thought and method of shape-number combination has become a turning point in the history of mathematics, with far-reaching significance. As stated by Lagrange, “as long as algebra and geometry went their separate ways, their progress would be slow and their applications would be narrow. But when these two sciences were combined into companions, they absorbed fresh vitality from each other, and have moved towards perfection at a rapid pace since then”. This thought and method of shape-number combination has brought great benefits to the study of geometry. It not only makes clearer some geometric problems that have not been solved for a long time (e.g., problems that cannot be constructed with ruler and compasses, etc.), but also makes it possible to unify various processing

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methods in geometry with this method of transforming geometric problems into algebraic problems, greatly expanding the research scope of geometry. The thought and method of shape-number combination has also brought many benefits to the research of algebra. It makes many algebraic topics clear and intuitive, and algebra acquires new vitality because it borrows geometric terms after analogy with geometry. For example, linear algebra borrows space, linearity, and other concepts in geometry and analogy methods to enrich itself and develop rapidly. The thought and method of shape-number combination also gives people very important enlightenment in methodology. During the development of mathematics, the thought and method of correspondence between points and numbers and between curves and equations in a plane has inspired mathematicians to further consider regarding each function as a “point” and regarding the whole of a certain type of functions as a “space”, and it gives rise to functional analysis and other mathematical branches. 3. Combining shape with number while teaching In the real world, shape and number are inseparable and integrated. This is the embodiment of the combination of intuition and abstraction and the combination of perception and thinking. The combination of shape with number is not only a need for the development of mathematics itself, but also a need for deepening our understanding of mathematical knowledge, developing intelligence, and cultivating ability. Superficially, traditional secondary school mathematics contents are divided into two parts: shape and number, secondary school algebra is the main discipline for studying numbers, secondary school geometry is the main discipline for studying shapes, and secondary school analytic geometry is the main discipline for studying the combination of number and shape. In fact, the combination of shape and number has infiltrated into the teaching of various disciplines of mathematics. For example, the study of the combination of real numbers and number axis, the study of the combination of complex numbers and coordinate plane, and the study of the combination of functions and their graphs. Algebraic equations can be expressed in various quantitative relations, so they can solve problems related to length, area, volume, etc. Linear equations and quadratic equations in two unknowns represent planar lines and quadratic curves, respectively, etc. On the contrary, the calculations related to length, area, and volume often can’t do without solving equations; and planar lines and quadratic curves also provide geometric representations for linear equations and quadratic equations in two unknowns, respectively. Just for this reason, the current secondary school mathematics curriculum standard no longer divides the content of mathematics into disciplines, which will be more conducive to implementing the principle of combining shape with number in teaching, and become another concrete manifestation of the new curriculum reform. For teaching on the principle of combining shape with number, students are required to master the thought and method of combining shape with number truly, dig into textbooks from the perspective of combining shape with number, understand related concepts, formulas, and rules in mathematics, and master the methods of using

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combination of shape with number to analyze and solve problems, to improve operational ability, logical thinking ability, spatial imagination ability, and problem-solving ability. Although the above teaching principles have different characteristics, they share the same spirit. Students’ understanding of objective things always starts from feeling and perception, and teaching should start from specific situations. This requires the implementation of the principles of concreteness, being practical and being realistic. Due to the characteristics of mathematics, it is rigorously logical, highly abstract, and widely applied. It is necessary to implement the principles of rigorousness, abstractness, and raising theoretical level. The above principles are also a dialectical unity. Learning is not simply for the purpose of accumulating knowledge and more importantly for cultivating ability. Therefore, in actual teaching, it is necessary to combine rigorousness with being realistic, concreteness with abstractness, theory with practice, imparting knowledge with cultivating ability, and shape with number. Finally, it must be pointed out that the above-mentioned principles also need to be further tested, developed, and explored in practice, in order to be more scientific and substantial in content, so as to gradually establish a new system of secondary school mathematics teaching principles with Chinese characteristics.

9.2 Teaching Methods of Secondary School Mathematics The teaching method is a mode of activity that teachers and students interact with each other to achieve the purpose of teaching and is a dynamic system composed of many specific teaching approaches (such as narration, description, explanation, presentation, demonstration) and means. The teaching method is summarized in the teaching practice and constantly developed and changed in practice. It varies with different teaching aims and tasks, teaching contents, and teaching objects. It is not only a reflection of different historical periods, but also is constrained by the conditions of teachers and the characteristics of students’ physical and mental development. Great importance has always been attached to the research and application of mathematics teaching methods in secondary school mathematics education in our country, which is still widely acclaimed by the world. We will introduce them separately as follows.

9.2.1 Heuristic Teaching Method The heuristic teaching method is a teaching method for teachers to follow the laws of cognition, proceed from the actual situation of students, and be good at stimulating students’ thirst for knowledge and interest in learning under the premise of giving full play to the leading role of teachers, and guide students to think actively to take

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the initiative to acquire knowledge. It is the most important, fundamental, and most widely used teaching method in secondary school mathematics teaching. However, some people don’t think that it has a set of fixed teaching model or several specific teaching links, so it is not a specific teaching method, but a kind of classroom teaching principle. In fact, in specific teaching, as long as the teaching method has the above basic heuristic characteristics, it can be called heuristic teaching method in general. Its opposite is “pouring-in” type of teaching, also known as cramming method of teaching. It is Confucius, an ancient educator in our country, who first proposed heuristic teaching. He advocated that “I will not enlighten my students until they have really tried hard but even so fail to understand, I will not instruct them until they have something to say but fail to make themselves understood, and if they fail to draw inference about other cases from one example, I will not continue giving examples”. This is where the term “enlightenment” originates from. After Confucius, Mencius also advocated heuristic teaching. In “Record on the Subject of Education”, he maintained that “’in his teaching, he (a superior man) leads and does not drag; he strengthens and does not discourage; he opens the way but does not conduct to the end (without the learner’s own efforts)”. So heuristic teaching is actually a traditional teaching method in our country. The heuristic teaching method is an effective teaching method that conforms to the materialist dialectics epistemology and methodology. Since ancient times, many educators have attached great importance to it. Comenius, a Czech educator in the seventeenth century, believed that “the acquisition of knowledge is subject to the learner’s aspiration, which cannot be forced”. Diesterweg, a German educator in the nineteenth century, advocated that teaching should be fascinating and interesting, etc. These illustrate the importance of inspiring and arousing students’ enthusiasm for learning in mathematics teaching. In recent years, people have regarded the heuristic teaching method as an emerging interdisciplinary subject of cybernetics, psychology, pedagogy, philosophy, and other subjects and have given special explanations to its basic concepts and principles. For example, cyberneticists believe that heuristics is an approach and method related to improving the problem-solving system; psychologists believe that heuristics is a chapter of the study of creative thinking in psychology; educators believe that heuristics is a science about the means and methods of solving problems; and philosophers believe that heuristics is a rule and judgement used to discover new things. There are many enlightenment ways, which can be divided into many basic methods, such as questioning enlightenment, situational enlightenment, intuitive enlightenment, analogy enlightenment, transformational enlightenment, and blackboard writing enlightenment. During the use of heuristics, teachers are required to prepare lessons carefully, have a thorough knowledge of students, be good at stimulating and inspiring students to think throughout the whole process of teaching, strive to achieve enlightenment “and guidance without chaos”, and maintain a friendly and harmonious learning atmosphere in order to achieve good teaching results.

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For example, a teacher designed three teaching methods in a lecture on the “classification of triangles” in middle school geometry. In order to save space, we select only one paragraph to illustrate the use of heuristic teaching methods here. Teaching method 1: “For the convenience of future research on triangles, we need to classify them. A triangle has three sides and three angles. We classify them by sides and angles respectively…”. Teaching method 2: The teacher shows the teaching aids and explains: “Here are six triangles made of iron wires. Let’s observe and see what their characteristics are”. Then the students speculate. Some say “some are big and some are small”, others say “some are regular and some are oblique”…The students can’t get the point of the problem, so the teacher has to explain it again finally. Teaching method 3: “We know that cars, boats, etc. are all means of transportation, and they are classified separately. This is our classification of means of transportation. If they are not classified, all cars are called ‘cars’, and all boats are called ‘boats’, which will cause great inconvenience for us to use them. Similarly, when studying triangles, we must classify them”. Then the teacher shows the wall charts, with six triangles marked with side lengths and interior angle degrees. Then the teacher asks students to “observe the sides or angles of these triangles, and see what their characteristics are”. The “teaching method 1” here does not create a problem situation, students have difficulty in thinking, and it is a typical “pouring-in” type of teaching; although in “teaching method 2” the teacher can create a problem situation in an attempt to cause students to think, it lacks correct guidance and has no direction for thinking, so it is an unsuccessful heuristic teaching method, while “teaching method 3” provides a more suitable problem situation, reaches the effect of guidance based on enlightenment, and makes better use of heuristic teaching method.

9.2.2 Several Common Teaching Methods 1. Explanation method This is a teaching method in which teachers make intensive and systematic explanation and analysis of the textbooks, and students are focused on listening attentively. In mathematics teaching, teachers often adopt this method to start a new unit, introduce concepts, draw propositions, and summarize knowledge. Generally, the teacher first introduces a new topic, then highlights the key points to solve the problem, clarifies the way to solve the problem, then solves the problem, and finally expounds the teaching content systematically. While using the explanation method, the teacher’s language must be inspiring, the explanation must be well arranged, focused, clear, and accurate; the teacher should be good at using analysis, synthesis, induction, deduction, analogy, and other scientific methods to inspire students to think independently; and teach students in accordance

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with their aptitude. If various links are used well, it can also achieve better teaching effect generally. The explanation method is currently a main teaching method that is widely used in secondary school mathematics, especially in higher grades. Its advantage lies in that it can maintain teachers’ initiative, fluency, and continuity in imparting knowledge, thus saving time, and the teaching process and teaching time can be easily controlled by the teacher. However, if it is misapplied, it will lead to “pouring-in” and “cramming” method of teaching, which is not conducive to the cultivation and improvement of students’ abilities. 2. Conversation method The conversation method, also known as the “small step” teaching method, is a method of teaching through the form of “dialogue” between teachers and students, that is, the teacher compiles the teaching content into a series of small questions with internal connection, raises them one by one in the classroom to enable the students to think, discuss, or answer, then corrects the errors in their answers to deepen their understanding gradually, and adjusts the teaching process and improves the teaching activities in time based on the feedback information. At present, it is widely used in mathematics teaching in primary schools and lower grades of secondary schools. To use the conversation method, teachers should plan carefully before class to enable the questions raised to be connected and logical, and all students in the class to be in a state of positive thinking; cherish students’ enthusiasm for answering questions; and prevent formalistic conversations. The conversation method is conducive to promoting students’ positive thinking, but teachers have to spend a lot of time preparing lessons, and it takes more time, and teachers are required to have a higher level of teaching art and be good at organizing classroom teaching. 3. Practice method This is a teaching method that allows students to master basic knowledge and carry out basic skills training through independent work under the guidance of teachers. It is suitable for solving exercises and can also be used to acquire new knowledge. When it is used for acquiring new knowledge, the teacher should first give students a certain amount of time to read the textbook and think, and then let the students discuss, practice, summarize, etc. under the guidance of the teacher. To use the practice method, the teacher should have a clear purpose, adopt flexible and diverse ways, and do a good job of individual tutoring; the exercises should be of distinct gradient, with appropriate difficulty level; the teacher should correct common mistakes in time, instead of leaving mistakes staying in the students’ minds. Its advantage is that it is conducive to promoting the development of students’ thinking

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and the cultivation of ability. However, this method is more time-consuming, and it is not easy to cover all students in the class with widely different levels. 4. Explanation-practice combination method This is a teaching method that guides students to acquire new knowledge, review and consolidate old knowledge, and cultivate basic abilities through the organic combination of explanation and practice under the guidance of teachers. For explanationpractice combination, we can focus on explanation, supplemented with appropriate practice; or focus on practice, supplemented with appropriate explanation; or focus on explanation and practice alternately, as the case may be. While using the explanation-practice combination method, the explanation must be properly detailed and concise and make a distinction between the primary and secondary one, and the exercises must center around two basics and be in various forms; the explanation and practice must be closely coordinated, with clear objectives and good planning. The advantages of explanation-practice combination method are as follows: it can give full play to the advantages of both the explanation method and the practice method, so that students can concentrate on listening to the lecture, and digest and consolidate what they have learnt through practice (solving problems), and continue to deepen their understanding to acquire knowledge in the process of combining explanation and practice. Therefore, it has become the most commonly used teaching method in secondary school mathematics teaching, and it is also a teaching method that achieves better teaching results. However, as students grow older, the proportion of explanation and practice varies. Generally speaking, the proportion of practice is preferably higher than that of explanation in middle schools and the other way around in high schools. 5. Teaching aid demonstration method This is a method of using audiovisual teaching aids for teaching. Because some mathematical concepts are very abstract, using teaching aids to combine abstract concepts with actual objects (or models) can stimulate students’ interest in learning, concentrate their attention, and achieve better teaching effect. For example, the demonstration of audiovisual teaching aids in solid geometry is conducive to cultivating students’ spatial imagination ability; the demonstration of related curve rulers in analytic geometry helps to deepen the understanding of relevant curves. With the modernization of teaching means, this ancient teaching method has gained new meanings and tends to expand its application gradually. To use the teaching aid demonstration method, the teacher should prepare the teaching aids in advance. The size and structure of the teaching aids should be convenient for use and students’ observation. It is not advisable to show the teaching aids before the demonstration, so as not to distract students; during the demonstration, the teacher should combine it with the explanation and pay attention to cultivating students’ observation ability and imagination ability, so that perceptual knowledge can be raised to rational knowledge, so as to enhance the demonstration effect.

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9.2.3 Several New Teaching Methods Various teaching theories have emerged in contemporary teaching field. The most prominent ones are three typical schools: Bruner’s “structure of the disciplines” in America, Zankov’s “new teaching system” in the former Soviet Union, and Wagenschein’s “example teaching” in Germany. They all believe that teaching is not only a process for students to acquire knowledge, but also a process for students to fully develop their abilities. With the exchange of international mathematics education after the reform and opening up and the change in people’s educational thought, the teaching methods in our country have also undergone great innovations and changes. The following are several new teaching methods that are quite prevalent in our country. 1. The reading-discussion-explanation-practice teaching method The reading-discussion-explanation-practice teaching method, also known as the “four-step teaching method”, was first summarized by Shanghai Yucai Secondary school in the 1970s. Among them, the so-called reading is to guide students to selfstudy textbooks and reference books and take notes, which is the basis of teaching; the so-called discussion is the discussion between students to explore problems actively, which is the key to teaching; the so-called explanation is to resolve doubts, either through the teacher’s explanation or through students’ discussion under the guidance of the teacher, which is the main link of teaching; the so-called practice is to let students practice by themselves, which is an important way to acquire and consolidate knowledge. The advantage of this method is that reading, discussion, explanation, and practice are conducted alternately, which can arouse students’ enthusiasm, improve the efficiency of classroom teaching, reduce students’ extracurricular burden, and help cultivate self-learning ability, expression ability, and innovative spirit, but it is not easy to control the teaching process. 2. Unit holistic approach This is a teaching method proposed systematically by Beijing Jingshan School in the early 1960s. It divides the textbook into several units according to the knowledge structure and the level of the students and is carried out in four steps: 1. Self-study and exploration: After the teacher reveals and guides the learning objectives and methods of this unit briefly, students are required to read the textbook, ask questions, and conduct a discussion; 2. Explanation of key points: The teacher explains the key points, difficult points, and confusing points of this unit briefly; 3. Comprehensive training: Students focus on studying those comprehensive and tricky exercises on the basis of doing imitated general exercises;

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4. Summary and improvement: The teacher deepens and improves it on the basis of students’ sorting and summarizing of this unit, that is, transition from the students’ “self-summary” to the “joint summary” of the teacher and students. The core of the unit holistic approach is, on the basis of the teacher’s thorough knowledge of the textbook and students, to find the knowledge structure of this part of content and students’ cognitive structure to acquire this part of knowledge actively and combine these two organically to find the best point, in order to achieve better teaching effect. The advantage of this method is that centering around the textbook, it is conducive to cultivating students’ self-learning ability and inquiry spirit and is conducive to students’ acquisition of more systematic and complete knowledge. 3. Six-lesson unit teaching method This is a teaching method proposed by Professor Li Shifa from Hubei University in the mid-1980s. It divides the textbook into several units, and the teaching is carried out through the following six lesson types in sequence: self-study lesson—students learn the textbook in the classroom themselves as required by the teacher; enlightenment lesson—the teacher explains the key points; review lesson—the teacher guides students to conduct independent reviews in the classroom; schoolwork lesson—the teacher guides students to do schoolwork independently in the classroom; correction lesson—the teacher and students correct the schoolwork jointly in the classroom: summary lesson—the knowledge and skills are generalized and integrated. This method can reduce students’ learning burden and can also reduce teachers’ workload of homework correction. 4. Research-centered teaching method This is a teaching method proposed by Liaoning Experimental Secondary School in the early 1980s. Based on the content of the textbook and the aim of teaching, the teacher first proposes provoking research topics and requirements, lets the students think and study freely to obtain preliminary knowledge, and then makes a summary. Its basic procedure is: the teacher raises questions—students think independently—conduct a study—answer the questions—the teacher makes a summary for improvement. The essence of the research-centered method is to enable students to explore problems actively, propose opinions, and develop thinking on the basis of independent thinking. It can be in the form of a research by desk mates or a research by four classmates; the research topics proposed should have clear requirements and can vary with lesson types. For example, in the case of a new lesson, the research should center around deep understanding of new knowledge; in the case of a practice lesson, the research should center around one or two problems so that students can use multiple methods to conduct research from different perspectives; in the case of a review lesson, the research should center around the systematicness of knowledge, etc. 5. Learning-guidance teaching method This teaching method is based on students’ self-study under the necessary guidance of teachers. It is another new teaching method that has emerged in the

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country in recent years. It includes students’ self-study, resolving doubts, teacher’s succinct explanation, and students’ practice. In the process of students’ mastering knowledge actively, it pays attention to the development of intelligence and is conducive to the organic unity of five elements of intelligence (attentiveness, observation, memory, thinking ability imagination) and seven elements of ability: selfstudy—exploration ability, expression—performance ability, physical strength— operational ability, social intercourse—management ability, innovation—creative ability, emotion—esthetic ability, and will—adjusting ability. In the learning-guidance teaching method, students learn before the teacher’s guidance; the teacher focuses on guidance and serves learning; the teaching method comes from the learning method and is determined by the needs of the learning method. The learning-guidance teaching method is conducive to teaching students in accordance with their aptitude and cultivating pioneering talents, and it is the negation of pouring-in type of teaching. The development from pouring-in to heuristics and from heuristics to learning-guidance marks a new stage of development of the teaching methods. 6. Discovery teaching method This is a teaching method first advocated by Bruner, a famous American psychologist, in the 1950s that allows students to discover problems on their own and acquire knowledge on their own initiative. Starting from adolescents’ psychological characteristics of being curious, studious, inquisitive, and with strong hands-on skills, Bruner proposes that under the guidance of teachers, through demonstrations, experiments, problem-solving, and other means, students are guided to discover knowledge as mathematicians discovered theorems in the past, so as to cultivate their ability to research, explore, and create. By different ways of thinking, the discovery methods are divided into analogy method, induction method, analysis method, learning transfer method, knowledge structure method, etc. The general steps of this teaching method are: creating a discovery situation; searching for answers to questions; communicating the findings; summarizing the findings; and applying the findings. The discovery teaching method allows students to acquire both knowledge and scientific thinking methods, but it is not conducive to students’ mastery of systematic knowledge and developing necessary skills. Generally speaking, it is difficult to apply it universally. 7. Inquiry teaching method This is a new teaching method that has emerged in recent years to meet the needs of implementing quality-oriented education and innovation education in our country. Because in conventional teaching, although training students’ ability to analyze and solve problems is also emphasized through the teaching of problem-solving, the scope of inquiry questions and the degree of inquiry ability are far from meeting the requirements for cultivating inquiry ability. To this end, the topic and teaching hours of inquiry learning were clearly stipulated in the Mathematics Curriculum Standard for Compulsory Education promulgated in 2001 and the High School Mathematics Curriculum Standard published in 2003.

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To implement the inquiry teaching method, we should have suitable inquiry topics first and then have appropriate teaching methods and means, to enable students to experience the process of mathematicians’ discovering the laws of mathematics through their own practice. A certain amount of experience has been accumulated in foreign countries since the 1980s. Due to the late start in our country and insufficient emphasis, it is still in the experimental stage. It is expected that it will play an important role in cultivating students’ innovative consciousness and ability in the near future. 8. Programmed teaching method This is a teaching method initiated and developed by Skinner, an American psychologist and educator, based on the principles of cybernetics in the 1950s that use teaching machines. The general process is as follows: the textbook contents are selected carefully to compile programmed textbooks including teaching materials, exercises, and keys, or electronic computers and other teaching apparatuses are used to present the textbook contents. Following the program, students read the textbooks, do exercises, check answers, and obtain feedback in time to continuously adjust their learning activities. When they encounter difficulties, the teacher will provide individual or group guidance. The programmed teaching methods can be divided into basic program (for everyone) and composite program (for top students or underachievers). The advantage of this teaching method is that it can fully arouse students’ enthusiasm for learning, help cultivate students’ self-learning ability, thinking ability and operational ability, and is conducive to teaching students in accordance with their aptitude. But it is not easy to program all contents; besides, if students’ activities are too programmatic, it will weaken the educability of teaching, which is not conducive to the development of students’ intelligence. In addition, there are discussion method, problem-based teaching methods, attempt teaching method, creative teaching method, situational teaching methods, mini-teaching method, example teaching method, etc. To sum up, they can be divided into explanation-listening mode, guidance-reading mode, inquiry mode, demonstration-practice mode, etc. In summary, we can see that these new (compared with traditional) teaching methods have highlighted the common characteristics of some contemporary teaching methods, which are laying emphasis on the development of students’ intelligence and the cultivation of their abilities; on the probe into students’ learning psychology and the study of learning methods; on the combination of teachers’ leading role and students’ dominant role; and on the proper retention and improvement of traditional teaching methods. In teaching, we should handle the relationship between traditional teaching methods and new teaching methods properly. On the basis of making good use of traditional teaching methods, we take the initiative to select new teaching methods cautiously and strive to summarize and create new system of mathematics teaching method suitable for our national conditions and with our own characteristics through practice.

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9.3 Determining Teaching Principles and Selecting Teaching Methods The teaching principle determines and has a direct guiding significance on the teaching method. It generally involves adherence and application in teaching. The teaching method is the implementation of specific teaching principle, and it generally involves flexible choice in teaching. In a sense, a proper teaching method is a guarantee for the implementation of teaching principle.

9.3.1 Application of Secondary School Mathematics Teaching Principles As we know, the teaching principles of secondary school mathematics are an objective reflection of the laws of secondary school mathematics teaching, a scientific summary of the practical experience of secondary school mathematics teaching, and the guiding principles and basic requirements that must be followed in the secondary school mathematics teaching work. For this reason, we should pay attention to the following points during the application of the secondary school mathematics teaching principles. First, we should make it clear that the teaching principles of secondary school mathematics play an important guiding role in secondary school teaching practice. The correct application of various teaching principles helps us act in accordance with the objective laws of teaching work consciously, give full play to teachers’ leading role and students’ dominant role, and create conditions for improving the quality of secondary school mathematics teaching comprehensively. Second, in secondary school mathematics teaching, we must implement both general teaching principles and the specific principles of secondary school mathematics teaching itself. Because of the interpenetration and interaction of various principles, generally speaking, the secondary school mathematics teaching practice is always the result of the interaction and unified implementation of various teaching principles. Third, all teaching principles must be implemented in all teaching activities, including every link of determining curriculum standards, compiling textbooks, formulating teaching work plans, and implementing classroom teaching. Of course, it does not mean that all teaching principles play an equal role under any circumstances, and it may involve the primary and secondary role or precedence order. Fourth, each principle must be implemented fully and dialectically to prevent absolutization and one-sidedness. For example, when emphasizing the abstractness, theoretical property, and rigorousness of secondary school mathematics, its concreteness, practicality, and being realistic must not be ignored; when emphasizing the teaching of basic knowledge, the cultivation of ability must not be ignored; when

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paying attention to the teaching of “numbers”, the teaching of “shapes” should not be ignored, etc.

9.3.2 Selection of Secondary School Mathematics Teaching Methods 1. Diversity of teaching methods As a very important issue in secondary school mathematics teaching, the teaching method is the mode of interrelated activities between teachers and students. The diversity of activity modes determines the variety of teaching methods. Each of the above teaching methods has advantages and disadvantages and has a certain scope of application. In actual teaching, several teaching methods are often used in a comprehensive manner, and a certain teaching method is rarely used in isolation. “There are methods for teaching, but there is no fixed method for teaching”. Any attempt to model or absolutize teaching methods and the tendency to use a teaching method as the only eternal teaching method are incorrect. We should select appropriate teaching methods flexibly according to actual needs and possibilities, so as to cooperate with each other. At the same time, we should work hard to continuously summarize experience and create effective teaching methods with more Chinese characteristics and in line with our own teaching practice. 2. Basis for selecting teaching methods Generally speaking, the teaching method is determined after comprehensive consideration of the teaching aims and requirements, teaching content, actual situation of the students, and the quality of the teacher. Different teaching methods may be selected for different teaching aims and requirements, teaching contents, actual situations of the students, and the qualities of the teacher. For example, if the purpose is to enable students to acquire new knowledge, the explanation method or discovery method can be selected; if the purpose is to cultivate students’ abilities, the practice method or research-based method can be selected; the explanation method is often selected for higher grades, while the conversation method is often selected for lower grades in secondary schools, and so on. At the same time, different teaching methods have different scope of application and conditions for application. For example, the conversation method requires ample teaching time and a strong ability to organize teaching; the discovery method requires more teaching time and designing of a certain discovery space; the discussion method is suitable for class of leaders, and the micro-teaching method is suitable for occasions with more teachers. Therefore, different teaching methods can be selected for

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different teaching contents, and different teaching methods can be selected even for the same content under different conditions. 3. Evaluation of teaching methods With the continuous deepening of teaching reform, people’s understanding of a good mathematics lesson has changed. In the past, the emphasis was on the teaching of knowledge, so it is a good lesson where the teacher explains clearly and smoothly and students can understand and accept easily. Now the emphasis is on the teaching of knowledge and abilities, it is a good lesson that can fully arouse students’ enthusiasm, with a high degree of students’ participation and a large amount of thinking activities. This also provides us with a reference for evaluating teaching methods. A good teaching method generally has two major characteristics: the unity of purpose, method and effect, and the high efficiency of teaching. This is because certain teaching methods are used to achieve the teaching purpose, and the effect is the sole criterion for testing the teaching method. It makes a good teaching method only when the purpose, method, and effect are unified. At the same time, a good teaching method should involve less manpower, less material resources, and shorter time; i.e., it must be highly efficient. Review Questions and Exercises (VIII) 1. What is the teaching principle of secondary school mathematics, how is it determined, and what is the practical significance of studying the teaching principles of secondary school mathematics? 2. What should the teaching principles of secondary school mathematics be, how to apply these principles, and what is the relationship between these principles? 3. How to understand the rigorousness and being realistic of mathematics, and how to combine rigorousness and being realistic in teaching? 4. How to understand the concreteness and abstractness of mathematics, and how to combine concreteness and abstractness in teaching? 5. How to understand the theoretical and practical nature of mathematics, and how to combine theory with practice in teaching? 6. What are knowledge, ability, and skill, what is the significance of cultivating ability, and how to combine imparting knowledge and cultivating ability in teaching? 7. Why are number and shape two basic concepts in mathematics, and how to combine shape and number in actual teaching? 8. What is heuristic teaching method, what is the significance of researching heuristic teaching method, and how to apply heuristic teaching method correctly in teaching? 9. What is a teaching method, what are the common teaching methods in secondary school mathematics teaching, and what are the new teaching methods? Illustrate them with examples. 10. What is the relationship between teaching principles and teaching methods, and how to apply teaching principles and select teaching methods in teaching?

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11. Try to analyze the application of teaching principles and teaching methods for two probation periods of secondary school mathematics. 12. Choose a concept, theorem, an example, or exercise in the secondary school mathematics textbook, and develop your tentative idea of applying specific teaching principles and teaching methods.

Chapter 10

Teaching Work of Secondary School Mathematics

The teaching process of secondary school mathematics is a two-way unified activity process of teaching and learning. In the course of this activity, students are supposed to master mathematical knowledge and skills, develop mathematical abilities and intelligence, and form certain ideological qualities. Teachers are busy with a series of main tasks, such as preparing lessons, giving lessons, correcting homework, tutoring, performance appraisal, and teaching research. Among them, preparing lessons and giving lessons well are particularly important, which will be the focus topic of this chapter, and the last two issues will be described in details in the following two chapters.

10.1 Lesson Preparation of Secondary School Mathematics A series of preparatory work carried out by teachers before class is called lesson preparation. Lesson preparation is the process of learning, analyzing, researching, and processing textbooks. It is the foundation of the entire teaching process, and it plays a decisive role in the quality of classroom teaching. The specific work and requirements of lesson preparation are introduced as follows.

10.1.1 Develop a Teaching Work Plan The key to lesson preparation is to “have a thorough grasp of both ends”, namely to have a thorough grasp of not only the syllabus and textbooks, but also the situation of students. Only by having a thorough grasp of the syllabus and textbooks can lesson preparation have a basis; and only by having a thorough grasp of the situation of students can lesson preparation be targeted.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_10

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Lesson preparation follows a top-down and coarse-to-fine working procedure. First of all, to prepare lessons, one must read through the syllabus and textbooks for overall preparation, then reread the textbooks to prepare lessons for the unit, and finally delve into the textbooks of each lesson to prepare lessons for the class hour. Although there are provisions on the teaching progress, teaching content, teaching requirements, and teaching arrangements for each semester in the syllabus, it is necessary for teachers, after careful investigation, research, and consideration, to work out a feasible teaching work plan to implement the stipulations and requirements in the syllabus in combination with the actual situation of the students in the class, so as to carry out the teaching work in an orderly and planned way. The teaching work plan generally includes the following five aspects: (1) (2) (3) (4) (5)

Teaching objectives and requirements of this semester; Analysis of the students’ situation in the class; Measures to improve teaching quality; Teaching schedule (including review, examination, etc.); Arrangements for extracurricular activities, etc. The teaching schedule can be printed and filled out in the following format.

Week

Date

Class

Chapter

hour

Teaching Implementation Remarks content

10.1.2 Prepare Lessons for the Class Hour In order to prepare each lesson well, the following work should be completed in general. 1. Read the textbooks intensively In intensive reading, study the definitions, axioms, theorems, formulas, rules, etc. in the textbook word by word carefully, grasp the key words that reveal their essential attributes, clarify the logical structure between each other, master the scientific nature of textbooks; clarify the cohesive relationship between chapters of the textbook, figure out the introduction of concepts, the relationship between applied knowledge examples and reality, master the practicality of textbook; explore and dig into the patriotism and dialectical materialism ideas in the textbook, master the ideological

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nature of textbook; distinguish the primary and secondary relations of knowledge, estimate the difficulty level of knowledge, and master the acceptability of textbook. 2. Consult relevant information Consult relevant materials, literature, and theoretical books as extensively as possible. It can not only deepen our understanding of textbooks and enrich the teaching content, but also absorb others’ teaching experience, to reduce or even avoid detours in teaching. At the same time, it provides reference and accumulates experience for the exploration of educational reform. For example, regarding the definition of equation, we can find more than six different narrative methods from the secondary school mathematics textbooks, secondary school mathematics textbook teaching method textbooks, mathematics dictionaries and related teaching reference books, etc. Through comparison, we can not only clarify the meaning of equation, but also deepen the understanding of textbook to put forward appropriate teaching requirements for students. 3. Set teaching objectives Teaching objectives (including cognitive objectives, ability objectives, and emotional objectives) are the basis for selecting lesson types and teaching methods, and also the benchmark to check teaching effects. The depth and breadth of teaching objectives should be appropriate. If it is too wide, it will not show the teaching requirements and characteristics of this lesson. If it is too narrow, it will lose the greater for the less. If it is too low, it will not meet the requirements stipulated in the syllabus. If it is too high, it will be divorced from reality and fail to complete the teaching tasks. In short, to set teaching objectives, we must consider all aspects appropriately and comprehensively, making them in appropriate measure, just the right range and difficulty level. The teaching objectives are usually set based on the analysis of knowledge points and ability points of the teaching content, then describing the teaching requirements for mathematical knowledge (cognitive objective), mathematical ability (ability objective), ideological education (emotional objective), etc. in generalized and concise language. 4. Clarify the focal points, difficult points, and key points The focal points are the points that run through the course and play a leading and central role in the textbook. They are determined by the position and role of the textbook. Usually, the definitions, theorems, formulas, rules and their derivations and important applications in the textbooks, the cultivation and training of various techniques and skills, the essentials and methods of solving problems, the sketching and description of graphics, and so on can all be determined as focal points. For example, the sine and cosine formulas of the sum or difference of two angles in trigonometry are a focal point; the definitions, standard equations, properties, and graphs, etc. of circle, ellipse, hyperbola, and parabola in analytical geometry are focal points. The focal point is relative. For example, similar figures are a focal point of plane geometry. In similar figures, similar triangles are a focal point. In similar

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triangles, the three decision theorems of similar triangles are a focal point. In the three decision theorems, the first decision theorem is a focal point, etc. The difficult points are those points difficult to understand, master, or apply in the textbook. The difficult points are also relative and specific to students. They are determined by the gap between students’ cognitive ability and knowledge requirements. Generally speaking, the relatively abstract knowledge, those with relatively complex structure, those with relatively hidden essential attributes, those requiring new viewpoints and methods, students’ lack of the necessary perceptual knowledge, and so on in textbooks can all be determined as difficult points. For example, discussion on the solutions to the equations with literal coefficients, formulating equations to solve application problems, and function concept, etc. are all difficult points in middle school mathematics. Sometimes the focal points in the textbook are also difficult points. For example, the locus of points and the concept of function in middle school mathematics, the concepts of curve and equation, establishment of equation of locus, and application of permutations and combinations in secondary school mathematics are both focal and difficult points. The key points are breakthrough points to understand and master a certain part of knowledge or solve a certain problem. They are also the points to overcome difficult points and highlight focal points, often acting as a turning point. Once the key points are mastered, other parts will be readily solved. For example, in the section of “Sum of Interior Angles of a Triangle” in plane geometry, the mastery of the theorem is the focal point, the proof of the theorem is the difficult point, but the addition of auxiliary lines in the proof is the key point. In the application of coordinate method, choosing the appropriate coordinate system is the key point. When using parameters to find a curve equation, choosing the right parameters is the key point. When taking the derivative of a composite function, analyzing the composite relationship of the function and mastering the derivation operational rule of the composite function is the key point. 5. Calculate exercises, and select problems Teachers must calculate all the practice questions, exercises, review questions, and general review questions in the textbook one by one and be familiar with the solution to each question. Teachers must not only master multiple solutions to one problem and shortcuts, but also understand the type and difficulty level of each question. Teachers need to distinguish between the primary and the secondary, between the unitary and the comprehensive; analyze the possible mistakes that students may make in solving problems; and select and self-edit a certain number of exercises from other reference books for students to choose when they do targeted exercises. After a thorough study of exercises, teachers should select them carefully according to their purpose, gradient, typicality, diversity, and pertinence. They should

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make overall arrangements for the examples, exercises, homework in and out of class, etc., to avoid blindness and randomness. 6. Determine the lesson type and teaching method The lesson type and corresponding teaching methods should be determined according to the content of textbook, teaching objectives, requirements, and the age characteristics of students. There are many lesson types and methods. For all teaching methods, the heuristic teaching method should be followed. The lesson type is the structure and form of the lesson, which is obviously clear. The teaching methods are often comprehensive. How to determine the teaching method? We can determine it according to the main teaching task to be completed and the main teaching methods used in the study and research of the main task and try new teaching methods as much as possible. 7. Understand the situation of students It is an effective experience in mathematics teaching over the years to prepare lessons and learn about students. To learn about students, we must have a thorough knowledge of students’ political thoughts, learning attitudes, hobbies and interests, basic knowledge, receptivity, life experience, health status, etc. To learn about students, we can obtain information through classroom questions, exercises, demonstration on the blackboard, discussions, and homework completion, get feedback through seminars or individual conversations, or consult experienced teachers for inspiration. 8. Prepare teaching aids Preparing or making relevant teaching aids, especially using modern teaching means such as slide, TV, video, recording, and the Internet, is very important to enhance students’ perceptual knowledge and improve teaching effects. We should attach great importance to them. 9. Develop teaching plans The teaching program for a lesson is called teaching plan. It can reflect the overall profile of classroom teaching. Due to different lesson types of various courses and different levels of detail, it is difficult to have a unified format for teaching plans, but it must include such three items as the subject, teaching objectives, and teaching process. Several common teaching plan formats are listed below (taking a new lesson as an example) for reference. The methods of developing teaching plans include text method and card method. As for the text method, the teaching plan is expressed in the form of text. As for the card method, the outline of teaching plan, key points of explanation, and demonstration of problem solving are written on cards. This kind of cards is not only a teaching plan, but also a complete blackboard writing and class hour allocation plan. As the card method is flexible, convenient for teaching, and helps to improve the teaching art of teachers, some teachers also develop the teaching plans by means of card method in addition to those developed by means of text method for future use.

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Format of detailed teaching plans

Format of teaching plans for demonstration lessons

I. Subject

I. Subject

II. Teaching objectives

II. Teaching time, location, class, and the teacher

III. Textbook analysis (focal points, difficult points, key points)

III. Teaching objectives

IV. Lesson type and teaching methods

IV. Textbook analysis (focal points, difficult points, key points)

V. Teaching aids

V. Lesson type and teaching methods

VI. Teaching process 1. Review and lead-in; 2. Teaching the new lesson; 3. Practice and consolidation; 4. Summary; 5. Assigning homework

VI. Teaching aids

VII. Teaching process (the same as left column) Format of simplified teaching plans

Format of teaching plans for practice lessons

I. Subject II. Teaching objectives III. Teaching process (ditto)

Same as the format of detailed teaching plans, but the teaching process is more specific and detailed, with time allocation and blackboard writing plan, etc.

10. Conduct trial lecture The process of rehearsing to get familiar with the prepared teaching plans is called trial lecture. For personal trial lectures, paper can be used to take the place of the blackboard. One can talk while thinking, draw while writing, ask questions, and answer them oneself. Through trial lectures, it helps one to be familiar with the teaching content, estimate the teaching time in class, and check the coherence between each link, so as to further modify and improve the teaching plan. Some senior professors and special-grade teachers have persisted in trial lectures for decades and have made outstanding contributions to cultivating a large number of talents for the construction of the Four Modernizations. Young teachers and interns should persist in conducting trial lectures conscientiously during lesson preparation. Only through trial lectures can one be familiar with the teaching plans and lay a solid foundation for improving the quality of classroom teaching.

10.2 Giving Lessons of Secondary School Mathematics From the point of view of modern education, teaching of mathematics is the teaching of mathematical thinking activities. Giving lessons is the process of implementing teaching plans and also the process of guiding students to fully engage in thinking

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activities. While giving lessons, the teaching should be organized well, paying attention to the creation of teaching situations, the explanation and practice of key content, the revelation of mathematical thinking methods and the cultivation of mathematical ability, the correct use of language and blackboard writing, classroom questioning and the use of textbooks, and handling several relationships properly. Among them, the teaching language and blackboard writing of mathematics have been elaborated in Chap. 8, and other related links are now specifically introduced.

10.2.1 Organize Classroom Teaching Well Classroom teaching is the main form of secondary school mathematics teaching. It came into being and gradually improved with the needs of social production development. As advocated by Ushinsky, a Russian educator, “Only when students give play to their own thinking skills, when teachers enable students to observe and develop their memory, imagination, interest and attentiveness, when teachers try to encourage students’ rational understanding activities can the purpose of lessons be achieved and the whole class benefits”. Classroom teaching is completed through bilateral activities between teachers and students, and teachers are both organizers and collaborators in a dominant position. Therefore, whether it is possible to maintain a stable learning environment, attract the students’ attention, and keep students full of enthusiasm for learning and thinking from beginning to end is the guarantee for improving the quality of teaching. For lower grade students, it is particularly important to organize classroom teaching well. To organize classroom teaching well, teachers are required to dress plainly and decently, teach seriously and conscientiously, be kind and natural, be punctual, and observe students’ learning condition at any time in teaching, especially pay attention to observing several poor students who are absent-minded in class and get up to little tricks. Teachers should have the adaptability to changes and can deal with contingencies in classroom teaching decisively. To be good at organizing classroom teaching, teachers are required to have superb ability and skills to harness classroom teaching on the basis of preparing lessons adequately and having a thorough knowledge of students. While teaching, teachers should be good at enlightening students, attracting the students’ attention, letting them think, and do more with a free hand, but should not be disturbed by the students’ thinking mistakes and confusion, so as to take complete control of the class, arrange lectures and exercises appropriately, and teachers and students act in perfect unison.

10.2.2 Pay Attention to the Use of Textbooks In addition to listening to lectures and completing certain exercises, students mainly rely on reading textbooks to gain mathematical knowledge. In the process of teaching,

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teachers should get rid of the habit of thinking little of or not being good at using textbooks, pay attention to using textbooks, and teach students how to read and use textbooks. Only by enabling students to master the skill of using textbooks can they improve their self-study ability. Emphasizing textbooks does not mean repeating what the textbooks say. We should guide and cultivate students’ reading ability in a purposeful and planned way step by step. For example, for middle school students, teachers can guide them to preview first, read selectively in class, and review after class. At the same time, teachers provide guidance on reading methods. While reading a concept, you must connect it with reality to clarify the connotation and extension of the concept. While reading a theorem, you must distinguish the conditions and conclusions and understand the train of thought and method of theorem proving. While reading a formula, you must make clear the application range of the formula and the internal relations between formulas. While reading an example, you must understand the problem-solving method of the example and its role. To attach importance to textbooks, teachers should provide guidance on the methods of delving into textbooks for higher grade students. They can not only have a preliminary understanding of the content of the textbooks, but also be good at discovering problems, asking questions, studying the changes after the generalization or extension of the problems, studying other solutions to theorems or examples, and researching methods of self-summary, thus converting the reading process of “thickening” into “thinning”.

10.2.3 Be Particular About Classroom Questioning Classroom questioning is a means of classroom teaching, an important way to inspire thinking, and a concrete embodiment of teaching art. If used properly, it can play a good role in guiding students to review and consolidate previous knowledge, discover and understand new knowledge, inspire students’ thinking, and cultivate their abilities. Classroom questioning should be clear, enlightening, and oriented to all. In other words, the question must be clear, specific, concise, and expressed clearly. It is not allowed to be vague and ambiguous, to make students at a loss. The questions asked should be meaningful and aimed at “critical junctures” that can stimulate students’ positive thinking. In terms of form, they are often important concepts and theorems, formulas, rules, as well as ideas and techniques for solving problems. In terms of content, they are often the key words in concepts, key conditions in theorems and formulas, and key variable relationships and relevant points for attention in solving problems. On the contrary, if the questions are only aimed at some individuals, or the questions asked are not clear, too simple or too complicated, they will not achieve the expected teaching effect of classroom questioning. x−1 = 1?” is not clear. Students For example, the question “How about x in |x−1| don’t know whether they are required to find x or to determine the value range

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of x. The question “How many circles can be drawn through three points that are not in a straight line?” is too easy. The question “Why do two negatives make a positive?” is too difficult. The question “What are the types of exponential equations and logarithmic equations?” is too broad. These are all undesirable. For another example, in a review lesson about circles, the teacher asks the question “If a round pot cover has been made, how do you determine the handle position?” After discussion, the students put forward the following methods. ➀ Choose two points on the cover side, the perpendicular bisector of the line formed by connecting these two points is the position to install the handle. ➁ Choose a point on the cover, fix one end of a string to this point, and move the other end along the cover edge, and the position where the string is the longest is the position to install the handle. ➂ Put the pot cover upright on the table, and the straight line position perpendicular to the table passing through the contact point is the position to install the handle. ➃ Lean the right angle vertex of a carpenter’s square (or large set square) against any point on the pot cover edge, and the position of the line formed by connecting the two points where two legs of the carpenter’s square intersect the cover edge is the position to install the handle. Through this questioning, students review the properties of perpendicular bisector of the chord, and the diameter is the largest among the chords of the same circle, the definition and properties of the tangent of the circle, and the relevant theorems of angle of circumference, which are very meaningful. In addition, when asking questions in class, we should proceed from the actual situation of the textbook and the students, the questions should be oriented to all students, to enable them to listen to the questions and answer them attentively in a state of positive thinking, and we should pay attention to correcting the errors in answers at any time. At the same time, we must pay attention to individualized teaching and choose the form of answer and the questioning object according to the difficulty level of the question. Teachers must consider in advance what questions are suitable for students to answer collectively, what questions are suitable for students to answer individually, what questions are suitable for “top students”, what questions are suitable for “underachievers”, and what questions are suitable for students to answer after discussion, to fully arouse students’ enthusiasm for learning. Do not turn classroom questioning into an individual activity between the teacher and a certain student or punish students in disguise of questioning to hurt their self-esteem and enthusiasm for learning.

10.2.4 Strengthen the Teaching of Mathematical Thinking Methods Mathematical thinking methods are the core and soul of mathematics. Studying and researching mathematics thinking methods are of great significance to the discovery

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of mathematics, to bring mathematics into full play, or to improve mathematics teaching and the quality of teaching. As pointed out in the new curriculum standard of secondary school mathematics in China: “The content, thoughts and methods of mathematics play a major role in the learning, research and application of natural sciences and social sciences”. “In teaching, students should be guided to master the laws of mathematics (including rules, formulas, theorems, mathematical thoughts and methods) on the basis of grasping concepts”. It is thus clear that strengthening the teaching of mathematical thinking methods has become a clear requirement for mathematics teaching in secondary schools today. To strengthen the teaching of mathematics thinking methods in secondary schools, we are required, in various activities of teaching mathematics knowledge and skills, to fully demonstrate the thinking process of the formation of concepts, the discovery of propositions, the exploration of ideas and problem solving, and to abstract mathematical thinking methods therefrom, and reveal the laws of development of mathematics, discovery, and invention. To strengthen the teaching of mathematical thinking methods, we are required to dig into the curriculum standards and textbooks carefully, pay attention to discovering and revealing the mathematical thinking methods therein, and have a more comprehensive consideration and a more thoughtful teaching arrangement of the penetration, revelation, and application of mathematical thinking methods. Only in this way can the teaching of mathematics thinking be implemented, and at the same time, can the teaching effect of mathematics thinking methods be improved effectively. To strengthen the teaching of mathematical thinking methods, teachers are required to clarify the basic knowledge of mathematical thinking methods and truly understand the significance of the teaching of mathematical thinking methods. Only by improving their professional competence and knowledge structure can they improve their ability to control textbooks and make positive contributions to the reform of teaching content, teaching methods, and the teaching of mathematical thinking methods.

10.2.5 Handle Several Relationships Properly In order to strive to improve the quality of classroom teaching, it is important to handle the following relationships properly. 1. Handle the relationship between new and old knowledge Secondary school mathematics is very systematic. New knowledge is all developed from old knowledge. Therefore, when teaching a new lesson, it usually starts with reviewing old knowledge, and then through comparison and association, we introduce the topic and teach new knowledge. At the same time, in the process of teaching new knowledge, we associate old knowledge as much as possible. We should follow the principle of “introducing new knowledge by associating old knowledge, and

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teaching new knowledge by associating old knowledge”. However, when to review and associate old knowledge and its depth and breadth should be determined flexibly according to the teaching objectives of a lesson, the relationship between the new and old knowledge, the students’ mastery of the old knowledge, and the situation of the teaching process at that time. 2. Handle the relationship between the deep and the shallow Secondary school mathematics teaching must start from the students’ practical experience and existing knowledge, conform to the laws of understanding from perceptual knowledge to rational knowledge, from the concrete to the abstract, from the particular to the general, and from external to internal relations. Among them, it’s very important to proceed from the shallower to the deeper. However, some teachers do not delve into the textbooks sufficiently, do not know about the students’ situation, so they cannot distinguish the depth of the content. Or because of the lack of work experience, some teachers mistakenly believe that the students feel bored if the contents are shallow, and only the deep contents can catch students’ attention. As a result, the shallow ones are only involved tangentially, so students cannot lay a solid foundation. While the deep ones are too difficult to learn, or students gulp down without thought or understanding. The deep is the development of the shallow. Only by starting from the shallow can we base ourselves on the deep, and neither can be neglected. The degree and ratio of the deep ones should be based on the teaching requirements and the actual situation of the class flexibly. At present, we must base ourselves on the shallow ones, facing the majority of students, in order to strive to improve the quality of teaching on a large scale. 3. Handle the relationship between many and few Many and few are a pair of contradictions, which are interconvertible under certain conditions. For example, some secondary schools are speeding up the process blindly. The teaching contents for six semesters are finished in five or even four semesters, using excessive assignments tactics, and increasing the number of exams. Students are putting in extra hours and learning mechanically. As a result, students are satisfied with a smattering of knowledge, and they don’t grasp the key points and lack the ability. However, experienced teachers implement the teaching plan strictly. On the basis of digging into the textbooks and selecting examples and exercises carefully, they give lessons by focusing on the focal points and grasping the key points, improve teaching methods, and make full use of the 45-min classroom teaching time to make students learn lively and actively and be able to draw inferences by analogy. In this way, although superficially the teaching progress has slowed down and students have learnt and practiced less and taken fewer exams, it has actually yielded twice the result with half the effort. In this regard, we hold that classroom teaching must be “fewer but better” in order to “use the few to defeat the many”. We must not be “innumerable and disordered” to “use the many to defeat the few”. How to be “fewer but better”, we must take pains to

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“have a thorough grasp of both ends”, emancipate out mind, learn from educational reform experience, and study educational science theories. 4. Handle the relationship between leniency and strictness Mathematical science is one of the rigorous sciences. We must be strict in classroom teaching. Firstly, during teaching, teachers should be serious and conscientious, use accurate and rigorous teaching language, and write on the blackboard correctly and reasonably. Secondly, teachers should adopt appropriate teaching methods, require students to listen attentively, strive to understand, digest, and consolidate what they have learnt in class, do their homework seriously, without perfunctory calculations or sloppy writing, and strive to meet the prescribed teaching requirements. But to “combine leniency with strictness”, we must strive to match our words with deeds. Teachers are required to conduct necessary inspections on students, not to overshoot the mark or to be unrealistic. At the same time, more care and encouragement should be given to underachievers or students with special circumstances, so that they have the confidence to catch up. 5. Handle the relationship between explaining and practicing In mathematics teaching, while explaining knowledge, some necessary problemsolving demonstrations and exercises must be carried out. The demonstrations should be typical, representative, and standard. Students’ exercises include reading textbooks, discussing, solving problems, etc. Sometimes, teacher’s explanation comes before practice, to deepen the understanding and consolidation of what they have learned through practice. Sometimes teacher’s explanation comes after practice, to discover the laws through practice, and raise them to the level of theory to guide practice again. Therefore, explaining and practicing always run through the mathematics classroom teaching. We hold that only by “teaching only the essentials and ensuring plenty of practice” and “combining explaining with practicing” can good teaching effects be achieved. The arrangement of explaining and practicing should be dependent on the selection of textbooks and teaching methods. The proportion of explaining and practicing also varies with the grades and different teaching contents. Generally speaking, the proportion of practicing for middle school students is relatively large, often accounting for 1/2–1/3 of classroom teaching time, while the proportion of practicing for secondary school students is relatively small, often accounting for 1/3–1/4 of classroom teaching time. 6. Handle the relationship between advancement and retreat In mathematics teaching, we should strictly follow students’ law of cognition, stimulate students’ learning motivation, inspire them to think positively, and gradually understand and master the knowledge. However, sometimes since students do not grasp the old knowledge well, lack due production and life experience, or there exist defects in the cognitive structure, they have difficulty in acquiring new knowledge according to the conventional procedures. At this time, it is necessary to slow down the progress of teaching appropriately, review the old knowledge further, start from

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the analysis and research of special cases or introduction of the application in production and life, summarize the rules further, get inspiration from it, and then acquire new knowledge, which can always achieve better teaching effect. It is the so-called retreat in teaching. For example, when the sine theorem is explained, students often find it difficult to study arbitrary triangles at the beginning. If we retreat from arbitrary triangles, study right triangles or equilateral triangles before general triangles, it is easier for students to accept it. In this regard, we believe that advancement and retreat form a relationship of dialectical unity. “Beating a retreat to advance again after understanding the truth in retreat” is also a basic teaching procedure. It is conducive to linking the knowledge structure with the students’ cognitive structure, to transform contradictions, and improve the ability to analyze and solve problems. 7. Handle the relationship between living things and dead things The classroom is a sacred palace of knowledge, which must be a quiet teaching environment, mathematical knowledge is rather theoretical, and these seem to be “dead” things. However, if students want to learn well and turn knowledge into abilities, they must be taught by means of good methods, be given the key to learning to master the laws of mathematical science. Therefore, only by “seeking living things in dead things” can good teaching results be achieved. It requires teachers to use teaching methods flexibly to make students learn lively and actively, enable students to be in a positive and active learning state, inspire them to use their brains for positive thinking, and improve their ability to analyze and solve problems. The “living things” do not mean the bustling on the surface, but students’ active thinking and open-mindedness under the enlightenment and guidance of teachers. 8. Handle the relationship between whole and parts In secondary school mathematics teaching, it is a contradiction to take into account the different development of top students and underachievers, so that all students can develop, achieve good academic performance, and overcome the differentiation among students. A good teacher usually starts from the training objectives, fully arouses the students’ enthusiasm for learning, improves teaching methods effectively, and takes effective measures to prevent or reduce differentiation as much as possible, so that all students in the class can meet teaching requirements and achieve good teaching results. To this end, it is necessary to “base ourselves on the whole, and grasp two ends”. We should consider the teaching progress, classroom capacity, and teaching requirements with all students as the object, and at the same time, we should pay close attention to individual tutoring of “top students” and “underachievers”, to enable the top students to improve further and the underachievers can keep up with the pace of the whole class without losing their confidence in learning. Handling the relationship between whole and parts correctly is a reflection of modern education thought, an embodiment of teachers’ revolutionary sense of responsibility, and also the key to

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improving the quality of mathematics education on a large scale. This is a problem worthy of attention in current secondary school mathematics teaching. The above has expounded several relationships to be handled properly during class. In view of the current secondary school mathematics teaching practice, how to improve the teaching and strive to improve the quality is a topic to be studied in depth.

10.3 Extracurricular Work of Secondary School Mathematics In order to improve the quality of teaching and strive for a bumper harvest, we should also make efforts to do the following extracurricular teaching work in secondary school mathematics teaching.

10.3.1 Correct Homework in a Timely and Serious Manner Homework is the continuation of classroom teaching, and it is a form of activity for students to complete teaching tasks independently. Completing a certain amount of homework can help students deepen their understanding and memory of what they have learned, improve their ability to analyze and solve problems, and learn to do self-check. It can also help teachers to understand the situation of teaching in order to improve teaching in a targeted manner. The homework of mathematics includes regular homework in and out of class, periodic review homework, winter and summer vacation homework, etc. The homework in and out of class includes written homework, oral homework, practical activity homework, etc. Assigning homework does not mean assigning exercises. Mathematics homework usually includes three basic items: “review, problem-solving and preview”, with certain purpose, pertinence, and specific requirements. The contents and requirements of review and preview should be specific and clear. The exercises should be selected and arranged carefully according to the curriculum standards and textbook requirements, in moderate quantity. The teachers should provide guidance on the methods of homework, understand the situation of homework in time, and correct and comment on it seriously. Teachers should persist in checking students’ preview and review homework. They should correct the assigned exercises meticulously as much as possible, especially the exercises aimed at new concepts, new theories, and new methods. Instead of simply putting a tick or cross, the teachers should point out where the error is. The teachers had better judge whether the problem is examined clearly, whether the problemsolving ideas are clear, whether the steps are complete, whether the operations are reasonable, whether the language is concise, whether the writing is standard, whether

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the graphics are appropriate, etc., and be good at analyzing the errors and guiding students to correct them. We oppose not only the “excessive assignments tactics” which increase the burden on students, but also the wrong practice of only assigning without checking, or checking without correcting and commenting. At present, the common phenomenon is that students are overburdened with too much homework. The teachers spend a lot of time and energy in correcting homework and cannot provide timely feedback to teaching. At the same time, it seriously affects teachers’ digging into textbooks and the deepening of educational reform. To this end, many schools have taken measures to strengthen the management of standardized homework and strive to improve the quality of practice homework. What’s more, some reform attempts have been made to address the shortcomings of traditional homework correcting. For example, some schools have compiled the “Homework Correcting Procedure Manual” for students to check the homework against it; some classes have experimented with the practice of correcting each other’s homework before random check by teachers. We should pay attention to learning from this experience. On the one hand, we must control the quantity of students’ homework strictly; on the other hand, we must actively explore ways to correct homework and strive to improve the efficiency and quality of correcting homework.

10.3.2 Strengthen Extracurricular Tutoring It is the sacred duty for teachers to be concerned about students comprehensively. In order to make up for the deficiency of uniform requirements of classroom teaching and cater to students’ individual needs, teachers should also do a series of tutoring work after class. There are two forms of extracurricular tutoring: group tutoring and individual Q&A. The work content includes analyzing the factors of students’ success or failure, guiding learning methods in a targeted manner, addressing students’ difficulties in understanding textbooks and specific problem solving, guiding extracurricular reading and conducting individual tutoring, etc. Through tutoring, top students can “learn to the best of their abilities” and improve their abilities further, and underachievers can also remove learning obstacles in time, enhance their confidence in learning, and “be willing to learn”, so that they can be appropriately improved and developed. Therefore, teachers should strengthen lesson preparation and improve teaching methods, and students should pay attention to listening and digesting in class and avoid the wrong practice of “making up for deficiencies in class”. Only by improving the quality of classroom teaching as much as possible can teachers’ extracurricular tutoring work be reduced and can the learning burden on students be reduced at the same time. The teaching process is also a process where teaching is learning. Strengthening extracurricular tutoring is also conducive to teachers’ understanding of learning situations of students, so as to obtain information, adjust teaching plans in time, improve teaching methods, and promote the improvement of teaching quality. In short, strengthening extracurricular tutoring is an important measure to improve the

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quality of teaching on a large scale. On the basis of careful lesson preparation and lecturing, teachers must also do a good job of extracurricular tutoring.

10.3.3 Carry Out Extracurricular Activities of Mathematics Actively In order to arouse students’ enthusiasm for learning mathematics, strengthen the connection between theory and practice, broaden their knowledge horizons, develop their intelligence, and improve their ability to analyze and solve problems, nowadays many secondary schools attach great importance to offering extracurricular activity courses. They not only incorporate the inquiry research topics in teaching into the content of extracurricular activities, but also set up a variety of mathematics extracurricular activity groups based on students’ wishes, such as reading group, mathematical research group, computer group, mathematical modeling group, teaching aid production group, measurement group, wall newspaper editing group, etc. At the same time, they also organize seminars, mathematical games, mathematics parties, mathematical competitions, visits, social surveys, and write mathematics essays, etc. Among them, the contents of seminars on mathematics include the history of mathematics, mathematicians, application of mathematics in production practice, learning methods of mathematics, mathematics in life, recreational mathematics, introduction of mathematical modeling, etc. There are deepening and broadening of key mathematical content, introduction of new mathematical ideas and methods, and introduction of mathematical problem-solving methods and skills, etc. For example, in addition to some topics specified in the curriculum standards, materials can be appropriately selected to make the special reports on the following topics based on the specific target students: Mathematical achievements in ancient China; stories of mathematicians at home and abroad; mathematics in our lives; essence of mathematics; characteristics of divisible natural numbers; development of mathematics; transformation of expressions; system of notation of positive integers; same solution to equations; solution to equations with literal coefficients; positive integer solution to indeterminate equations; coincidence points of triangles; how to add auxiliary lines; how to draw solid figures; on locus; on coordinates; on transformation; on set and correspondence; on functions; all kinds of curves; the shortest route problems, extreme value problems; application of system of curves; on mathematical language; problem examining of secondary school mathematics; thinking methods of secondary school mathematics; thoughts of secondary school mathematics; problem-solving methods of secondary school mathematics; problem-solving skills of secondary school mathematics; characteristics and application of mathematics esthetics in secondary schools; cultivation of secondary school mathematics ability; dialectics in secondary school mathematics; and so on.

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10.4 Performance Appraisal in Secondary School Mathematics As a regular work, performance appraisal has always been an important link in secondary school mathematics teaching and also an important research topic that people have been concerned about for a long time. 1. Significance of performance appraisal Appraisal is a working method adopted to examine and assess students’ academic performance. Correct understanding and mastery of the purpose and methods of appraisal are of great significance to promoting students’ learning, improving teaching work, enhancing teaching quality, and selecting talents. At the same time, it is also conducive to summarizing and promoting teaching experience and help the administrative departments for education to make decisions to guide teaching work. However, the appraisal also has certain limitations. In particular, the appraisal is often limited to the content of the syllabus and textbooks. In addition, the test questions and scoring are unified, and it is likely to guide students to adopt rote learning and pursue scores, resulting in “high scores and low abilities”, which will affect all-round development and hinder students from giving play to their personality and talent. 2. Types and methods of performance appraisal There are two types of appraisal: internal appraisal and external appraisal. The external appraisals include entrance examinations and mathematics competitions. The internal appraisals also have two types. The first type is routine exams, including classroom questioning, demonstrations on the blackboard, checking written assignments, and unit tests, etc., and the main purpose is to understand students’ learning in a timely manner to determine the starting point and progress of teaching, the depth and breadth of teaching content. The second type is unified exams, including mid-term exams, final exams, and graduation exams, and this is a comprehensive and summative inspection of students’ learning condition and is the main basis for assessing students’ mathematics academic performance. In order to reduce the burden on students, the final exam in the second semester of the third grade of secondary schools is usually combined with the graduation exam to reduce the number of exams. In some areas where the universal Nine-Year Compulsory Education has been implemented, the graduation exam is combined with the entrance exam. The appraisal mainly includes three types: written exam, oral exam, and practical exam. The written exams are divided into two types: open-book exam and closed-book exam. The closed-book exam is the most common exam type in the current secondary school mathematics exams. It is used to evaluate the operations, demonstration, construction, and written expression ability within a specified time, to master the standards of performance evaluation. It is an appraisal method with a large number of examinees at the same time in the same area and to complete the evaluation task quickly.

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For the principles, standards, and methods of test question setting, as well as standardized test papers and standardized tests, please refer to the introduction in Chap. 12. Review Questions and Exercises (IX) 1. What is the teaching process of secondary school mathematics? What aspects of work are included in this process? 2. How to make a teaching work plan? Pay a visit to a secondary school teacher (or a teaching and research group) to learn about his (their) making a mathematics teaching work plan. 3. What is the significance of lesson preparation in teaching? What are the specific tasks and requirements during lesson preparation? 4. How to develop a teaching plan? What contents should be included in the detailed plan format, simplified plan format, plan format for demonstration lessons, and plan format for practice lessons? 5. Try to write out the teaching objectives, focal points, difficult points, and key points of the following teaching content: (1) (2) (3) (4)

1.14 Power of rational numbers; 2.7 Decision of congruent triangles; 1.13 Logarithmic function; 2.1 Curve and equation.

6. Choose a section in algebra and geometry textbooks, develop detailed teaching plans, and draw up a detailed blackboard writing plan and class time allocation plan, respectively. 7. Choose four sections in textbooks, and develop simplified teaching plans separately according to the requirements for four different types of lessons. 8. What is the significance of trial lectures in lesson preparation? Rewrite the teaching plans of the above four lessons into plans of card type, and choose two of them for trial lectures. 9. Why is it said that classroom teaching is both a science and an art? What aspects of work should be paid attention to during class, and why? 10. What is the significance of paying attention to the organization of classroom teaching? How can we organize classroom teaching well in mathematics teaching? 11. What is the significance of paying attention to the use of textbooks? How can we do a good job in the use of textbooks in teaching? 12. How should we handle it if students are found talking or sleeping during class? 13. Which aspects of relationship should be handled properly during class and why? 14. What should be observed for classroom questioning? What should be done if the student cannot answer the question momentarily? 15. What should we do if a student breaks in with an objection to your explanation during class?

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16. Try to design more than five different forms of opening and closing words in mathematics class? 17. How should we correct homework? Participate in the work of homework correcting personally once or twice in a secondary school. 18. How should we carry out extracurricular tutoring? Participate in the work of extracurricular tutoring personally once or twice in a secondary school. 19. Try to develop the lecture outline of a seminar on mathematics for students in a certain grade of a middle school or a high school. 20. How should we carry out extracurricular activities in mathematics? Try to develop a plan for publishing a math wall newspaper and holding a math evening party. 21. What is the significance of performance appraisal? What are the commonly used appraisal methods in secondary school mathematics teaching? 22. Try to formulate an appraisal plan for the teaching of a certain unit, and make a summary of the appraisal with the guidance and support from secondary school teachers according to the probation or internship situation.

Chapter 11

Teaching Research of Secondary School Mathematics

Secondary school mathematics teaching research is a purposeful and planned creative activity to study the laws of mathematics education in the field of secondary school mathematics education. For a long time, countries all over the world have attached great importance to the research and reform of secondary school mathematics teaching. For current and future secondary school mathematics teachers, it is necessary to master the methods of secondary school mathematics teaching research systematically, understand some of the past and current reforms and the development trend of future reforms, so as to throw themselves into teaching research and reforms actively. In this regard, we will make some brief introduction.

11.1 Teaching Research Methods of Secondary School Mathematics Secondary school mathematics teaching research is a highly ideological and theoretical work. First of all, it must be guided by correct philosophical thought, that is, it must be guided by materialistic dialectics, using the standpoints, viewpoints, and methods of dialectical materialism to analyze and solve problems. Proceeding from reality, we should analyze the views and lessons of mathematics teaching theories at home and abroad in a realistic manner and absorb the useful things through the test of teaching practice, to make the past serves the present and foreign things serve China. At the same time, we are also required to use the standpoints, viewpoints, and methods of dialectical materialism correctly to research and solve some practical teaching problems encountered at present and try our best to implement them to improve the quality of secondary school mathematics teaching. Next, we introduce the general methods and specific methods of secondary school mathematics teaching research, respectively.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_11

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11.1.1 General Methods of Secondary School Mathematics Teaching Research 1. Plunge into practical work, research, and use the experience and lessons of modern secondary school mathematics teaching The secondary school mathematics teaching research is a highly theoretical and practical work, which needs continuous replenishment of fresh materials and constant update of original theories. The rich and colorful teaching practice activities of teachers will inevitably contain rich experience and useful lessons, some of which are sparks of thought that can be used to extract new theories. Therefore, being good at digging, researching, identifying, summarizing, accumulating, and making use of these experience and lessons are the key to studying secondary school mathematics teaching, and also the beginning of engaging in researches in this area. If he/she is engaged in secondary school mathematics teaching and hopes to make a difference, he/she must pay attention to and work hard to master it. 2. Be good at absorbing and processing new ideas, new theories and new methods of related subjects Secondary school mathematics teaching research is not only a regular job, but also a comprehensive job. It needs to apply the basic principles of related subjects, especially new ideas, new theories, and new methods related to philosophy, pedagogy, psychology, logic, cognitology, and methodology, to think about and solve a series of problems in teaching. However, when we apply these new ideas, new theories, and new methods, we should not be content with only using mathematical examples to illustrate the rationality of these ideas, theories, and methods, but we should make it become an integral part of secondary school mathematics teaching after “pedagogical processing” and implement them to improve the teaching quality and teaching effect. 3. Carry out experimental research activities in mathematics teaching vigorously The results of secondary school mathematics teaching research are based on plausible logic, and most of them are probable hypotheses. Therefore, they need to rely on a large number of experimental activities to establish and test. To carry out experimental research activities in mathematics teaching, it is necessary to use sampling and comparison methods. Sampling is to scientifically select a part of the population as the research object, so as to obtain reliable data that can indicate the population through partial research and infer the population situation more accurately. The common sampling methods include random sampling method, mechanical sampling method, stratified sampling method, cluster sampling method, etc. At the same time, when we carry out secondary school mathematics teaching researches, we must use another method: comparison. When selecting and determining the research objects, we seek typical cases through comparison. When collecting and sorting out the data, we identify the authenticity and judge the causal

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relationship through comparison. When expressing the results, we reveal the laws and illustrate the research significance through comparison. At present, there are important experimental research projects in secondary school mathematics, such as curriculum design, textbooks, teaching methods, cultivating ability, and reducing the burden on students. Although these experiments involve many factors, often with subjective factors, they can be overcome by processing data with statistical methods, and gratifying results have been achieved.

11.1.2 Specific Methods of Secondary School Mathematics Teaching Research (1) Literature method The literature method is a method to understand the development laws of mathematics teaching through the analysis and research of the rich mathematics teaching experience in human history. Almost all mathematics education research adopt this method, especially in the study of historical teaching problems. It includes four basic steps of literature collection, sorting, identification, and application. To apply the literature method, we must understand the literature condition and master the methods and procedures of document retrieval. For mathematics education research, it is extremely important to improve the ability and level of retrieving and using literature effectively. (2) Observation method The observation method is a research method to make systematic and continuous observations of mathematics teaching phenomena under natural conditions according to certain purposes and plans and make accurate, specific, and detailed records, so as to understand the relevant situation of secondary school mathematics teaching fully and correctly. Its steps mainly include preparations (making a plan, preparing necessary conditions), field observation, recording (filling in forms or resorting to audio and video equipment, etc.), and analyzing and sorting out. (3) Survey method The survey method is a research method to find out the laws of mathematics teaching by analyzing and synthesizing a large number of facts through direct observation and based on the systematic and thorough grasp of the first-hand data. The survey method is different from observation method. The observation method mainly relies on the researcher’s field observation of the phenomenon studied in person, while the survey method is to collect materials reflecting the actual situation without any interference on the research objects through various ways and methods. As for the survey method, the interview, questionnaire, survey meeting, quiz, evaluation, and other means are often used. It includes researching the respondents and scope, drafting questionnaires and plans, carrying out actual survey activities,

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drawing survey conclusions (writing survey reports), etc. The survey report includes three parts: introduction, main body, conclusions, and recommendations. In addition, sometimes it is necessary to list relevant documents, tables, data, pictures, and audio and video recordings as attachments. (4) Experimental Method The experimental method is a research method of observing the situation and results of mathematics teaching in a purposeful and planned way and then deriving the laws of mathematics teaching under controlled conditions. It has three basic organizational forms. Single-group method: For the experiment on a class or group, observe the difference in results between applying and not applying a certain experimental factor or applying another experimental factor in different periods. Equivalent-group method: Apply different experimental factors to two classes or groups with equivalent situations in all aspects and then compare their results. Cyclic method: Apply several different experimental factors to several different classes or groups in a predetermined sequence, add the effects of various factors several times together, and compare them. The general steps of experimental method are: determining the experimental method and organizational form, drawing up the experimental plan, making the experimental record, and writing the experimental report. (5) Statistical method The statistical method is a research method that statistically classifies a large number of data materials obtained through observation, test survey and experiment and makes quantitative analysis results of mathematical teaching phenomena. It is divided into two steps. ➀ Statistical classification: sort out the data into systems, do statistics, and make statistical tables or charts if necessary; ➁ Quantitative analysis: perform data calculations to obtain central tendency, dispersion tendency, or correlation coefficients, and find out improvement measures. In addition, there are comparison method, demonstration method, case method, and so on. Finally, it should be noted that the methods of mathematics education research are purposeful, pertinent, and comprehensive. The methods serve the purpose of research, so the selection of methods should be based on research examples. What’s more, the same research project often requires the application of multiple methods, that is, the application of methods is comprehensive. At the same time, the current research on mathematics teaching in secondary schools shows the development trend of combining macro-perspective with micro-perspective, dynamic analysis with static analysis, qualitative analysis with quantitative analysis, and theoretical research with experimental research.

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11.2 Teaching Research Work of Secondary School Mathematics 11.2.1 Routine Research Work 1. Collective lesson preparation Collective lesson preparation is a frequent and important teaching research activity. Putting heads together, teachers can discuss and research general problems in teaching together to ensure that teaching work is carried out in a purposeful and planned manner. The collective lesson preparation group is often composed of teachers utilizing the same textbooks, with the number of activities, contents, and requirements of the lesson preparation group in each semester specifically determined in accordance with the school’s teaching work plan. The collective lesson preparation at the beginning of the semester generally focuses on the research of the teaching work plan for the semester. The collective lesson preparation at the end of the semester generally focuses on the research of performance appraisal and a summary of the lesson preparation activities of the semester. The collective lesson preparations in between are carried out by chapters or units or at a fixed time each week. Every lesson preparation activity should be fully prepared in advance. The keynote speaker should be determined, and the focal points, difficult points, key points, and teaching precautions in the textbooks are discussed and researched together. However, there is no need to pursue uniformity too much, or it will hamper teachers’ initiative and hinder them from giving play to their creative talents according to the specific teaching situation, which is not conducive to the improvement of teaching quality. But how to carry out collective lesson preparation activities is a topic worth studying. 2. Open class Open class is also an important teaching research activity. It facilitates giving play to collective intelligence to study some common problems in teaching, summarize and exchange experience, improve teaching methods, and improve teaching quality. The school’s mathematics group or lesson preparation group should organize several open classes in a purposeful and planned manner in each semester. Each open class should be fully prepared, lesson observation should be organized carefully, and reviews should be conducted in time. The open classes may be organized according to the following procedures: (1) Clarify the nature and purpose of open class In terms of nature, they can be divided into demonstration and experimental classes. Obviously, the former has the function of summarizing and generalizing, and the latter has the significance of exploration and communication.

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In terms of purpose, for example: ➀ Summarize and exchange the teaching experience of the focal and difficult points in the textbook; ➁ Summarize and exchange the teaching experience of training ability and developing intelligence; ➂ Explore the teaching experience of implementing new textbooks, improving teaching methods, and reducing the burden on students; ➃ Explore the teaching experience of different types of lessons (such as new lessons, practice lessons, initial lessons, research lessons, review lessons for graduating classes, etc.); ➄ Explore the experience of carrying out extracurricular activities; ➅ Explore the experience of strengthening the training of basic skills to improve the teaching level of young teachers, etc. (2) Select the type of open class There are many types of open classes, mainly new lessons. Different lesson types have different teaching requirements and teaching steps. The corresponding lesson type and teaching contents should be selected according to its nature and purpose. (3) Determine the grade and instructor of open class Which grade will be responsible for the open class, and who will teach it is determined by the nature, purpose, lesson type, and teaching content of the open class? Conversely, the specific situation of the grade and the instructor can also affect the determination and selection of the nature, purpose, lesson type, and teaching content of the open class. (4) Make a teaching plan jointly The teaching plan of open class should be finally researched and formulated by the teaching and research group after collective discussion on the basis of respective lesson preparations. For large-scale open classes, the organizers and leaders in charge should participate in the research. Avoid making it become a personal behavior of the instructor by all means. (5) Organize trial teaching seriously and revise the teaching plan in time In order to perfect the formulated teaching plan, trial teaching should be organized in a parallel class, especially for different teaching plans, whose strengths should be absorbed through trial teaching comparisons. Teachers in the teaching and research group should participate in the trial teaching and make suggestions for revision promptly. Finally print and distribute it to the observers in advance, so that the observers can make preparations and research as early as possible and have a definite objective in view during lesson observing, so as to facilitate the review. It is inadvisable to hold large-scale open class frequently as it is time-consuming and labor-intensive, while the small-scale open class within the teaching and research group and lesson preparation group is flexible and simple, so open class activities

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can be carried out in a planned way according to different grades, different lesson types, different teaching methods, and the needs of training young teachers. In this way, on the one hand, it is conducive to a comprehensive and systematic summary of teaching experience; on the other hand, it also facilitates new teachers’ growing up as soon as possible to be competent for secondary school mathematics teaching. However, how to carry out open classes to further improve their quality is a topic worthy of research. 3. Review of open class After the open class, review should be made in time. Review itself is an important teaching research work. However, how to carry out a correct and in-depth review is a problem that has generally not been resolved for many years. We believe that at the review meeting, the organizer should first introduce the purpose of open class and the preparation process, and then the instructor presents the lesson, introduces the situation of the class, the teaching situation in the previous stage, especially the first one or two classes, and his/her own experience in open class. Here, whether a frank and detailed introduction can be made is the key to the quality of review and also a sign to test whether the organizer and the instructor are sincere. The content of the review can be roughly divided into the following six parts. (1) Mastery of textbooks Mastering the textbooks is a prerequisite for a good mathematics lesson. Here, it is necessary to make comments on whether the teacher has made sufficient lesson preparation through subjective efforts, and whether the syllabus and textbooks have been thoroughly studied. In other words, we should comment on the content of teaching, the status, role in secondary school mathematics, and the relationship with the production practice, whether the embodiment of political and ideological education is clear, whether the requirements are appropriate, whether the focal points are prominent, the difficult points are scattered, the key points are grasped, whether ideological and scientific mistakes are made or not, etc. (2) Application of teaching methods Classroom teaching is the embodiment of comprehensive teaching art, in which the application of teaching principles and teaching methods is particularly important. After lesson observation, comments should be made on whether the teacher has followed the teaching principles and applied relevant teaching methods in a targeted manner. Among them, there are such teaching principles as the combination of theory and practice, abstractness and concreteness, rigor and capacity, knowledge and ability, and shape and number, etc. The teaching content must be fewer but better, and the heuristic teaching method must be implemented in teaching. In terms of heuristic method, we should observe whether the teacher can guide students to think more and ask more according to the actual situation of teaching and learning, inspire students to ask questions, analyze problems, and solve problems; when explaining new knowledge, whether the teacher can inspire students to have a strong thirst for knowledge and inspire students to better understand and master knowledge; when

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explaining examples and exercises, whether the teacher can inspire students to grasp the key points, conduct comprehensive analysis, and obtain the ability to analyze and solve problems. As for fewer but better and heuristic teaching, they should be more prominently reflected in the focal points and difficult points of the teaching contents and in the arousing of students’ enthusiasm for learning and the cultivation of ability. (3) Embodiment of educational reform The reform of secondary school mathematics lesson is prominently reflected in the reform of textbooks and teaching methods. In a sense, the reform of teaching methods is particularly important, while the reform of teaching methods is also focused on arousing students’ learning enthusiasm, strengthening the foundation, cultivating abilities, and reducing the burden on students. At present, it should be focused on the high degree of students’ participation, increased thinking activity, and the mastery and application of mathematical thinking methods. An attempt of teaching reform needs to concentrate the wisdom of many educators; and a teaching reform achievement also needs to be introduced in accordance with the specific conditions of the region. Of course, one or two open classes cannot solve all the problems in the teaching reform, nor can they be perfect, or even lead to failure. During the review, we should carefully analyze and study the exploration and reform attempt of the open class and distinguish between mainstream and tributary, essence and non-essence. As long as there is a little achievement and improvement in one or two aspects, we should provide support and encouragement enthusiastically, strive to put forward feasible suggestions for improvement, and encourage the teacher to continue exploration. (4) Teaching skills During mathematics teaching, the teacher must have good skills in using mathematics teaching language, blackboard writing (graphing and drawing on the blackboard) and presentation, etc. The mathematical language should be clear, accurate, concise, popular, understandable and interesting, and in Mandarin. The mathematical language should be logical and enlightening. The necessary rhythm, repetition, and gestures are important. Do not use unfamiliar terms or vulgar and low-level language. The correct use of mathematical language should also be reflected in reading formulas and mathematical symbols correctly. The blackboard writing should be focused, planned, demonstrative, and complete and is conducive to inspiring thinking, memorizing, and application. The welldesigned writing also includes clear, accurate, and orderly drawing, with necessary block diagrams. It is also important to use colored chalk appropriately. During the review, the instructor’s language and blackboard writing should be reviewed, especially the narration of key contents and blackboard writing. (5) Teaching organization In classroom teaching, teachers are guiders and collaborators in a dominant position. The ability to maintain a stable learning environment, catch the students’ attention, and keep students full of enthusiasm for learning and thinking from beginning to end

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is the guarantee for improving the quality of classroom teaching. For lower grade students, it is particularly important to organize classroom teaching. In an open class, both the teacher and students are nervous to a certain extent, the classroom atmosphere is not active, and students’ cooperation is not satisfactory. This requires the teacher to have more experience in organizing classroom teaching. During the review, the teachers’ ability to organize classroom teaching should be analyzed. (6) Teaching efficiency Teaching efficiency is the reflection of teaching input and teaching effect. We should achieve good teaching effect through appropriate classroom capacity with less manpower, material, and financial input. Good teaching effect is reflected in five aspects that students can “understand, explain, memorize, work out, and apply flexibly”. This can be comprehensively examined from the classroom atmosphere, the state of students’ thinking, the number of answers, blackboard writing, exercises and discussions, the quantity and quality of questions, and the responses of top students, average students, and underachievers. Of course, it is also important to talk with students after class and check their homework. The above six aspects are an interrelated organic whole, which cannot be completely separated. Among them, the mastering of textbooks is the prerequisite, the application of teaching principles and methods is the measure, the vigorous implementation of teaching reform is the means, the correct use of language and the careful design of blackboard writing are the conditions, the proper organization of teaching is the guarantee, and the high teaching efficiency is the goal. They complement each other and determine the teaching quality of the open class. In addition, in the organization and review of open classes, the following aspects should be noted. ➀ Holding open classes is an important part of teaching research and a collective teaching research activity. It should be conducted in a purposeful and planed manner, under the guidance of the leaders, and be well prepared and organized. It mustn’t be a mere formality. ➁ For open classes, we must organize the review in a timely and serious manner. The review must be aimed at the formulated teaching plan and the reality of teaching and learning and strive to be objective, accurate, and realistic. The review must be justified, well-grounded, and analytical, without talking in generalities and empty verbiage. Here, the instructor’s initial introduction is very important. Don’t make blind comments on it at the beginning. ➂ During the review, different viewpoints, systems, and methods should be allowed in the processing of textbooks. Here, we must strictly distinguish the boundaries between scientific errors and the opinions of various schools. We should neither regard the opinions of various schools that are different from textbooks as errors, nor regard the scientific errors as the opinions of various schools. ➃ During the review, in terms of teaching principles, it can only be a matter of compliance and application; in terms of teaching method, it can be diverse,

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as “there is no fixed rule of teaching”. We should advocate “all flowers blooming together” on disagreements instead of rushing to unify them. We should encourage continued discussion and research. During the review, we should be strict with the instructor’s attitude toward work and sense of responsibility, but the requirements for the teaching level and teaching effect should vary with person (different experiences and educational background) and place (village and town, key, and non-key secondary schools). During the review, it is advisable to touch slightly in passing instead of focusing on the individual ill-formed sentences, slip of pen and related shortcomings that have been realized during teaching due to nervousness, carelessness, and other reasons. Teachers should concentrate on exploring how to prevent such phenomena. During the review, the actuality and feasibility of the open class should be analyzed. The undesirable phenomena such as “showing off”, “putting on a hollow display”, and “falsification” should be exposed and be criticized appropriately. During the review, our words should be clear and cordial and be approachable to the instructor and enlightening to the participants. We should strive to prevent irresponsible attitude or opinions based on personal grudge, resulting in unrealistic comments or praises or criticism of the instructor. During the review, the moderator should be good at concentrating and summarizing everyone’s opinions and constantly leading them to go further, not only inspiring everyone to focus on the practical problems of the open class, but also inspiring everyone to explore the universal problems in the current educational reform, so as to raise the practice of open class to the theoretical level and point out the direction for future practice.

The above review content is aimed at a new lesson at large. In fact, the reviews should have different focuses for different lesson types. The above reviews are also applicable to normal teaching in general, to the assessment of practical lessons by normal colleges, and to teachers’ daily self-evaluation. In particular, if young teachers can persist in self-evaluation, prepare and give lessons according to the requirements of the review, it is undoubtedly beneficial to improve their teaching level quickly. 4. Lesson presentation (1) Meaning of lesson presentation In recent years, with the rapid development of mathematics education reform in primary and secondary schools in our country and the continuous deepening of teaching research activities, the teaching research activity in the form of “lesson presentation” has received increasing attention from teachers. At present, “lesson presentation” has become not only a commonly used form of mathematics teaching research activities in primary and secondary schools, but also an important form of mathematics teaching skills competition and an important form and method of assessing and recruiting teachers.

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Generally speaking, the so-called lesson presentation means that the teacher talks about how to give a certain lesson and why on a certain occasion. It is the design and analysis of the teaching. Its content involves the analysis of textbooks, the determination of teaching objectives, the design of teaching process, the selection of teaching methods, the evaluation of teaching effects, and the analysis of the above items. It is thus clear that “lesson presentation” is a pre-class behavior and a component of preclass preparations. It is not only a new form of teaching research activities developed on the basis of open class, but also a specific content of teaching research activities. Carrying out lesson presentation activities is conducive to promoting teaching research activities, the construction of the teaching team, and improving the quality of classroom teaching. (2) Content and requirements of lesson presentation Generally speaking, a lesson presentation includes the following contents. ➀ Textbook analysis A detailed analysis of this lesson in four aspects: teaching status and role, knowledge connection, teaching objectives, and focal and difficult points. ➁ Teaching methods A detailed description of the teaching methods used in this lesson and the corresponding teaching means, especially whether it is necessary to use modern teaching means. ➂ Teaching process This is the specific design of the entire teaching process and the focus and core of the lesson presentation. The introduction should be specific, clear, and definite. Taking the “five steps” of a new lesson as an example, it should explain in order: How to introduce the new lesson and create the teaching situation? How to present the topic and teach the new lesson? How to highlight the focal points, scatter the difficult points, and emphasize the points for attention in teaching? How to conduct stage exercises and consolidation exercises? How to conduct in-class check and assign homework? Explain not only the teaching methods for each step of the above items, the form, and degree of students’ participation in activities, but also the general time arrangement and blackboard writing plan for each step. The supplementary teaching content (theorems, formulas, rules, examples, etc.) should be explained in detail or written on the blackboard clearly. ➃ Points for attention in teaching Finally, some supplementary explanations should be made on the understanding of textbooks, the application of teaching methods, the tentative ideas of teaching reform, the handling of possible situations in teaching, how to teach students in

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different classes, etc., so as to fully demonstrate the teachers’ teaching ability and teaching experience. (3) Specific procedures for conducting lesson presentation activities The specific procedures of lesson presentation activities are similar to those of open classes Firstly, clarify the content, form, time, and place of the lesson presentation activities. In principle, lesson presentation can be carried out for any teaching content, and it is generally controlled within 20 min. In the process of lesson presentation, blackboard writing, demonstration, or modern teaching means can be used. However, lesson presentation activities for the purpose of competition or observation are often aimed at a specific teaching form and content, so the way, method, and time of the lesson presentation are strictly regulated and are informed in a relatively short period in advance, to assess the knowledge and ability of the presenters. Secondly, according to the lesson presentation activity plan, the presenter presents the lesson on his/her own. During the lesson presentation, the presenter not only introduces the design of teaching form, teaching purpose, teaching content, and teaching procedures, but also analyzes the above content in detail, that is, analyzes the teaching content, teaching status, teaching conditions, actual situation of students, and the problems that may occur, etc., so as to achieve the purpose of explaining clearly why it is done in this way. Only by focusing on analysis can the quality of lesson presentations be guaranteed. Thirdly, after lesson presentation, we should organize reviews in time. All participants should conduct serious discussions on the teaching design and analysis of the lesson presentation based on the principles of being educational, scientific, and practical, affirm his/her achievements, and put forward suggestions for improvement or discussion. Only through review can we achieve the purpose of mutual learning and common improvement, so as to create conditions for improving the quality of mathematics teaching. As lesson presentation is a new type of open class activities, some precautions for the review of open classes are also applicable to the review of lesson presentation activities. In addition, attention should also be paid to the following points. (4) Points requiring attention while carrying out lesson presentation activities ➀ The teaching design requirements should be appropriate, with moderate capacity Since the lesson presentation is not on-the-spot teaching, the time required to implement the teaching design is rather flexible. Therefore, the teaching design requirements should be appropriate, neither too high nor too low. The teaching capacity should be moderate, neither too much nor too little. How to deal with the relationship between improving teaching efficiency, classroom teaching requirements, and teaching capacity, on the one hand, it is necessary to fully improve the efficiency of classroom teaching; on the other hand, classroom teaching is an important part of students’ learning process, and we should walk with a firm step to basically realize what we have said. In fact, learning is a process of continuous accumulation, and it is

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impossible to achieve “quick success”. If the progress and requirements deviate from the actual situation of students, students may not be able to digest, so haste makes waste. As there is still a tendency to “speed up the process to start reviewing early” in teaching all over the country, the above-mentioned problems in “lesson presentation” deserve more attention. In addition, as it is a “lesson presentation” appraisal, some teachers think that “the more content they have taught in a lesson, the higher the teachers’ level is”. Actually, this is a misunderstanding. In fact, the quality of teachers lies in the accurate grasp of the difficulty level in classroom teaching and the “degree” of classroom capacity. ➁ The specific introduction should be properly detailed and concise, giving prominence to the key points For a certain class, whether the key problems to be solved are the introduction of problems, the explanation of concepts, the derivation of formulas or the analysis of examples, the consolidation, and application of knowledge are often different. Therefore, during lesson presentation, on the basis of a comprehensive introduction of the situation, the presenter should focus on the problems or key issues that teachers are more concerned about and eager to understand and fully demonstrate the presenter’s understanding, analyzing, and handling the issues. Only in this way can we give play to the role of communication and research of lesson presentation. On the contrary, if the presenter makes the introductions evenly without focus or only emphasizes the elaboration of the teaching process rather than the analysis, etc., he/she will not be able to achieve good results. ➂ The teaching means and methods should be based on the lesson, incorporating skills in the lesson Teaching means and methods all serve certain teaching objectives and tasks. In order to show the intensity of teaching reform or teaching skills, some teachers emphasize the use of modern teaching means or the function of lecture skills too much, resulting in gilding the lily and loss of the reality and feasibility of teaching. Lesson presentation is different from giving a lesson. While giving a lesson, one is facing the students, and its effect depends not only on the design of classroom teaching, but also on the level of teaching ability, which involves language, blackboard writing, observation, teaching gesture, use of teaching means, etc. As for “lesson presentation”, although the art of lesson presentation has a lot to do with the effect of lesson presentation, its focus is still mainly on the teaching design and analysis of teaching content. Different from basic teaching skills competitions and teaching skills performances, lesson presentation must be based on the “lesson” itself. The means and methods used in the lesson presentation are utterly subject to the needs of good lesson presentation, and the teaching ability should be incorporated in the lesson presentation. This not only guarantees the quality of the lesson presentation, but also

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makes people feel the teaching level of the presenter and his ability to grasp a lesson accurately. 5. Work of teaching and research section (group) At present, teaching and research groups are usually set up by subject or grade in primary and secondary schools, and their duties are roughly composed of the work in the following four aspects. (1) Organizing routine teaching and research activities For example, we organize teachers to study educational policies and regulations, study documents and literature of teaching reform, organize the formulation of teaching work plans, organize collective lesson preparation, open class, and lesson presentation and other activities, prepare test papers and organize examinations, organize mathematics lectures, mathematics competitions, and interest group activities, and assist schools in assessing and appraising teachers’ teaching and research work. (2) Organizing teachers to carry out curriculum construction Teaching work is the central task of primary and secondary schools. Curriculum construction is one of the most fundamental constructions of school teaching. The schools with a long history or schools with distinctive characteristics all attach great importance to curriculum construction. The content of curriculum construction is extremely rich, including the determination of curriculum, the selection of textbooks, the application of teaching means and methods, teaching experiments, teaching guidance, performance appraisal ,and the staffing, training, and improvement of teachers, teaching, and research achievements, etc. It embodies all the practices and experiences of teaching and management work and should be constructed and summarized by grade. The complete curriculum construction files should be stored. (3) Organizing teachers’ professional development With the deepening of teaching reform, the requirements for teachers’ quality and ability are getting higher and higher. Teachers can be competent for teaching work only by continuously improving their political and professional level, teaching and scientific research abilities. In addition to participating in various training classes and refresher courses, teachers should, more fundamentally, be engaged in spare time on-the-job training. Therefore, the teaching and research group should actively organize teachers to study teaching reform documents, advanced teaching reform experience, recommend related educational theories, and professional books; actively make plans for improving teachers’ professional level and teaching ability effectively; and provide possible support and help. (4) Organizing teachers to participate in educational research Mathematics teachers in primary and secondary schools should actively participate in educational science research. Mathematics teaching researches often involve many aspects and require extensive cooperation, and some valuable topics, in particular, need collective brainstorming. The teaching and research group should pay attention

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to guiding and organizing teachers to care about and participate in teaching research actively and providing support for the development of research work, the appraisal, exchange of achievements, etc. where possible.

11.2.2 Special Research Work Special research can be conducted on common issues in secondary school teaching. There are a wide range of issues that need special research in secondary school mathematics teaching. Here, we only illustrate researches in several main aspects. 1. Curriculum research What is curriculum research? It is very narrow to regard the curriculum research as formulating curriculum standards and compiling textbooks only. In fact, it should also include the purpose of mathematics education, the research of mathematics teaching content, means, methods and evaluation methods, etc. With the development of mathematics, as well as the needs of mathematics teaching reform, the research on mathematics curriculum has gained due development in the past half century. At present, some countries have established curriculum research centers, some have established curriculum research networks, and others have established specialized research institutions in regions or universities. China formally established the National Institute of Mathematics Curriculum Research in 1983. Regarding the curriculum research approaches, there are the behavior school approach, the new-math approach, the configuration school approach, the formalization approach, the integrated approach, etc. How to evaluate curriculum research approaches is an extremely complex problem. There are social factors, teachers’ factors, publicity factors, national conditions factors, and so on. In addition, we need to study whether the curriculum design is of “linear type”, with the main curricula learnt before branch curricula, or of “leap type”, without core curriculum arranged, what alternatives teachers have, etc. No matter whether teachers are willing or not, actively or passively, in fact it is impossible to avoid the research of mathematics curriculum. An outstanding mathematics teacher should have a certain understanding of curriculum research, summarize the experience at home and abroad carefully, and try to participate in reform experiments. 2. Research on textbooks At the turn of the century, the biggest event in China’s mathematics educational circles as nothing but the formulation of Mathematics Curriculum Standards. The official publication of Mathematics Curriculum Standards for Full-Time Compulsory Education (Draft for Comment) took the lead in 2000. Meanwhile, the Secondary School Mathematics Curriculum Standards is also being actively formulated and tested.

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With the formulation of the national mathematics curriculum standards, quite a few provinces and municipalities have compiled their own middle and high school mathematics textbooks, resulting in the gratifying situation of “one syllabus with multiple books” across the country. Each of these new textbooks has certain characteristics, but also with many imperfections. This requires us, in the teaching practice, to strive to study new educational theories, establish new educational concepts, understand the characteristics of new textbooks correctly, reflect the teaching requirements of new textbooks, and make efforts to make up for the drawbacks in new textbooks through researches. Only through research and practice can we push forward the mathematics teaching reform and promote the improvement of mathematics teaching quality effectively. 3. Research on teaching methods The broad masses of mathematics educators in our country have done a lot of researches on how to carry out classroom teaching, accumulated extremely rich and valuable experience, and produced many teaching methods with different styles. According to the statistics from Liaoning Province in 1984, they summarized up to 24 different teaching methods in the teaching reform in the whole province. At present, the experience and methods of secondary school mathematics teaching in our county are still under continuous development and improvement, which has attracted more and more attention from the international mathematics educational circles. Nowadays, the establishment of new teaching theories, the application of information theory, cybernetics and system theory in teaching, and the extensive application of modern teaching means have broken new ground in researches on teaching methods both in theory and practice. On the basis of summing up various teaching methods conscientiously, we should strive to create and summarize effective teaching methods suitable for our own characteristics through practice and contribute to the researches on mathematics teaching methods in our country. 4. Research on learning methods The process of mathematics teaching is the process of information exchange between persons and between person and textbook, in which students are the principal part. One of the characteristics of modern teaching method is to attach importance to the study of learning methods while studying teaching methods. It should be mentioned that the students’ methods of learning mathematics at present are still in a relatively backward state. For example, they are not accustomed to using textbooks, with poor self-learning ability, unable to grasp the essentials in taking lessons and taking notes, content with rote memorization, and neglect the teaching of basic knowledge and the training of basic skills, lack the ability to analyze and solve problems, especially the ability to connect with reality, etc. On the one hand, we must be good at constantly summarizing good learning methods among students, improving and promoting them in time. On the other hand, it is necessary for us to proceed from the characteristics of students’ psychological activities, understand the laws of things, and reveal the cognitive structure of students’ mathematics, so as to achieve the optimal combination

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of knowledge structure and cognitive structure and yield twice the result with half the effort. 5. Research on teaching forms The current form of classroom teaching is class teaching system, which has basically been gradually developed after the first industrial revolution, and it still plays an indisputable role. Its advantages are high economic efficiency, sufficient flexibility, good coordination, and integrity of internal structure. However, with the rapid development of science and technology, the scattered time, too unified teaching time and teaching requirements, and focusing on theory and neglecting practice of traditional classroom teaching are not conducive to the development of students’ personality. It has become increasingly important and urgent to attach equal importance to the first classroom (that is, traditional classroom teaching) and the second classroom (that is, the purposeful and planned extracurricular mathematics teaching activities), and open up the third classroom (that is, mathematical statistics, cartography, field measurement, social survey, etc.) actively. At present, there are so-called group teaching, non-class instruction, and open classrooms, etc. in foreign countries. The good news is that some schools in our country have accumulated certain experience, but how to further develop in depth and strive to achieve results is also an important research topic. 6. Research on teaching means In recent decades, countries all over the world have attached great importance to the development and promotion of modern teaching technologies, which is recognized as the Fourth Revolution in world education. At present, the Ministry of Education has audiovisual education department, and each province and municipality has audiovisual education center and research institute, which are responsible for leading and promoting audiovisual education. The modernization of teaching means is a revolution in the field of education. We should get rid of superstitions, emancipate our mind, carry forward the spirit of self-reliance, adapt to local conditions, and apply modern teaching means as much as possible in mathematics teaching, to make our own contributions to the modernization of mathematics teaching means in secondary schools in China.

11.3 Teaching Reform of Secondary School Mathematics 11.3.1 A Brief Historical Review Mathematics is an ancient science and also the oldest teaching content. As is known to all, ancient Babylon, Egypt, Greece, and China have all made outstanding contributions to the birth and development of mathematics. However, since ancient times, in mathematics teaching, there have been two courses: one taking

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mathematics as solving practical problems and the other taking it as training thinking. They are two totally different tendencies. The former focuses on arithmetic, algebra, and practical geometry, while the latter focuses on demonstrative geometry. The Nine Chapters on the Mathematical Art and other works in ancient China and the Elements of Euclid in ancient Greece are typical textbooks of these two tendencies. For a long time, although the mathematics teaching content in secondary schools has undergone thousands of reforms, the modern reforms began with the “Encyclopedists” in the eighteenth century. Its representative figure D’Alembert wrote in an article about geometry: the Elements of Euclid was by no means written for the children of our times. To meet the needs of social development at that time, another geometry textbook should be compiled, the new geometry should combine plane geometry and solid geometry properly and enable the thought of analytical method and calculus to take a place in the new textbook, etc. Later, Bezout wrote a new algebra textbook in 1770, and Legendre wrote a new geometry textbook in 1794. These textbooks were short, didn’t pursue rigor, and rarely included some cumbersome contents over-emphasizing skills. They once promoted secondary school mathematics teaching. However, by the end of nineteenth century and the beginning of twentieth century, textbooks have gradually became cumbersome and complicated, calling for urgent reform again. The reform this time was led and initiated by Klein, a German mathematician, and Beili, a British mathematics educator, it was also called the Klein-Beili Movement, and it was tried out in UK. They advocated using the standpoints of modern mathematics to transform the traditional secondary school mathematics teaching content. Klein wrote a set of Elementary Mathematics from an Advanced Standpoint, which strengthened the teaching of functions and calculus, used this to reform and enrich the content of algebra, incorporated geometry into the teaching content of secondary school mathematics, and reformed the traditional geometric content from the viewpoint of geometric transformation. Their viewpoints had a profound impact on secondary school mathematics teaching, and the direction was correct. However, the teaching content reform is related to school structure, teaching thoughts, teaching theories and methods, and the level of teachers. Due to the limited conditions at that time and some valuable research and reform experiments were interrupted by two world wars, the modernization movement of mathematics education initiated by Klein and Beili failed to achieve better results.

11.3.2 Modernization Movement With the rise of modernization movement and mainly due to the increasing status of mathematics, people realized that the mathematics teaching contents couldn’t keep up with the requirements of the times, coupled with poor teaching quality, low work efficiency, poor teaching theories, and improper teaching methods. It was urgent to

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carry out a comprehensive and thorough reform in the field of secondary school mathematics. The direct cause was the scientific and technological competition between the United States and the Soviet Union. In 1957, the Soviet Union launched the first artificial satellite, which caused a sense of crisis in the United States. The reason was insufficient talents and backward education. Therefore, this massive modernization movement first began in the United States. In 1958, the US government assisted in the establishment of the “School Teaching Research Group”, which was funded by the Department of Defense with additional appropriation and was in full charge of experimental research and textbook compilation. In 1959, the National Academy of Sciences held a special meeting to focus on the reform of secondary school mathematics and sciences and put forward four new thoughts: structural thinking, the thought of early education, the thought of focusing on cultivating intuitive thinking, and the thought of learning motivation lying in enthusiasm. Later, under the impetus of the Conference of the States of the European Community and a series of conferences of UNESCO, this reform movement has almost swept the globe. The United States, Britain, France, the Netherlands, and other countries have compiled a series of new textbooks. These new syllabuses and new textbooks generally tended to be filled with many modern mathematics courses, such as set theory, mathematical logic, modern algebra, probability statistics, and elementary calculus and arranged for a single comprehensive subject. At the height of this reform, Anthony put forward the amazing slogan “Let Euclid Get Out”, and the United States has proposed a plan for middle school students to complete the content of traditional secondary school mathematics and high school students to reach the third grade level of science and engineering mathematics. It is remarkable that in the early 1960s, the process and scale of this reform in our country were not inferior to foreign countries. There were many unique creations, which were quickly interrupted due to historical reasons. But the Soviet Union has always paid attention to this trend, and the reform has proceeded quietly. Japan has adopted a more prudent approach, taking a step-by-step approach and implementing gradual reforms and adjustments. Later, practice first proved that such reforms in European and American countries were difficult, producing very little effect. After the 1970s, they had to readjust their reform plan. Only France was still persisting in the large-scale reform because of its effective measures. At present, the modernization reform movement worldwide is in the stage of adjustment and consolidation. It has profound enlightenment to us. “There is no prospect without reform”. With the innovation of teaching theories, teaching methods and means, and the maturity of capability to process textbooks, the modernization movement of secondary school mathematics teaching will inevitably deepen increasingly and continue to develop extensively.

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11.3.3 Reform of Secondary School Mathematics Education in China China’s mathematics education system began in the Sui Dynasty. The Imperial Academy of the Tang Dynasty set up an arithmetic department, with Ten Classics of Ancient Chinese Mathematics as the main textbooks. During the reign of Emperor Wanli of the Ming Dynasty, Matteo Ricci, an Italian missionary, and Xu Guangqi, a Chinese scientist, jointly translated and imparted western mathematics, which had a significant influence on the birth and development of mathematics education in China. The founding of the People’s Republic of China has brought life to mathematics education in China and become a new starting point in the history of the development of mathematics education in China. Since the founding of the People’s Republic of China, our mathematics education has roughly experienced the following six development stages. The first stage: the establishment of secondary school mathematics education (1949–1957). The characteristic of this stage was to learn from the Soviet Union in an all-round way, transform the old education completely, and create a new system of mathematics education in China’s socialist secondary schools. In 1952, the Ministry of Education compiled new national unified secondary school mathematics syllabus and teaching plan. In 1956, new textbooks were issued, the purposefulness, scientificity, systematicness, and ideological content of secondary school mathematics education were clarified, and mathematics teaching methods were gradually improved. Thus it laid the foundation of China’s socialist secondary school mathematics education system and created good conditions for the future reform of secondary school mathematics education. However, because the old education was completely negated, and the ways and methods of learning from the Soviet Union were divorced from the actual situation of China, for example, the mathematics curriculum and textbooks of the Soviet Union’s 10-year system were copied indiscriminately in 12-year schools in China, thus extending the learning time, analytic geometry and other courses were canceled, and the level of secondary school mathematics education declined. The second stage: the preliminary reform of secondary school mathematics education (1958–1960). The characteristic of this stage was the emerging upsurge of educational reform. Reform experiments on secondary school mathematics teaching were carried out across the country, and various reform plans were proposed. At the beginning of this stage, influenced by the “Great Leap Forward” and the international modernization movement of mathematics education, a mass movement was launched, various reform experiments on mathematics education were actively carried out, and the practice of copying the Soviet Union indiscriminately that was not in line with our country’s national conditions was corrected. The phenomena that the textbooks were outdated and backward, divorced from reality and politics, isolated and tedious were criticized. The useful attempts to establish a new

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secondary school mathematics education system were made, and nine-year system textbooks were compiled. However, due to the lack of correct ideological guidance, the excessive pace of reform, hit by the struggle against Right deviations, and the exceptional economic hardship suffered, this massive reform movement was soon forced to discontinue. In fact, due to the influence of the ultra-Left trend of thought, there were also some major problems in educational revolution at this stage. For example, excessive emphasis on productive labor weakened classroom teaching, and even the normal teaching order was disturbed and battered. The improper additions and deletions to the content of textbooks, to some extent, weakened the scientificity and systematicness of knowledge, weakened double-basics teaching, and affected the quality of secondary school mathematics teaching. The third stage: adjustment, consolidation, and development of secondary school mathematics education (1961–1966). The characteristic of this stage was to summarize the lessons learnt from mathematics education reform for adjustment, consolidation, and development. In this stage, on the basis of summarizing the experience and lessons of the first two stages, we implemented the policy of “adjustment, consolidation, enrichment and improvement” by the Party Central Committee, formulated the regulations on the work of universities, secondary schools, and primary schools, emphasized that schools should focus on teaching and attach importance to “double basics”. In 1963, the Full-Time Primary and Secondary School Mathematics Syllabus was further revised, and a complete set of high-quality secondary school mathematics textbooks was compiled. At the same time, we pointed out the importance of learning mathematics, strengthened teaching research, continuously accumulated teaching experience, and steadily improved the quality of secondary school mathematics teaching, so that the mathematics teaching quality reached a higher level in the mid- to late stage. The fourth stage: serious damage to secondary school mathematics education (1966–1976). This stage was the ten-year turmoil of the “Cultural Revolution”. The socialist education in the first seventeen years before the Cultural Revolution was completely negated and stigmatized as “revisionist education”, the classrooms were smashed, textbooks were discarded as illicit materials, and teachers were regarded as sinners and were generally criticized. Later, at the call of “resuming classes to carry out revolution”, the school system of secondary schools was changed from six years to four years, the system of 2-year middle school and 2-year high school was implemented, and workers, peasants, and soldiers were encouraged to take the stage. Although various provinces and municipalities organized the compilation of some textbooks, they were too pragmatic and full of views of class struggle, which greatly weakened the teaching of basic knowledge and the training of basic skills. The order of teaching was chaotic, resulting in a substantial decline in teaching quality, and mathematics education in secondary schools suffered extremely severe damage. The fifth stage: the restoration, initial reform, and development of secondary school mathematics education (1976–1998).

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The characteristic of this stage was that mathematics education was rejuvenated after breaking out of the crisis of ten-year turmoil and achieved vigorous development and improvement based on reforms to meet the needs of the construction of the Four Modernizations. In this stage, under the guidance of the principle and policies of the Third Plenary Session of the 11th Central Committee of the Communist Party of China, the national education working conference was held in 1978, the then Ministry of Education formulated the Full-Time Ten-Year Secondary School Mathematics Syllabus (Draft for Trial Use), compiled new national textbooks, and put an end to the chaos in the education front for more than a decade. In 1983, the Ministry of Education decided to implement “basic” and “higher” teaching requirements for secondary schools across the country based on the trial situation of the state-compiled textbooks, and two kinds of textbooks (class A and class B) were used, while the revised uniform textbooks were still used in middle schools. In 1986, the National Education Commission strengthened the leadership of educational work, established the National Primary and Secondary School Textbooks Examination and Approval Committee, revised the syllabus formulated in 1978, reissued the Full-Time Secondary School Mathematics Syllabus in 1987, compiled a complete set of secondary school mathematics textbooks, and further clarified the teaching purpose, teaching principles, and teaching methods of secondary school mathematics. According to the requirements of the Compulsory Education Law of the People’s Republic of China and “three-orientations”, with the adjustment of structure and school system of secondary education and the further deepening of the reform of secondary school mathematics education, the State Education Commission promulgated the Middle School Mathematics Syllabus for Nine-Year Compulsory Education in September 1993, issued supporting textbooks successively, proposed adjustments to the secondary school mathematics teaching in connection with compulsory education, and stepped up the formulation of Full-Time Secondary School Mathematics Syllabus at the same time. The sixth stage: the further reform and development of secondary school mathematics education (1999-present). The characteristic of this stage is to put forward new mathematics education concepts and implement new mathematics curriculum standards to meet the needs of cultivating innovative talents. In the past ten years, with the acceleration of technological revolution, the advancement of economic globalization, and the intensification of talent competition, countries all over the world are vigorously researching and formulating mathematics curriculum standards for primary and secondary schools and promoting mathematics teaching reform comprehensively to meet the needs of world competition. In 1999, the Ministry of Education in China initiated the formulation of national curriculum standards. In 2001, the Curriculum Standards for Nine-Year Compulsory Education was promulgated, and the curriculum standards for high schools were also being formulated. Subsequently, many provinces and municipalities compiled corresponding new textbooks. A new curriculum reform and the implementation of new standards have been fully carried out in China.

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However, the reform of mathematics education is a long and arduous process. It not only involves people’s scientific understanding of the essence of mathematics and mathematics education, but also involves many factors of China’s national conditions. Many problems will inevitably arise in the reform process. The current curriculum reforms and new textbooks are still in the active experimentation and exploration stage and need to be adjusted appropriately through practice and research to achieve gradual improvement. Therefore, through study and research, we should understand the past of mathematics education reform (understand the reform); we should increase the enthusiasm for reform (think about reform); and we should learn the method of carrying out the reform (be good at reform). Only by understanding, thinking about, and being good at reform can we continue to put forward the reform of mathematics education and play an important role in cultivating talents. Review Questions and Exercises (X) 1. What is teaching research? What is the significance of carrying out secondary school mathematics teaching research? 2. What are the general and specific methods for carrying out secondary school mathematics teaching research? 3. What are the routine and special research work of secondary school mathematics teachers? Participate in a specific teaching research activity of a secondary school in person. 4. What are the procedures for organizing open classes? How to make reviews in general? What aspects should be noted? 5. Write detailed comments on a mathematics open class. 6. What is lesson presentation? What are the content and requirements of lesson presentation? What are the specific procedures for lesson presentation? What are the points requiring attention? 7. What is the history and current situation of the international secondary school mathematics education reform? What are the experience and lessons? 8. How many stages of development has the reform of secondary school mathematics education in China experienced, and what are the characteristics of each stage? 9. What is the development trend of the current reform of secondary school mathematics education? What problems should be paid attention to in the reform of secondary school mathematics education in China? 10. Understand the situation of mathematics teaching of a secondary school, and put forward your relevant suggestions or tentative ideas about secondary school mathematics teaching reform in a certain aspect.

Chapter 12

Educational Measurement of Secondary School Mathematics

Educational measurement of secondary school mathematics is a science taking the examination and evaluation of secondary school mathematics learning as the main research object, also known as the secondary school mathematics examination science. The examination and evaluation of academic performance has always been an important link to do a good job of teaching work and improve the teaching quality. With the development and improvement of secondary school mathematics education, it has become increasingly important and urgent to evaluate the quality of secondary school mathematics education from quantitatively to qualitatively. To this end, this chapter will introduce the basic knowledge related to educational measurement of secondary school mathematics.

12.1 Meaning of Educational Measurement of Secondary School Mathematics 12.1.1 Birth and Development of Educational Measurement of Secondary School Mathematics Since ancient times, once educational activities came into being, there was a way to appraise the effectiveness of educational activities. As stated in Records on the Subject of Education: “Every year some entered the college, and every second year there was a comparative examination. In the first year it was seen whether they could read the texts intelligently, and what was the meaning of each; in the third year, whether they were reverently attentive to their work, and what companionship was most pleasant to them; in the fifth year, how they extended their studies and sought the company of their teachers; in the seventh year, how they could discuss the subjects of their studies and select their friends. They were now said to have made © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_12

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some small attainments. In the ninth year, when they knew the different classes of subjects and had gained a general intelligence, were firmly established and would not fall back, they were said to have made grand attainments”. This is a set of systematic method of appraisal. The foreign educational measurement circles believe that educational measurement and evaluation originated from the ancient imperial examination system in ancient China, but the long-term feudal system in China restrained this subject from obtaining due development. The term “educational measurement” was first proposed by Thorndike, an American educator, in Introduction to Society and Psychology in 1904, but developed rapidly only in the recent half a century as a science. This is because, on the one hand, there was mathematics education modernization movement internationally, whose effectiveness needed to be scientifically appraised correspondingly; on the other hand, the society needed to investigate the quality of education, and the education itself needed to put forward the evidence of the achievements. At the same time, with the popularization of computers and the development of statistics and psychology, this subject can continuously develop and become increasingly mature. The development of appraisal has generally undergone two stages: one is the development from the old style test to objective test; the other is the development from norm-referenced test to target-referenced test, from absolute score to relative score or combination of absolute and relative scores. Nowadays, educational measurement has developed rapidly in the international world and been widely used and has become universal, legalized, and scientific. In some countries, one must take an exam to be enrolled, recruited, or employed, and even in a few cities, one must take an exam to obtain the right of dwelling. With the birth and development of educational measurement, the educational measurement of secondary school mathematics has come into being and developed accordingly and has become an extremely important part of educational measurement.

12.1.2 Characteristics of Educational Measurement of Secondary School Mathematics As we know, physical measurements deal with some phenomena that are easy to be quantified objectively (such as length, weight, temperature, and time) and can be measured directly, and as long as the measuring tool itself is standardized (i.e., unified unit and reference point) and the use method of measuring tool is standardized (i.e., unified operating procedures), the measurement error can be completely controlled within the permissible range. However, educational measurement is a measurement of learning outcome (psychological phenomenon), dealing with objects that are rather difficult for objective quantification (such as memory ability, reasoning, operations, and thinking

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ability) and cannot be measured directly. Obviously, in order to reduce errors, it must also solve the problem of standardization of measuring tool itself and standardization of use method of measuring tools. But for educational measurement, both are rather complicated problems. For example, educational measurement does not have a unified scale, and it cannot be used to measure different disciplines, or even the same scale cannot be used to measure two different measurement objects in the same discipline. Educational measurement does not have relative zero point either, and it is determined by teachers by adjusting the difficulty of test questions, which makes it difficult to compare different test scores. For another example, the error of educational measurement comes from three aspects: inside the test paper, test process, and examinees themselves. This means that the sampling of the test questions, the climatic environment during the test process, the examiner’s degree of cooperation, the mastery degree of scoring standards, and the influence of the examinee’s own psychological and physiological factors, and so on will affect the accuracy of educational measurement directly or indirectly, and these are difficult to control. Therefore, the educational measurement of secondary school mathematics is much more complex and difficult than general physical measurements. It is often influenced and controlled by many subjective and objective factors and is characterized by subjectivity, randomness, and relativity of results. However, high and low quality of education exists objectively. Since the difference is obvious, then it can be completely expressed by an order of magnitude, that is, secondary school mathematics education is completely measurable. As stated by Thorndike and McCall, “Everything that exists has quantity”, and “everything that has quantity can be measured”. Not only can academic performance be measured, but people also make very specific moral education measurements (actually evaluation). For example, they compile some measurement items, including moral quality, learning attitude, learning discipline, interests and hobbies, etc., and test the students by asking students to answer them orally. Of course, it is infeasible to carry out moral education measurement by only depending on written tests, requires a scientific evaluation technology, and also requires strong sense of responsibility to implement it. It should be noted that because educational measurement is closely related to the social system and social ideology of a country, it has not only important school educational significance, but also extensive social significance. Obviously, the educational measurement of secondary school mathematics in China is closely linked with China’s educational system, educational policies, secondary school mathematics teaching objectives and tasks, etc.

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12.1.3 Significance of Educational Measurement of Secondary School Mathematics In traditional examinations, the teacher generally makes out a few questions, requires the students to answer them in a certain period of time, then gives scores according to the standards formulated in advance, and thus brings an end to the task of an exam. In fact, in these traditional examinations, not only the sampling of question setting is subjectively random, the scoring standards are not objective, but also the implementation of examination is often affected by the surrounding objective environment and the examinees’ physiological, psychological, and other factors, so that the resulting scores lack stability and reliability. For the scoring of traditional tests, because there is no objective, accurate, and unified standard, the scoring error is quite surprising. Si Taiqi, a famous educator, once conducted an experiment several decades ago. He gave a Chinese test paper to 142 secondary school teachers for scoring, and surprisingly, there were 35 different scores, ranging from 50 to 98. He gave another geometry test paper to 116 teachers for scoring, and there turned out to be 60 different scores, ranging from 28 to 92. As discovered in a sampling of college entrance examination papers for review by the relevant authorities of China in1994, the mean error was also above 15 points. At the same time, this traditional examination does not pay attention to the post-test analysis, even if there is a little analysis, it is only a mere formality, which cannot be well fed back to the teaching practice. It is thus clear that the traditional examinations are not scientific enough to examine and evaluate the quality of mathematics teaching in secondary schools well, are not conducive to the selection and training of talents, and need to be reformed urgently. To study and research educational measurement of secondary school mathematics is not only the need of scientific, reasonable, and accurate evaluation of academic performance, but also the urgent need of the education process itself, because it can provide correct feedback information, is conducive to teachers’ targeted teaching, and is conducive to students’ improving learning methods. At the same time, it can also provide the basis for the administrative departments for education to make decisions, revise the courses and textbooks, and adjust the teaching requirements. Therefore, studying and researching the basic knowledge related to educational measurement of secondary school mathematics and making educational measurement of secondary school mathematics scientific and modern are of great significance to guide the current secondary school mathematics teaching practice, promote the reform of secondary school mathematics teaching, accelerate the construction of mathematics education science, improve the quality of secondary school mathematics teaching, and train and select talents.

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12.2 Test Question Setting of Secondary School Mathematics The performance appraisal of secondary school mathematics mainly consists of two basic types: evaluation and examination. The examinations can be divided into oral examination, written examination, and practical operation, and the written examinations can be divided into open-book examination and closed-book examination, which have been elaborated in Chapter Nine. The key of closed-book examinations lies in the test question setting. For this reason, the type of test questions, the principles, standards, steps, and methods of test question setting will be elaborated concretely here.

12.2.1 Types of Secondary School Mathematics Test Questions Secondary school mathematics test questions can be divided into two categories by answer mode: free-response questions and fixed-answer questions. Free-response questions are questions that allow students to answer in their own language or action, such as short answer questions, exercise problems, essay questions, open-ended questions, inquiry questions, and experimental operation questions. Free-response questions are also known as subjective questions. It is convenient to set this kind of questions, but the scoring standard is difficult to master. Fixed-answer questions are questions that require students to identify the correct answer from the options predetermined by the test maker, such as yes–no questions, multiple choice questions, error correction questions, and matching. Fixed-answer questions are also known as objective questions. It is rather difficult to set this kind of questions, but the scoring standard is easy to master.

12.2.2 Principles of Test Question Setting of Secondary School Mathematics At present, the secondary school mathematics examinations in China are mainly of mixed type (that is, including free-response type and fixed-answer type). The main test question setting principles are: 1. Being in line with the current standard and teaching materials The teaching contents are clearly stipulated in the current Secondary School Mathematics Curriculum Standard, and the teaching requirements for compulsory courses and optional courses respectively are also put forward for high schools. Therefore,

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the test question setting must be in strict accordance with the spirit of the Standard. For example, the test questions set for the college entrance examination should not exceed the scope of the basic requirements, except for bonus questions. The teaching contents such as determinants, system of linear equations, preliminary calculus, unary polynomials, equations of higher degree, tangents of conic curves, and discussion on general binary quadratic equations should not be required for all candidates. Of course, sometimes it is difficult to tell the mathematical knowledge and method involved in a mathematical question is the basic or higher requirement, so we can only resort to specific analysis of specific problems. 2. Being conducive to selecting talents and promoting the improvement of mathematics teaching quality Test question setting should be conducive to the training and selection of talents, but also to promoting the improvement of mathematics teaching quality. These two aspects are complementary. Without the improvement of teaching quality, it would be difficult to train and select talents; without the training and selection of talents, the improvement of teaching quality would have no motivation. In the current secondary school mathematics teaching, some malpractices still prevail, such as ignoring the basics, ignoring the ability, guesstimates, excessive assignments tactics, and overwhelming materials. At the same time, most of the existing test question setting are not very scientific and not reasonable, so the way and direction of the test question setting have a lot to do with reversing the above malpractices, promoting the examination reform, and improving the teaching quality. 3. Combination of examining knowledge, skills, and ability Knowledge, skills, and ability are interrelated but different from each other. People have gradually paid attention to the cultivation and improvement of ability, but it is far from adapting to the needs of teaching reform and training talents. The combination of knowledge, skills, and ability in test question setting is not only the requirement of training objectives, but also the need of teaching reform. For example, for the test question: Let α = 43 π , then the value of the arccos (cos α) is () 2 2 1 4 (A) π ; (B) − π ; (C) π ; (D) π. 3 3 3 3 and this question: The necessary and sufficient condition for arccos (−x) greater than arccos x is. () (A) x ∈ (1, 0); (B) x ∈ (1, 0]; (C) x ∈ [−1, 0]; (D) x ∈ (0, 1].

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Only a single concept is examined in the former, which can be solved simply with conventional thinking, while both knowledge and ability are examined in the latter. 4. Striving to be purposeful, scientific, enlightening, and pertinent The test question setting must be based on the purpose of examination. For example, the purpose of a stage examination is to focus on the grasp of the knowledge in this stage and the ability to apply knowledge. With the clear purpose, the test question setting will be objectives-oriented. The test question setting must have scientific basis, can correctly reflect the requirements of the standard and the actual situation of the examinees, and should not include the contents that are obviously beyond the standard or ambiguous. While test question setting, determine the layout of the test questions and their proportions in the score scientifically according to the requirements of the knowledge and ability examined. The test questions should inspire students to use the acquired knowledge flexibly and solve the problems through positive thinking and reasoning. Generally speaking, do not set or reduce the test questions with narrow knowledge scope, single method, and special content as far as possible. At the same time, the test question setting should also have strong pertinence, appropriate difficulty, good discrimination, appropriate gradability, in order to attain the goals of being conducive to not only checking the teaching quality, but also to arousing students’ enthusiasm for learning.

12.2.3 Standards for Test Question Setting of Secondary School Mathematics Traditionally, the test question setting of secondary school mathematics only focuses on qualitative requirements, and its standards can be summarized as paying attention to the foundation, rigorous design, broad thinking, and conforming to no conventional pattern. Later, reflection of ability is added. According to modern theories, test question setting of secondary school mathematics should pay attention to quantitative requirements, and its standard can be summarized as gentle slope, distinct gradation; comprehensive coverage, giving prominence to key points; moderate difficulty, strong discrimination; and high reliability and good validity.

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12.2.4 Steps of Test Question Setting of Secondary School Mathematics 1. Determine the purpose of the examination Before starting the test question setting, the first thing is to make clear the purpose of the examination. Generally, there are different types of tests: the predictive test (namely assessment test) at the beginning of teaching, the formative and diagnostic tests (namely unit test or stage test) during the teaching process, and the summative test (namely final test, graduation test) at the end, they vary in purposes, so the way and requirements of test question setting vary. For example, the purpose of an assessment test is to find out students’ grasp of the acquired knowledge, so the discrimination requirement for the test is not high, the requirement for difficulty is rather low, and single question type is often selected. The final exam or graduation exam should have a certain difficulty and a broad coverage, and they focus on examining the students’ knowledge and ability, etc., so mixed question types are often selected. Only by clarifying the purpose of the examination can the test question setting be objectives-oriented. 2. Make clear the specific requirements of the examination Different disciplines, different grades, different types of schools, and different classes all have different teaching requirements. For example, while middle school students grasp the basic knowledge related to plane figures, the requirements for logical thinking ability, especially spatial imagination ability and reasoning ability, are rather low, while middle school students grasp the basic knowledge related to numbers, formulas, functions, equations, inequalities, etc., the requirements for the transformation of formulas and reasonable, correct, and quick operations of numbers and formulas are rather high. Only by clarifying the specific requirements for the examination can the test question setting have a basis. 3. Prepare a test question setting plan The test question setting plan prepared according to the purpose and requirements of the examination is the blueprint to design the test paper. It is usually a two-way detailed table which indicates the knowledge and ability to be measured in the test paper, their respective proportion in the whole test, the type and number of questions, etc. The so-called knowledge refers to the knowledge points to be measured in a certain subject; the so-called ability refers to the purpose to be attained in cognitive behavior through teaching. Only by preparing the test question setting plan as comprehensive, thoughtful, and accurate as possible, can we lay the foundation for the preparation of a good test paper. 4. Formulate specific test questions The basis of formulating the test questions is the test question setting plan and the basic principles and standards of test question setting. Before the formulation, do a

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good job of collecting data to make the data as complete and universal as possible, and during formulation, pay attention to the following points. ➀ The scope and requirements of the test question contents should be consistent with the two-way detailed list prepared in advance; ➁ The number of test questions is greater than those required for screening; ➂ The difficulty and discrimination of the test questions must meet the needs of the examination; ➃ The test questions must start with the easier ones, with a certain gradient; ➄ The instructions of the test questions must be clear, concise, and to the point and will not cause students’ misunderstanding or confusion. 5. Prepare model answers and give scoring standards After formulating the test questions, the teacher must answer them personally to estimate the time, speed, difficulties in the test paper, and multiple solutions rather accurately, make necessary adjustments according to the specific situation, and provide model answers and scoring standards on this basis. The above work should be done together with the formulation of test questions to prevent excessive infiltration of human factors to affect the reliability of the test.

12.2.5 Methods of Test Question Setting of Secondary School Mathematics Formulation of test questions is a creative labor, which often requires teachers to have high mathematical accomplishment and rich test question setting experience and go through repeated deliberation. The test question setting methods vary with the examination purposes or question types. In terms of free-response questions, there are the following main question setting methods. 1. Changed condition method This is a method of making appropriate changes in the conditions in the question to constitute a new proposition, and when it is correct, a new question is constructed. For example, there is an example in the textbook Analytic Geometry “It is known that a curve is above the x-axis, the difference between the distance from each point on it to point A (0, 2) and its distance to the x-axis is 2. Find the equation of this curve”. If we remove the condition of “above the x-axis” or change the “x-axis to y-axis”, they can become two new questions. 2. Causality interchange method This is a method of interchanging the conditions of a proposition with the conclusion to constitute a converse proposition, and when it is true, a new test question is constructed.

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Fig. 12.1 .

For example, change an example “Sketch the graph of sine function y = 3 sin(2x+ in the Secondary School Algebra to a multiple choice question: “As shown in Fig. 12.1, the analytic expression of the sine function is ()”. π )” 3

(1) (2) (3) (4)

  y = 3 sin 2x + π3 ; y = 3 sin x2 + π3 ; ; y = 3 sin x2 + 2π 3 y = 3 sin x2 − π3 .

3. Proper combination method This is a method of organically combining several relevant concepts, operations, or demonstrations to construct a new test question.  √ √  3 6 For example, combine “Find the value of 2 − 5 · 9 + 4 5” with “the cubic root of− 1” to construct a multiple choice question: if complex number x satisfies √  √ 3 6 3 χ = 2 − 5 · 9 + 4 5, the number of value of χ that meets the condition is (). (A)0; (B)1; (C)2 (D)3; (E)4. 4. Generalization method This is a method of generalizing the conditions of a known proposition to obtain a new conclusion and construct a new test question. For example, transform “It is known that θ is a real number, the two roots of quadratic equation x 2 − 2(sin θ − 2)x + 3 cos 2θ = 0 of x is α, β. Evaluate the maximum and minimum of α 2 + β 2 ” into “It is known that the two roots of quadratic equation x 2 − 2(sin θ − 2)x + 3 cos 2θ = 0 of x are α, β”. Evaluate the maximum and minimum of α 2 + β 2 in the following cases, respectively. π π   π  ; (3)θ ∈ . (1)θ ∈ R; (2)θ ∈ 0, , 4 4 2

12.2 Test Question Setting of Secondary School Mathematics

331

Fig. 12.2 .

For another example, for “Prove the sum of the distances of any point inside a regular triangle to three sides is a fixed value”, we can extend the regular triangle to: isosceles triangle =⇒ arbitrary triangle =⇒ regular ngon, and the point inside the triangle can be extended to a point outside the triangle. In general, the plane can be extended to space, straight line to curve, finity to infinity, binary to multivariate, acute angle to arbitrary angle, and so on. 5. Shape-number conversion method Simply put, this is a method to convert a geometric problem into an algebraic problem, or vice versa, to construct a new test question. For example, “What’re the values of x and y when 

(x + 1)2 + (y − 2)2 +

 (x − 4)2 + (y − 2)2

is minimum?” can be converted into a geometric test question. For another example, as shown in Fig. 12.2, in rectangular ABCD, it is known that. Ai E//AB, A A1 = A1 A2 = A2 A3 = A3 A4 = A4 D = a √ AB1 = B1 B2 = B2 B = 3a Prove: ∠B1 A1 E 1 + ∠B2 A1 E + ∠B DC = 90◦ It is a geometric question, but algebraic proposition and trigonometric proposition, etc. can also be constructed from it (left to the readers).

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12.3 Standardized Test Questions and Standardized Tests 12.3.1 Standardized Test Questions The so-called standardized test questions are mainly a kind of single, multiple, large number of choice questions. Since its trial in Japan in the 1960s, it has developed rapidly in the world. At present, it is most prevalent in Japan, the United States, and Western Europe, with a trend of development, and it is a direction of examination reform. Some standardized questions were tried out in the college entrance examination in China in 1983. In 1985, standardized tests were tried out in mathematics and foreign languages of the college entrance examination in Guangdong Province. After obtaining good test results, it has been used in a wider range of the country since 1986. At present, standardized test questions have become one of the basic question types of various mathematics tests in China. The characteristics of the standardized test questions are: ➀ Large number of questions, broad coverage, small random factor, high reliability of the test; ➁ Mechanization and automation of rating and scoring to be fair, reasonable and fast, and to save manpower, material resources, and financial resources as well; ➂ Because the examination syllabus, question types, number of questions, and answering method are published in advance, with no exercises or review questions provided, it can extricate the teachers and examinees from excessive assignments. The disadvantages of the standardized test questions are: ➀ It is not conducive to examining and understanding the examinees’ expression ability and demonstration ability completely; ➁ It is difficult to distinguish between different levels of examinees, especially difficult to examine students’ comprehensive application ability; ➂ Because the guesstimate scoring method has not been ruled out completely at present, some examinees may generate speculation psychology; ➃ However, after weighing the pros and cons, this kind of test questions does more good than harm, which is the very reason why it is widely used at present.

12.3.2 Formulation of Choice Questions 1. Structure of choice questions Standardized test questions are dominated by choice questions. Choice questions derive from yes–no questions, there are three to five conclusions for choice after each question, generally only one is correct or appropriate, and you are required to select this correct conclusion.

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The standardized test question generally consists of three parts: directions, stem, and options. The directions are usually written after the general question number and before all choice questions, mainly explaining the basic requirements for solving problems and scoring method. The stem refers to the sentences, forms, or illustrations indicating the examination content, and the sentences here can be incomplete or special questions. The options are the alternative conclusions after the stem. Whether there is only one correct conclusion or not must be made clear in the directions. The erroneous conclusions in the options are usually called disturbance terms. A good choice question must make the disturbance terms specious and adulterate as far as possible, so that one can choose the correct one only after thinking them over. 2. Basic requirements for formulating choice questions First, the overall structure should be arranged reasonably. A group of good choice questions mix difficult ones and easy ones properly, start with the easier questions, and are difficult but not complicated. Second, the overall structure should be arranged reasonably to achieve the purpose of high selectivity, high flexibility, and strong practicality. Generally, various options should be incompatible. Third, the examination content should be expressed clearly, the description should be clear, accurate, coordinated, and the codes of correct answers should be randomly distributed, to minimize the possibility of suggestion or speculation. 3. Basic methods of formulating choice questions Generally speaking, it is more complex and difficult to formulate choice questions than short answer questions and essay questions. Next, we’ll introduce how to formulate such choice questions as concept questions, calculation questions, proof questions, and construction questions, respectively. (1) Formulating concept questions For important concepts that are difficult to understand or easy to be confused in teaching, we can formulate choice questions aimed at the ambiguity in students’ understanding, ignorance of the internal connection between concepts or errors in the use of mathematical language and symbols. For example, aiming at the characteristics of the complex representation z = √ a + bi = r (cos θ + i sin θ): r = a 2 + b2 ≥ 0: the principle value of argument may or may not be taken for θ, cos θ comes before isin θ, they are connected with “ + ”, and we can formulate the following choice questions: Example 1 The trigonometric form of complex number 3x 2 − 7x + 2 = 0 is (…).     5π 5π 5π 5π + i cos ; (B)2 cos + i sin ; (A)2 sin 6 6 3 3

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  2π π π

2π + sin ; (D)2 cos + i sin ; (C) − 2 cos 3 3 3 3 π π

(E)2 cos − i sin . 3 3 (2) Formulating calculation questions Starting with answering ready-made questions, we can formulate choice questions aimed at students’ thoughtlessness in examining the questions, unclear algorithm, errors in calculation and solution, etc. For example, solving equation (x 2 − x − 1)x+2 = 1 in integer range should be considered in three cases: ➀ Since a nonzero real number to the power of 0 equals 1, we get 

x +2=0 ⇔ x1 = −2, x 2 − x − 1 = 0

➁ Since 1 to the power of arbitrary number equals 1, we get x 2 − x − 1 = 1 ⇒ x2 = 2, x1 = −1; ➂ Since −1 to the power of even numbers equals 1, we get 

x 2 − x − 1 = −1 ⇒ x4 = 0, x5 = 1 (Unsuitable) x + 2 = 2k k ∈ Z

But students are often thoughtless when solving the problem, resulting in extraneous root or loss of root, so we can formulate the following choice questions. Example 2 How many integers are there to satisfy equation (x 2 − x − 1)2+2 = 1. (A) One; (B) two; (C) three; (D) four; (E) five (3) Formulating proof questions Starting with answering ready-made questions, we can formulate choice questions aimed at students’ ambiguities that are likely to occur in demonstration, logical errors, or neglected logical relationship in the reasoning. We can also formulate choice questions aimed at the specific requirements for improving students’ logical reasoning ability and developing students’ creative thinking ability. Example 3 Out of the following four propositions concerning the positional relations between straight lines and planes, how many are true? ( ). (I) Two lines parallel to the same plane are parallel to each other; (II) Two planes parallel to the same line are parallel to each other;

12.3 Standardized Test Questions and Standardized Tests

335

Fig. 12.3 .

(III) Two lines perpendicular to the same line are parallel to each other; (IV) Two planes perpendicular to the same plane are parallel to each other. (A) None; (B) one; (C) two; (D) three; (E) all. (4) Formulating construction questions Starting with answering ready-made questions, we can formulate choice questions aimed at the errors that students are likely to make, neglected conditions, or positional relationship during construction. We can also formulate choice questions through the questions that are readily solved through graphing based on direct or indirect needs. For example, we can formulate such a choice question aimed at the fact that students are prone to get the graphic shapes wrong when sketching a function graph with absolute value signs: Example 4 As shown in Fig. 12.3, of which figure solid line part the is the approximate shape of the graph of function y = 1 − x − x 2 ?

12.3.3 Methods of Solving Choice Questions 1. Direct method For some choice questions, the logical relationship between the conditions and the conclusion is not clear, inference or calculation is needed to obtain a correct judgement. Analysis method or synthesis method, deduction method or induction method, forward induction or backward induction, direct proof method or indirect proof method, etc., can be used in the deduction and calculation process. Example 5 If circle x 2 + y 2 + Dx + E y + F = 0 is tangent to the x-axis at the origin, then (). (A) E = 0, D = 0, E = 0; (B) E = 0, F = 0, D = 0; (C) D = 0, F = 0, E = 0; (D) D = 0, E = 0, F = 0. In fact, since the circle is tangent to the x-axis at the origin, we get x 2 + y 2 ±2r y = 0. that is x 2 + (y ± r )2 = r 2

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Fig. 12.4 . √ 2 4 −4F Since D2 = 0, E2 = 0, F2 = 0, and D +E = E2 . 2 We get D = 0, E = 0, F = 0, so the answer is (C).

Example 6 The trigonometric expression of complex number S 2 +S 2

1 2

−

√

3 i 2

(A) cos π3 + i sin π3 ; (B) − cos π3 + i sin r = 0 S 2 C = 1 −     + i sin 5π ; (D) cos − π3 + i sin − π3 . (C) cos 5π 6 6

is: (). S E2 S2

;

In fact, after transforming the√options into algebraic expressions,     cos − π3 +i sin − π3 = 21 − 23 i meets the definition requirements, so the answer is (D). 2. Screening method Narrow down the selection range by excluding wrong answers. Especially when only one answer is correct, we can first judge that the others are wrong, then the remaining one is correct. Example 7 It is known that y = ax + b and y = ax 2 + bx + c, then their graphs (Fig. 12.4) are (). First compare a, weed out (A) and (D), and then compare b. When a > 0, b > 0, b < 0, weed out (B), indicating that only (C) is correct. the vertex abscissa − 2a 3. Special value method Verify the alternative conclusions with special values or special formulas to make the correct judgement, or substitute the special values into the known conditions for identical transformation, and then compare it against the alternative conclusions to find the correct answer. Example 8 It is known that (a + l) (b + l) = 2, then the value of arctan a + arctan b is equal to (). π (A) , 2

π (B) , 3

π (C) , 4

π (D) . 5

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337

Substitute a = 1, b = 0 or a = 0, b = 1 into the original formula, we get arctan a + arctan b = π4 , so the answer is (C). 4. Checking method This is a method of substituting the provided options into the known conditions to make correct judgement directly. √ √ Example 9 The solution to 7x − 3 + x − 1 = 2 is: (). (A) x = 3; (B) x = 23 ; (C) x = 2; (D) x = 1. Substitute the above values into the original formula, only x = 1 fits, so the answer is (D). 5. Graphic method For some choice questions, we can sketch the function graphs and make judgement intuitively by observing the graphs. Example 10 Let; A = {(x, y)|(x − 1)2 + y 2 = 1},

 y = −1 , B = (x, y)| xy · x −2 

kπ C = (ρ, θ )|ρ = 2 cos θ, θ = ,k ∈ z , 4  

x = 1 + cos θ D = (x, y)| θ = kπ, k ∈ Z , y = sin θ then (A) A = B; (B) B = D; (C) A = C; (D) B = C. By sketching four graphs (Fig. 12.5), we know that B = D, so the answer is (B). Several common basic methods are introduced above. In solving specific problems, these several methods should be used flexibly and crosswise. Generally speaking, when solving choice questions, we should first consider the checking

Fig. 12.5 .

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method, the special value method, and the screening method and then consider the direct method or the graphic method. If none of the above methods works, a special method is required.

12.3.4 Standardized Tests Standardized tests are the tests organized in accordance with scientific procedures, with unified standards and strict control of errors. It includes four aspects: standardization of test question setting, standardization of testing, standardization of scoring, and standardization of score interpretation. 1. Standardization of test question setting The standardization of test question setting is the most important link in standardized tests. For traditional tests, the test questions are generally set by the subject specialists and teachers in a centralized way in a short time, the number of questions is less, and the test questions also reflect the thought and will of the question setters to a larger extent. The test question setting group of standardized tests is generally composed of education statistics and measurement experts and teachers. Before the test question setting, the test syllabus is developed, and the test question setting plan (two-way detailed list) is formulated first to ensure the examination of knowledge and ability. After the test questions are formulated preliminarily, a pre-test is carried out among some students. The pre-test paper is analyzed to get the difficulty, discrimination, and other data of the test questions. For choice questions, each option is even statistically analyzed, to enhance the covertness, confusability, reliability, and other functions of each option. The test questions are then revised and screened on these grounds, and those meeting the requirements are stored in the test questions database. Before the test, the test questions are taken from the test questions database to generate the test paper according to the test requirements. 2. Standardization of testing For traditional tests, because there is no continuity of test question setters, the types and contents of test questions are often different every time. To adapt to the college entrance examinations, teachers often guess questions, and organize students do excessive assignments to strive for high scores, and this directly affects the normal teaching of secondary school mathematics. For standardized tests, the test syllabus and question types, number of questions, and answering methods are generally published six months or three months before the tests, enabling teachers and examinees to have a clear idea about the tests and to know better than to affect their normal level due to inappropriate learning and reviewing and no knowledge of the testing. At the same time, there are clear and specific requirements for the setting of examination rooms, examination rules, and even the pencils, erasers, and other tools used,

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so that the examinees can take the examination under relatively unified and objective conditions. 3. Standardization of scoring As mentioned above, for traditional tests, there may be large errors in scoring generally. For standardized tests, the choice questions are marked directly by electronic computers, to ensure the accuracy and speed of marking. There are also a complete set of tiered scoring methods for other questions such as free-response questions and mixed questions, and after marking, they are controlled and counted by electronic computers to ensure the accuracy and reliability of scoring. Therefore, standardized tests can minimize the error in scoring. 4. Standardization of score interpretation At present, for the score of each subject of the college entrance examination, we are used to adding them directly to decide the admission to a college based on the total score, but this is not reasonable. Each examinee has six or seven examination subjects, and the requirements and difficulty of test questions of each subject are not the same, so the value of the scores is also different. If one get是 70 points in Chinese and 80 points in mathematics, we will think that the student is better at mathematics than Chinese. However, if the average score of Chinese is 60, then 70 is a rather high score; if the average score of mathematics is 85, then 80 is a rather low score. Thus, one point in Chinese is not equal to one point in mathematics. Therefore, it makes sense to make a comparison by converting the scores of different subjects into a scale with unified reference point and same unit. It is more reasonable and more conducive to the selection of talents to calculate the total score of examinees than the original score. How to convert original scores into standard scores will be described in detail in the next section.

12.4 Sorting and Analysis of Test Scores In the educational measurement of secondary school mathematics, the academic examinations are usually divided into norm-referenced examinations and objectivereferenced examinations. The former is mainly used to explain students’ relative grade (relative score), namely the position in a group, its main function is to queue students’ scores, and it is often used for selection or grouping; the latter is mainly used to describe the degree of achieving the teaching goals (absolute score), and it is not compared with others, but with the goals proposed in the standard and textbooks. Obviously, these two tests have different requirements for formulating the test questions and have a different interpretation of the test results. Next, we’ll introduce how to sort and analyze the scores of norm-referenced examinations.

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12.4.1 Sorting of Test Scores 1. Mean and median The mean is the average value of a group of test scores, and the median is the number in the middle of a group of test scores. For example, there is a group of scores of 76, 80, 88, 90, 96, its mean is 86, while the median is 88. The median is calculated as follows: ➀ Arrange the scores from high scores to low scores; ➁ If the number of scores is odd, the score in the middle is the median; ➂ If the number of scores is even, the average of two adjacent scores in the middle is the median. Usually, it is easier to determine the median than the mean, and when too high or too low scores occur in the test, the median can reflect the actual situation of the test more than the mean. 2. Standard deviation and variance Although the average and median can reflect the academic performance of a class or group, only these two characteristics are obviously not enough. For example, the mathematics scores of 5 students of Group A of a class are: 20, 40, 90, 100, 100, 100, while the math scores of 5 students of Group B are 65, 68, 71, 72, 74, their average score is both 70, but the situation of these two groups is very different: the scores of Group A differ greatly, while the scores of Group B differ slightly. To this end, the quantity (standard deviation and variance) reflecting the degree of dispersion of the scores must be considered. Suppose there are n scores x1 , x2 , x3 , . . . , xn , their average or median is x, then   n 1  s= (xi − x)2 n i=1 is called the standard deviation of theses n scores, while s 2 =

1 n

n

i=1 (x i

− x)2

is called the variance of these n scores. Obviously, the variance is the square of the standard deviation. It is not difficult to obtain that the standard deviation of the scores of Group A above is 33.4, while that of Group B is only 3.16. 3. Distribution diagram of scores The above mean and median, standard deviation, and variance indicate the mean level and dispersion degree of the test scores, respectively, and they are important characteristic quantities reflecting the scores from two different aspects. The score distribution diagram can be made for intuitive representation as follows: ➀ Calculate the mean or median x of scores xi (i = 1, 2, . . . , n);

12.4 Sorting and Analysis of Test Scores

341

Fig. 12.6 .

➁ Starting from x, divide scores xi (i = 1, 2, . . . , n) into x ± σ, x ± 2σ, x ± 3σ , … , etc. groups (generally 3–5 points is taken for σ ); ➂ Calculate the frequency of scores xi in the above groups yi = (1, 2, . . . , ); ➃ Establish the rectangular coordinate system with scores xi as the horizontal axis and frequency yi as the longitudinal axis. Construct the score distribution diagram, and the ideal score distribution should be a normal distribution (Fig. 12.6). If the peak is skewed to the left, it is a skewed positively distribution (Fig. 12.7), indicating the test questions are rather difficult; if the peak is skewed to the right, it is a skewed negatively distribution (Fig. 12.8), indicating that the test questions are too easy. Fig. 12.7 .

Fig. 12.8 .

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12.4.2 Item Analysis of Test Scores 1. Difficulty Difficulty is a quantitative index that reflects the difficulty of the test questions. (1) Difficulty of test question By different scoring method, mathematics questions can be divided into two types: dichotomy scoring question and non-dichotomy scoring question. For the former, its scoring has only two cases: “right” and “wrong”, and the formula to calculate its difficulty is p=

n × 100% N

where p indicates the difficulty, N is the number of all examinees, and n is the number of examinees answering this question correctly. For example, among the 50 examinees in a class, 26 people have answered a certain question correctly, so the difficulty of this question is 0.52. The latter is applicable to calculation questions, proof questions, construction questions, essay questions, etc. As long as a part is answered correctly, a certain score is given. The formula to calculate its difficulty is p=

x × 100% x

where p indicates the difficulty, x means the full score of the test question, and x means the average score obtained by the examinees to answer this test question. For example, a certain question is 20 points, the average score obtained by students is 12 points, and then the difficulty of this question is 0.60. In practice, for the sake of convenience, the mean of the scores of high and low score groups can be divided by the full score of this question. For example, among the 50 examinees in the above case, for a certain question of 20 points, 10 people from the high score group get 180 points for answering this question, and 10 people from the low score group get 85 points, then the difficulty of this question is   180 85 1 + = 0.66. p= 2 20 × 10 20 × 10 (2) Difficulty of test paper Combine the local difficulty of each test question together to get the overall difficulty of the test paper. The overall difficulty of the test paper can also be approximately expressed by the average difficulty of the test questions.

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It should be noted that the greater the value of p mentioned above, the easier the test questions; the smaller the value of p, the harder the test questions. It is just the opposite of the “difficulty” in the usual sense. Therefore, it is argued that we should use q = 1p to indicate difficulty, so that the large value of q indicates high difficulty (test questions are difficult); small value of q indicates low difficulty (test questions are easy). In practical application, pay attention to the meaning of difficulty p and q, so as to avoid confusion. The selection of difficulty should be consistent with the purpose and nature of the examination. Generally, the difficulty of the examination should preferably be at 0.6–0.8. 2. Discrimination The so-called discrimination (differentiation) is a quantitative index to differentiate the degrees of various questions answered between top students and underachievers. A high discrimination indicates that top students get high marks and underachievers get low mark. There are many methods to calculate the discrimination, mainly: (1) Correlation coefficient method This applies to “dichotomous variables”, that is, each question is expressed as correct or wrong answer, and the criterion is the condition of continuous variables. The calculation formula is where r pq represents the relative coefficient, x p and xq represent the average scores on the criterion of the examinees whose answer is correct and wrong, respectively, p and q represent the ratio of the number of examinees whose answer is correct or wrong to the total number of examinees, respectively, and s represents the standard deviation of all examinees’ scoring on the criterion. The 15 students answering a question in the following table are taken as an example. Here, 8 people answered correctly, and 7 people answered incorrectly, Student

1

2

3

4

5

6

7

8

9

10 1 l 12 13 14 15

Criterion score (total score) 65 70 21 49 80 50 35 20 81 69 78 55 77 90 42 Answer situation

w

R

w

R

R

w

R

w

w

R

R

w

8 = 0.5333, q = 1 − 0.5333 = 0.4667; 15 548 334 xp = = 68.50; xq = = 47.71; 7  8  xi = 882; xi2 = 58936;

Because :

p=

R

R

w

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     n  ( x i )2 2 1   − x _ i n s= (xi − x)2 = , n i=1 n  2 58936 − (882) 15 So : = = 21.72; 15 (After checking it again, we get to know whether its discrimination meets the requirements. Omitted). (2) Two-end grouping method This method facilitates calculation and is generally applied. Its calculation method is: the proportion of the number of the people of the high score group who answered correctly to the examinees PH minus the proportion of the number of the people of the low score group who answered incorrectly PL (preferably each accounting for about 27% of the total number of examinees) is the discrimination: D = PH − PL where 0 ≤ D ≤ 1, and when D = 1, it indicates those of high score group all answered correctly and those of low score group all answered incorrectly; when D = 0, it indicates the percentage of those of two groups who answered correctly was the same. For example, among 52 examinees of a class, the number of those who answered a certain question correctly of high score group and low score group (10 people per group) was 8 and 3 respectively, and then the discrimination is 3 8 − = 0.5 10 10 Generally, the discrimination is preferably at about 0.4. The main reason for low discrimination is the lack of hierarchy of difficulty of test questions. Its solution is dividing the test questions into basic questions, comprehensive questions, complicated comprehensive questions, difficult questions, and other types, and the difficult questions should be divided into several hierarchies.

12.4.3 Overall Analysis of Test Scores Here, we begin with the concept of measurement error. All measurements have errors, with no exception to educational measurement of secondary school mathematics. Its errors are divided into two types: systematic error and non-systematic error. Systematic errors, also known as conditional errors, are errors that occur rather stably and are caused by variables independent of the purpose

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of the test. For example, the examinees’ reading ability and comprehension ability affect their performance, and the error caused by this variable is a systematic error. Non-systematic errors (also known as random error, accidental error, or measurement error) are errors caused by accidental factors. For example, the errors caused by the diversion of examinees’ attention, change of mood, and other temporary external factors. Due to the errors, the relationship among the actual score obtained by each examinee in a test x and effective score x0 , systematic error xt , and random error xε is as follows: x = x0 + xt + xε By mathematical statistics, we know that the variations (or variance) of their group scores have the following relationship: S 2 = S02 + SC2 + S E2 where S02 , SC2 , S E2 are called valid variance, invalid variance, and random variance, respectively. With the above mathematical model, we can introduce two important quantitative indexes in the overall analysis. 1. Reliability The so-called reliability refers to the degree of difference between the measured value and the true value, which is a quantitative index that reflects the stability and reliability of the test questions. It contains two meanings: ➀ When we repeat the test in the same way, we can obtain the same results to maintain the measurement stability; ➁ The effect of random errors is reduced to maintain the accuracy of the measurement results. The smaller the percentage of random error in the test score, the more reliable the measurement. Therefore, reliability can be defined as the proportion of true score   variance S02 + SC2 in the obtained score variance, that is, r=

S02 + SC2 S E2 = 1 − . S2 S2

In practice, reliability can be defined as the correlation coefficient of the measured scores in the same measurement two times obtained by the same batch of examinees or as the correlation coefficient of question groups of odd numbers and even numbers of a test, that is,    yi N xi yi − xi  r=     2 ∗   2 N xi2 − xi · N yi2 − yi

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where xi is the obtained score of examinee i in the first test; yi is the obtained score of examinee i in the second test; and N is the total number of examinees. The actual score is the total number of candidates. For example, 10 students are tested 15 days apart using the same scale, and the data obtained is as follows. Examinee

xi2

yi2

xi

yi

01

8

9

6 4

81

72

02

10

10

100

100

100

03

9

10

81

100

90

04

6

6

36

36

36

05

10

10

100

100

100

06

7

8

49

64

56

07

5

4

25

16

20

08

7

8

49

64

56

09

9

9

81

81

81

10

4

4 

3x 2

− 7x + 2 = 0

yi = 78

16  2 x i = 601

16  2 yi = 139.4

xi yi

16  x i yi = 627

Substitute them into the formula, we get 10 × 627 − 75 × 78 r=√ √ 10 × 601 − 752 · 10 × 658 − 782 = 0.96. The most commonly used method of calculating reliability is: arrange the test questions of a test from the more difficult to the easier, halve them according to odd and even numbers, and apply the above formula (*) to calculate the correlation coefficient of the scores obtained by the examinees in the two halves, which is called split-half reliability coefficient. The calculation formula for the reliability of the whole test paper is r=

2r2 . 1 + r2

where r2 is the correlation coefficient of the scores in two halves and r is the correlation coefficient of the whole test paper.

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347

For example, a test paper has 20 questions, and the average scores of even number and odd number questions are shown in the following table. Odd number questions (xi )

4.9

4.6 4.1 4.3

4.6

4.0

3.8

4.1 3.1 2.5

Even number questions (yi )

4.8

4.7 3.9 3.8

4.2

3.6

3.2

3.3 3.3 1.4

The reliability is generally required to be above 0.9. Under the conditions of controlling the   2  2  xi = 164.8, yi = 139.4, x i yi = So we get x i = 40, yi = 36.2, 150.6. Then r=

2 × 0.913 = 0.954 1 + 0.913

difficulty and discrimination of the test questions, if we can increase the number of questions appropriately, reduce the complicated questions as much as possible, master the scoring standard strictly, etc., we can improve the reliability of the test. 2. Validity The so-called validity is a quantitative index that reflects whether the measurement can reach the characteristic value or functional degree to be measured, so that it can reflect the correctness of the test. The validity here also includes two meanings: ➀ Validity has particularity, namely any measurement is valid only for some particular purpose. For example, we cannot use a Chinese or English test to reflect the mathematical level of examinees or cannot use a test of algebraic knowledge to reflect examinees’ level of geometric knowledge; ➁ Validity is relative, namely that any measurement is only an indirect judgement of the characteristics to be measured and can only achieve some degree of correctness. In test scores, the proportion of systematic error is an important indicator to measure the test validity, that is, the smaller the factors unrelated to the test objectives, the higher the test validity. Thus, validity can be defined as r=

S02 S2

In practice, validity can be defined as the correlation coefficient of the test score and the criterion data (e.g., the examinee’s previous performance), that is, the above formula (*) is still applied   yi xi yi − xi  r=     2 .   2 N xi2 − xi · N yi2 − yi N



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where N is the total number of examinees andxi and yi are the test score and the score of criterion data of examinee i, respectively. For example, the obtained scores of 12 students in one test and their average scores in mathematics over the years are shown in the following table: Test score (xi )

35

42

55

69

66

68

73

78

82

88

94

97

Average score over the years (yi )

61

45

60

58

63

61

89

69

85

92

90

98

    where N= 12, so we can get xi = 847, yi = 871, xi2 = 63881, yi2 = 66475, xi yi = 64544, substitute them into the above formula, we get r=

12 × 64544 − 847 × 871 (12 × 63881 − 8472 )(12 × 66475 − 8712 )

= 0.84

The validity is generally required to be above 0.8. There are also many factors affecting the validity, and it is particularly important whether the test questions are appropriate. For example, if the test questions are beyond the syllabus, or there are tricky, strange, or obscure questions, the validity will often be reduced. On the contrary, if these problems can be avoided, the validity will be increased. As mentioned above, it is not hard to see that reliability is the necessary condition for validity, and if a test is not reliable, it must be invalid, but if it is reliable, it is not necessarily valid, that is, if a test has high reliability, its correctness is not necessarily high, but if a test has high correctness, its reliability must be high.

12.4.4 Standard Score As mentioned earlier, the score obtained after the test paper is marked is called the original score. Original scores cannot reflect the student’s learning situation scientifically, so the original scores are required to be converted into standard scores. There are two common conversion methods. The z-score is the score obtained by means of the following conversion formula zi =

xi − x s

where xi is the original score, x is the average of original scores, and S is the standard deviation. When the z-value is positive, it indicates this score is above the average; when the z-value is negative, it indicates this score is below the average. Seen from the normal distribution table, almost all the data are included within z = − 4 to z = + 4. Therefore, it is generally assumed that the value range of standard scores is between − 4 and + 4.

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For example, the average score, standard deviation, and scores of a test of two students in a class in the third grade are as follows: Average score

Standard deviation

Student A

Student B

Chinese

80

8

70

80

Mathematics

50

5

60

50

The total score of Chinese and mathematics of both student A and student B is both 130 points, which is difficult to compare. But when converting the original scores into standard scores, i.e., Student A: Chinese − 1.25, mathematics 2, with a total score of 0.85; Student B: Chinese 0, mathematics 0, with a total score of 0. Obviously, student A’s performance is better than that of student B. T-score is the score obtained by means of the following conversion formula: T = 10z + 50. This conversion is made because the values of z-scores are within [− 4, 4], often with decimals, and are inconvenient for use. For example, the average score of a mathematics test in a grade is 60, the standard deviation is 5, and the original scores of four students are converted to z- and T-scores as shown in the table below.

Original score

Student A

Student B

Student C

Student D

80

70

60

40

z-score

4

2

0

−4

T-score

90

70

50

10

In general, the original score (usually a percentage score) can be used to indicate the quantitative and qualitative level of students’ mastery of knowledge and ability, and it should be used in tests that examine the quality of teaching. The z-score and T-score can be used to indicate student’s position in the examinees, so they should be used for selective examinations. With the above basic analysis, after obtaining the relevant quantitative indexes, we can make a rather scientific and reasonable evaluation for the test to feed back to teaching, in order to improve the quality of mathematics teaching, which will not be discussed in detail due to space limitation. Review Questions and Exercises (XI) 1. What is educational measurement of secondary school mathematics, what are its characteristics, and what is the significance of learning and studying its basic knowledge?

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2. What are the types of secondary school mathematics questions, what are the principles, standards, and steps of test question setting, and what are the common methods of formulating free-response questions? Illustrate them with examples. 3. Try to make a structural analysis of a slightly complex problem, and formulate a set of exercises for this question. 4. Formulate a test paper, provide answers for the teaching contents of a chapter or a semester of secondary school mathematics, and indicate the scoring standard. 5. What are standardized test questions, and what are the commonly used solution methods? Illustrate them with examples. 6. Prepare a standardized test paper, and provide answers for the teaching contents of a chapter or a semester of secondary school mathematics. 7. What is the standardized examination, and what is its development prospect? 8. What is the difficulty, discrimination, reliability, and validity, and what is their respective significance in educational measurement? 9. What is the standard score, and what is the significance of researching standard scores? Illustrate the conversion methods between original and standard scores with examples. 10. How to sort and analyze the test scores of secondary school mathematics? Illustrate it with specific examples. 11. The following is the scores of question 5 in a geometry test of 30 students in a class. Try to work out the difficulty and discrimination of this question.

Total score

85

73

62

68

53

82

79

64

53

84

Student

1

2

3

4

13

14

15

16

17

18

1

1

1

0

1

0

1

0

0

1

Total score of question 5

24

23

22

21

20

19

8

7

6

5

Student

32

97

64

51

65

47

76

42

35

76

Total score

1

1

1

0

1

0

0

0

1

1

Total score of question 5

30

29

28

27

26

25

12

11

10

9

Student

92

53

76

81

88

46

94

96

85

84

Total score

1

1

0

1

1

0

1

1

1

1

Total score of question 5

12.4 Sorting and Analysis of Test Scores 351

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12. Calculate the reliability of a test of 16 people based on the scores of odd and even number questions. Score

Examinees

Odd number questions

18 17 18 15 14 17 17 13 13 13 16 14 15 16 13 16

Even number questions

18 17 16 16 13 18 17 14 13 13 14 14 14 16 15 17

13. A teacher formulates a set of test questions to test the students’ academic performance. The test results of 16 students and their academic year performance data are shown below. Estimate the validity of the test questions. Examinees Scores of a test

18 17 18 15 14 17 17 13 13 13 16 14 15 16 13 16

Academic year test scores

18 17 16 16 13 18 17 14 13 13 14 14 14 16 15 17

14. Try to make a scientific analysis of a mathematics test of a class in a secondary school, and put forward your suggestions based on what you have learnt.

Chapter 13

Teaching and Research Practice of Secondary School Mathematics

The teaching and research of secondary school mathematics are a complex activity process. Teachers need not only to master the basic knowledge and methods of teaching and research well, but also to have good teaching and research abilities, which can only be improved through continuous and repeated practice. This chapter introduces the teaching and research practice.

13.1 Teaching Skill Training of Secondary School Mathematics 13.1.1 Teaching Skills of Secondary School Mathematics Teaching skills are a teacher’s vocational skills, and the mastery of basic teaching skills is the basis for the implementation of teaching. For this reason, the Department of Normal College Education of the State Education Commission issued the Basic Requirements for Teachers’ Vocational Skills Training for Students of Normal Colleges in 1992 and the Program of Teachers’ Vocational Skills Training for Students of Normal Colleges (Preliminary Draft) in 1994, which standardized the teaching skill training for normal college students and set high requirements. Mathematics teaching skills are the teaching behaviors or activity modes of teachers applying mathematical expertise and educational theories to promote students’ learning in the process of teaching mathematics. Mathematics teaching skills run through the whole process of teaching and has a wide meaning, including both in-class skills and extracurricular ones. We mainly introduce classroom teaching skills here. In general, classroom teaching skills mainly include teaching language skill, explaining skill, blackboard writing and drawing skill, lead-in skill, questioning skill, variety skill, presentation skill, feedback strengthening skill, abstraction and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3_13

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generalization skill, example inquiry skill, teaching organization skill, and closing skill, etc. Teaching language skill is the language skill that teachers use in the classroom to clarify problems, impart knowledge, organize students to learn, and stimulate students’ enthusiasm for learning, it is the most fundamental teaching skill and the most widely used teaching skill. The teaching language can be divided into three categories: explanatory language, organizational language, and evaluative language. Explanatory language is mainly used to explain mathematical concepts, formulas, principles, laws, rules, methods, reveal the internal connection of knowledge, and demonstrate the propositions, theorems, etc. While using explanatory language, pay attention to giving appropriate examples and applying metaphors, analogies, etc. Organizational language is mainly used to organize students to carry out various learning activities, and it fully reflects the leading role of teachers. While using organizational language, strive to be clear, short, attractive, and approachable. Evaluative language is mainly used to evaluate the teaching content and students’ learning behaviors, and it should be factual, appropriate, centered on teaching objectives, and clearly oriented. Explaining skill is the teachers’ skill to use systematic and coherent language to articulate and clarify the teaching content and impart knowledge, it is an important aspect of teaching language skill. However, teaching language skill focuses on the formation of language, whereas explaining skill focuses more on the organization of language and the procedures of expression. The explaining skill should follow the principles of being inspirational, scientific, and pertinent. Your explanation should have a clear purpose, give prominence to focal points, grasp the key points, and be hierarchical and stage-specific. Pay attention to the connection between theory and practice, and combine the explaining skill with other teaching skills. Blackboard writing and drawing skill refer to the skill of writing characters, symbols, or sketching graphs and tables on blackboard, projector film, or other media for auxiliary explanation in teaching. It is of great significance in mathematics teaching to use blackboard writing and drawing to stimulate the interest in learning, show the teaching process, and guide and develop the intelligence. They can be of word type, outline type, clue type, table type, graph type, and so on. During their use, it must be carefully designed, so that the blackboard writing and drawing are purposeful, planned, standardized, feasible, and clear. With the popularization of modern education technology, some complex and time-consuming blackboard writing and drawing can be replaced by multimedia courseware, thus greatly improving classroom efficiency. However, teachers’ blackboard writing and drawing skill cannot be weakened. Lead-in skill is a teaching skill to introduce students to a certain learning situation at the beginning of a new learning activity. There are usually direct lead-in, review lead-in, presentation lead-in, question lead-in, analogy lead-in, activity leadin, and other types and generally have four basic links: “attracting attention—stimulating motivation—organizing and guiding—establishing links”. During its use, we should follow the principles of being purposeful, pertinent, enlightening, and highly efficient.

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Questioning skill is a kind of teaching ability for teachers to guide students to learn by setting and asking questions, which is characterized by the joint participation of teacher and students and the transmission of information in an inquisitive, timely, and targeted manner. It can be divided into recall questioning, comprehension questioning, application questioning, analytical questioning, creative questioning, evaluative questioning, etc. In its specific application, attention should be paid to designing questions meticulously to make the questions inspiring, pertinent, and purposeful. Variety skill refers to a teaching skill that teachers use various stimulations to attract students’ attention, impart knowledge vividly, communicate emotions, and promote learning. The main types include variety in the eyes, variety in the voice, variety in the facial expressions, variety in the body movements, etc. Pay attention to using these skills with ease and appropriately in order to enhance the teaching effect. Other teaching skills also play an important role in mathematics teaching. Presentation skill is a teaching skill for teachers to effectively guide students to grasp the essential features of things through observation of visual objects. Feedback strengthening skill is a teaching skill that enables both teachers and students to adjust their teaching and learning activities in a timely manner in order to help achieve teaching goals. Teaching organization skill is a teaching skill that ensures the smooth execution of teaching activities to a certain extent. Closing skill is a teaching skill that helps students to summarize a certain teaching content and play a role of bringing out the crucial point. Abstraction and generalization skill and example inquiry skill are classroom teaching skills that are closely related to classroom teaching. Readers are expected to explore these skills on their own.

13.1.2 Training of Teaching Skills Teaching skills must be acquired and improved through repeated study and practice. To this end, as pointed out by Makarenko, an educator of the former Soviet Union, “The normal colleges should train our teachers in other ways, such as how to stand, how to sit … how to raise their voice, how to laugh and how to look, and other minor details … they are all necessary for a teacher, and without these skills, one cannot make a good teacher…”. With regard to the training of teaching skills, there are the following procedures and approaches in general. 1. Defining training objectives Training objectives are the final learning outcomes that the students should achieve at the end of the training and include knowledge and mastery of the principles of each teaching skill. The training objectives are the starting point of training activities and restrict the direction of training; they are also the destination of training activities

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and provide standards for evaluating training. But the training objectives should be not only objective and comprehensive, but also concise and specific. 2. Learning relevant theories The form of teaching skills is usually closely related to and based on the acquisition of theoretical knowledge. Therefore, both theoretical and methodological principles should be learnt prior to training, to address not only how to do it well, but also why it should be done this way, in order to prevent simple imitation. 3. Providing internship examples In order to enable students to understand various characteristics of normative teaching behavior, positive examples or negative examples can be analyzed depending on various teaching skills, mainly positive examples. Examples can be designed and demonstrated by instructors; they can be demonstrated by secondary school teachers; or they can be chosen from videos or audio recordings related to teaching. Generally speaking, audio and video demonstrations are more convenient and economical and can be used repeatedly. When using audio or video demonstrations, the instructor’s explanation on the spot should be timely, accurate, concise, and to the point for students to watch and understand. 4. Implementing training practice On the basis of the above, students should carry out imitation practice or corrective practice for specific teaching skills according to the training objectives and training plan and strive to meet the training requirements. To facilitate the training practice, groups can be formed to help and teach each other. Conditions permitting, micro-teaching can be used. Micro-teaching is a training system where the teaching skills are developed through micro-lesson practice based on audiovisual technologies. Due to small class, less content, and short class hours, it is known as micro-teaching, micro-standard teaching, and so on. In fact, it is a kind of miniature teaching, which is a mode widely used in professional skills training since the 1960s. 5. Conducting training feedback Training feedback should be conducted to communicate and check the training effect. This feedback is usually a reteaching, in which students give report-back presentations and speeches on teaching skills and are reviewed by instructors. Of course, replaying audio and video recordings in micro-lessons is the best way to provide feedback, which can not only facilitate the instructors’ review and peer observation, but also facilitate the students’ self-analysis and review, resulting in better effect.

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13.2 Basic Teaching Skills of Secondary School Mathematics Classroom teaching is the main form of mathematics teaching in secondary schools, with characteristics of being scientific, visual, affective and creative, etc., and it is not only a science, but also an art. In order to be competent for classroom teaching and to improve the quality of classroom teaching, mathematics teachers in secondary schools must have certain basic teaching skills based on the formation of teaching skills.

13.2.1 Basic Skill of Organizing Textbooks Classroom teaching is a process in which the teacher organizes and guides the students to master the knowledge and form their own abilities, with the textbooks as the main focus, giving full play to the teacher’s leading role and students’ principal role. In order to achieve a good teaching effect, the teacher must first be good at organizing the textbooks and have the basic skill of organizing the textbooks. To have strong ability to organize textbooks, teachers are required to, on the basis of being familiar with the standard, delving into the textbooks, and having a thorough understanding of the students and in view of the spirit of the standard, the textbooks and the actual situation of students make clear the teaching objectives, focal points, and difficult points in teaching, put forward appropriate teaching requirements, and adopt appropriate teaching approaches, methods, and means according to the cognitive rules of secondary school students. The essence of the classroom teaching process is the teacher’s continuous organization of textbooks and gradual implementation of teaching plans, and the quality of organizing textbooks is the key to improving the quality of classroom teaching. To have strong basic skill of organizing textbooks, teachers are required to be familiar with the standard, master the textbooks, have a solid basic knowledge of mathematics, have theoretical knowledge of pedagogy, psychology, mathematics pedagogy, etc., read teaching reference books, study educational science theories from as many materials as possible through extensive reading, and absorb teaching reform experience, so as to constantly enrich and improve themselves. In addition, they should improve their ability to dig into textbooks, improve their reading ability, and enhance their ability to summarize and express textbooks by observing teaching and strengthening actual teaching practice.

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13.2.2 Basic Skill of Solving Mathematical Problems The mathematics teaching in secondary schools cannot be separated from problem solving, and the development of problem solving ability is an important task of mathematics teaching in secondary schools. A mathematics teacher in secondary schools must be equipped with good basic skill of solving mathematical problems. To have good basic problem solving skill, teachers are required not only to solve the exercises in the textbooks correctly and skillfully, but also to be familiar with specific problem-solving ideas, understand the mistakes that students are prone to make in thinking and solving problems, and be familiar with the common problemsolving skills, multiple solutions to related exercises, deformation, generalization and extension of exercises, and the connection between exercises. For those relatively unfamiliar exercises, teachers are required to be able to point out the thinking methods and points for attention. Besides, teachers should also pay attention to careful thinking, complete problem solving, and be able to express it reasonably, clearly, and concisely. In order to improve teachers’ problem-solving ability, they are required to research the thoughts and methods of problem-solving in secondary school mathematics and solve the problems themselves to accumulate experience and skills in problem-solving while mastering the basic knowledge of secondary school mathematics systematically. Only by developing a problem-solving habit and purposeful and planned problem-solving practice with perseverance can teachers have good problem-solving ability to meet the needs of classroom teaching and extracurricular tutoring.

13.2.3 Basic Skill of Mathematical Language Also known at symbolic language, mathematical language is an improved natural language using words, symbols, and graphics widely. It is characterized by conciseness, accuracy, clarity, succinctness, and being “variable”. The language of secondary school mathematics teachers often refers to their mathematical language, including verbal language and nonverbal language. The verbal language refers to the teaching language used by teachers in the classroom, and the nonverbal language refers to the teachers’ presentation, teaching manner, gesture, dress, and blackboard writing during teaching. The language competence often refers to verbal and writing skills. For mathematics teaching in secondary schools, the expression ability of verbal language is more important. The correct use of mathematical language is an important basic skill for mathematics teachers. In the specific application, efforts should be made to master mandarin, make the language purposeful, scientific, intuitive and inspiring, and grasp the relevant methods and skills of using language.

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In order to improve secondary school mathematics teachers’ basic skill of mathematical language, we should strengthen the mathematical language accomplishment, participate in the teaching practice vigorously, analyze our use of mathematical language, absorb the peers’ strengths, overcome our own shortcomings, improve the skill of using mathematical language, and improve our writing skills further on this basis.

13.2.4 Basic Skill of Mathematical Blackboard Writing Blackboard writing is of special significance in secondary school mathematics teaching. Blackboard writing is also a basic teaching skill for mathematics teachers in secondary schools. In order to improve the quality of blackboard writing, efforts should be made to meet the requirements of being planned, intuitive, demonstrative, inspirating, etc. In order to improve the basic skill of mathematics blackboard writing, teachers are required to consider the blackboard writing plan while preparing lessons, strengthen the practice of writing and drawing with chalk, master the rules of blackboard writing, and determine the layout, details, and speed of blackboard writing flexibly according to the teaching content, teaching time, and classroom conditions, so as to improve their mathematical blackboard writing ability. The correct use of mathematical language and blackboard writing will not be detailed here as they have been specifically introduced in Sect. 13.2, Chap. 7.

13.2.5 Basic Skill of Organizing Teaching Classroom teaching is a process of collective activity of teachers and students and also a controllable process. In order to achieve the optimal teaching effect, teachers must have strong basic skill in organizing classroom teaching. Attention should be paid to the following aspects in specific practice. Teachers should be able to take the appropriate teaching steps flexibly, arrange the teaching structure reasonably, and arrange the time scientifically according to the lesson type, teaching content, and students’ actual situation. Teachers should be good at controlling the process of classroom teaching, pay attention to having a good beginning and artistic end, make full use of the law of alternating voluntary attention and involuntary attention, and pay attention to adjusting students’ attentiveness at any time, to enable students to feel vivid, interesting, and not tired. Teachers should be particular about the art of gesture language, pay attention to the expression role of correct use of eye language, facial expressions, and gestures. The gaze and look should be cordial and natural, full of trust and expectation for students. The facial expression should be gentle, elegant, and solemn with smiles.

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Teachers should pay attention to observing the situation of students at all times. For those unpredictable events caused by natural or human factors, teachers should not only be good at detecting, but also take effective measures to deal with them according to the circumstances. Teachers should be aware of the need to create a good classroom atmosphere and to promote democratic teaching, encouraging students to express their different opinions to create a situation of psychological compatibility between teacher and students. In order to have good basic skill in organizing teaching, teachers are required to cultivate and improve their organizing and leadership ability gradually, go deep into the class to have a thorough knowledge of students’ learning situation, and absorb the methods and experience of their peers in controlling classroom teaching through observation. Among them, it is very important to strengthen lesson preparation, master textbooks, and use appropriate teaching methods. The above is a brief discussion of five basic skills of classroom teaching of secondary school mathematics, and these are essential for an excellent mathematics teacher in secondary schools. Admittedly, each of these aspects has its own emphasis, but they are an integrated whole, neither can be neglected, and will have a direct impact on the quality of mathematics teaching in secondary schools. In order to be competent for mathematics teaching in secondary schools and strive to improve the quality of classroom teaching, it is necessary to practice the basic skill of teaching well. For future mathematics teachers in secondary schools, it is of great importance to advocate practicing basic skill frequently, early, and better.

13.3 Teaching Practice of Secondary School Mathematics As an important part of higher normal education, teaching practice is a comprehensive practice course with normal education characteristics and a special educational and teaching practice activity carried out by normal colleges to realize the training objectives. It has the characteristics of teacher training, duality, comprehensiveness, and practicality. Whether normal college students can complete the teaching practice task successfully and meet the basic requirements for teaching practice is of great significance for laying the foundation for their future work.

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13.3.1 Purpose of Mathematics Teaching Practice As a comprehensive practical activity, teaching practice is an important professional practice course in normal colleges, whose purpose can be summarized as follows: 1. To consolidate the professional thinking of normal college students and to cultivate in them the spirit of conscientious dedication to the cause of socialist education through teaching practice; 2. To test the specialized knowledge and basic skills acquired by normal college students through teaching practice, and to further develop the ability to analyze and solve problems and the ability to work independently in teaching practice; 3. To understand and become familiar with secondary school mathematics education, teaching, and management, to enhance normal college students’ adaptability to teaching in secondary schools, and to shorten the distance from students to teachers through teaching practice; 4. To review the educational philosophy, training objectives, standards, and teaching quality of normal colleges, and to promote the reform of teaching in normal colleges through teaching practice.

13.3.2 Tasks of Mathematics Teaching Practice For secondary school mathematics teaching practice, students must undertake classroom teaching, extracurricular work, class teacher work, educational survey, etc., be familiar with the whole process of mathematics education in secondary schools, do a good job in teaching, thinking, and management work, and improve the corresponding abilities gradually in practice. 1. Classroom teaching Classroom teaching is the main form of learning for students and the main task of secondary school mathematics teaching practice. Through the teaching practice, normal college students further study the curriculum standard and existing textbooks and further understand the teaching contents and teaching requirements of current secondary school mathematics. Through lesson observation and with the help of advisers, they initially master the teaching methods and skills required for classroom teaching, have the ability to dig into textbooks, divide class hours, and prepare teaching plans; and have the skills to organize teaching, implement teaching plans, and impart knowledge to students, have the ability to educate people, strengthen the ideological education, and cultivate the mathematical accomplishment; and have the ability to master the whole teaching process by giving lessons, tutoring,

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commenting, correcting homework, performance evaluation, and extracurricular activities of mathematics. 2. Extracurricular work Extracurricular education is another important channel for students to acquire knowledge and is the deepening of and supplement to in-class education. The main extracurricular work in secondary schools at present includes: ➀ Extracurricular tutoring is mainly aimed at students who have difficulty in learning and those who have the ability to learn more, respectively; ➁ Conducting mathematical lectures, mathematical competitions, mathematical games, mathematical parties, and other activities as a supplement and extension of classroom teaching to further broaden students’ knowledge; ➂ Guiding students’ interest groups of mathematics, and opening up mathematical wall newspapers, to cultivate the students’ ability to apply what they have learned and develop their innovative spirit; ➃ Assisting in the establishment of various science and technology groups, art groups, sports groups, to cultivate students’ interest and improve their intelligence, accomplishment, and physical fitness. Normal college students should gradually master the basic methods and skills of organizing extracurricular education by guiding extracurricular education activities and constantly explore the general rules of extracurricular education. At the same time, they can also steel and improve themselves in participating in activities. 3. Class work The class teacher work practice is an important part of teaching practice and an important way to cultivate the ability of educational work of normal college students. The class teacher practice enables interns to deepen their understanding of the significance of the class teacher work, establish the awareness of becoming an excellent class teacher, further understand and master the basic rules of secondary school education, explore new models of education and teaching management, cultivate the ability to manage classes, and develop a close relationship between teachers and students. 4. Educational survey Educational survey is one aspect of educational practice for normal college students, and it is also one of the teachers’ basic vocational skills that interns must master. By conducting educational survey, interns can deepen their understanding of the Party’s educational policy and the current state of education. Through drawing up plans, holding survey meetings, interviewing, writing survey reports, etc., interns can learn how to conduct a survey, improve their ability to teach and research specialized courses, and also provide basis for schools to make education and teaching development plans and improve teaching work.

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13.3.3 Process of Mathematics Teaching Practice 1. Preparatory stage Under the school’s unified arrangement for practice work, the more important thing for normal college students is to make preparations in professional knowledge, textbooks, and teaching methods and strengthen the training of basic teaching skills in addition to making mental and material preparations for teaching practice. Generally, the schools also take two major measures: teaching probation observation and trial teaching training. (1) Teaching probation observation Teaching probation observation is to go to a secondary school to observe an experienced teacher’s lesson. Through the teacher’s demonstration teaching, they can be familiar with the classroom teaching links and understand the methods and skills of organizing classroom teaching, giving lessons, questioning, doing exercises, and consolidation. At the same time, when conditions permit, students should be organized to listen to the introduction of the current secondary school teaching, so as to have a more comprehensive understanding of secondary school education and teaching situation. (2) Trial teaching training Trial teaching training refers to teaching training for educational practice contents under the guidance of didactics teachers. Its purpose is to train students how to analyze textbooks, prepare lessons, prepare teaching plans, organize teaching, give lessons, etc. and further improve their teaching skills by practicing courage, expression, teaching methods, and blackboard writing. When conditions permit, microteaching can also be used to make students feel about themselves and carry out targeted training. 2. Implementation stage After the intern enters the practice school, he/she usually first gets to know the teaching progress and arrangement of the practice class and the learning situation of the students in the class through the introduction of the adviser. Then, on the one hand, he/she observes the lessons and carefully learns the teaching experience and teaching methods of the adviser; on the other hand, he/she assists the adviser and the class teacher in carrying out extracurricular activities, especially strengthening extracurricular tutoring and correcting homework. After 1–2 weeks, the intern should conduct independent lesson preparation and trial teaching with the help of the adviser and begin to undertake teaching tasks when conditions mature. When conditions permit, he/she can also observe the adviser’s demonstration teaching first and then teach lessons in another class. At the same time, he/she must strengthen summarizing after class and listen to the adviser’s instructions carefully. Here, careful and repeated preparation for the first lesson, timely classroom review, and intensive tutoring of students after class are especially important.

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During the practice, the intern should take the initiative to assist the class teacher in carrying out class work and organize class theme activities to the best of his/her ability. In the mid- to late period of the practice, with the approval from the adviser, he/she may choose to observe mathematics lessons of the same grade or different grades, so as to have a broader understanding of the secondary school teaching situation and learn from secondary school teaching experience comprehensively. 3. Summary stage Teaching practice summary refers to a comprehensive review and specific analysis of the completion of various tasks in the process of teaching practice to affirm the achievements and examine the deficiencies and lay a solid foundation for further improvement in the future. In this regard, interns must do a good job of summing up the work. For interns, there are two tasks: personal summary (fill out the personal internship summary form) and completing the educational survey report (if an educational survey has been carried out). When necessary, with the adviser’s help, a teaching summary should also be carried out to quantitatively evaluate the teaching work and class work assumed.

13.3.4 Some Points for Attention During Teaching Practice 1. It is necessary to attach great importance to teaching practice in guiding philosophy. Teaching practice is an important part of normal education and a necessary way for normal college students to obtain the job qualification. Normal college students must attach great importance to teaching practice, try their best to complete various teaching practice tasks, and strive to make remarkable progress in teaching practice. 2. Normal college students must complete all practice tasks actively, conscientiously, and completely. There are four basic tasks in teaching practice: classroom teaching, extracurricular work, class teacher’s work, and educational survey. The one-sided phenomenon of emphasizing classroom teaching practice and neglecting extracurricular work practice and emphasizing theory teaching practice and neglecting class teacher’s work practice should be corrected. 3. The normal college student should play a dual role as a good teacher and a good student. Acting as a teacher before the students, the intern should be a model of virtue for others and handle the relationship between teaching and guidance well. However, acting as a student before the adviser and the peers, the intern should be modest and studious and accept guidance humbly. 4. Normal college students should pay attention to the relationship between various practice tasks and be in a tense and orderly state from beginning to end. The teaching practice is short, with many contents and high requirements. Therefore, it should be arranged in sequence, and with priorities, overall planning and reasonable arrangement. Especially for the students carrying out independent

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practice, it is necessary to prevent a tight schedule in early period and a loose schedule in late period and the phenomenon of caring for this and losing that. 5. Normal college students should not only focus on the practice tasks undertaken by themselves, but also have an understanding of the whole teaching situation. An outstanding intern can not only accomplish his/her own practice tasks satisfactorily, but also have a deeper understanding of secondary school teaching and management through practice. Only by fanning out from one point to an area and overall improvement can make them become an excellent educator.

13.4 Writing of Papers on Secondary School Mathematics Education Writing papers on mathematics education is an indispensable part of mathematics teaching research. By writing papers on mathematics education, teachers can not only summarize their own research results or teaching experience in time to inspire others, but also further improve their mathematical accomplishment and teaching ability by referring to a large number of documents and carrying out certain discussions and research, in order to promote the improvement of teaching quality. Therefore, teachers in secondary schools have traditionally attached greater importance to the writing of papers on mathematics education, and the publication of papers has been regarded as a demonstration of their mathematics accomplishment and research ability, an important symbol to measure the quantity and quality of their completed research work, and a necessary condition for professional promotion. Writing a paper on education is a creative labor process. Due to different nature, subjects, and objects of secondary school mathematics education papers, there are differences in the structure, style, writing methods, and skills. The following is a brief introduction to their common issues.

13.4.1 Determination of the Subject of Papers on Mathematics Education To write a paper on mathematics education, we must first determine the subject. For optional subjects, we can start with the following six aspects. (1) Mining from the teaching work itself Combining with the practical work, we can mine subjects from the teaching work itself, with many sources and benefits. First, there are rich sources of subjects, so there is large room for selection. Second, the research results can directly serve the teaching work and promote teaching through scientific research. Third, the research conditions can be guaranteed, because teaching research and lesson preparation are often combined together. In order to prepare lessons well, we have to consider the

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determination of teaching contents, selection of teaching methods, use of teaching means, and the need to answer students’ questions. This requires us to learn how to mine and be good at mining subjects and be good at solving them through research. (2) Breaking down “planned” subjects The state, provinces, and municipalities announce the planned subjects on education and scientific research in different periods, such as the recent researches on innovative education, researches on the reform of new curriculums and new textbooks, researches on new teaching forms and means, and researches on learning and teaching methods. These subjects are often the first- and second-level research subjects. We can break down them in combination with our actual work, for example, select a subject for a certain type of schools, a certain grade, and some textbooks, to carry out the research on innovative education, curriculum reform, teaching and learning methods, etc. in depth and in detail. (3) Extracting from practical problems As an abstract science, mathematics is the foundation of science and technology and also a widely used science. As mathematics educators, we should have a strong sense of mathematics. While keeping a close eye on daily production and life and related things actively, we should be good at extracting mathematical models from practical problems and strive to solve them through certain research. If we can persist in doing so, it will be of great benefit to cultivate our mathematics application consciousness and improve the ability to analyze and solve problems. (4) Participating in discussion of hot issues At present, there are many hot issues under discussion in secondary school mathematics education. Newspapers and magazines often start special columns to call on people to participate in the research and discussion. There are macro- and microissues, discussions on educational viewpoints and educational ideas, teaching of a certain knowledge point, and application of problem-solving methods. If we choose one or two of them to participate in the discussion based on our own practical work and research foundation, it is also a channel of subject selection. (5) Paying attention to comprehensive application of disciplines Transplantation method is a common method used in scientific research subject selection. It is to transplant the research methods of one or several disciplines to the research of another discipline to form a new subject. This is especially true of mathematics and its teaching research. Shapes and numbers, elementary mathematics and advanced mathematics, and mathematics and other disciplines are often interpenetrating and interdependent. We should pay attention to subject selection from the comprehensive application of disciplines. (6) Carrying out research on macro-issues People usually start with micro-problems when they carry out educational researches. But after accumulating certain experience and possessing certain data, we may as well

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consider the discussion on macro-issues, including an overview and evaluation of macro-researches and a systematic research on a specific issue in theory and practice. Only by going from the micro- to the macro- and combining them organically can our research develop to a higher level. Generally speaking, it is advisable for young teachers who start their career to select subjects from the first three aspects above, especially the first aspect, on the basis of mastering the knowledge and methods related to education research. The process of delving into the syllabus and textbooks and discussing the teaching means and methods can help them to master the knowledge system of textbooks and improve their teaching ability, so as to be competent for teaching work. For teachers who have a certain research foundation and have been teaching for many years, it is appropriate to select subjects from the last three aspects above, so that they can make full use of their teaching experience and possession of more materials to make new breakthroughs in research.

13.4.2 Components of a Paper on Mathematics Education In general, a research paper on mathematics education consists of the following main parts. 1. Title The title is also known as the heading and general title. The paper title should not only summarize the central content of the whole paper and grasp the basic argument and idea of the paper, but also be compelling, so that the readers can preliminarily estimate it is worth reading or not therefrom. Therefore, its wording should be precise, appropriate, clear, and concise, generally not exceeding 20 Chinese characters. Sometimes, in order to express the main content more fully, extend the theme, or to explain a fact in the title, a subtitle can be added to the title to facilitate the publication of a series of papers. 2. Authorship A paper must be signed with the author’s real work unit and name, which not only shows the author’s serious attitude toward sole responsibility for the paper, but also reflects the attribution of the research results, indicating the author’s unit and the author own intellectual property rights of the paper. 3. Abstract The abstract is a short statement of the paper content without annotations or comments, and it is a miniature of the basic idea of the paper and can serve as a brief introduction to the paper. It generally includes a high degree of “concentration” of the significance, objectives, methods, results, and conclusions of the subject research, to enable the readers to have a general understanding of the main content and results of the paper after reading the abstract in a short time, so as to absorb

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information and provide convenience for the compilation of secondary literature. The abstract is an objective introduction to the paper without comments and can make an independent essay, generally not more than 300 Chinese characters. 4. Keywords Keywords are words extracted from the title, abstract, and text of the paper, which have practical meanings to describe the subject content, and are used to index the subject content characteristics. Because these words with practical meanings play a key role in retrieving this paper, they are called keywords. It is produced to meet the need of catalogue indexing process automation and is extensively used in computer information retrieval. There should be no more than eight keyword phrases. 5. Foreword As also known as the introduction and preface, the foreword is the opening remarks of the paper, generally including the background of the subject research, practical significance, and value of researching this subject. Its wording should be concise, simple, factual, and accurate. 6. Text The text of the paper is the main body and core of the whole paper. It reflects the quality and academic level of an academic paper. The text must make a comprehensive exposition and demonstration of the research content, including the materials observed, tested, surveyed, and analyzed in the whole research process, as well as the views and theories formed by these materials. The thesis, arguments, and demonstration in the paper should be fully displayed in the text. In order to make the exposition well-organized, the text is generally divided into several sections, and each section should have a title. The basic requirements for writing the text are having materials, viewpoints, and expositions, with clear concepts, explicit thesis, sufficient arguments, rigorous, and logical demonstration, without scientific errors; with clear, smooth, and fluent presentation, using accurate, distinct, and vivid words and expressions. 7. Conclusion and discussion The conclusion part is the summary of the whole paper. It reflects the author’s research results, expresses the author’s views and opinions on the research subject, and plays a role of bringing out the crucial point for the whole paper. It enables readers to have a general understanding of the paper and master its core ideas only through the conclusion, so as to preliminarily determine the value of this paper. The conclusion part can summarize what issues have been studied, what achievements have been made, how to conduct in-depth research on existing problems, what problems remain to be solved, etc. The conclusion must play the role of summarizing the full text, deepening the theme and revealing the law. At the same time, the conclusion must

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be written discreetly, with rigorous, concise and specific wording, strict logic, clear sentences, and echoing each other in the beginning and at the end. 8. Appendix and references The appendix refers to some important materials which are not convenient to be incorporated into the text because of its content and length but must be explained clearly to the readers. The main reason is that some of the contents have not been given full expression, and they will affect the prominence of the text if included in the text; therefore, they are included in the last part of the paper as a supplementary appendix. It mainly includes the symposium outlines, questionnaire forms, test questions, scoring standards, various charts, photographs, etc. The references refer to the contents of the books, periodicals, and materials cited by the author in the process of writing a paper, including the materials, data, theses, etc. referenced or directly quoted, whose references must be given in the paper, such as Chinese and foreign books, periodicals, academic reports, dissertations, scientific and technological reports, patents, and technical standards. Giving references reflects the serious scientific attitude of the author, the scientific basis of the paper, and the author’s noble writing style of respecting the research results of predecessors and others, and provides readers or peers with some literature or materials that can be referred to in the study of similar issues. The source of the listed references must be accurate and the format is as follows: [Book] Main editors. Title. Edition. Place of publication: Publisher, Year of publication. [Journal] Author’s name. Title. Periodical Name, Year of Publication, Volume (Issue): Page number.

13.4.3 Process of Writing a Paper on Mathematics Education 1. Selection of subject and theme The author should first have a clear idea about whether your paper is about theoretical discussion, textbooks and teaching methods, problem solving methods and skills, teaching experience summary, contention and review, the depth and breadth of the exposition, etc. with a clear purpose and theme. No matter what content and specific theme are selected, we must strive for its advancedness, pertinence, and practicality. To this end, we should not set out to write the paper immediately after we come up with an idea, but once we have the idea, we should, according to the document retrieval method, read as many books as possible, possess materials, and keep abreast of the latest research trends at home and abroad, that is, look before you leap. Do a thorough survey to understand whether similar papers have been published or not, what are their contents and characteristics, what achievements have been made, what detours have been made, what journals they have been published in, etc. Only when we understand these conditions first and

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then carry on the scientific research design, can we avoid detours and highlight the characteristics of subject to increase its academic value. It would certainly be heartening to discover that this is a subject that has not been explored by others after consulting reference materials. In this case, it is necessary to obtain information in a larger scope, probe into and study it seriously, and further verify and analyze the achievements we have made after calmly analyzing why this is a “gap”. If it is really a breakthrough, we should have the courage to proceed with it. If this is an old subject, which have been discussed and expounded by many people after consulting reference materials, we should not negate it easily and lose confidence. After delving into these materials, we ponder over whether we can get further inspiration, whether there are new opinions, whether it is necessary to write a review, and whether it is necessary for a further discussion. In fact, most subjects of papers on secondary school mathematics education are repeated, which is nothing new now. The question is whether we can write something new to make the views clearer, the methods more effective and make it more advanced, pertinent, and practical from different aspects in the similar theme, in combination with different examples, and according to the needs of different objects. In short, the selection of subject and theme should be based on reality. The size of the subject and the depth and breadth of the theme should be appropriate and must have new ideas, opinions or methods. Don’t select subject arbitrarily and write whatever you want, which often leads to failure. It usually takes a long time to think about and give birth to a good subject, and it is the key to the birth of a good research paper on school mathematics education. For those who aspire to write papers and lack experience, please do not rush for quick results, but make more efforts to possess materials and select subjects and themes. 2. Drawing up outline and writing After determining the subject and theme, how to do the actual writing? This requires certain methods and skills, as well as writing ability. The first step is to lay out the content and structure, which is the same as writing an ordinary article. We should first make an outline: how many parts are included, what problems are introduced in each part, and how do these parts relate to each other, all of which need to be designed meticulously so that they are well-structured, well-arranged, scientific, and logical. Preparing a good outline is the foundation for writing a good paper. Of course, in the specific writing process, we often need to make some revisions and additions to the prepared outline. But preparing an outline is indispensable. Once the outline has been drawn up and the material has been organized, we should “make hay while the sun shines” if time permits, losing no time to complete the paper at one fling. Secondly, we should pay attention to the characteristics of all kinds of papers. For example, for theoretical papers, it is better to be able to determine the title and subtitles, with clear thesis and sufficient arguments. For papers related to experience, the experience provided should be specific, reliable, and easy for others to learn

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from. For papers related to analysis of textbooks, there should be comparison, with suggestions for improvement or subjects worthy of in-depth research, etc. 3. Revision and finalization Revision is a process after the first draft of the paper is completed. It includes the revision of text, the verification of materials, the deliberation of scientificity, and so on. After the first draft of the paper is formed, the author should read it from beginning to end repeatedly and deliberate word by word to check whether the thesis is clear, the argument is sufficient, the demonstration is reasonable, the structure is rigorous, and the calculation is erroneous, etc. A good paper on mathematics education should have both good mathematical content and good verbal expression. Therefore, writing ability is also very important for mathematical papers. Mathematical papers should be simple, with few pompous words, so as not to weaken the main idea of the paper. The text should be easy to understand and concise, generally about four or five thousand words. The narration should be accurate, brief, complete, and standardized. In particular, the central content must be clearly stated. In addition, the writing should be standardized, using standard simplified characters. The title number, table number, figure number, graphics, and punctuation should be correct, and the handwriting should be neat and clear, it should be preferably printed on special manuscript paper, and stored in the backup floppy disk. The format of writing (numbers and formula) is especially important here. Many beginners’ writing fail to meet the requirements. It is desirable that beginners get inspiration from the classical works and papers published in official journals. Do not scribble any alterations, which should be recopied or patched appropriately. The accompanying drawings in the original manuscript should preferably be drawn with a plotter formally. If this is not possible, formal accompanying drawings can also be drawn in the traditional way, with the appropriate text or letters for reference during type setting. Revision is a delicate task. Only by deliberating the text repeatedly, can we stop up some loopholes and reduce unnecessary mistakes. After the first draft is completed, if possible, we can ask for comments within a certain range and invite experienced colleagues to review or ask peers to put forward suggestions for revision from different perspectives and then make detailed revisions. If there are no conditions at the moment, we can also “cool” the written manuscript, that is, put it aside for a few days, and then take it out to revise and finalize it by ourselves. In this way, it is easier to find problems and revise it without difficulty. Only through repeated revision and processing, can the quality of the paper be improved continuously, which is especially important for beginners. The above is an introduction to the writing of papers on mathematics education. For general mathematics research papers, especially elementary mathematics research papers, the components of a paper are relatively simple, usually consisting

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of only the title, authorship, abstract, keywords, text, references, etc., or even sometimes of only the title, authorship, and text, and its writing process is also relatively simple, which will not be introduced here. Review Questions and Exercises (XII) 1. What is the significance of studying mathematics teaching skills in secondary schools, what are the main aspects of mathematics teaching skills in secondary schools, and how to train them? 2. What is the significance of studying the basic skills of secondary school mathematics teaching, and what are these basic skills? 3. What is the purpose, significance, and task of secondary school mathematics education practice, and what are the main work and points for attention during the practice stage? 4. Make an analysis of your teaching skills, and make a training plan. 5. Write out five different types of subject lead-in examples, and conduct the explanation rehearsal. 6. Write out five different types of subject blackboard writing and drawing plans, and carry out specific rehearsal. 7. What is the significance of writing papers on secondary school mathematics education, and how to determine the research subject? 8. What are the parts of research papers on mathematics education, and what are the specific requirements for each part? 9. What is the writing process of papers on mathematics education, and what problems should be paid attention to? 10. Select a subject, draw up an outline of a research paper on mathematics education, and preliminarily complete the writing of the paper.

Appendix A

Examples of Teaching Plan of Secondary School Mathematics

I Subject: Factorization (Lesson 1) Teaching objectives Cognitive objectives: To understand the concept and meaning of factorization, to understand the correlation between factorization and integral multiplication, that is, reverse deformation, and to seek the methods of factorization by using their correlation. Ability objectives: To explore the way to solve problems by students themselves, cultivate students’ observation, analysis, judgement, and innovation ability; develop students’ intelligence; and deepen students’ reverse thinking ability and comprehensive application ability. Emotional objectives: To cultivate students’ spirit of independent thinking and exploration and the scientific attitude of seeking truth from facts from the perspective of the unity of opposites of contradictions. Focal point and difficult point: The focal point is the concept of factorization, and the difficult point is the understanding of the relationship between integral multiplication and factorization. Teaching methods: The questioning-inquiry teaching method is adopted, with questioning–perception–generalization–application as the teaching procedure, fully following students’ cognitive rules, so that they can master the focal points, break through the difficult points, and improve their ability. At the same time, students actively participate in teaching practice by using their brains, hands, and language, to fully arouse their enthusiasm for learning.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3

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Teaching process I Asking Questions to Create Situations Question: Who is the fastest calculator? (Questions displayed by the computer) ➀ If a = 101, b = 99, then a 2 − b2 = (a + b)(a − b) = (101 + 99)(101 − 99) = 400; ➁ If a = 99, b = −1, then a 2 − 2ab + b2 = (a − b)2 = (99 + 1)2 = 10000; ➂ If x = −3, then 20x 2 + 60x = 20x(x + 3) = 20(−3)(−3 + 3) = 0. II. Exploring New Knowledge Through Observation and Analysis ➀ Ask the fastest calculator to explain his/her train of thought to obtain the best way to solve the problems (at the same time, the computer displays the answer). ➁ Observe the formulas on the left and right sides of the following equalities. What forms are they in?

➂ Draw an analogy between them and the concept of integer factorization leaned in primary schools to introduce the concept of factorization. Write the subject on the blackboard: §7.1 Factorization. Concept of factorization: The conversion of a polynomial into the product of several integral expressions is called factorization, also known as resolving into factors. III Independent Practice to Consolidate New Knowledge (1) Which of the following transformations from left to right are factorizations, which are not, and why? (Computer demonstration). ➀ ➁ ➂ ➃ ➄ ➅ ➆

(x + 2)(x − 2) = x 2 − 4; x 2 − 4 = (x + 2)(x − 2); a 2 − 2ab + b2 = (a − b)2 ; 3a(a + 2) = 3a 2 + 6a; 3a 2 + 6a = 3a(a + 2); x 2 − 4 + 3x = (x − 2)(x + 2) + 3x; 18a 3 bc = 3a 2 b · 6ac;

(2) Relationship between factorization and integral multiplication: Factorization → a 2 − b2 = (a + b)(a − b) ← Integral multiplication

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Note: The process from left to right is factorization, which is characterized by the conversion from the form of sum and difference (polynomial) into the form of integral product. The process from right to left is integral multiplication, which is characterized by the conversion from the form of integral product into the form of sum and difference (polynomial). Conclusion: Factorization is the opposite of integral multiplication. Question: Can you give some examples of factorization taking advantage of the fact that factorization is the opposite of integral multiplication? For example: From (x + 1)(x − 1) = x 2 − 1, we get x 2 − 1 = (x + 1)(x − 1); From (x + 2)(x − 1) = x 2 + x − 2, we get x 2 + x − 2 = (x + 2)(x − 1). IV Example Teaching to Apply New Knowledge Example 1 Factorize the following formulas: (computer demonstration) (1) am + bm; (2) a 2 − 9; (3)a 2 + 2ab + b2 ; (4) 2ab − a 2 − b2 . Example 2 Fill in the blanks (computer demonstration): (1) Since 2x y() = 2x 2 y − 6x y 2 , then 2x 2 y − 6x y 2 = 2x y(); (2) Since x y() = 2x 2 y − 5x y 2 , then 2x 2 y − 5x y 2 = x y(); (3) Since 2x() = 2x 2 y − 6x y 2 , then 2x 2 y − 6x y 2 = 2x(); Example 3 Evaluate: (1) If x 2 + mx − n can be factorized into (x − 2)(x − 5), then m =, n =; (2) If x 2 − 8x + m = (x − 4)(), then m = . 5. Review and summary to form the structure ➀ The concept of factorization: Factorization is an identical transformation of integral expressions. ➁ Factorization and integral multiplication are two opposite identical transformations and also two ways of thinking in opposite directions. Therefore, the thinking process of factorization is the reverse thinking process of integral multiplication. ➂ By taking advantage of the relations in (2), we can search the result of factorization from integral multiplication. ➃ The dialectical materialist thinking method of unity of opposites and maintaining the status quo is penetrated in teaching.

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6. Reading and homework in class (omitted) II Subject: Curve and Equation (Lesson 1) Teaching Objectives To enable students to understand the relationship between curve and equation to lay a theoretical foundation for finding the equations of known curves. To further train students’ logical reasoning ability and abstract thinking ability and receive materialist ideological education from the combination of shape and number. Lesson type: New lesson Teaching method: Explanation method Teaching process: I Review and Questioning 1. What are the basic forms of propositions and how are they related? (Blackboard writing: The original proposition is equivalent to the conversenegative proposition, and the converse proposition is equivalent to the negative proposition.) 2. What is locus of points, and what is the relationship between the locus of points and the conditions met by the locus of points? Blackboard writing: Locus ⇔ Condition

3. In quadrants i and iii, what is the locus of points that have the equal distance from two axes? What are the conditions for its locus? Blackboard writing (Fig. A.1 in appendix). Angular bisector of quadrants I and III l ⇔ equal x-coordinates and y-coordinate s of points (x = y). Locus (curve) ⇔ Condition (equation) In analytic geometry, the locus (set of points) is usually called a curve, and the condition met by each point on the locus is usually expressed by an equation. Therefore, what should be the corresponding relationship between “locus and condition” in middle school geometry? This is the subject for this lesson. (Blackboard writing: Subject “Curve and Equation”).

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Fig. A.1 In appendix

Fig. A.2 In appendix

II Teaching the New Lesson Example 1 Given two points A(−2, −1) and B(3, 5).What conditions are met by the coordinates of the points on the perpendicular bisector of line segment AB? Solution As shown in Fig. A.2 in appendix, let the moving point be P(x, y), then from the known conditions, we get

that is,

square both sides, and we get

Simplify it, we get

Equation ➃ here is the conditions satisfied by the coordinates of the points on the perpendicular bisector of AB. In fact, we can find that: (1) The coordinates of arbitrary point P on the perpendicular bisector of AB are equidistant from two points A and B, it must satisfy condition ➀, that is, the

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Fig. A.3 In appendix

coordinates of point P are the solutions to Equation ➁ and must also be the solutions to Equation ➃. (2) A set of solutions to Equation ➃ is also solutions to Equations ➂ and ➁ respectively, that is, satisfying condition ➀, point P with coordinates (x, y) must be on the perpendicular bisector of AB. In other words, the coordinates of the points on the perpendicular bisector of AB are solutions to Equation ➃; and vice versa, the points with the solutions to Equation ➃ as the coordinates are all on the perpendicular bisector of AB. Example 2: The slope of the connecting line between a moving point and the origin is equal to its distance from the y-axis. Find the condition satisfied by this moving point. Solution As shown in Fig. A.3 in appendix, let the moving point be P(x, y). From the known conditions, we get

that is,

simplify ➀ and ➁, and we get

You might think that equation ➂ is the condition the moving point satisfies. In fact, from ➀ → ➁ → ➂, the coordinates of the points on the curve are all solutions of equation ➂. However, ➀ cannot be derived from ➂ through backward inference, that is, the points whose coordinates satisfy Equation ➂ are not necessarily on the curve. For example, (−2, 4), (2, −4), and other points are not on the curve. In other words, the coordinates of all points on the curve are solutions to Equation ➂, whereas points whose coordinates are solutions to Equation ➂ are not necessarily on the curve.

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Fig. A.4 In appendix

In this way, after conversion of the relationship from locus ⇔ condition to “curve ⇔ equation”, there should be the following corresponding relationship between the points on the curve and the solution of some binary equation f (x, y) = 0:  1. The coordinates of points on the curve are all solutions to this equation. 2. The points with the solution of this equation as coordinates are all points on the curve.

Then this equation is called the equation of the curve, and this curve is called the curve of the equation. Therefore, Eq. ➃ in Example 1 is the equation of perpendicular bisector of AB, which is the curve represented by Eq. ➃. Equation ➂ in Example 2 is not the equation satisfied by the moving point; that is, the curve shown in Fig. A.3 (dotted lines) is not the curve represented by Eq. ➂. Only equation y 2 = x 4 (x y > 0), namely the curve shown in Fig. A.3 (solid lines), is the condition (equation of locus) satisfied by the moving point. Since a converse proposition is equivalent to its negative proposition, the above definition can also be changed to: 

1. The coordinates of all points on the curve are solutions to this equation; 2. The coordinates of a point that is not on the curve are not a solution to this equation.

III Practice (1) Describe the curves represented by the following equations. (Students are required to answer orally.) ➀ x = a; ➁ y = b. (2) Determine whether the following two points are on the curve represented by equation x 2 + y 2 = 25, and why? (Students are required to answer orally.) √ P1 (3, −4), P2 −2 5, 2 . (3) Prove that the equation of a circle of radius R with C(a, b) as its center is (x − a)2 + (y − b)2 = R 2 . Proof (Inspired by the teacher, students and the teacher complete the proof together, while the teacher writes on the blackboard simultaneously.)

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➀ As shown in Fig. A.4 in appendix, let a point on circle C be P(x1 , y1 ), then |PC| = R,  that is, (x − a)2 + (y − b)2 = R, square both sides, we get (x − a)2 + (y − b)2 = R 2 , that is, the coordinates of point P (x1 , y1 ) is the solution of the given equation. Figure A.4 in appendix ➁ Let a set of solutions of the known equation be (x1 , y1 ), i.e., (x1 − a)2 + 2 2 (y1 − b)2 = R 2 , extract the square root, and  we get (x − a) + (y − b) = ±R. 2 2 Since R ≥ 0, then “+” is taken, we get (x − a) + (y − b) = R, which means that the point with (x1 , y1 )as the cordinates is on the circle. Step ➁ can also be changed to:  Let the point which is not on the circle C be Q (x1 , y1 ), then |QC| = R, that is, (x − a)2 + (y − b)2 = R, square both sides, and we get (x − a)2 +(y − b)2 = R 2 , that is, the coordinates of the point which is not on the circle C are not solutions to the known equation. IV Summary Under certain conditions, the moving points form a locus (curve) and the coordinates of the moving points are constrained by an equation. As mentioned above, on the basis of locus, we convert the corresponding relationship between “locus and condition” into corresponding relationship between “curve and equation”. When we say a particular equation is the equation of a curve and a particular curve is the curve represented by an equation, it means that the requirements in the above-mentioned two aspects are all met. Only the requirements in these two aspects are met can study the curves, that is, geometric problems, be converted into the study of equations. This idea of “addressing shapes from the perspective of numbers” is a remarkable feature of this course and a basic method of solving problems. V Homework (1) Read the textbook P.105 − 106; (2) Exercises 17: 1 and 3 on P.112 (3) Reference question. Prove: The equation of locus of the moving point (x, y) whose distance to fixed point M(−a, 0) is equal to a(a > 0) is x 2 + y 2 + 2ax = 0. (Excerpted from the Fifth Volume of Secondary School Mathematics Textbook Research and Selected Teaching Plans by Zhang Shizao and Zhou Jing, Beijing Normal University Press).

Appendix B

Examples of Lesson Presentation of Secondary School Mathematics

I Subject: § 12.3 Relations between roots and coefficients of quadratic equations with one unknown (Class hour 1) I Textbook Analysis This section is taught on the basis of introducing the concept of quadratic equations with one unknown and learning the square root extraction method, factorization method, completing the square method, and formula method of solving quadratic equations with one unknown. The study of this section can not only deepen the understanding of quadratic equations with one unknown and improve the ability of solving quadratic equations with one unknown, but also is of great significance to the study of equations of higher degree and equation theory in the future. The teaching objectives of this section are: Cognitive objectives: ➀ To keep in mind that two roots of equation ax 2 + bx + c = 0(a = 0) have the relations: X 2 − (X 1 + X 2 )X + X 1 X 2 = 0 − ab ,x1 · x2 = ac . ➁ To keep in mind that two roots of equation x 2 + px + q = 0 have the relations: x1 + x2 = − p, (x − a)2 + (y − b)2 = R 2 = q.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3

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Skill objectives: ➀ To be able to prove the relations between roots and coefficients of a quadratic equation with one unknown; ➁ To be able to find the sum and the product of two roots of a given quadratic equation with one unknown correctly; ➂ To be able to find another root and the unknown coefficient when one root of a quadratic equation with one unknown is given Emotional objectives: To enable the students to be able to study always in a full, warm, and happy mood; to conduct observation, analysis, comparison, and summarization according to the teaching requirements; and to think and answer questions with teacher’s inspiration. Focal points: The relations between roots and coefficients of quadratic equations with one unknown and their application. Difficult points: Derivation and flexible application of the relations between roots and coefficients of quadratic equations with one unknown. II Teaching Methods Description of teaching method: The discovery teaching method is used with the teaching objectives as the framework. Instruction of learning method: Teach students to master the observation, generalization, expression, and demonstration methods in the process of acquiring knowledge. Ability cultivation: Cultivate students’ observation, analysis, thinking abilities, and dialectical materialism. Teaching means: The projector is used to increase the classroom capacity and improve the efficiency of practice. III Teaching Process 1. Review and lead-in Solve the equations: x 2 −5x + 6 = 0, x 2 +8x + 7 = 0, x 2 −9x = 0,3x 2 −7x + 2 = 0. ➀ Students are required to choose the appropriate method to find two roots of the above equations correctly and quickly (among them, 4 students demonstrate calculations on the blackboard, and other students solve the problems by themselves). ➁ The teacher and students evaluate the demonstrations on the blackboard jointly.

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➂ Students observe the sum and the product of two roots of each equation and discuss their relations with the coefficients of the equation. 2. Teaching the new lesson Guide students to demonstrate that two roots of ax 2 + bx + c = 0 (a = 0) have the relations x1 + x2 = − ab , x1 · x2 = ac . Simple exercise: Exercise 1.2 on p.48. 3. Comprehensive exercise Example l (p.47) It is known that the root of 5x 2 + kx − 6 = 0 is 2. Find the value of another root and k. (Two solutions). Example 2 (p.48) It is known that 6x 2 − 13x + 6 = 0. Evaluate x12 + x22 , (Two solutions).

1 x1

+

1 . x2

Example 3 (Supplementary) Find a new equation which takes the product of two roots of ax 2 + bx + c = 0 as the sum of two roots and takes the sum of two roots as the product of two roots. 4. Summary The teacher and students summarize together and read the textbook P. 47–49. 5. Homework. IV Points for Attention During Teaching 1. There are many contents in this section. If students have poor receptivity,   x 2 − x1 + x2 x + x1 x2 = 0 can be taught next time. 2. During the review and lead-in, it is necessary to create an atmosphere of students’ flexible problem solving, discussion, and discovery. For this purpose, time should be guaranteed, with appropriate inspiration from the teacher. 3. When discovering and inferring the relations and doing the comprehensive exercises, let students participate in thinking, analysis, and comparison with a free hand. 4. The difficulty of this section is moderate, so it should be based on average students, enabling them to learn actively and easily, so as to truly understand the relevant thinking methods and problem-solving skills.

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Attachment: Blackboard writing design and time allocation. Solve ➂ x2 + 8x + 7 = 0 equations: (students’ demonstration on the blackboard, 16 ) ➀ x 2 −5x +6 = 0 x 2 −5x +6 = 0 (6 ) x1 + x 2 = 5, x1 · x2 = 6; 10x + 12y − 29 = 0 x1 + x2 = −8, x1 · x2 = 7;

➂ x2 − 9x = 0

§12.4 Relations between roots and coefficients of quadratic equations with one unknown (8 ) Suppose that two roots of ax 2 + bx + c = 0 (a = 0) are x1 and x2 , then

Example 1 It is known that the root of 2 √ √ −b + . . . −b − . . . 2b b 5x + x1 + x2 = + =− =− kx − 6 = 2a 2a 2a a 0 is 2. √ √

−b + . . . −b − . . . c Find the = · · · = · x = x · 1 2 x 2 − 9x = 0 2a 2a a value of x1 + x2 = another x 2 + px + q = 0, 9, xi x2 = 0; root and + x = − p, x 2 3x 2 −7x +2 = 1 k (6 ) x1 x2 = q 0 Solution x 2 − 73 x + 23 = 1: 20 + 2k − 6 = 0 0, x1 + x2 = K = −7, 7 2 ,x · x = 1 2 3 3 x1 = − 53 Solution 2: x1 + 2

➂ 3x 2 − 7x + 2 = 0

Example 2: It is known that 6x 2 − 13x+6 = 0. Evaluate x 1 2 + x 2 2, x11 + 1 x 2 (3) Solution: (1) x12 + x22 = (x1 + x2 )2 − 2x1 x2  2 = 13 − 6

2·1= (2)

133 36 ;

1 1 x1 + x2 = x 1 +x 2 x 1 ·x 2 = 13 13 6 1 = 6

= − k5 , 2x 1 = − 6 5, x 1 = − 35 , k = −7; (continued)

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(continued) Example 3 // Find a new equation which takes the product of two roots of ax 2 + bx + c = 0 as the sum of two roots and takes the sum of two roots as the product of two roots (3’) Solution: From ax 2 + bx +c=0 we get x 1 + x 2

//

Summary and homework (Omitted, 3 )

b = - a

x 1 + x 2 = ac The new equation is x2 x 2 − ac x + ab = 0, that is, ax 2 − cx + b = 0

II Subject: § 7.5 Curve and Equation (Class hour 1) I. Textbook Analysis 1. Status and function: The concept of curve and equation is one of the most important concepts in secondary school mathematics. Its understanding and mastering not only affects the study of conic curve directly, but also relates to the study of subsequent courses such as analytic geometry and advanced mathematics. The concept of curve and equation is the theoretical basis of the combination of shape and number. The understanding and mastering of the concept of curve and equation will directly affect the understanding and application of the thinking method of combination of shape and number and will undoubtedly be of great significance to improve mathematical ability. Therefore, this section is of great importance and great effect. 2. Knowledge connection: Students have learned locus, one-variable quadratic functions, and graph in middle schools; have learned exponential, logarithmic, and trigonometric functions in senor secondary schools; and have just learnt

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linear equations in this chapter. These are important basis for the teaching of this section. At the same time, the teaching of this section will create conditions for the subsequent study of conic curve. 3. Teaching objectives ➀ To enable students to preliminarily understand the concept of curve and equation and know the internal relations between curve and equation through teaching; ➁ To enable students to preliminarily master the basic method of using coordinate system to study simple geometric problems through teaching; ➂ To realize the combination of shape and number through the use of the concept of curve and equation, and to enable students to further understand the combination of shape and number and receive ideological education of dialectical materialism. 4. Focal point and difficult point in teaching: The focal point is the understanding of the concept of curve and equation; the difficult point is that students are used to discussing numbers and shapes in terms of number and shape, respectively, but not used to studying curves through equations. II. Teaching Methods By means of heuristic teaching method, the concept of curve and equation is revealed, and the steps and methods of establishing curve equations are explained by recalling the concept of locus and in combination with examples. Teaching aids: Ruler, compass, and blackboard. The projector or multimedia teaching can be used if possible. III Teaching Process 1. Lead-in (6 min) First, locus learned in middle schools Locus (F) ⇔ Condition (A) ➀ Every point on the graph satisfies condition A. ➁ Every point that satisfies condition A is on graph F. Angular bisector of quadrants I and III l ⇔ x–y = 0. 2. New lesson (28 min) Example 1 Given two points A (−2, −1) and B (3, 5). Find the equation of the perpendicular bisector of segment AB. As shown in Fig. B.1 in appendix, let P(x, y). From the given conditions, we get

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387

Fig. B.1 In appendix

Fig. B.2 In appendix

that is,

Simplify it and we get, 10x + 12y − 29 = 0 (simplest equation). Example 2 The slope of the connecting line between a moving point and the origin is equal to its distance from the y-axis. Find the locus of the moving point. Solution As shown in Figure B.2 in appendix, let the moving point be P(x, y) From the known conditions, we have kop = |PS| that is,

y x

= |x| (original equation)

Through equivalent transformation and simplification, we get y2 = x 4 (xy > 0) (simplest equation) The locus is the two curves represented by solid lines in Fig. B.2. The concept of curve and equation: If the points on the curve and the solution of f (x, y) = 0 satisfy that: ➀ the coordinates of the points on the curve are all the solutions of this equation; ➂ the points with the solutions of this equation as coordinates are all on the curve. Then this equation is the equation of the curve, and this curve is the curve represented by this equation. Original equation ⇔ Simplest equation (Equivalent transformation)

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This means that when finding the locus equation, we obtain the original equation from the geometric conditions and then simplify the original equation. We must pay attention to the equivalence of equations. Otherwise, we may obtain only part of solutions, making the curve equation lose its completeness, or may obtain extraneous solutions, making the curve equation lose its purity. Simple exercise: (1) and (2) on P150 3. Consolidation exercise (6 min) Prove that the equation of a circle of radius R with C(a, b) as its center is (x − a)2 + (y − b)2 = R 2 (Example 2 on p.149). 4. Summary (3 min) 5. Homework (2 min). iV Points for Attention During Teaching 1. The description in the textbook is rather generalized and concise, but it is not fully developed. However, it is difficult to teach this important concept, so teachers should attach great importance to it and make preparations meticulously. 2. Different requirements can be adopted according to the actual situation. For top classes, it should be thoroughly explained theoretically (focus on theory and methods), while for average classes, the requirements can be lowered appropriately (focus on methods, not theory). 3. The status and function of the content in this section are expounded in teaching, to enhance students’ enthusiasm for learning and facilitate the connection between knowledge. 4. This section focuses on the understanding of the concept of curve and equation, and there is no need to rush to deal with complex locus problems.

Appendix C

Examples of Secondary School Mathematics Research

I Several Theorems on Symmetric Curve and Surface Equations The research on symmetric curves and surfaces is often used in special cases in geometry. In this paper, several theorems are deduced and generalized to general cases, so that the symmetric curve equations of plane curves, the symmetric surface equations of space surfaces, and the symmetric curve equations of space curves can be easily obtained. Theorem 1 Let a plane curve be F(x, y) = 0, then the symmetric curve equation with regard to fixed point (a, b) is F(2a − x, 2b − y) = 0. Proof Let the symmetric point of any point P1 (x1 , y1 ) on the curve F(x, y) = 0 with regard to fixed point (a, b) be P(x, y), then a = x12+x , b = y12+y , so x1 = 2a − x, y1 = 2b − y. While F(x1 , y1 ) = 0, so the desired symmetric curve equation is F(2a − x, 2b − y) = 0. Theorem 2 Let a plane curve be F(x, y) =  0, then the symmetric curve equation  with 2 A(Ax+By+C) 2B(Ax+By+C) regard to fixed line Ax + By+C = 0 is F x − = 0. ,y− A2 +B 2 A2 +B 2 Proof Let the symmetric point of any point P1 (x1 , y1 ) on the curve F(x, y) = 0 with regard to fixed line Ax + By + C = 0 be P(x, y), and P1 P intersects this line at P0 (x0 , y0 ), then x0 = x12+x , y0 = y12+y , and Ax0 + By0 + C = 0. Therefore,

and

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Zhang, The Pedagogy of Secondary-School Mathematics, https://doi.org/10.1007/978-981-99-1248-3

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k P1 P = that is,

B , A

so

y−y1 x−x1

=

B , A

After solving ➀ and ➁, we get x1 = x − 2 A(Ax+By+C) A2 +B 2 2B(Ax+By+C) . While F(x , y ) = 0, so the desired symmetric 1 1   A2 +B 2 2B(Ax+By+C) = 0. F x − 2 A(Ax+By+C) , y − A2 +B 2 A2 +B 2

and y1 = y − curve equation is

Theorem 3 Let a space surface be F(x, y, z) = 0, then the symmetric surface equation with regard to fixed point (a, b, c) is F(2a − x, 2b − y, 2c − z) = 0. Proof Let the symmetric point of any point P1 (x1 , y1 , z 1 ) on the surface F(x, y, z) = 0 with regard to fixed point (a, b, c) be P(x, y, z), then a=

x1 +x , 2

b=

y1 +y , 2

c=

z 1 +z , 2

therefore, x1 = 2a − x, y1 = 2b − y, and z 1 = 2c − z. While F(x1 , y1 , z 1 ) = 0, so the desired symmetric surface equation is F(2a − x, 2b − y, 2c − z) = 0. Theorem 4 Let a space surface be F(x, y, z) = 0, then the symmetric surface equation with regard to fixed plane Ax + By + C z + D = 0 is

2 A(Ax + By + C z + D) 2B(Ax + By + C z + D) ,y− , 2 2 2 A + B +C A2 + B 2 + C 2

2C(Ax + By + C z + D) =0 z− A2 + B 2 + C 2

F x−

Proof Let the symmetric point of any point P1 (x1 , y1 , z 1 ) on the surface F(x, y, z) = 0 with regard to plane π : Ax + By + C z + D = 0 be P(x, y, z), and P1 P intersects plane π at P0 (x0 , y0 , z 0 ), then x0 =

x1 +x , 2

y0 =

y1 +y , z0 2

=

z 1 +z , 2

and Ax0 + By0 + C z 0 + D = 0, so

And since P1 P⊥ plane π , we obtain

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391

From ➁, we get y1 = y − BA (x − x1 ) and z 1 = z − CA (x − x1 ). Substituting them into ➀ and after simplification, we get x1 = x −

2 A(Ax + By + C z + D) . A2 + B 2 + C 2

z+D) z+D) Similarly, we get y1 = y − 2B(Ax+By+C and z 1 = z − 2C(Ax+By+C . A2 +B 2 +C 2 A2 +B 2 +C 2 While F(x1 , y1 , z 1 ) = 0, so the desired symmetric surface equation is

2B(Ax + By + C z + D) 2 A(Ax + By + C z + D) ,y− , F x− A2 + B 2 + C 2 A2 + B 2 + C 2

2C(Ax + By + C z + D) = 0. z− A2 + B 2 + C 2 By Theorems 3 and 4, similarly we can get.  F1 (x, y, z) = 0 , then the symmetric curve Theorem 5 Let a space curve be (C) : F2 (x, y, z) = 0  F1 (2a − x, 2b − y, 2c − z) = 0 . equation with regard to fixed point (a, b, c) is F2 (2a − x, 2b − y, 2c − z) = 0  F1 (x, y, z) = 0 Theorem 6 Let a space curve be (C) : , then the symmetric curve F2 (x, y, z) = 0 equation with regard to fixed plane Ax + By + C z + D = 0 is ⎧ 2 A(Ax + By + C z + D) 2B(Ax + By + C z + D) ⎪ ⎪ F1 x − ,y− , ⎪ ⎪ 2 + B2 + C 2 ⎪ A A2 + B 2 + C 2 ⎪

⎪ ⎪ 2C(Ax + By + C z + D) ⎪ ⎪ =0 z− ⎨ A2 + B 2 + C 2 2 A(Ax + By + C z + D) 2B(Ax + By + C z + D) ⎪ ⎪ ,y− , ⎪ F2 x − ⎪ 2 + B2 + C 2 ⎪ A A2 + B 2 + C 2 ⎪

⎪ ⎪ 2C(Ax + By + C z + D) ⎪ ⎪ =0 z− ⎩ A2 + B 2 + C 2 At the same time, applying the above theorems, we can obtain the following deductions. Deduction 1 The symmetric curves of plane curve F(x, y) = 0 with regard to the origin are F(−x, −y) = 0, and its symmetric curves with regard to straight lines x = a, y = b, y = x, and y = −x are F(2a − x, y) = 0,F(x, 2b − y) = 0, F(y, x) = 0, and F(−y, −x) = 0. Specifically, the symmetric curves of plane curve F(x, y) = 0 with regard to the x-axis and y-axis are F(x, −y) = 0 and F(−x, y) = 0, respectively.

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Deduction 2 The symmetric surfaces of space surface F(x, y, z) = 0 with regard to the origin are F(−x, −y, −z) = 0, and its symmetric surfaces with regard to planes x = a, y = b, z = c, x = y, y = z, and z = x are F(2a − x, y, z) = 0, F(x, 2b − y, z) = 0, F(x, y, 2c − z) = 0, F(y, x, z) = 0, F(x, z, y) = 0, and F(z, y, x) = 0, respectively. Specifically, the symmetric surfaces of space surface F(x, y, z) = 0 with regard to coordinate planes x O y, y Oz, and z O x are F(x, y, −z) = 0,F(−x, y, z) = 0, and F(x, −y, z) = 0.  F1 (x, y, z) = 0 with Deduction 3 The symmetric curves of space curve F2 (x, y, z) = 0  F1 (−x, −y, −z) = 0 , and its symmetric curves with regard to the origin are F2 (−x, −y, −z) = 0 regard  to places x = a,y = b, z = c, x = y, y = z, and z = F1 (2a − x, y, z) = 0 F1 (x, 2b − y, z) = 0 F1 (x, y, 2c − z) = 0 x are , , , F2 (2a − x, y, z) = 0 F2 (x, 2b − y, z) = 0 F2 (x, y, 2c − z) = 0    F1 (x, z, y) = 0 F1 (z, y, x) = 0 F1 (y, x, z) = 0 , , and , respectively. F2 (y, x, z) = 0 F2 (x, z, y) = 0 F2 (z, y, x) = 0  F1 (x, y, z) = 0 with regard Specifically, the symmetric curves of space curve F2 (x, y, z) = 0 to coordinate planes x O y,y Oz, and z O x are   F1 (−x, y, z) = 0 F1 (x, −y, z) = 0 F1 (x, y, −z) = 0 , , and , respectively. F2 (x, y, −z) = 0 F2 (−x, y, z) = 0 F2 (x, −y, z) = 0 (Selected from Mathematics Bulletin, Issue No. 11, 1984, Author: Zhang Shizao). II On the Research Directions of Elementary Mathematics Abstract: This paper expounds eight research directions of elementary mathematics and introduces the latest research trends at home and abroad as well as the specific research significance and research methods in each direction as detailed as possible. Keywords: Elementary mathematics, research directions, significance, methods. The elementary mathematics discussed in this paper refers to the elementary mathematics in the traditional sense. In the past two or three decades, with the development of production, the application of modern mathematics, and the deepening of mathematics teaching reform, people have reemphasized the research into elementary mathematics, resulting in a large number of research results. Hilbert said: “As long as a branch of science can provide a large number of problems, it is full of

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vitality”. Now I share my opinions on the research problems of this still dynamic discipline here for your comments. I Continuing Our Research into Famous Classical Mathematical Problems During the establishment and development of elementary mathematics, many famous problems have been raised successively. For example, Darrie, a German mathematician, once edited a monograph to collect 100 famous mathematical problems, most of which were popular for a while and played an important role in the history of mathematics. Some are still of great interest to this day, and people continue to seek solutions or improve the original researches. For example, the research into integer solutions to indeterminate equation x 3 + y3 + z3 + w3 = 0 has attracted many people since the sixteenth century. Hua Luogeng, a late famous mathematician in our country, once wrote: “This is an interesting problem. But unfortunately we have not been able to find its integer solution so far”. However, this problem was completely solved by Mr. Fan Shaoling in our country by means of elementary methods in May 1987 (see the introduction in Nature, September 1989). For another example, for the research into the well-known Mordll problem, the , although some people abroad used elliptic solutions to equation y 2 = x(x+1)(2x+1) 6 function and the theory of quadratic form to prove it and drew the conclusion that there were only two solutions of (1, 1) (24, 70) in 1979 and 1952, but the theoretical knowledge they applied was relatively advanced, and the research methods were relatively cumbersome. The proof obtained by Mr. Xu Zhaoyu and Cao Zhenfu using elementary mathematics method in 1985 was far simpler than theirs (see the introduction in Science Bulletin for details, Issue 7, 1985). On the other hand, with the development of modern mathematics, or because of the adoption of new viewpoints, new methods, and new means to study elementary mathematics, people have reemphasized the research into some ancient themes and famous mathematical problems. For example, the development of optimization method prompts people to reemphasize the research into golden section, Fibonacci sequence, and continuous fractions. The rise of combinatorics has prompted people to reemphasize the research into magic squares and the seven-bridge problem. The development of the science of mathematical thinking has prompted people to dig into ancient arithmetic treasures and carry out research into Jia Xian’s triangle, Chinese remainder theorem, Zu Chongzhi’s evaluation of π, Liu Hui’s cyclotomic method, volume of accumulation, arithmetic–geometric mean inequality, etc., resulting in important discoveries from time to time. II Exploring New Fields and Carrying Out Research on New Topics Constantly Although elementary mathematics has been tempered repeatedly and it is relatively mature and complete, it still has many areas to be explored. There are many problems in elementary algebra, elementary geometry, or trigonometry that need to be further summarized, deepened, extended, and generalized. At the same time, because elementary mathematics is a comprehensive mathematics, it combines shapes and

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numbers, with various methods and superb skills. Whenever new problems arise in theoretical or practical research, people often try to solve them by using elementary mathematics methods. In addition, relevant theorems, formulas, and rules in elementary mathematics are continuously absorbed, transformed, and applied by advanced mathematics, and the related topics in advanced mathematics tend to be elementary and popularized, so this also provides a new source for the development of elementary mathematics. What’s more, the development of the International Mathematical Olympiad (IMO) has given a strong impetus to the study of elementary mathematics. Some questions of domestic and foreign mathematical competitions are often the combined wisdom and efforts of a large number of mathematicians, and they are enlightening, directional, and pioneering, also pose new problems, and open up new fields for the study of elementary mathematics. In this regard, if we leaf through dozens of common secondary school mathematics magazines at home and abroad, we will find that progress is made in the research into elementary mathematics at home and abroad almost every month, resulting in achievements in every issue. Taking the situation in our country in recent years for example, the research into autogenic numbers, transcendental numbers, and multivariate numbers, the research into the determination of prime numbers, the research into special equations and special inequalities, the research into special sequences, the research into the properties of special curves and surfaces, the research into curve similarity and symmetry, the research into function periodicity, the research into triangle and polygon equations, the research into polyhedrons, the research into cubic functions, fractions, and irrational functions, the research into inequalities with absolute value, the research into solutions to trigonometric equations and transcendental equations, as well as the high-dimensional generalization and application of Menelauss theorem, Ceva theorem, Routh theorem, sine theorem, Neubrg-Pedoe inequality, etc., are the outstanding research results in recent years, and some of them have drawn worldwide attention. III Carrying Out Research into Elementary Mathematics Thoughts and Methods During the process of the emergence and development of mathematics, mathematical problems, mathematical knowledge, mathematical thoughts, and mathematical methods have always been combined with each other, interrelated and developed in a coordinated manner. In order to solve various mathematical problems in practice and theory, people are bound to create various mathematical thoughts and methods, and corresponding mathematical knowledge will come one after another. For example, in order to seek formula solutions of algebraic equations of higher degree, Galois created the “group theory” method of thinking, which led to the famous fundamental theorem of algebra; Descartes created the method of combining shapes with numbers and established analytical geometry; Newton proposed the method of thinking of “calculus of fluxion” and created calculus; and so on. The basic thoughts of elementary mathematics are the thoughts of letter algebra, logical reasoning, decomposition and combination, conversion and transformation,

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etc. The basic methods of elementary mathematics can be divided into discovery method, logical method, problem-solving method, esthetic method, etc. People’s understanding of the thoughts and methods of elementary mathematics is far from over. Carry out the research into the thoughts and methods of elementary mathematics to, on the one hand, lay the foundation for the establishment of new disciplines and provide a prototype; on the other hand, to explore using the thoughts and methods of modern mathematics in elementary mathematics to communicate elementary mathematics with advanced mathematics. The thoughts and methods of modern mathematics here are the thoughts and methods of motion, change and transformation, shape-number combination, set and correspondence, mathematical model, mathematical axiomatization, using logical calculus of propositions for demonstrations in elementary mathematics, etc. For example, studying equations and inequalities; studying the distance between two straight lines in different planes from the point of view of functions; studying permutations and combinations, parametric equations, and necessary and sufficient conditions from the viewpoint of set and correspondence; solving problems with the principle of RMI, etc., are specific applications of the thoughts and methods of modern mathematics in elementary mathematics researches. In addition, with the popularization of computers and the development of computing technology, people have paid attention to how to use computers to study elementary mathematics. It is known to us to use computers to perform numerical calculations and solve equations, but using computers for geometric proof has also come true. IV Carrying Out Research into Elementary Mathematics Propositions As stated by Halmos, an American mathematician, “It is true that mathematics is composed of concepts, theories and methods. Without these components, mathematics would not exist, but the real components of mathematics are problems and solutions”. The study of mathematical propositions can not only deepen our understanding of mathematical propositions, but also promote the development of mathematics. For example, through the research into the test question “Let the sides √ and area of ABC be a, b, c and  respectively, and verify: a 2 + b2 + c2 ≥ 4 3” of the second IMO, people have found that there √ are more than 10 proofs, and the conclusion can be enhanced to a 2 +b2 +c2 ≥ 4 3+(a −b)2 +(b −c)2 +(c −a)2 ; through the research into another IMO test question “Let x, y and z be real numbers, then for any ABC, there is x 2 + y 2 + z 2 ≥ 2x y sin C + 2yz sin A + 2zx sin B, people have found that it can include a large number of famous triangle inequalities, and it has become a unique and powerful tool for studying triangles. The study of mathematical propositions is, on the one hand, to study how practical problems are abstracted into mathematical problems, and this abstraction is often in multilevels and on the other hand, to study the change, generalization, and extension of the propositions for established mathematical problems. The main methods are: (1) Exchanging the conditions and conclusions of propositions to study their converse propositions and partial converse propositions;

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(2) Retaining the conditions and deepening the conclusions to study the resulting new conclusions further; (3) Retaining the conclusions and weakening the conditions to study whether the propositions are established or generate new conclusions; (4) Generalizing the propositional conditions and studying the generalization or extension of the propositions. Elementary mathematics problems are ever-changing and voluminous. If all the existing propositions are studied in the above way, wouldn’t it be a huge project? Its topics and achievements are almost endless in this sense. V Carrying Out Research into Elementary Mathematics Problem Solving Problem solving has special significance in mathematics research and mathematics teaching. When a mathematical problem is solved, the theory of mathematical science advances one step, followed by the discovery and solution of new mathematical problems. For many mathematicians in history, it is because of difficult problems that they can exploit their talents, and it is precisely because of solving difficult problems that they leave a name in history. The propositions of elementary mathematics are vast and endless. There are many different ways and methods to solve problems. Specifically, there are nearly a hundred problem solving methods, and they are still developing constantly. For example, trigonometric method, complex number method, construction method, nonstandardized method, etc. are some emerging specific problem solving methods. To carry out the problem solving research on elementary mathematics, we need, on the one hand, to summarize, process, and extract specific problem solving methods systematically to form general methods with guiding significance, clarify the principles of general methods, and reveal the differences and connections between methods in order to better guide the problem solving practice; on the other hand, we need to analyze and compare all possible solutions to various problems in order to extract simpler solutions. Using elementary mathematics as the material, George Polya devoted himself to the study of mathematical discoveries and mathematical problem solving methods, published a series of monographs, and summed up the general problem solving methods and the general laws of mathematics systematically, having a huge impact on the study of elementary mathematics, also setting an example for our research into problem solving methods. The research into mathematical problem solving methods has become a “hot” topic in elementary mathematics research in recent years, and almost every secondary school mathematics journal has research articles in this aspect. Some have also created columns to extensively solicit and collect solutions in order to explore the optimal solution. For example, for “Lehmns-Stiner Theorem: in a triangle, if two angle bisectors are equal, then the triangle is an isosceles triangle”, the Journal of Japan Society of Mathematical Education has publicly solicited more than 80 solutions since 1979, and the solicitation is still under way.

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Fig. C.1 In appendix

For another example, about the principle that the light refraction theorem is a sufficient condition for the travel of light at the highest speed, some people have asserted that it cannot be proved with elementary mathematics methods. However, the good news is that more than 10 elementary mathematics proofs have been solicited in a short period of time. There is a lot of work to do for the problem solving research of elementary mathematics. If the study of propositions is endless, then it is safe to say that the research into problem solving is also endless. For example, ⎧ 2 1 ⎨ x + x y + 3 y 2 = 25, 1 2 It is known that positive numbers x, y, and z satisfy y + z 2 = 9, ⎩ 23 z + zx + x 2 = 16. Find the value of x y + 2yz + 3zx. People usually solve it algebraically. But if the above formula is transformed into x2 +

2 1 1 √ y − 2x · √ y · cos 150◦ = 52 3 3

2 1 √ y + z 2 = 32 3

z 2 + x 2 − 2zx cos 120◦ = 42 , As shown in Fig. C.1 in appendix, it is very ingenious to regard the above three equations as the relations between the sides and angles of the triangle in Fig. C.1 and use the graphical method to solve them, that is, SABC = 6, and SABC =

1 1 1 1 1 x · √ y sin 150◦ + z · √ y + x z sin 120◦ 2 2 2 3 3 1 = √ (x y + 2yz + 3zx) 4 3

√ so, x y + 2yz + 3zx = 24 3. For another example, for the competition problem “Find a point P in quadrilateral ABCD to make SA P B = SB PC = SC P D = SD P A , and point P must be in AC”,

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Fig. C.2 In appendix

this problem can be proved by calculating the areas using cosine theorem. Establish the skew coordinate system as shown in Fig. C.2 in appendix, then From SA P B = SD P A , we get c =

y0 . x0

From SA P B = SB PC , we get

From SA P B = SC P D , we get

Taking ➀ and ➁ as the system of equations for a and b, it is obvious that when 2x0 − 1 = 0, we have 0 (2x 0 −1) 0 (2x 0 −1) a = 2x(2x = 2x0 and b = 2y(2x = 2y0 , so point P is in AC and is its 0 −1) 0 −1) midpoint. When 2x0 − 1 = 0, we have P( 21 , y0 ) and D(0, 2y0 ), so point P is in BD and is its midpoint. In this way, not only the proof is completed, but it is further found that P must be the midpoint of AC or BD. This is said to be out of the question setters’ expectation.

VI Carrying Out Application Research of Elementary Mathematics Elementary mathematics is widely applied in production and life. With the progress of society, the development of production, and the deepening of the construction of Four Modernizations, it will be applied increasingly deeply and extensively. For example, through research, people have compiled special books on the application of elementary mathematics in metalworking, carpentry, architecture, water conservancy, meteorology, transportation, military, etc. At the same time, there are also a lot of problems in daily life and production practice to be solved applying elementary mathematics knowledge. In solving practical problems, it will also promote the development of elementary mathematics. Regarding application research, the key is to extract a concrete mathematical model from the intricate practical problems through observation, analysis, conversion, and simplification and then give practical meaning to the obtained results.

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Fig. C.3 In appendix

Undoubtedly, in the process of application research, we will deepen our grasp of elementary mathematics knowledge and improve our ability to analyze and solve problems by applying elementary mathematics knowledge comprehensively. For example, there is a typhoon center 300 km to the due west of city A, which is moving northeast at a speed of 40 km per hour, and the area within 250 km of the typhoon center will be affected by it. When will the city be affected by the typhoon, and how long will it last under the condition of constant wind? For this practical problem, the area affected by the typhoon is actually the area formed by the system of circles with the moving typhoon center as the circle center with a radius of 250 km, so the problem of whether the city is affected by the typhoon depends on the relationship between point A and the system of circles. Solution I As shown in Fig. C.3 in appendix, select the rectangular coordinate system, and assume that the typhoon reaches B (x, y) after t hours, then √ x = −300 + 40t cos 45◦ = −300 + 20 2t √ y = 40t sin 45◦ = 20 2t √ √ − 20 2t)2 = √ 2502 . Equation of system of circles: (x + 300 − 20 2t)2 + (y √  2 2 When A(0, 0) is within B (x, y), we have √ (0 + 300 − 20 2t) + (0 − 20 2t) ≤ 2 2 250 , and after sorting, we get 16t − 120 2t + 275 ≤ 0, so the solution is 1.9 ≈

√ √ √ √ 15 2 + 5 7 15 2 − 5 7 ≤t ≤ ≈ 8.6 4 4

Therefore, the city will be affected by the typhoon in about 2 h, which will last for about 6.7 h. Solution II As shown in Fig. C.4 in appendix, it is obvious when the typhoon center is between EF, the city will be affected by it. In AB E and AB F, by the cosine theorem, we√get AE 2 = AB 2 + B E 2 − 2 AB · B E cos 45◦ , that is, 250◦ = 3002 + B E 2 − 300 2B E. Therefore, the above conclusion can also be obtained similarly.

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Fig. C.4 In appendix

Vii Carrying Out Research on the Guidance of Advanced Mathematics on Elementary Mathematics Elementary mathematics is the foundation of advanced mathematics, and in turn, advanced mathematics also has good guiding significance for elementary mathematics. On the one hand, advanced mathematics provides a theoretical basis for the relevant contents of elementary mathematics. For example, the continuity and denseness of real numbers in algebra the compilation of numerical tables, the derivation of formulas of circumference and area of a circle and volume of a sphere, etc. in geometry, the continuity and smoothness of elementary function graphs, etc. can only be clarified by the application of advanced mathematics knowledge; on the other hand, advanced mathematics will also provide solutions for elementary mathematics problems. For example, the use of advanced mathematics knowledge will generalize the processing of extreme value problems in elementary mathematics, and using the viewpoint of group of transformations and the knowledge of advanced geometry simplifies elementary geometry problems further. For example, for the problem “In a non-right ABC, let the orthocenter be H, the pedals on three sides be D, E and F respectively, and EF intersect AD at K. Prove: AK·HD = AD·KH”, from the point of view of projective geometry, the above problem is essentially the harmonicity of a complete quadrilateral, and the condition of three heights is unnecessary. As long as three lines AD, BE, and CF intersect at point H, the conclusion is still true. VIII Carrying Out Research on the Teaching of Elementary Mathematics In “basic” status and with universal educational value, elementary mathematics has become the main content of mathematics in primary and secondary schools. Carrying out elementary mathematics teaching research is a major issue having a bearing on improving the quality of mathematics teaching in primary and secondary schools, improving the quality of the whole nation, and providing talents to higher-level schools. For nearly half a century, with the rise of the “New Mathematics Movement”, the application of mathematics has become increasingly extensive, and to meet the need of modernization and scientization of mathematics education, mathematics teaching research has always been an extremely active research field and is flourishing at present. There are many academic organizations, frequent professional conferences, various new theories, new viewpoints, and new methods that are constantly emerging, the research team is constantly expanding, with increasingly rich results, and it is faced with the new task of creating mathematics pedagogy.

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Regarding the research on the teaching of elementary mathematics, this is certainly the task of a large number of professional researchers in research institutions, and it is also the glorious mission of the broad messes of frontline primary and secondary school teachers. For this research field with a very promising prospect and a wide range of topics, the theory needs to be deepened, and the results need to be identified, generalized, and applied, and the current research can focus on the following topics. Research on the birth, development, and application of elementary mathematics; research on the educational value, teaching purpose, significance, and task of elementary mathematics; research on the curriculum, syllabus, and textbooks of elementary mathematics; research on the logical structure and form of elementary mathematics; research on the thinking form, method, process, and quality of elementary mathematics; research on the viewpoints, thoughts, and methods of elementary mathematics; research on the essence, methods, approaches, means, and psychology of elementary mathematics learning; the research on the form of elementary mathematics ability and its training; research on the principles and methods of elementary mathematics teaching; research on the forms, methods, and means of elementary mathematics teaching; research on the development history and educational history of elementary mathematics; research on comparative education of elementary mathematics; research on the educational measurement and evaluation of elementary mathematics; research on the professional qualities and further education for improvement of mathematics teachers in primary and secondary schools, etc. The above eight main research directions of elementary mathematics may not be able to describe the complete picture and status of its research fully and accurately, and the topic itself is a problem worthy of serious discussion. For a long time, many mathematicians, mathematics educators, professional researchers, and the broad masses of frontline secondary school teachers, amateurs, and even primary and secondary school students have made great contributions to the study of elementary mathematics. Many great masters of mathematics in history have been engaged in the research of elementary mathematics all their lives. The famous contemporary mathematicians Hua Luogeng, Su Buqing, Xu Lizhi, Wu Wenjun, Zhang Jingzhong, etc. have attached great importance to, actively participated in and guided the research work of elementary mathematics, and have made numerous achievements. Some of these achievements are made by the masters and experts, and more are made by mathematics teachers, even amateurs and secondary school students. For the broad masses of mathematics educators, as long as they are enthusiastic about this research work, concerned about the mathematics teaching reform, keep abreast of the research trends, master scientific research methods, choose appropriate research topics, and make unremitting efforts, they will definitely be able to make gratifying achievements in elementary mathematics research and make their due contributions. (Selected from Journal of Yancheng Teachers University, Issue No. 7, Author: Zhang Shizao).