Intellectual Development and Mathematics Learning 9789811987564, 9789811987571

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Chongde Lin

Intellectual Development and Mathematics Learning

Intellectual Development and Mathematics Learning

Chongde Lin

Intellectual Development and Mathematics Learning

Chongde Lin Beijing Normal University Beijing, China

ISBN 978-981-19-8756-4 ISBN 978-981-19-8757-1 (eBook) https://doi.org/10.1007/978-981-19-8757-1 Jointly published with China Light Industry Press Ltd. The print edition is not for sale in the Mainland of China. Customers from the Mainland of China please order the print book from: China Light Industry Press Ltd. © China Light Industry Press 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 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 publishers, 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 publishers 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 publishers remain 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

Foreword

I am honored to write the foreword for the English version of this highly impactful and influential work by Prof. Lin on Intellectual Development and Mathematics Learning. The earlier edition (the 1984 edition) of this book has been widely adopted as a theoretical foundation and implementation guide for research in developmental psychology and mathematics education in China for the past 40 years. The author has successfully outlined the landscape of research questions and best practices in Intellectual Development and Mathematics Learning. Specifically, the author offers unique views by integrating (a) the oriental tradition and philosophy of teaching and learning (Confucius) and (b) research (cognitive psychology) and best practices (mathematics teaching and learning) observed in China over the last 40 years. I am amazed by the breadth and depth of the content area in both mathematics learning and intellectual development and benefited greatly from reading the theoretical discussions, experiment results, and examples. As a cognitive psychologist educated in the Western tradition and familiar with the current research on the science and technology of learning, I enjoy and appreciate the special angle of this book, which combines the traditional and classical Oriental perspectives of learning, practice, intelligence, and ability. Their association with modern research and best practices in teaching and learning mathematics in China and other Asian countries made perfect sense to me. Should you be familiar with mainstream accounts in the West, such as How People Learn: Brain, Mind, Experience, and School: Expanded Edition (National Research Council, 2000), How People Learn II: Learners, Contexts, and Cultures (National Research Council, 2018), and Top 20 Principles from Psychology for PreK–12 Teaching and Learning (Coalition for Psychology in Schools and Education of APA, 2015), you will likely have a greater appreciation of the unique views of the book. Although it is relatively short it does a superb job of clearly elaborating on the theories, models, and frameworks with an abundance of empirical studies and demonstrative examples. • The author presents a theoretical framework for six factors of mental/intellectual activities: goal, process, content, quality, monitoring, and others (psychological/environmental), specifically highlighting the importance of the last one.

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Psychological/environmental factors are not necessarily cognitive but play nontrivial roles in intellectual development. This framework has been elaborated throughout the book, with examples and considerations of mathematics learning for all ages (up to high school). • When comparing Western and Eastern approaches to intelligence and creativity, the author proposes the “T” Type Talents Model, which integrates Western and Eastern approaches on intelligence and creativity. Specifically, this model emphasizes the breadth and depth of knowledge and its respective impacts on intelligence and creativity. • The author is also instrumental in developing Chinese “common core” standards. The theoretical foundation of the previous and current editions plays a key role in this effort. For example, the author attempts to make intellectual development “measurable” with several quantifiable variables, such as qualitative change, time/course of development, and coordination/relation among other relevant psychological/physiological/environmental aspects. The author himself is an educator in psychology and mathematics education. As such, he offers an amazing number of examples and deep insights into mathematics learning. The book can be read as an accumulation of research on intellectual development over the last 40 years in China. Indeed, it is a masterpiece in mathematics education, with examples ranging from basic exercises in elementary mathematics to challenging problems in mathematical Olympic competitions. Through these examples, the author is especially successful in demonstrating the relationship between mathematics learning and intellectual development. The author makes a strong case for why “mathematics is the gymnastics of thinking” and proposes a “complete structure of mathematical thinking.” The author occupies a unique position and roles as a distinguished professor at Beijing Normal University, Editor-in-Chief of Psychological Development and Education, and an active researcher and educator of 60 years. As a result, this work reflects the author’s own research in the field of mathematics education and intellectual development while also providing readers with a perspective on modern Chinese educational research on science, technology, engineering, and mathematics (STEM). In addition, this book will help you understand the crucial and important roles Chinese academics have played in the last 40 years in China. Xiangen Hu The University of Memphis Memphis, USA

Xiangen Hu is currently a professor of psychology, electrical and computer engineering, and computer sciences at the University of Memphis (UofM). He is the director of the Advanced Distributed Learning (ADL), University of Memphis Partnership Lab, and serves as one of seven members of the Organization for Economic Co-operation and Development (OECD) Program for International Student Assessment (PISA) Research and Innovation Group.

Preface

I have been engaged in research on intelligence development for a long time. Intellectual Development and Mathematics Learning is the product of decades of research on the intelligence development of children and adolescents in mathematics teaching in kindergarten and elementary and secondary schools. The book details my theory of the “Triangular Pyramid Structure of Thinking,” which I have applied for more than 40 years to guide mathematics experiments that seek to develop elementary and secondary school students and young children’s mental abilities at more than 3000 experimental sites. The experiments cover 26 provinces, autonomous prefectures, and municipalities across China and have produced a series of valuable research findings. In recent decades, this theory has become a systematic and structured guiding concept in mathematics teaching in the field of basic education in China and has played a positive role in helping mathematics teachers cultivate students’ mathematical abilities and improve their teaching effectiveness. I believe that the development of mathematical ability should be centered on the development of thinking abilities. Such an approach is conducive to clarifying the purpose of school teaching; that is, teachers develop students’ intellectual abilities while imparting knowledge. It also contributes to understanding that the core component of intelligence and ability is thinking. Indeed, mathematics is a form of mental gymnastics, and learning cannot succeed without thinking. It is essential to acknowledge that a core literacy of mathematics involves mathematical abstraction, logical reasoning, mathematical modeling, intuitive imagination, mathematical operations, and data analysis. Based on this, this book discusses four questions about thinking, all of which are critical issues in mathematics learning and crucial to enhance students’ interest in mathematics, improve the efficiency of mathematics learning, and develop young learners’ mathematical abilities. • How many kinds of logical thinking are there? Three: action logic/practical thinking, such as manipulative mathematical abilities; image logic thinking, such as the intuitive or spatial imagination ability in mathematics; and abstract

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logic thinking, such as the mathematical abstraction, logical reasoning, and mathematical modeling qualities described above. • What is the primary characteristic of thinking? Generalization. The example of “combining like terms” in mathematics learning is a typical generalization. The essence of mathematical ability is the ability to generalize in mathematics. • How can we improve students’ thinking abilities? Developing students’ quality of thinking is key to developing their thinking ability. Thinking quality reflects the thinking characteristics of individual differences in intelligence and mainly includes profundity of thinking, e.g., solving mathematical problems through logical deduction; flexibility of thinking, e.g., solving multiple problems in mathematics learning; creativity of thinking, e.g., creatively answering mathematical problems; criticality of thinking, e.g., excelling at criticizing and questioning during mathematics learning; and agility of thinking, e.g., answering mathematical problems quickly and accurately. • How is thinking structured? It is a system comprising six components: the purpose of thinking, the process of thinking, the materials of thinking, the quality of thinking, the monitoring of thinking, and the non-cognitive elements of thinking (see Chap. 4, Subsection “The Triangular Pyramid Structure of Thinking”). Mathematics is all about structure, and learning mathematics is about building systems of thinking. The book is divided into three parts that focus closely on the above four issues. Part I, “The Mystery of Intelligence,” consists of three chapters (1–3) that provide an overview of intelligence and intellectual activity and explain in detail the importance of intelligence development in mathematics teaching. Chapter 1 explores the essence of intelligence, which I consider a psychological phenomenon. In this chapter, I discuss the definition, components, and levels of intelligence and explore the relationship between intelligence and knowledge and skills. In addition, I highlight international perspectives on intelligence and the Chinese intelligence view of “The Six Arts” (see Chap. 1, Subsection “The Main Views of Intelligence in the International Psychological Community” and Chap. 4, Subsection “The Triangular Pyramid Structure of Thinking”) to inform readers of current research advances in and Chinese cultural interpretations of intelligence. Chapter 2 reveals the relationship between the laws of intellectual development and mathematics learning. The basic laws of children and adolescents’ psychological development are divided into four issues based on the research results on the topic: nature and nurture, internal and external causes, education and development, and age and individual differences. These four relationships also provide the theoretical basis for the psychological development of mathematics learning in kindergarten and elementary and secondary schools. In particular, in mathematics learning, the contradiction between the new needs in each stage of students’ intellectual development and their previous levels becomes a driving force for the development of their mathematical abilities.

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Chapter 3 describes the relationship between intelligence and creativity and explores the similarities and differences between the two factors and their connections. Importantly, intelligence is a necessary but insufficient condition for creativity, while the cultivation of creative talent requires attention to both creative intellectual and nonintellectual factors. In addition, I propose the “T”-type talent model (see Chap. 3, Subsection “Characteristics of Creative Talents and Growth Environment”), which integrates the Eastern and Western education models. While I believe that both educational models have their own advantages, we should take advantage of their strengths and avoid their weaknesses in educational practice to successfully shape “T” type talent into creative talent. Cultivating “T”-type talent is also the fundamental goal of creative education in China in the 21st century. Part II, “Mathematics is the Gymnastics of Human Thinking,” likewise consists of three chapters (4–6) that introduce how to promote the development of mathematical abilities in children and adolescents based on a structural view. Chapter 4 discusses the complete structure of mathematical thinking. I first present the history of my proposed theory of the Triangular Pyramid Structure of Thinking and then apply this structure as a basis for exploring the accomplishment of mathematical entirety. I argue that student’s mathematical ability is a complete structure of thinking or intelligence and that teachers should teach mathematics from a comprehensive perspective of thinking. At the same time, I provide examples of relevant mathematics teaching approaches, methods, and requirements. Chapter 5 examines the development of student’s thinking ability in arithmetic. In mathematics learning, student’s thinking ability is often not only the manifestation of a process of thinking activity but also the complete realization of many processes. Students’ thinking ability in arithmetic is mainly manifested as generalization, spatial imagination, and propositional and logical reasoning abilities, which form the core literacy in students’ mathematics learning. Chapter 6 explores the differences and development of students’ intellectual qualities in arithmetic. The intellectual quality of thinking or thinking quality is the manifestation of intellectual characteristics in individuals in thinking activities, with certain individual differences and characteristics. Based on the five intellectual qualities of profundity, flexibility, creativity, criticality, and agility, I provide a number of examples in mathematics teaching to demonstrate the importance of teaching based on aptitude. Part III, “The Development of Mathematical Abilities in Children and Adolescents,” consists of three chapters (7–9), which describe early mathematics education and arithmetic thinking ability of preschool students; learning and intellectual development for elementary school students; and learning and intellectual development for secondary school students. This part discusses the age and individual differences in the development of the mathematical ability of children and adolescents from a practical point of view and in conjunction with theory. I believe that everyone possesses mathematical ability, but when it arises and how it develops remains the subject of academic discussion. The development of children and adolescent’s mathematical

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ability shows the unity of commonality and individuality but is constrained by mathematics education. Therefore, age-specific instructional design and aptitude-based learning should be the most basic principles in mathematics education. This part provides recent research on the development of children and adolescents’ mathematical abilities, supported with a wealth of examples and analyses of mathematics teaching. The methods and techniques to be observed in teaching mathematics to children and adolescents are vividly described through the analysis of specific cases in this book. In addition, I offer my views and suggestions on some important issues of concern in the teaching and learning of mathematics today; for example, the problem of combining traditional and modern mathematics as seen in the Mathematical Olympiad and its development in China. We cannot deny the value of the Mathematical Olympiad in cultivating adolescents’ intellects or deny its potential; however, we should not be so eager to reap the immediate benefits that we see the Mathematical Olympiad as a mere tool to obtain extra scores and ignore students’ interest in development. At the same time, the compilation of problems in the Mathematical Olympiad should balance the difficulty and focus on the examination with training of students’ thinking methods, which will be more conducive to the cultivation of students’ flexible thinking quality and the scientific spirit of questioning and exploration. Finally, I would like to express my sincere thanks to the recommender of this book Dr. Xiangen Hu (Professor at University of Memphis); and two translators, Dr. Jianzhong Hong (Distinguished Professor at Central China Normal University, Ph.D. in Psychology from University of Eastern Finland) and Can Xiao (Ph.D. student at the School of Psychology, Central China Normal University, M.Ed. from Washington State University); Nick Melchior (Editorial Director), Grace Ma (Senior Editor), Melody Zhang (Editor), Alice Xie (Associate Editor), Sophie Li (Editorial Project Manager), Gayathri K. (Production Project Manager), Poojitha Ravichandran (Production Editor) from Springer Nature Publishing Group; Tie Shi, chief planner of “Wanqian Education” and “Wanqian Psychology,” and Hong Wu, editor-in-charge from China Light Industry Press. Without these people, the English edition of Intellectual Development and Mathematics Learning would not have been possible. Beijing, China October 2022

Chongde Lin

Contents

Part I

The Mystery of Intelligence

1 The Nature of Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intellectual Activity is a Mental Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions, Compositions, and Levels of Intelligence . . . . . . . . . . . . . . . . . Relationships Between Knowledge, Skills, and Intelligence . . . . . . . . . . . The Main Views of Intelligence in the International Psychological Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Laws of Intellectual Development and Mathematics Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature and Nurture in Intellectual Development . . . . . . . . . . . . . . . . . . . . . The Internal and External Causes in Intellectual Development . . . . . . . . . Education and Intellectual Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Relationship Between Age Characteristics and Individual Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Intelligence and Creativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics of Creative Talents and Growth Environment . . . . . . . . . . . Implementation of Creative Education and the Training Model of “T”-Type Talents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approaches to Developing Students’ Creativity Through Creative Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approaches to Developing Students’ Creativity in Mathematics Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

3 3 4 13 16 25 27 27 34 39 44 49 51 52 57 63 68 75

Mathematics: The Gymnastics of Human Thinking

4 The Complete Structure of Mathematical Thinking . . . . . . . . . . . . . . . . The Triangular Pyramid Structure of Thinking . . . . . . . . . . . . . . . . . . . . . . .

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Accomplishment of Mathematical Entirety . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Students’ Mathematical Abilities Are the Structural Integrity of Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Mathematics Teaching Should Start from the Integrity of Thinking . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5 The Development of Student’s Thinking Ability in Arithmetic . . . . . . Mathematics Learning and the Development of Students’ Generalization Abilitys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics Learning and the Development of Student’s Spatial Imagination Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics Learning and the Development of Student’s Proposition Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics Learning and the Development of Students’ Logical Reasoning Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Differences and Development of Students’ Intellectual Qualities in Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profundity of Students’ Thinking and Its Cultivation . . . . . . . . . . . . . . . . . Flexibility of Students’ Thinking and Its Cultivation . . . . . . . . . . . . . . . . . . Creativity of Students’ Thinking and Its Cultivation . . . . . . . . . . . . . . . . . . Criticality of Students’ Thinking and Its Cultivation . . . . . . . . . . . . . . . . . . Agility of Students’ Thinking and Its Cultivation . . . . . . . . . . . . . . . . . . . . . Importance of Studying Students’ Thinking Qualities . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 120 125 130 138 139 141 145 149 154 156 158 160

Part III Development of Mathematical Abilities in Children and Adolescents 7 Preschool Children’s Arithmetic Thinking Ability and Early Education of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Characteristics of Thinking and the Developmental Profile of Arithmetic Thinking Abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Development of Level of Thinking Activity in Acquisition of Number Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Research on the Development of Mathematical Abilities in Children Aged 0–6 Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Teaching of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Early Education to Early Mathematics Teaching . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 166 173 178 183 186 193

8 Mathematics Learning and Intellectual Development of Elementary School Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Development of Mathematical Intelligence in Elementary School Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Contents

Recent Studies on the Development of Elementary School Students’ Mathematical Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Improving Elementary School Students’ Abilities to Solve Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From “Wormy Formula” to Thinking Training Questions . . . . . . . . . . . . . . Key Points that Elementary School Mathematics Teaching Should Pay Attention to . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Mathematics Learning and Intellectual Development of Secondary School Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Mathematical Intelligence in Secondary School Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Studies on the Development of Secondary School Students’ Mathematical Abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emphasis on the Development of Mathematical Ability Before Intellectual Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modern Mathematics Helps the Development of Abstract Thinking . . . . . The Mathematical Olympiad and Intellectual Development of Secondary School Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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202 207 217 225 231 233 233 241 243 252 258 267

About the Author

Chongde Lin gained his Ph.D. in psychology from Beijing Normal University; he is the professor of the Faculty of Psychology, Ph.D. tutor, the president’s professor of Beijing Normal University, the former chairman of the Chinese Psychological Society, the committee member of the Social Science Committee of the Ministry of Education of China and the convener of the Faculty of Education and Psychology, the director of Mental Health Education Expert Guidance Committee for Students in Ordinary Colleges and Universities of Ministry of Education of China. After graduating from Beijing Normal University in 1965 with a degree in psychology, he worked in elementary and middle schools for 13 years (1965–1978), serving as a school leader and a teacher (teaching including but not limited to mathematics). He was admitted to Beijing Normal University in 1978 to study psychology and received his Ph.D. in 1984. He has been a teacher at Beijing Normal University since May 1980. He was promoted to full professor in June 1986. From May 1985 to December 1999, from May 1985 to December 1999, he served as deputy director and director of the Institute of Developmental Psychology at Beijing Normal University. From 1998 to 2011, he also served as the director of the Teaching Steering Committee of Beijing Normal University. Over the past 40 years, he has conducted a number of studies on intelligence promotion around the cognitive development of children and adolescents. These studies have strongly contributed to the basic education reform in China, improved Chinese education quality, and promoted a major breakthrough in the field of thinking theory. The theory of “Triangular Pyramid Structure of Thinking” is one of them, and the related paper “Multiple Intelligence and the Structure of Thinking” was published in the international journal Theory and Psychology. He has presided over nearly 20 major or key projects of the National Social Science Foundation of China and the National Natural Science Foundation of China and has published more than 30 articles in SCI and SSCI, and nearly 400 articles have been published in CSCI and CSSCI; he has published 16 monographs, edited The Comprehensive Dictionary of Psychology with more than 6.3 million words, organized and edited 12 textbooks of applied psychology, and presided over the

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translation of the Handbook of Child Psychology with more than 8 million words. In October 2020, The Collected Works of Chongde Lin (12 volumes) was published. For his outstanding contributions, he has been awarded the titles of “National Advanced Worker,” “National Outstanding Teacher,” “National Outstanding Scientific and Technological Talent,” and 26 other government academic awards.

Part I

The Mystery of Intelligence

With the development of science, technology, and production, intellectual development has become an increasing concern. All the material and spiritual wealth created by human beings crystallizes their intellectual activities in practice. Intelligence is the subjective energy and inner foundation of human creativity and invention; thus, there is a growing desire to explore its mysteries. In fact, we are no strangers to intelligence. When we say “smart” or “stupid,” it is an evaluation of intelligence. In traditional Chinese culture, intelligence is often understood in multiple manifestations and fields. For example, the qualities of being smart and talented, able to recognize laws, master rules, distinguish right and wrong, deal with problems, create and execute strategies, and so on are all perceived as wisdom and intelligence. Of course, the scientific definition of intelligence is not that simple. Among the international academic community, there are more than 140–150 well-known definitions of intelligence, thus indicating the complexity of clarifying the problem. Mathematics learning by students, especially children and adolescents, is an intellectual activity. The core component of intelligence is thinking, and mathematics is a form of mental gymnastics. In the foreword of the book Mathematics Thoughts Outlines, Ningzhong Shi, the former president of Northeastern Normal University and a renowned mathematician, writes that mathematical development depends on the realization of three essential elements: abstraction, reasoning, and modeling (Shi, 2008). This illustrates the complementary relationship between mathematics learning and intellectual activity: Mathematics learning must be based on intelligence, at the same time it promotes intellectual development. Professor Shi also highlights the importance of abstraction in mathematical thinking: “abstraction is the most critical part. Through abstraction, the concepts and algorithms of mathematics could be obtained from reality; through reasoning, the development of mathematics could be promoted; then, through modeling, the connection between mathematics and the external world is established.”

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Reference Shi, N. Z. (2008). Mathematics thoughts outlines. Northeast Normal University Press. (in Chinese).

Chapter 1

The Nature of Intelligence

Intellectual problems, primarily psychological problems, are closely related to other subjects (epistemology, genetics, neurophysiology, pedagogy, logic theory, etc.). Therefore, psychologists should explore them theoretically and practically in relation to related subjects. To explore the mystery of intelligence, it is necessary to reveal its nature of psychology.

Intellectual Activity is a Mental Activity Intelligence is a mental phenomenon. To clarify what intelligence is, we should first understand the nature of mental phenomena. In fact, we are familiar with mental phenomena in daily life. We often have contact with the things around us, pay attention to or remember certain things, contemplate various problems, imagine future scenarios, and so on. Here, sensation, attention, memory, thinking, imagination, etc., are all forms of mental phenomena, that is, the mental activities we perform to understand the objective world. This is called cognitive processing. However, while objectively recognizing the things around us, various emotional experiences, such as happiness, anger, sadness, joy, etc., are also aroused. This is known as affective processing. At the same time, for some specific needs, we propose goals, formulate plans, overcome difficulties, and complete tasks, all of which represent a third form: volitional processing. These cognitive, affective, and volitional processes have a process of generation, development, and completion, collectively referred to as the mental process. Based on the relationship between nature and nurture, especially under the influence of social life conditions and education, an individual will form some relatively fixed characteristics in terms of mental activity. For example, some people are interested in the literature and art, while others are more attracted to mathematics, science, and chemistry; some individuals are selfless, while others are selfish and self-interested. Human interests, hobbies, motivations, purposes, ideals, beliefs, © China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_1

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desires, values, etc., are manifestations of human needs, which are usually referred to as individuality consciousness trends. Some people are lively and enthusiastic, some are quiet and silent, some are impatient and temperamental, some are gentle and friendly, some are strong and brave, some are timid and cowardly, some are decisive, and some are indecisive. These are the characteristics of temperament, trait, intelligence, and ability, often called the psychological characteristics of individuality. Collectively, individuality consciousness trends and psychological characteristics of individuality make up one’s personality. In short, personality can be considered the overall mental representation of an individual. People’s mental processes and individualities differ in practice but are inseparably united in human activities. Individuality is formed through and expressed in the mental process and simultaneously regulates it. Thus, all mental phenomena occurring in an individual relate to these processes and personality. Although everyone is familiar with mental phenomena, it is not easy to interpret them correctly. Mental phenomena is an organic function of the brain, which is a reflection of objective reality. By saying that the mind is a function of the brain, the human brain becomes the organ of mental activity. Without the brain, there will be no mental activity, and if the brain is damaged, mental activity will be severely disrupted. Thus, in the case of brain damage, even though you still have eyes and ears, you may lose your hearing and vision. The mind reflects objective reality; that is, objective reality is the source of the mind. If there is no objective world, there will be no impression of objective reality, and there will be no mind. The mind comes from objective reality and reflects it, but the reflection of the human mind is a dynamic reflection. Individuals’ reflection activities are carried out in social practice through speech. Therefore, individuals’ minds and consciousnesses are socially and consciously active. Once formed, the mind and consciousness play a regulating and orienting role in human activities. Individuals transform objective reality through their practical activities based on their cognition of objective reality. Therefore, the mind is stimulated during such activities while also reacting to them. In short, human mental activity, in terms of its production method, is a high-level neural activity produced by objective things acting on the human brain; in terms of its content, it is the active response of the human brain to exposure to objective reality, also expressed through behavior. Intellectual activity is one kind of mental activity.

Definitions, Compositions, and Levels of Intelligence Scholars in China and abroad hold different understandings of intelligence. Intelligence is the research object in psychology. Chinese psychology comes from the West, but psychological thoughts existed in China long before Western psychology was introduced. Hence, the interpretation of intelligence here refers to ancient Chinese psychological thoughts. This chapter seeks to understand intelligence from

Definitions, Compositions, and Levels of Intelligence

5

the perspective of the relationship between intelligence and ability. Ability and intelligence cannot be separated; they are both distinct and intrinsically linked.

The Definition of Intelligence and Ability Intelligence and ability are well-adapted psychological characteristics that manifest as successful problem solving (or task completion). How can we interpret this definition? First, both intelligence and ability are individual psychological characteristics of personality. The Chinese educator Confucius (551–479 B.C.) proposed the theory of “the wisest, ordinary, and dumbest person,” a series of IQ or aptitude tests that measure differences in intelligence or ability among individuals. Psychologists and other great men define intelligence and ability as personality-related, illustrating that they are individual differences. Second, the first adherent adjunct in the definition of intelligence and ability is to “successfully solve a problem (or complete a task).” Why is that? What is the difference between intelligence and ability in terms of psychological personality characteristics and other factors (such as temperature and personality) of psychological characteristics? The difference is that the fundamental function of intelligence and ability is to solve problems or complete tasks successfully. Therefore, in a certain sense, the level of intelligence and ability depends on the level of problem solving or task accomplishment. This characteristic could explain why schools must cultivate students’ abilities to analyze and solve problems. The second adherent adjunct in the definition of intelligence and ability is to be “well-adapted.” This definition derives from the mission of intelligence and ability, which is actively adapting, coordinating with individuals and the environment, and completing the mission of understanding and changing the world. Piaget (1981) insisted that the function of psychology is adaptation, and intelligence is the thinking related to adaptation to the environment. This means that the nature of intelligence and ability is adaptation, with the aim of balancing the individual and environment. Currently, there is consensus on this conception among the members of the international psychological community. At the same time, Chinese educators are also impressed by some graduates’ great adaptability after entering society. This reaction suggests that being “well-adapted” plays an essential role in people’s minds. Then, how can we understand the difference and connection between intelligence and ability? There is a certain distinction between intelligence and ability. Generally, intelligence tends to be cognition, which aims to solve the problem of distinguishing between the known and unknown. It is a combination of stable psychological characteristics that guarantee effective cognition of objective matters. Ability tends to be more activity-based, aiming to solve the problem of being able and unable. It is

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a combination of stable psychological characteristics that ensure the smooth implementation of actual activities. However, cognition and activity are always unified— cognition is the basis of activity, and activity must be accompanied by cognition. Therefore, intelligence and ability are cross-relationships of mutual constraints and presuppositions. From the international perspective of intelligence and ability, some scholars believe in “belonging theory,” in which intelligence belongs to an ability that tends to cognition; some scholars hold to “involving theory,” wherein intelligence involves various abilities such as sensation, perception, thinking, memory, attention, etc. In actuality, the intersection of intelligence and ability reflects both “belonging” and “involving.” The essence of teaching is the unity of cognition and activity. The development of intelligence and cultivation of ability is inoperable in teaching. The “intelligence training” of mathematics learning I proposed includes teaching intelligence and the training of ability. This is because intelligence and ability are mutually connected. The combination of intelligence and ability is intellectual ability. Because intelligence and ability are closely related, ancient Chinese thinkers generally regarded intelligence and ability as two concepts that are both different and connected, transforming and improving each other. For example, in the book Master Lü’s Spring and Autumn Annals, Jiuzhou Chunqiu and Lunheng combine intelligence and ability. In essence, they combine intelligence and ability as a standard of talent assessment.

The Composition of Intelligence In traditional Chinese culture, the composition of intelligence is a complete structure. More than 4000 years ago, there was a passage in The Book of Documents: The Great Plan which contained the “Five Matters”: appearing,1 speaking, seeing, hearing, and thinking, among which speech, perception (seeing and hearing), and thinking were proposed as the main components of intelligence. Subsequently, the crucial roles of memory, imagination, and skill in determining intelligence and ability were also suggested. In the Han Dynasty, Zhongshu Dong (179–104 B.C.) discussed the relationship between intelligence and thinking in his work, Chunqiufanlu: Birenqiezhi, in which he clearly states that thinking is the core of intelligence. Today, as a result of work, the components of intelligence can be divided into the factors of speech, sensation and perception, memory, imagination, thinking, and manipulation skills (see Fig. 1.1). Thinking is the core of intelligence. Speech. In our daily lives, “language” and “speech” are often indistinguishable. However, scientifically speaking, they are two different concepts that are related and distinct. Language is a communication tool—a social–historical phenomenon created by people, produced as societies emerge, and developing as they progress. 1

Appearing: Intending the meaning of modest, respectful, decent, and polite.

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7

Sensation and Perception Memory

Language Thinking

Imagination

Manipulation skill

Fig. 1.1 The composition of intelligence

Language is composed of three parts: sounds, symbols, and grammar. For a language to achieve a communicative function, the three parts must be applied comprehensively. In contrast, speech refers to the process in which an individual acquires and uses language, which is a psychological phenomenon. Children learn speech when communicating with adults and then incorporate various languages into the speech. Speech is the use of language in the process of communication. By using the same language, individuals may formulate a number of different kinds of speech. The language system is preserved in a variety of speech communication forms, which are divided into three categories: oral, written, and internal. Oral speech is the speech we hear and speak, while written speech is what we write and see. Both oral speech and written speech are external speech that others can perceive through analytical organs. On the contrary, internal speech is a form of speech that does not need to be expressed externally. Concealment is a characteristic of internal speech. It is this speech that becomes the matter of our thoughts when individuals think quietly. These three types of speech are both distinguishable and closely related. The levels of all three types of speech vary from person to person, and the different levels of speech ability are often interpreted as the concrete expression of the individual being “smart” and “not that smart.” Sensation and Perception. Red, black, blue, white, yellow, and other colors are captured by the eyes; the high, low, strong, weak, and other sounds we hear are perceived by our ears; cold, heat, pain, itchiness, and other feelings are felt by the body; the sour, sweet, bitter, spicy and other flavors are tasted by the tongue; and various fragrances and odors such as moldy, fishy and other smells are sensed by the nose. All of these are the human brain’s cognition of certain individual properties of things through what we call sensation. When we look at the heavens, there are clear skies or heavy clouds; when we look at the sea, there are calm waters or roaring waves; when we look at the land, there is an isolated tree or a picturesque landscape. All of these are examples of the overall cognition of things produced by the human brain that we refer to as perception. Sensation and perception have much to do with

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abilities such as visual, auditory, and kinesthetic abilities. In particular, observation ability is a conscious, planned, and sustained perceptual activity that is a component of intelligence. Perceptual ability varies from person to person. This becomes evident when individuals look at the same piece of cloth and see different colors, which reflects different color discrimination abilities, or when individuals listen to the same part of music and hear different timbres, which reflects different auditory discrimination abilities. In the process of sensation and perception, different individuals are shown to be “smart” and “not that smart.” Memory. Memory is the re-recognition or reappearance of what we have perceived or experienced in the past. Its content is wide and can be summarized into four types: appearance, verbal, concepts, emotion and affection, and bodily kinesthetic. After visiting a park, we recall the image of the lake and trees. This is the memory of image. After reading a book, we recall the abstract concepts, formulas, and laws. This is the memory of verbal and concepts. Sporadically, or perhaps prompted by a certain cue, we recall an exciting moment and feel inspired. This is the memory of emotion and affection. After learning how to swim at a very young age, we remain skilled in swimming many years later. This is bodily kinesthetic memory. Individual differences in memory ability were also significant. For instance, memorization methods differ, as do the abilities to recognize and recall. The quality, speed, correctness, and other features of memory vary among individuals. After A and B meet for the first time and then meet again after a few months, A may recognize B at once, but B cannot remember who A is; when reading the same novel, some people can tell the story clearly while others may forget the name of the main character. Hence, the ability of memory distinguishes between those who are “smart” and “not that smart.” Imagination. In addition to recalling images of things we have perceived in the past, we can also create images of things we have never experienced. This is the mental process of transforming and combining existing images in the human brain under the influence of objective things and mediation of language to create new images. This process is the imagination process. For example, in the story “Journey to the West,” children like the monkey Wukong Sun, who is an imaginary character. The imagination ability plays an essential role in an individual’s practical activities. Without imagination, individuals lack innovation and foresight. There are also differences in imagination between individuals. In particular, the degree of creativity, spatial imagination, reality, and foresight differ. For example, in technological innovation, some people excel at using the original machines and equipment to produce new machines through transformation and innovation. Meanwhile, others only follow routine and are lazy in their thinking, thus making it impossible for them to innovate and create. The different imagination abilities indicate the difference between those who are “smart” and “not that smart.” Thinking. Thinking is the cognition of the human brain about the nature of and regular relations between things. It is based on perception and memory, mediated

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by existing knowledge, and achieved by language. Thinking belongs to rational cognition and is a core aspect of intelligence. The main reason thinking is rational cognition is that it involves the process of abstraction and generalization. Thinking is the reflection of generalization. The process of abstraction seeks to distinguish between the essential and nonessential properties or characteristics of something in mind and then discard the latter and keep the former. The process of generalization functions to combine things with common characteristics or combine the common properties or characteristics that have been separated due to some aspect(s) and then match the essential properties of individual things to the essential properties of similar things. The process of abstraction is the process of thinking from the single to the general. Generalization is the primary characteristic of thinking. For instance, when a secondary school student asks his or her mathematics teacher, “How can I improve my mathematical ability?” the teacher replies without hesitation, “Remember to combine like terms.” “Combining like terms” is a form of generalization. In addition, the intelligence we usually evaluate, or the levels of various thinking abilities, for example, analytical ability, comprehensive ability, judgment ability, reasoning ability, etc., are all expressions of logical thinking ability. Humans cannot understand objective things, learn basic knowledge, master the fundamental laws, or create and invent without thinking ability. The abstraction and generalization processes of thinking and logical thinking ability are inseparable from teaching the next generation to absorb the knowledge of our predecessors in a short time and avoid repeating the longer, outdated cognitive processes. People have different levels of abstraction ability, generalization ability, and logical thinking ability. For example, in school, students from a class do the same mathematical problems. Some students have clear concepts, correct judgment, and logical reasoning. Some students are the opposite. The different abstraction, generalization, and logical thinking abilities represent the “smart” and “not that smart” students. Manipulation Skills. Intelligence is not exclusively about using the brain but also includes handwork, manipulation, and practice. One of the most important factors is skill. Skill is a complex system of intellectual and physical movement developed through practice using existing knowledge and experience. Skill includes basic skills based on knowledge and experience, repeated practice in a certain way or imitation, and advanced skills that are developed in a certain way with repeated practice to achieve a level of automation, which is also a technique or technique skill. Skills can be classified by their properties and characteristics into mental (intellectual) skills (e.g., mathematical operation skills) and movement skills; however, skills are commonly referred to as movement or manipulation skills. Skills differ from knowledge. Knowledge is an experience system formed in the human brain due to the generalization of experiences, while skills are generalizations of movements and are a complex movement system belonging to a particular individual. However, skills and knowledge are interconnected and transform each other. Knowledge is a prerequisite for mastering skills, and it governs and restricts

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the speed, depth, difficulty, flexibility, and consolidation of skills. Meanwhile, the formation and development of skills help to improve the level and depth of knowledge development. The different levels of manipulation skills can be used to evaluate whether an individual is “smart” and “not that smart.”

Levels of Intelligence Intelligence has not only various factors but also different levels. A survey conducted in Beijing and Shanghai (Zhang & Wen, 2017) found that approximately 3% of the population had very poorly developed intelligence, the socalled subnormal children. This is a significant number and is a noteworthy issue concerning the country’s development, especially when considering the quality of human resources. People with supernormal intelligence, also known as geniuses, are also in the minority. The supernormal child or genius is perceived as “just a little bit smarter,” meaning that one is highly capable in several aspects that make up overall intelligence or is abnormally outstanding in a certain aspect. It is formed under certain material and spiritual conditions; some such characters that appeared in the past and present are not a mystery. Except for subnormal and supernormal intelligence, most people belong to the normal intelligence level. In statistical terms, it is called a normal distribution, which is a curve with two small ends and a flat peak in the middle, that is, a symmetrical bell curve (see Fig. 1.2). Whether a person’s intelligence is normal, subnormal, or supernormal is primarily determined by the intellectual quality. Intellectual quality is the performance of intellectual characteristics in cognition-related activities; it is called intellectual thinking quality or thinking quality. In fact, quality is a personality characteristic of human thinking. It is both an indicator for evaluating the level of intelligence and a breakthrough for developing intelligence and ability. Thinking quality reflects the different levels of individual thinking and intelligence and differences in ability. One of the main purposes of school education and other teaching is to improve students’ learning quality and thinking ability. Therefore, in the cultivation of intelligence and ability, it is often necessary to grasp breakthroughs in the quality of students’ thinking so as to teach students in accordance with their aptitude.

Fig. 1.2 The normal distribution of intelligence level

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There are many components and manifestations of thinking quality. These mainly include five aspects: profundity, flexibility, creativity, criticality, and agility. • Profundity refers to the degree of abstraction and logical level of thinking activities, as well as the breadth, depth, and difficulty of thinking activities. It is manifested by the individual’s ability to think deeply about problems, generalize and categorize, be logical, have a high level of abstraction, and anticipate intellectual activity. People with supernormal intelligence have high abstraction and generalization abilities, while people with subnormal intelligence tend to think about problems directly without depth. • Flexibility refers to the degree of flexibility in thinking activities. It reflects the level of intelligence and ability to “transfer,” as we usually say, a problem with more than one solution, that is, learning by analogy. A person with high flexibility will be flexible in applying intellectual directions, will be good at thinking from different angles and aspects (from analysis to synthesis and vice versa), and will conduct “comprehensive analysis,” that is, think and solve problems from a big-picture or holistic approach. Western psychology calls this kind of thinking divergent thinking. • Creativity refers to innovative thinking or creative ability. In practice, in addition to identifying problems and thinking about problems, it is more important to solve them creatively. The development of society, the progress of science and technology, and even the innovations of individuals all depend on the creativity of thinking. Edison was able to think up thousands of inventions in his lifetime with neither equipment nor materials. Rather, his brilliant achievements were mainly due to his outstanding intellectual qualities of boundless creativity. Creativity is a higher intellectual quality shared by inventors and scientists throughout the ages. Accordingly, it is extremely important to develop creativity in younger generations. • Criticality refers to the degree of independent analysis and criticism in thinking activities. Western psychology refers to this concept as “critical thinking.” Should one follow the routine and maintain the status quo or think independently and question authority? This is a critical difference in thinking. Criticality is an important aspect of thinking. With criticality, humans are able to understand the thinking itself; that is, they are able to recognize not only the object but also the subject and change the subjective world in the process of transforming the objective world. Therefore, criticality is a manifestation of human reflective ability. • Agility refers to the speed of thinking activity, which reflects the degree of correctness and rapidity (acuity) of intelligence. People with supernormal intelligence are adept and quick at thinking and responding to problems; in contrast, people with subnormal intelligence tend to be slow. People with normal intelligence lie somewhere in between at an average level. The five aspects of thinking quality are the main basis for determining the level of intelligence and ability. In a certain sense, thinking quality is the manifestation of intelligence and ability concentrated on five aspects, namely: profundity, flexibility, creativity, criticality, and agility. Performance in these aspects of thinking quality

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Table 1.1 Comparison of mathematics comprehensive ability test scores for students at different grades Grade

Participants

Mean

First grade

Experiment

89.4

Control

85.7

Experiment

81.6

6.7

300

Control

77.2

10.6

300

Experiment

81.1

10.0

325

Control

74.0

15.6

300

Fourth grade

Experiment

87.1

8.6

345

Control

74.1

21.0

310

Fifth grade

Experiment

84.1

9.1

320

Control

67.3

24.5

310

Second grade Third grade

Standard deviation

Number

Significant difference test

8.8

300

p < 0.05

10.2

300 p < 0.05 p < 0.01 p < 0.01 p < 0.005

serves as the main indicator for determining whether a person’s intelligence is normal, subnormal, or supernormal. One pedagogical experiment combined the characteristics of various subjects in elementary and secondary schools and developed a set of specific measures to develop the students’ thinking qualities. As the experiment focused on the cultivation of thinking quality, the intelligence, ability, and creativity of the students in the experimental class developed rapidly, and the measurement indexes were much higher than those of the control class. Moreover, the longer the experiment was conducted, the more significant the difference. As an example, the performance of the mathematics comprehensive ability test was measured in the experimental and control classes in the aforementioned elementary school (Li et al., 1999), and the statistical analysis clearly illustrates this trend (see Table 1.1). As we can see from Table 1.1, students in the experimental classes generally scored higher than students in the control classes on the Mathematics Comprehensive Ability Test; the standard deviation of students’ performance in the experimental classes was generally lower than the standard deviation of students’ performance in the control classes. In addition, the significant difference in student performance between the experimental and control classes increased with grade level and the duration of the experiments; the comparison of deviation also increased year on year. In other words, the polarization in the control classes became more significant each year, while the polarization in the experimental classes remained approximately the same over the five years, with a marginal change in the polarization. From this experiment, we can see that developing thinking quality is a credible and feasible way to develop intelligence and ability and even reform all kinds of teaching, including mathematics instruction. This viewpoint will be demonstrated in more detail in Chapter Six to reveal differences in the intellectual qualities of students in algorithms and their development.

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Relationships Between Knowledge, Skills, and Intelligence Teaching in elementary and secondary school places great emphasis on developing students’ basic knowledge and basic skills (referred to as the “double basics”), which is regarded as an important task of school teaching. The concepts of intelligence and skill were discussed earlier in this chapter. What, then, is knowledge? From the viewpoint of psychology, knowledge is the summary of human social and historical experience that is mastered by people in the form of realized content. Knowledge comes from social practice, which is the basis and test of all human knowledge. Knowledge should be formed with human language as a tool, materialized into a form of experience from social practice products, and used for communication or transmission from generation to generation, for forming culture, and building the common spiritual wealth and civilization of humankind. The concepts of skills were briefly elaborated in the previous section: it is the manipulation technology mastered by people in the form of movement, be it action skills such as handwriting, gymnastics, etc., or skills in activities such as writing, calculation, etc. A technique may be developed until it reaches the stage of perfection in which the basic components of the activity have been automated, such as the aforementioned mastering of skills of swimming and cycling. The oft-quoted saying, “Practice makes perfect,” perfectly summarizes the relationship between skills and techniques. Knowledge, skills, and intelligence are closely interrelated. However, as knowledge and skills are not intelligence or a personality characteristic, they cannot represent a person’s level of intelligence. For example, when students take the same test and all receive 100, there may be great variation in the amount of effort invested or technique used to achieve the same score. Some spend a lot of time studying while others spend less; some rely on rote memorization, and others depend on personal tricks or techniques. Therefore, the evaluation of intelligence must not be simply limited to the examination of knowledge and skills. However, knowledge, skills, and intelligence complement each other, and a person who does nothing cannot progress without training or learning. At the same time, intelligence determines to a certain extent an individual’s possible achievement in the acquisition of knowledge and skills. For instance, in the process of practicing writing, individuals often use their thinking abilities. The more the brain is used, the stronger analysis, synthesis, judgment, reasoning, and other abilities are. Thus, those with a stronger thinking ability will have a faster acquisition of writing skills. Teaching in elementary and secondary schools (including kindergarten teaching) should develop students’ intelligence based on the improvement of “double basics.” At the same time, it is necessary to promote the improvement of “double basics” in the process of developing their intelligence. Improving “double basics” is the basis for developing intelligence, and developing intelligence is the purpose of enhancing “double basics.” “Double basics” and intelligence are unified in the complete teaching process. When discussing the psychological development of adolescents, Professor Zhixian Zhu, a Chinese psychologist, pointed out that the intermediate step from

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educational measures to the psychological development of children and adolescents is that they understand or master educational content, which is subject to a quantitative and qualitative change process (Zhu, 2018). It can be seen that it is important to impart knowledge to students through teaching, but this is only a quantitative change process of the development of thinking ability and intelligence. That is, it is an intermediate step and not the final goal. The important thing is the development of intelligence itself and thinking ability, which is the process of qualitative change or the ultimate goal. The focus should shift from the “quantitative change” of the development of knowledge and capabilities to the “qualitative change” of intelligence and capabilities, which is the intermediate step in the process of generalization. In the previous analysis of the components of thinking, “generalization” was mentioned, and it is this process of generalization that achieves the transformation of knowledge and skills into intelligence and capabilities. As mentioned earlier, the most significant characteristic of thinking is generalization. Generalization is a prerequisite for the formation of concepts and is the key to developing the thinking quality. Generalization ability is an important characteristic of the profundity of thinking. The process of learning and application is the process of generalization transfer. Without generalization, there can be no transfer; without generalization, there is no acquisition and use of knowledge, and one cannot acquire new knowledge; without generalization, there is no logical reasoning, and there can be no profundity and critical thinking; without generalization, there is no flexible transfer, and there is no flexibility and creativity of thinking; without generalization, there is no “reduced form,” and there can be no agility of thinking. Therefore, generalization is the basis of all thinking qualities. Mathematical ability should be considered a component of intelligence, which can also be called one of the forms of “special intelligence.” Generalization is the foundation of mathematical ability and the intuitive expression of the profundity of mathematical thinking. Generalization is a prerequisite for forming or mastering mathematical concepts. Students’ mastery of such concepts is directly governed by the generalization level. Mathematical concepts include the cognition of mathematical materials and their associated essential properties in the brain. To master mathematical concepts is to analyze, synthesize, compare, and engage in other types of abstract processes that explore and identify mathematical relationships, extract common and essential properties or characteristics from them, and then combine similar items and generalize them. Mathematical abstraction and generalization are related to the profundity of mathematical thinking, which is the ability to extract the most important things from a large number of complicated mathematical materials and distinguish the common elements from mathematical materials with different forms. The process of abstract mathematical generalization should include the following four aspects: first, the abstract generalization of mathematical concepts and mathematical laws; second, the abstract generalization of a thing; third, a broader and higher level of generalization based on the existing generalization; and fourth, the

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systematization of mathematical knowledge based on the generalization, which is the most advanced stage of generalization. The improvement of students’ mathematical ability is not unattainable, but it cannot be achieved overnight. It needs to be cultivated in a purposeful and planned manner in various teaching activities, such as classroom teaching, homework guidance, preview, review, examination, experiments, and extracurricular activities. Two simple examples illustrate the role of the generalization process in the development of mathematical ability and to further elaborate the relationship between knowledge, skills, and intelligence. In the first example, a student asks the teacher, “How much is 13 of 23 ?” The teacher did not answer directly, instead asking the student: “How much is 3 of 13 ?” The student answered: “is 1.” The teacher then asked: “How did you get that?” The student replied: “3 multiple 13 . ” The teacher then takes the opportunity to ask, “So how much, approximately, is 13 of 6, 13 of 9, and 13 of x?” The student replied, “Then, how much is 1 of 23 ?” because the student combined like items in the above question. The student 3 replied happily, “I get it!” This small example roughly illustrates that generalization in teaching imparts knowledge to students and cultivates their intelligence. Another example is that middle school students generalize the concept and solutions of a general binary linear equation system after understanding the resolution of a specific binary linear equation system, which is a generalization of mathematical concepts and mathematical laws. Students apply this solution to a variety of binary linear equation systems to concertize the generalized solution. Some equation systems are not seemingly composed of quadratic equations, but students can see the commonalities between them and a quadratic equation system and find their solutions, which is a broader and higher-level generalization. After students have studied various equation systems and their solutions, they analyze the connections and relationships between them and systematize these elements, which is an advanced stage of generalization. Some equation systems do not appear to be binary linear equation systems, such as fractional equation systems that can be transferred to binary linear equation systems or even ternary linear equations, but students can distinguish the commonalities between them and binary linear equation systems and find solutions, which leads to a broader, higher-level generalization. Thus, the process of generalization is the process of transfer; the higher the level of generalization, the wider the range of transfer, and the greater the “span.” Due to the generation, students grasp the nature of mathematical knowledge, the broader relationships and internal connections, and the regularity of mathematical knowledge. Due to generalization, students excel at discovering the connections between the mathematical knowledge they have mastered and new mathematical problems, become adept at using what they have learned to solve new problems, acquire new knowledge and skills, learn by analogy, and learn from past experience. Therefore, generalization ability is the foundation for all mathematical abilities. The improvement of generalization ability leads to a significant increase in students’ ability to learn mathematics and thus attracts special attention. In summary, the relationship between knowledge, skills, and the development of ability is as follows. After people acquire knowledge and skills, through the

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continuous process of abstraction and generalization, people will develop related intelligence and ability; at the same time, the development of intelligence and ability enables people to acquire knowledge and skills more effectively and quickly.

The Main Views of Intelligence in the International Psychological Community There are many perspectives on intelligence in the international psychological community. Below, I introduce a few that are relevant to this book.

Factor Theory and Structure Theory Factor Theory. “Factor theory” is the study of the factors that make up intelligence. As early as the late nineteenth and early twentieth centuries, Thorndike (1927) proposed a “specific factor theory” that posited four general dimensions of abstract intelligence: level, range, area, and speed. C. Spearman developed the “two-factor theory” in 1904, which claimed that intelligence consists of G, the general factors that run through all intellectual activities, and S, the special factors expressed in a special ability (Spearman, 1904). T. L. Kelly and L. L. Thurstone put forward the “multifactor theory” in the 1930s and 1940s, which stated that intelligence can be categorized into different primary mental abilities. However, Kelly and Thurstone’s ideas differ in one respect. Kelly believes that there are five factors: intelligence, number, shape, language, memory, and reasoning (Kelly, 1928). In contrast, Thurstone proposed that there are seven factors: associative memory, numerical ability, perceptual speed, reasoning, spatial imagination, verbal comprehension, and word fluency (Thurstone, 1931). Structure Theory. “Structural theory” is actually a kind of “factor theory,” but it clarifies the factors of intelligence from the perspective of structure. In other words, the structural theory emphasizes that intelligence is a structure. A British psychologist, P. E. Vernon, proposed a hierarchical theory of intelligence that claimed intelligence is a multilevel structure (Vernon, 1971) (see Fig. 1.3). Within this theory, the top level is the general factors of intelligence (g), and the second level includes two orthogonal group factors: verbal-educational factor (v:ed) and perceptual-mechanical skill factor (k:m). These two large group factors can be divided into several small group factors. The V:ed factor includes verbal, educational abilities, and numerical factors, and the k:m factor is divided into spatial, practical, and mechanical abilities.

The Main Views of Intelligence in the International Psychological …

17

Fig. 1.3 Vernon’s hierarchical theory of intelligence

American psychologist J. P. Guilford proposed a three-dimensional structure of the intellect model (Guilford, 1956). According to Guilford, intelligence is a threedimensional structure consisting of operations (cognition, memory, divergent production, convergent production, and evaluation), content (semantic, symbolic, figural, and behavioral), and products (units, classes, relations, systems, transformations, implications) comprising 120 components (see Fig. 1.4). American psychologists Schlesinger and Guttman proposed a two-dimensional radex of intelligence (Schlesinger & Guttman, 1969). This radex represents a twodimensional space divided into three polar and three modular regions. The first dimension is verbal, figural, and numerical (divide their range with straight lines). The second dimension is rule-inferring, rule applying, and school achievement (divide their range with curved lines). The variables are presented in Fig. 1.5.

Fig. 1.4 Guilford’s three-dimensional structure of the intellect model

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Fig. 1.5 Schlesinger and Guttman’s two-dimension radex of intelligence

They pointed out that as age increases, the perception and motor skills factors decrease, while cognitive factors increase for children and adolescents. In 1986, American psychologist R. J. Sternberg formulated the Triarchic theory of intelligence. The Triarchic theory of intelligence is composed of three subtheories: analytical, creative, and contextual (Sternberg, 1986). Sternberg emphasizes the relationship between intelligence, context, and experience. He believes intelligence can be divided by function and labeled metacomponents, performance components, and knowledge-acquisition components. This Triarchic theory of intelligence helps to understand the nature of intelligence from different perspectives.

Piaget’s Theory of Intelligence In Piaget’s writing, cognition, intelligence, thinking, and mind are synonymous. He always believed that the function of the mind is adaptation, and intelligence is an adaptation to the environment. In other words, the nature of intelligence is adaptation, which allows the individual to adjust to the environment. This adaptation is not passive and negative but active and positive. Piaget clearly emphasized that intelligence is an active and positive structure, as seen by this statement: The intellect at all stages assimilates the material into transformed structures, rising from primary structures of movement to higher structures of operation, which are constituted by

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organizing reality in action or in thought and not only by the depiction of reality (Piaget, 1981). In Piaget’s view, intelligence is a process of the continuous formation and reorganization of thinking structures. At each stage, there is a relatively stable cognitive structure that determines students’ behavior and illustrates the main behavioral mode; at that stage, education must adapt to this kind of cognitive or intellectual structure; that is, the teaching strategies, teaching materials, and teaching content should be adjusted and adapted according to the students’ cognitive or intellectual structure. If students’ cognitive or intellectual structures are unreasonable, they will have weak memory and poor thinking and lack flexible problem-solving skills. In this situation, any attempt by teachers to accelerate students’ development of intelligence will be a waste of time and energy. What is the cognitive or intellectual structure? Piaget initially emphasized the concept of the scheme (referring to the structure or organization of movement). The scheme could create a new scheme by accommodation, assimilation, and balance. In later years, he highlighted the integrity of the structure (logical structure of thinking), transformational (cognition is an active and developmental constructive process), and self-regulating (the balance of subject and object in the structure regulates the schema). The “construct” in Piaget’s “constructivism” differs from other structures— it is the result of the interaction between the subject and the object in which three aspects are emphasized: first, the interaction of subject and object; second, the unity of synchronicity and diachronique; and third, the category of central activity (regarding activities as the starting point and motivation for the occurrence and development of cognition).

The View of Cognitive Psychology of Intelligence In the late 1950s and early 1960s, cybernetic information theory and computer technology were developed. Behaviorism regards the human mind as a “black box.” Later, psychologists thought that this perspective was rather pessimistic and developed it into cognition psychology. In recent years, cognitive psychology has emerged. Generally, the American psychologist Ulric Neisser has been referred to as the “father of cognitive psychology.” Cognitive psychologist J.R. Anderson pointed out that cognitive psychology tries to understand what intelligence is and how people think (Anderson, 1989). Here, he specifies that cognitive psychology is the study of intelligence and thinking. However, modern psychology holds different understandings of cognition. In Neisser’s book on cognitive psychology, he claimed that cognition is the entire mental process of sensory input being transformed, coded, processed, stored, retrieved, and used (Neisser, 1967). Based on the definition by Neisser, S. K. Reed further suggested in 1972 that cognition is often simply defined as the acquisition of knowledge (Reed, 1972).

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Cognition actually includes many psychological skills, such as model identification, attention, memory, vision, image, language, problem solving, and decision making. In 1985, A. L. Glass also argued in his book “Cognition” that our collective mental abilities (perception, memory, reasoning, and others) form a complex system, and their combined functions are called cognition (Glass, 1985). J. P. Houston and his colleagues summarized various views on cognition and believed that there are five main opinions (Houston et al., 1981): (a) cognition is information processing; (b) cognition is symbolic operations in the mind; (c) cognition is problem solving; (d) cognition is thinking; and (e) cognition is a set of related activities such as sensing, memory, thinking, judgment, reasoning, problem solving, learning, imagination, concept formation, and language use. Here, there are only three opinions. The first and second are a narrower form of psychology that is information processing theory; the third and fourth believe that cognitive psychology is the study of thinking, and the last is a broader understanding of psychology. Cognitive psychology emphasizes that cognition should include three aspects: function (adaptation), process, and structure (Dodd, 1980). The most important point here is that cognition is the activity of information processing in a certain mental structure for a certain purpose. In a certain sense, intelligence is also the activity of information processing in a certain mental structure for a certain purpose. Cognitive psychology utilizes a developmental process in the study of intelligence. Contemporary cognitive psychology focuses on the study of perception and the higher cognitive factors. It examines the establishment of general cognitive models and networks of connection. The reaction time is a breakthrough in the analysis of information processing; it not only focuses on the exploration of physiological mechanisms but also on the improvement of computer design based on the features of human neural elements and neural networks. It is concerned with both the study of theory and the study of reality. Studies on intelligence and thinking in cognitive psychology have contributed to the following three main points: • Combining psychology, the psychology of thinking, and modern science and technology (cybernetics, information theory, computer science, etc.). For instance, A. Newell and H. A. Simon studied the basic model of computer simulation of human thinking. • Although it regards cognition as the main object, it is not limited to the range of cognition. It not only regards the low-level perception of high-level thinking as an indivisible continuum but also attempts to combine cognitive factors with noncognitive factors. In this way, it regards people’s mental, consciousness, cognition, and intelligence as a whole or system. • A new approach should be used to as the image of the transition from perception to thinking, explore it in a more rational way to better connect perceptual and rational cognition, and facilitate the study of the internal processes of the human mind and intelligence.

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Gardner’s Theory of Multiple Intelligences In 1983, Gardner, a professor at Harvard University, published a book, “Frames of Mind,” which introduced the concept of multiple intelligence. Gardner has been exploring this issue for 20 years. In 1993, he published another book, “Multiple Intelligence: The Theory in Practice,” which attracted much attention from Chinese readers after the publication of the Chinese version of “Multiple Intelligence” (Gardner, 1999). The six basic courses of the “Six Arts2 ” were introduced in the official schools of ancient China 3500 years ago and in the private schools of Confucius 2500 years ago: the Rite, the Music, the Archery, the Driving, the Writing, and the Numbers were designed to develop students’ interpersonal, intrapersonal, musical, bodily kinesthetic, spatial, linguistic, and logical-mathematical intelligence. How similar Gardner’s theory of multiple intelligences is to our “Six Arts.” This is the reason it has garnered the attention of many Chinese readers. Gardner proposed the theory of multiple intelligence. Initially, he listed seven components of intelligence. He believed that they were relatively different from each other and that every individual would have more or less of these seven types of intelligence. While admitting that there could be more than seven components of intelligence, he believed and supported this idea of seven components of intelligence for more than ten years. Linguistic Intelligence. Linguistic intelligence is the ability to use words. People with strong linguistic intelligence include those with strong oral skills (politicians, speech writers, storytellers, hosts, etc.) and people with strong writing skills (journalists, playwrights, poets, editors, etc.). Individuals with linguistic intelligence can manipulate the pronunciation, grammar, and semantics of a certain language and master its rules and conventions. Logical–mathematical Intelligence. Logical-mathematical intelligence is the ability to use and manipulate numbers (often the work of mathematicians, statisticians, and accountants) and the ability to apply logical reasoning (as needed for programmers, logicians, and scientists). Individuals with this kind of intelligence possess the abilities of perception, logical schema, identification, and understanding of logical relationships, statements, propositions, functions, and related abstraction. Interpersonal Intelligence. Interpersonal intelligence is the ability to quickly comprehend and evaluate others’ intentions, motivations, and emotions. Individuals with this form of intelligence are sensitive to others’ facial expressions, gestures, and tone of voice and are good at reading people (and are not offensive), thus reducing others’ negative emotions and inspiring them to take positive actions. Intrapersonal Intelligence. Intrapersonal intelligence is the ability to understand oneself and act appropriately. Individuals with this intelligence are able to honestly, accurately, and comprehensively reflect on themselves, know their strengths and 2

Six Arts: Refers to the six abilities in the aristocratic education system of the Zhou Dynasty in China.

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weaknesses, understand their motivations, desires, and mind, are self-disciplined, and have healthy self-esteem. Musical Intelligence. Musical intelligence is the ability to perceive music (enjoy music), identify and distinguish music (musical review), convert musical forms (composing), and express music (music performance). Individuals with this kind of intelligence are sensitive to rhythm, melody, etc. Bodily Kinesthetic Intelligence. Bodily kinesthetic intelligence is the ability to use the body to express thoughts and feelings and the ability to use the hands to create or transfer things (as seen in the work of craftsmen, painters, mechanics, artists, surgeons, etc.). Individuals with this type of intelligence have strong muscle coordination, balance, agility, and grace and are sensitive to touch. Spatial Intelligence. Spatial intelligence is the ability to perceive the spatial world accurately (often seen in the work of guides, hunters, scouts, etc.). Individuals with this kind of intelligence are able to perceive colors, lines, spaces, and the relationships between them, as well as visualize, represent visually or spatially and understand their location spatially. In 1993, Gardner added a new component of intelligence: naturalistic intelligence. Naturalistic intelligence is the ability to understand, relate, categorize, and interpret things in the natural world. Farmers, herders, hunters, gardeners, animal feeders, etc., all demonstrate aspects of naturalistic intelligence. At the end of the nineteenth century, Gardner added another component of intelligence: existential intelligence. Existential intelligence refers to the exploration and consideration of ultimate issues, such as the self and human nature. Theologians and philosophers have demonstrated the ability to develop existential intelligence.

Sternberg’s Theory of Successful Intelligence Sternberg, a professor at Yale University, after working on intelligence research for many years, finally proposed the theory of “successful intelligence.” This theory makes people realize that success in life is not primarily based on intelligence quotient (IQ) but on successful intelligence. Sternberg engaged in theoretical research on successful intelligence and conducted numerous practical experiments on its application. The Chinese version of his book Successful Intelligence (Sternberg, 1999) has also attracted much attention from a significant number of Chinese readers. The Concept of Successful Intelligence. Sternberg argues that we should pay less attention to traditional intelligence norms, especially the concept of IQ, and more attention to successful intelligence (Sternberg, 1999). In his book, “Successful Intelligence,” he describes that after failing IQ tests in elementary and secondary school, he encouraged himself that if he succeeded in the future, it would not be because of his IQ. For this reason, he eventually set himself on the path to exploit intelligence

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and tried to find the intelligence that would truly predict the future success of an individual. Intelligence is the ability to adapt, change, and choose one’s environment in order to accomplish the goals of the individual and the group or culture. If individuals have successful intelligence, they know when to adapt to the environment, change the environment, and choose the environment. In addition, they can balance the three. People with successful intelligence can recognize their weaknesses and disadvantages. Knowing how to develop their strengths and overcome their weaknesses is one way for people to succeed. The Components of Successful Intelligence and Its Mission. Analytical thinking, creative thinking, and practical thinking are three kinds of thinking abilities that are essential for successful intelligence. The mission of analytical thinking is to analyze and evaluate the choices faced in life, including identifying existing problems, defining the nature of the problem, determining problem-solving strategies, and monitoring the problem-solving process. The mission of creative thinking is to be the first to conceive a solution to a problem. A creative person is good at “buying low and selling high” in the world of thinking. Research shows that these abilities are at least partially different from traditional IQ, and they roughly belong to a specific field of competence. The task of practical thinking is to implement choices and make them work; if intelligence is applied to the real world, then practical thinking comes into play.

Goleman and Mayer’s Theory of Emotional Intelligence The concept of emotional intelligence was developed by J. D. Mayer and his colleagues at the University of New Hampshire in the United States in 1990. In 1995, journalist Goleman published a book, “Emotional Intelligence,” which played an important role in promoting this theory (Goleman, 1995). The concept of “emotional intelligence” that we hear often comes from this theory of emotional intelligence. What is emotional intelligence? What are its components? Meyer and Goldman proposed their theories of emotional intelligence (see Table 1.2), which are interpreted as follows (Goleman, 1995; Mayer et al., 2000). Two of the most influential theories of emotional intelligence are summarized in Table 1.2. Both theories define emotional intelligence from the meaning of its contents, but the difference is that Goleman defines it as a mixture of ability and personality (or personality trend). For example, personality traits such as enthusiasm and persistence have been added in addition to ability. However, Mayer and his colleagues oppose the definition of emotional intelligence as a mixture of abilities, dispositions, and other factors and insist on defining it as one of the components of traditional intelligence. The two theories share some similarities, believing that emotional intelligence involves multiple factors, although the amounts differ. In conclusion, emotional intelligence is a new field for psychological research that needs to be further investigated in the area of concepts, theories, etc.

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Table 1.2 Comparison of the two emotional intelligence models Theory

Mayer, etc.

Goleman, etc.

Definition

Emotional intelligence is defined as illustrating people’s ability to perceive and understand emotion, assimilate emotion in thinking, and comprehend and analyze emotion

Emotional intelligence includes self-control, enthusiasm, perseverance, and the ability of self-motivation; this kind of emotional intelligence was originally called personality

Content

1. The perception and expression of emotion 2. Assimilation of emotion in thinking 3. Comprehension and analysis of emotion 4. Reflective monitoring of emotion

1. Recognition of one’s emotions 2. Emotion regulation 3. Self-motivation 4. Recognition of others’ emotions 5. Relationship handling

Type

Capability

Combination of capability and personality

What is referred to in today’s society as an emotional quotient is a development based on emotional intelligence.

References Anderson, J. R. (1989). Cognitive psychology (Q. Yang, & S. Zhang, Trans.). Jilin Education Press. Dodd, D. H. (1980). Cognition: Mental structures and processes. Allyn & Bacon. Gardner, H. (1999). Multiple intelligences (Z. L. Sheng, Trans.). Xinhua Publishing House. Glass, A. L. (1985). Cognition (2nd ed.). Addison-Wesley. Goleman, D. (1995). Emotional intelligence. Bantam Books. Guilford, J. P. (1956). The structure of intellect. Psychological Bulletin, 53(4), 267–293. https:// doi.org/10.1037/h0040755 Houston, J. P., Bee, H., & Rimm, D. C. (1981). Essentials of psychology (2nd ed.). Academic Press. Kelly, T. L. (1928). Crossroads in the mind of man. Stanford University Press. Li, Q. A., Xin, T., & Lin, C. D. (1999). The experimental research on manifestation key words method and English words memory. Psychological Development and Education., 15(1), 34–38. (in Chinese). Mayer, J. D., Salovey, P., & Caruso, D. (2000). Models of emotional intelligence. Cambridge University Press. Neisser, U. (1967). Cognitive psychology. Appleton-Century-Crofts. Piaget, J. (1981). Pedagogy and psychology of children development (T. Fu, Trans.). Culture and Education Press. Reed, S. K. (1972). Pattern recognition and categorization. Cognitive Psychology, 3(3), 382–407. https://doi.org/10.1016/0010-0285(72)90014-X Schlesinger, I. M., & Guttman, L. (1969). Smallest space analysis of intelligence and achievement tests. Psychological Bulletin, 71(2), 95–100. https://doi.org/10.1037/h0026868 Shi, N. Z. (2008). Mathematics thoughts outlines. Northeast Normal University Press. (in Chinese). Spearman, C. (1904). “General Intelligence”, objectively determined and measured. The American Journal of Psychology, 15(2), 201–292. https://doi.org/10.2307/1412107 Sternberg, R. J. (1986). A triangular theory of love. Psychological Review, 93(2), 119–135. https:// doi.org/10.1037/0033-295X.93.2.119

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Sternberg, R. J. (1999). Successful intelligence (G. H. Wu, & W. Qian, Trans.). East China Normal University Press. Thorndike, E. L. (1927). The measurement of intelligence. Bureau of Publications. Thurstone, L. L. (1931). Multiple factor analysis. Psychological Review, 38(5), 406–427. https:// doi.org/10.1037/H0069792 Vernon, P. E. (1971). The structure of human abilities (2nd ed.). Methuen. Zhang, L. F., & Wen, F. (2017). Intelligence development screening results and influencing factors of 2889 infants at eight months of age in Beijing. Chinese Journal of Woman and Child Health Research, 28(2), 105–107. (in Chinese). Zhu, Z. X. (2018). Children psychology (6th ed.). People’s Education Press. (in Chinese).

Chapter 2

The Laws of Intellectual Development and Mathematics Learning

As a part of psychological development, the intellectual development of children and adolescents has its inherent and essential connections, which are the basic laws of their psychological development. As early as the 1960s, Professor Zhixian Zhu summarized the basic laws of children and adolescents’ psychological development into four issues based on the research results on the topic: nature and nurture; internal and external causes; education and development; and age and individual differences (Zhu, 2018). These four questions systematically revealed the fundamental laws of the psychological development of children and adolescents. The four laws constrain the whole process from birth onward, providing a theoretical basis for both research and education on psychological development in China and mathematics learning in kindergartens, elementary schools, and secondary schools.

Nature and Nurture in Intellectual Development The relational issue of nature and nurture is concerned with the conditions of intellectual and psychological development of children and adolescents. Is their intellectual development determined by genetic predisposition, or is it determined by the environment and education? This is an endlessly debated issue and a high-cost evaluative project both in China and in other parts of the world. In ancient China, there was a “Theory of Xing (nature, 性) and Xi (practice, 习).” The term “Xing” refers to the natural disposition, which is the feature with which one is born; “Xi” refers to the habits and characteristics which is one’s acquired social nature. The Theory of Xing and Xi was introduced in The Book of Documents 4000 years ago, and Confucius further made it clear in the Analects of Confucius: Yanghuo that “By nature, men are nearly alike; by practice, they get to be wide apart (Confucius, 540 BC 400 BC/2011, p. 283).” This issue of Xing and Xi has been debated in China for thousands of years. The relationship between nature and nurture is one of the most

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expensive research projects in the world. There are also different views in the educational community. The nativist theory, theory of maturation potency, and theory of environmental determination have all been developed. In the following sections, I will present my own perspectives while reviewing these three opposing viewpoints.

Three Opposing Viewpoints The Nativistic Theory. The nativistic theory states that a human’s abilities, intelligence, mental content, and psychological individuality are not formed by nurture but given by nature. The extent and expressions of innate factors that the nativistic theory adopted are distinct. There are explanations for innate talents employing mysterious powers outside the individual. For example, Plato believed that the innate ideas of humans are a reflection of the intelligible world. Descartes believed that the idea of God, the idea of eternity, and the idea of essence were all gifts. The soul and the body were determined by a third entity, that is, by God, and the innate ideas he believed could only be explained by God (Stich, 1975). Some explain innate talent by genetic qualities, arguing that human intelligence, ability, and even mental content are mainly inherited. For example, in the nineteenth century, a British eugenicist, F. Galton, believed that human fame and achievement were derived from superior intelligence, which was inherited (Fancher, 2004). An American psychologist W. McDougall believed that all human behavior is driven by instinct and instinctual differentiation and that this instinct is an innate mind–body tendency (Krantz & Allen, 1967). Gestalt believed that perception and thinking are structured and that an individual, when perceiving or thinking, is constantly seeking and reorganizing these structures in his mind until he discovers the perfect structure. This discovery is “insight” (Weisberg, 2015). Gestalt’s theory considers the human ability to explore and discover new structures to be innate, so it is also a theory of innate ideas. Until 1969, American psychologist A. R. Jensen still discussed race and intelligence and even heritability of intelligence from innate ideas (Jensen, 1970). Theory of Maturation Potency. The American psychologist A. L. Gesell’s perspective on children’s psychological development is based on the theory of maturation potency (Gesell, 1929; Gesell & Thompson, 1934). He believes that there are two primary factors governing children’s psychological development: maturation and learning. He focuses most on maturation and believes that it is related to the internal environment, which is genetically determined, while learning is related to the external environment, which is the acquisition of experiences and behavioral changes and only contributes to maturation. Gesell illustrated the role of maturation with identical twin stair-climbing experiments (Gesell & Thompson, 1929). Child T performed 10 min of stair climbing per day for 48 weeks, while child C did not. On the 54th week, after six weeks of training, C started stair-climbing training every day for six weeks. Ultimately,

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C proved that climbing stairs with the same ability as T could be achieved after only two weeks of training. C started training later than T, reducing training time by two-thirds. It is clear that there is no learning without maturity, and learning is merely a facilitator of maturation. He points out that the form concept of experimental age and the function concept of maturity level are indispensable for common sense in reality and the science of children’s development. Therefore, it is necessary to consider the appropriate age at which behaviors should appear when guiding children’s development. Gesell observed, recorded, and filmed a number of infants and children. The data he obtained were relatively objective, characteristic, and valuable. Gesell’s maturational theory became the dominant view of early development. His argument that children’s psychological development depends on heredity and their natural growth ability, while environmental factors are only facilitators and have been controversial and debated by others. Theory of Environmental Determination (Environmental Determinism). The theory overstates the role of the environment and education in children’s psychological development, believing that the intelligence and even psychological development of children and adolescents is mechanically determined by the environment and education, denying the role of intelligence, psychological characteristics and genetics, and denying their initiative and self-awareness (Livingstone, 2011). American Behaviorism is a typical example of environmental determinism, from J. B. Watson, its founder, to B. F. Skinner, the proponent of reinforcement theory, to A. Bandura, the advocator of social cognition theory. Although many innovations exist in their theories, they all claim that the environment determines intellectual and even psychological development. In other words, they exaggerate the role of the environment, assuming that education is omnipotent and that it is possible to arbitrarily train children and adolescents into whatever is needed by society as per the will of educators. Simultaneously, they ignore or deny the existence of objective laws such as external behaviors from internal causes and the law of step by step. This view contrasts nature and nurture, heredity and environment, physiology and psychology, maturity and education, affirming one factor and denying another, and hence is metaphysical. These opposing factors should be viewed comprehensively. Any manifestation of the development of individuals’ psychology is the common product of the interaction of nature and nurture, heredity and environment, physiology and psychology, maturity, and education. Nature comes from nurture, and nature promotes nurture. The developments of modern science confirm that genetic qualities and physiological maturity are the biological prerequisites for the psychological development of children and adolescents. The social environment and education could make the possibility a reality based only on biological prerequisites.

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Heredity and even Physiological Maturity Are the Biological Prerequisites for Intellectual Development The Role of Heredity. Heredity is a biological phenomenon that transmits many biological characteristics from ancestors. Inherited biological characteristics mainly refer to the anatomical and physiological characteristics that are inherent to the body, such as its structure, morphology, and sensory and nervous system characteristics. Good genetic factors are undoubtedly the material basis and prerequisites for the normal development of intelligence. Hence, heredity is the biological prerequisite for psychological development. People are only given physiological and anatomical traits, which are mainly expressed in the structure and function of the brain and neurological system, but genetic qualities are the only biological prerequisites for the formation of the individual’s intelligence and certain temperamental traits and do not determine the mental content and tendencies. The nativistic theory overstates the role of heredity. Heredity influences intellectual development by talent. For example, someone born deaf cannot be a singer, and someone born color-blind cannot be a painter. Evidence suggests that when comparing dizygotic twins with different genetics, identical twins share similar levels of mental ability, memory ability, language development, intellectual acuity, flexibility, and abstraction ability. We recognize that some of the characteristics of genetic qualities are conducive to the formation of certain abilities. For example, are the achievements of young painters, musicians, and arithmeticians who show early talent and super intelligence related to genetic qualities? This is entirely possible. Since some features of the nervous system may provide favorable conditions for forming certain intelligence, it is entirely reasonable that the collection of these features promotes the emergence of certain intelligence at an early stage. It is also beneficial for us to understand this to do a better job in school education. Elementary and secondary school education in China is basic, and it is important to pay more attention to the selection and training of students from their childhood onward. For example, some students have physiological advantages or gifts in music, and their fingers are longer and more flexible; if possible, would it not be advisable to raise them to play the piano? Some students have good voices and clear tones; would it not be advisable to cultivate them to be singers? Some students have physiological advantages in developing sports; would it not be advisable to foster them in sports consciously? Some students are sensitive to the relationship between numbers and space, and if educators intentionally develop their mathematical abilities, they may become mathematicians. The Role of Maturity on the Emergence and Development of Thinking. The emergence and development of intelligence must be based on the development, changes, and maturation of physiological abilities. The regularity of physiological changes in children and adolescents, such as the procedure and process of changes in the brain’s weight, the gradual development of brainwaves, the level of neurological

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connections established in the brain, etc., is the physiological basis for age-specific characteristics of intellectual development. 1. The relationship between brain weight and intellectual development: 390 g for newborns; 660 g for infants at eight or nine months; 990–1010 g for babies between two and three years of age; 1280 g for children between six and seven years of age; and 1350 g for nine-year-old children. The average brain weight of 12- or 13-year-old teenagers is similar to that of an adult, which is 1400 g (Dekaban & Sadowsky, 1978). It is interesting to note that in a study about the formation of mathematical concepts and the development of arithmetic ability in children, it was found that developmental changes in children’s mathematical and arithmetic thinking abilities were consistent with changes in brain weight. The transitional periods of brain weight changes, as mentioned above, eight to nine months, two to three years, and nine to ten years (third and fourth grade), are also accelerated periods of development of mathematical and arithmetic thinking abilities (Lin, 1980, 1981), and this is not a coincidence. 2. The relationship between brainwaves and intellectual development: The electroencephalogram (EEG) is a record of variations in electrical activity recorded from different regions of the head by electrodes. These variations are generated spontaneously by certain nerve cells or induced by stimulation in the cerebral cortex. These variations are displayed through the amplifier on the screen or recorded by an ink pen controlled by output voltage on the continuous moving paper, forming regular brain waves. Frequency (expressed as “period/second”) is the most important parameter in brain development and is one of the most important indicators for studying children’s brain development. The study found that the general trend of the brain waves in Chinese participations aged 4–20 years was a gradual increase in the frequency of the alpha (α) wave (8–13 Hz). The development of the brain occurs through the struggle between the α wave and theta (θ ) wave (4–8 Hz), which ends with the θ wave gradually giving way to the α-wave. There are two significant acceleration periods or “leaps” in the brain development of these Chinese participations. The first significant acceleration period occurs at the age of 5–6 years, marking the most intense struggle between the occipital α wave and θ wave. The second significant acceleration period occurs at the age of 13–14 years, marking the end of the struggle between the α wave and θ wave in almost the entire cerebral cortex, except the frontal lobes (Liu, 1962). It is also interesting to note that research found that the ages of 5–6 years and 13–14 years (eighth grade) coincide with the critical ages for the development of thinking, especially logical thinking, in children and adolescents (Lin, 1980, 2013). This critical age issue will be elaborated in chapter four. 3. Study on the neural system mechanism of mathematics learning: The State Key Laboratory of Cognitive Neuroscience and Learning of Beijing Normal University studied the neural system mechanism of mathematics learning. By

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comparing the ERP and fMRI, Qi Dong, Xinlin Zhou, and others effectively tested their arithmetic equation coding differential hypothesis, stating that due to the influence of learning experiences, people mainly use visual-number representations for addition and subtraction Eqs. (9 + 7 = 16, 8–3 = 5) and auditory-verbal representations for multiplication Eqs. (3 × 7 = 21). In ERP experiments, it was found that one-digit multiplication evoked a larger N300 component that labeled speech processing on the left frontal scalp electrodes than addition and subtraction; and in fMRI experiments, it was found that addition equations activated more regions of the brain related to visual and spatial sensory processing on the right parieto-occipital of the brain, and multiplication equations activated more regions of the brain related to speech output on the left motor area, supplementary motor area, and posterior superior temporal gyrus. These results were published in the journals NeuroImage and Neuropsychologia (Zhou et al., 2006b; Zhou et al., 2007a). Furthermore, two recent ERP experiments on number processing proved the above hypothesis once again. Qi Dong, Xinlin Zhou, and others have discovered some cognitive and brain mechanism characteristics of number processing in Chinese groups from multiple perspectives. By using the Stroop-like task of numbers (comparing two numbers, e.g., 2 and 7), they found that automatic processing of numerical magnitude was present in Chinese children around the age of five, whereas it did not occur in Western children until after the age of seven. The problem size effect (such as 9 + 7 is harder than 3 + 2) in solving simple addition problems is common among Western adults. However, they found that participants who responded quickly had a reversed problem size effect, subjects who responded slowly had a normal problem size effect, and there was a significant positive correlation between the problem size effect and the reaction speed. For Chinese college students, the addition and multiplication equations in the Chinese multiplication table could produce stable, sustained EEG difference patterns and cortical activation difference patterns. There are no similar reports in studies on Western adults. As Chinese students need to memorize the Chinese multiplication table, research on ERP found that multiplication inside and outside the table leads to entirely different EEG activity patterns in college students and that stable, sustained differences occur after 120 ms of second digit sound presentation and are not present in Western adults. The results above were published in the journals Memory and Cognition, Neuroscience Letters (Zhou et al., 2007b, 2007c). Qi Dong, Xinlin Zhou, and their colleagues proposed an event-related hypothesis of cortex localization of digital semantic processing using the numerical repetition experimental paradigm and the spatial-numerical association of response codes (SNARC) effect and found that children as young as four years old have a spatial representation of numbers (mental number line) and that the spatial representation of two digits was holistic rather than compositional, but the processing method was a parallel processing target for both holistic and compositional. These results were published in the journals Cognition and NeuroReport (Zhou et al., 2006a, 2008). From the above, we should properly estimate the role of maturation of genetic and physiological factors in psychological development. It is incorrect to deny the role of

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heredity and physiological maturation, but it is also wrong to overstate them. Teachers should properly address the differences in students and psychological development caused by genetic and physiological maturation factors and adopt particular and appropriate approaches to educate them in a timely manner.

The Role of Environment and Education Genetic qualities only provide the possibility of intellectual development, while environment and education define the actuality of human intellectual development. The environment is a decisive condition for intellectual development, especially when it is dominated by planned and purposeful education. The difference in the intellectual development of most people with little difference in genetic quality is due to the role of environment and education, which is the socio-historical constraint of intelligence. It proves that human intellectual development is in accordance with the development of social productivity, science, and technology, acquiring different contents and forms in accordance with the changes in culture and education, and acquiring different components, levels, and intellectual qualities, as discussed above. Therefore, intellectual development depends largely on accepting and mastering the achievements of human history. This acceptance and mastery, in turn, are conditioned by the position and activities of the individual in the social system. For example, some young genius college students, if they are not raised in families that value early childhood education and do not have access to good teachers, will not reach the level of intelligence they have the potential to achieve. Some psychologists have asserted that “talents are shaped by early education” and state that one of the reasons for the success of great scientists is the emphasis on early education, especially mathematical training. They would not achieve high levels of attainment without the influence of the environment, education, and training. We often say that in the development of intelligence, an important method is “temperament attention.” Temperament represents the nerve type of an individual, and it has the distinction between strong and weak, flexible and inflexible, and balanced and unbalanced. Temperament itself is neither good nor bad. It is always expressed in the social activity of an individual, acquires a particular social significance, and becomes an individual’s positive or negative intellectual characteristic. Individuals with a choleric temperament are impatient, which can develop into lively and agile intellectual qualities and may also be impulsive; individuals with a sanguine temperament are flexible, who can develop into lively and witty persons with outstanding intellectual qualities of divergent thinking and may be hesitant and indecisive; individuals with a phlegmatic temperament are calm, which can develop into meticulous, independent, and calm intellectual qualities and may be stubborn and dull; and individuals with a melancholic temperament are sensitive, which can develop into thoughtful and profound intellectual qualities and may be suspicious. This requires teachers to understand the students’ temperament types and actively guide them to

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behave appropriately and thus develop their temperaments into excellent intellectual qualities. Education plays a dominant role in intellectual development. The decisive role of social conditions of life in the development of human intelligence is often realized through education. Education is made by some educators who select environmental influences based on specific educational purposes, form educational content, and adopt educational methods to systematically influence the mental development of the educated persons. The dominant role of education is not separate from the dynamic role of the teacher. In a sense, the dominant role of education is mainly reflected in the dominant role of the teacher. The investigation found that the level of intellectual development of elementary and secondary school students largely depends on teachers’ teaching. For example, in the mathematics teaching at the experimental site, I highlighted the cultivation of the intellectual quality of students’ thinking. The results showed that the teaching time was shortened, and the intellectual development of the students was also outstanding (Lin, 2006). The purpose of understanding the decisive role of environment and education on intellectual development is to create an environment that is conducive to the intellectual development of children and adolescents, to change those unfavorable factors in the environment and to guide them in a purposeful and planned manner to promote their healthy growth. The great English mathematician and physicist J. C. Maxwell’s outstanding achievements in mathematics and physics were due to his father’s insight and guidance. Maxwell’s father asked his young son to draw on a vase full of golden chrysanthemums, and when he saw his son’s drawing, he could not stop laughing. The drawings were full of geometric figures: the vase was a trapezoid, the chrysanthemum was a cluster of large and small circles, and the triangles of different sizes probably represented the leaves. This drawing made his father discover his son’s mathematical talents, fostered them, guided him, and young Maxwell became involved with mathematics from then on. Although environment and education play a decisive role in the intellectual development of children and adolescents, they should not be interpreted mechanically and simplistically. This effect is always realized through the internal causes of the intellectual development of children and adolescents.

The Internal and External Causes in Intellectual Development The issue regarding the relationship between internal and external causes is not only a philosophical topic but also a matter on which Chinese culture places great emphasis. As early as 1500 years ago, Yan Chen in the Song Dynasty used the “three causes” theory of internal, external, and noninternal/nonexternal causes to discuss the traditional Chinese medicinal methodology. He stated that the external causes of wind, cold, summer heat, dampness, dryness, and fire are affected by

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the “seven emotions”: internal causes of joy, anger, worry, anxiety, grief, fear, and fright. Similarly, ancient Chinese educational thoughts emphasize the realization of Confucius’ “introspection” (a reflective looking inward) rather than merely preaching the critical point for talent cultivation, which also illustrates the relationship between internal and external causes. In education, the teaching materials, teaching methods, and teaching aids used by teachers are all external causes, and these external causes must work through the internal causes of students.

The Internal Causes in Intellectual Development Psychology refers to the internal causes of intellectual development as “motivation,” which is the driving force of mental development. Professor Zhu (1981) believed that, in the process of people’s active activity, the contradiction between the new needs caused by the demands of society and education and their current psychological level or structure is the motivation for their psychological development. The motivation of intellectual activity or psychological development is first realized through an individual’s activities or practices. Intelligence is generated or produced in the process of activity and is also manifested in activities or practices. There is no intellectual motivation at all without activity or practice. Moreover, the prerequisite for promoting students’ intellectual development is creating conditions for various activities or practices. The motivation for intellectual development is new needs and the limitations of the original level or structure. Need is the reflection of the objective demands of people. It is expressed as motivation, purpose, interest, preferences, aspirations, desires, ideals and beliefs, and other forms. Traditional Chinese culture uses “will (Zhi, 志),” aspiration, ambition, and the ideal to express needs and their constructs. The original level is an open, self-organized, dynamic system structure that can both assimilate and adapt, including the original knowledge level, intellectual level, and intellectual (or thinking) structure. In terms of knowledge level, this includes what students have already understood, what they have mastered, what they do not understand, what they have mastered but are not comprehensively understood, what they have not performed, etc. The teacher should have an idea about these aspects but should do a good job of mapping to acquire basic information and take this “basic knowledge” into account when preparing the lesson. In terms of intellectual level or structure, it refers to the student’s original verbal, perceptual, memory, imagination, thinking, and manipulative skills; the student’s intellectual quality; what the student’s intellectual qualities are; what kind of attention status the student has at the time, whether they are concentrating or distracted, etc. Teachers’ methods of teaching must be based on the students’ initial levels. Otherwise, it will be challenging to foster intelligence.

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In one aspect, students achieve a certain level of knowledge and intelligence; in another, they may generate certain motivations and needs. These two aspects are the internal causes or internal contradictions of intellectual development, and they are interdependent and mutually conditional. Without a certain level of knowledge and intelligence, it is impossible to generate new motivations and needs. It is impossible to make progress on the current level with a certain level of knowledge and intelligence without guidance or new motivations and needs for learning. At the same time, the current level represents the “old” side of things, and the motivation and need to learn new things to develop intelligence represents the “new” side of things; “new” and “old” are a pair of contradictions. The task of teaching is to constitute this pair of contradictions, and the teacher’s duty is to guide the movement of this pair of contradictions in accordance with the objective laws, gradually and progressively, so that the student’s intelligence improves. The teacher uses appropriate methods to stimulate the students’ internal contradictions or cognitive conflicts, thus mobilizing their internal motivation, and the students solve their problems through the realization of their potential self-improvement and, ultimately, psychological development. Mathematics is an important field of study that is logical, systematic, highly organized, and has a high degree of abstraction. Mathematics learning entails possessing the corresponding logical thinking ability and intellectual development as the subjective foundation; at the same time, mathematics learning can also stimulate new needs for logical thinking ability and intellectual development, and this new need keeps deepening. It constantly constitutes a contradiction with the original level of thinking and intellectual level and becomes the most active and dynamic motivation to improve students’ intellectual development. The following two examples illustrate this. Second-grade students are proficient in one-step arithmetic word problems, reflecting their arithmetic thinking ability of direct reasoning. If they are presented with two-step arithmetic word problems, they may feel embarrassed. Because the two numbers needed to solve two-step arithmetic word problems are often directly stated in the question, the other must be found through indirect reasoning to complete the calculation task. In this way, the need to learn to solve two-step arithmetic applications arises. The feeling of being “embarrassed” is the contradiction between the new needs and the old level. According to the second-grade mathematics textbook, teachers explain the knowledge of two-step arithmetic equation problems, students understand and master them, the contradiction is temporarily resolved, and their initial arithmetic thinking ability of indirect reasoning is also developed. Another example is that students initially mastered some basic knowledge of algebra and geometry after seventh grade. In their thinking ability components, logical, abstract components gradually dominate and have a larger specific image component. This is their intelligence level at this time. From the eighth grade, the school assigns more abstract and more proof-based knowledge of algebra and geometry, which creates a new need for students to develop higher levels of abstract logical thinking ability. This new need promotes the development of students’ empirical abstract logical thinking ability to theoretical, abstract logical thinking ability in the process of contradicting the previous level. Studies show that the intelligence level of eighth-grade students is the beginning of true logical thinking abilities.

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In mathematics teaching and learning, the internal conflict between the student’s intellectual development needs and their original levels and states of intellectual development constitutes an internal contradiction in their intellectual development, which is the driving force for their intellectual development in mathematics teaching and learning. The requirements of teaching suggest new intellectual development needs. How should we meet these requirements? Should the requirement be difficult or easy? What is the most appropriate requirement? In mathematics teaching, the easy requirements make students feel “unfulfilled,” which cannot stimulate their interest and curiosity and cannot form their desire to learn mathematics. That is, new needs are challenging to generate, and the motivation system has no power. In this way, not only do students fail to learn mathematics, but mathematics teaching fails to promote their intellectual development. Some teaching experts propose a high difficulty, fast speed, and heavy rational teaching method. Here, it is worth studying the appropriate level of “difficulty,” “speed,” and “rationality.” A level that is “too difficult” compared to the students’ original level causes them to feel “daunted” by the knowledge and skills to be learned and does not stimulate new needs; the result would be the opposite of what is desired, resulting in the “indigestion” problem. The most appropriate requirements are higher than the students’ original level of intelligence and knowledge but can be achieved through the students’ subjective efforts. In other words, the requirement of “jumping for picking” is the most appropriate. To promote the intellectual development of students, we should constantly propose requirements that are difficult but can be achieved with efforts to suit their original level and give them a direction to work hard, and such requirements can also make students have a strong need to learn new things, stimulate their cognitive conflicts, and thus promote their intellectual development. Teachers should pay attention to this issue.

The Role of Human Agency and Intelligence The development of intelligence depends on internal causes, and it is the motivation of psychological development (the contradiction between the new needs and the previous level), as mentioned above. This motivation is subjective, and it is reflected in objective practice, which is manifested as human agency. We cannot discuss intellectual development without human agency. S. J. Ceci, who proposed the “bioecological theory” in 1996 (Ceci et al., 1996), believes that intelligence is a function of the interaction between innate potential, environmental context, and internal motivation. This theory is worth studying and learning from. Regarding the issue of human agency, two aspects will be demonstrated here. Diligence Produces Genius. There are Chinese proverbs stemming from ancient times that are still prevalent today: “diligence produces genius” and “diligence makes

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up for deficiency.” The realization of human agency is highlighted by constant effort and hard work, which is summarized as diligence. Genius belongs to the category of ability, and psychology refers to it as a complete combination of highly developed abilities. The combination of multiple abilities is required for a person to perform a certain activity effectively. Psychologists study the elements of genius from the aspect of multiple intellectual factors and believe that genius consists of profound insight, good memory, quick thinking, rich imagination, deep practical ability, and unique creative ability. A young man asked Einstein to disclose the secret of his success. Einstein wrote the formula A = X + Y + Z . A is success, X is hard work, Y is the right way, and Z is talking less. It is impossible to achieve complete growth of intelligence without hard work and diligence. Only through subjective efforts to solve internal contradictions of intellectual development can we lead to intellectual development. Since ancient times, politicians, strategists, litterateurs, scientists, etc., all succeeded only after long and persistent hard work and tenacious struggle. It is impossible to succeed or promote intelligence without indomitable will. This is the truth that diligence produces genius. There is an article about the famous British mathematician Gauss. Gauss still devoted himself to research while his wife was seriously ill. One day, the servant told him that his wife was getting worse, and Gauss seemed to have heard him, but he kept his head at work. Shortly afterward, the servant came to him again and told him that his wife was very ill and asked him to see her immediately. Gauss only replied that he would, but he was still thinking. The servant came a third time to tell him that his wife was dying and asked him to see her one last time, but Gauss replied: “Tell her to wait.” I object to such propaganda that is contrary to people-oriented thinking and shows a lack of humanity. My intention in presenting this anecdote is to illustrate that Gauss’s intelligence came precisely from his dedication and indomitable will to his work. Interest Is a Critical Opportunity to Develop Intelligence. The role of human agency should be based on interest. This is mainly focused on internal motivation. Despite the complexity of the motivational system, it is recognized that interest is the most direct, active, and long-lasting component of motivation in humans. Let us look at some biographies of scientists or outstanding people. We find that the inventions and achievements of many of them are often inseparable from their interests in certain areas. Interest is a cognitive tendency with emotional color and is an important part of human agency. Interest is a form of expression of the new needs of intellectual development and an important component of the motivation system. With interest, one’s intellect can be activated; with interest, one’s wisdom can shine through. Confucius stated more than 2000 years ago: “Those who know it are not as those who love it; those who love it are not as those who find their joy in it” (Confucius, 2011, p. 11).

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Anyone who has achieved success is passionate about his career or profession, even to the extent of obsession. We are all familiar with the story of the famous British scientist Isaac Newton. Once, Newton invited his guest to dinner, but he stayed in the laboratory to do experiments. Suddenly, he remembered that the guests had not yet eaten and immediately went to the living room. When he saw that there was no one at the table and saw fish spines, meat bones, empty glasses, and bottles scattered on the table, he tapped his head and said to himself: “Hey, I do have a bad memory, did I not just have dinner with the guests?” There are countless scientists such as Newton who are obsessed with their pursuit and enterprises. The secret of genius lies in strong interests and hobbies, which are an important motivation for diligence. Intellect, interests, and hobbies are mutually dependent. Interests and hobbies attract people to engage in activities, which in turn promote the development of intelligence. The successful performance of the activity further develops interest and hobby. Teachers should take students’ interests and hobbies as an opportunity to develop their agency and a certain kind of intelligence. Interests and hobbies are catalysts constantly motivating students to explore and practice discovering their intellectual development path. I frequently hear from society the call to “burden reduction,” that is, to reduce students’ excessive burden of learning and mental burden. Experienced mathematics teachers are good at cultivating students’ interest in solving mathematical problems, gradually leading them to love mathematics, thus developing their thinking abilities and intelligence and laying a solid intellectual foundation for their future in-depth study of science and technology. The famous mathematician Jingrun Chen’s interest in the “Goldbach Conjecture” came from the inspiration and training of his high school mathematics teacher, Mr. Yang. However, some mathematics teachers and parents simply evaluate students by their exam scores, which is not a comprehensive judgment. The scores are important, but for students with poor test scores, teachers should analyze their situation in detail, promote their strengths, and guide them, especially in developing their interest in learning mathematics. For those students who have strengths in other areas but are not as good in mathematics, we should also pay attention to cultivating and caring for them while recognizing their strengths and try to enhance their interest in learning mathematics and never suppress or trample on their interest and hobbies.

Education and Intellectual Development Since education plays a leading role in developing thinking and intelligence, what is the relationship between appropriate education and the development of thinking and intelligence? What are the manifestations of the leading role of education? How is this achieved?

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Children’s and Adolescents’ Parameters Regarding Thinking and Intellectual Development There is an indicator issue in the thinking and intellectual development of children and adolescents. This issue refers to the parameters of development. Let us refer to the research on the indicators and patterns of physical growth and development of children and adolescents in China (Ye, 1980). Considering, for example, Chinese children and adolescents’ thinking and intellectual development, the parameters of intelligence and even psychological development are summarized as follows. • Psychological development transits from quantitative change to qualitative change. The development of thinking and intelligence in children and adolescents is complex, from tiny, unobtrusive quantitative changes to fundamental, sudden qualitative changes. Their thinking goes from intuitive action thinking to concrete image thinking to abstract logical thinking through a series of more evident and stable qualitative change processes. • The time of development. Children’s and adolescents’ thinking and intellectual development have a certain procedure, both continuous development and stages, which are characteristic of each age. Time is needed from the germination of thinking to the emergence of logical thinking, the emergence of dialectical thinking, and the maturation of thinking. • The speed of development. The speed of development of thinking and intelligence is not a straight line increasing over time but rather wavelike and unequal, with a steady rate of development and a period of acceleration (that is, the critical years discussed in the next section). • The coordination of development. The entire psychological development process is unified and coordinated, but sometimes it is unbalanced. The development of thinking and intelligence in the mental process has a relationship between commonality and individuality. The development of thinking and intelligence, despite its specificity, depends on its relationship with the whole mental structure. • The relationship between physiological and psychological development. As mentioned earlier, physiological maturity can be used as a reference indicator for thinking and intellectual development. • Differences in development. The difference in the development of thinking and intelligence shows that the individual’s thinking and intellectual development have their own peculiarities. By comparing the indicators of thinking and intelligence of each child and adolescent with the norm, the level of thinking, intelligence, and even psychological development can be determined. The six parameters mentioned above can also be regarded as indicators of students’ development of mathematical abilities.

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The Role and Function of Education in Intellectual Development What is the role and function of education in intellectual development? As mentioned earlier, first, education becomes a necessary condition for the possibility of intellectual development to become a reality; second, education can accelerate or delay the process of intellectual development; third, education enables intellectual development to express specific forms and individual differences. Experienced mathematics teachers will understand this well. There are many ways to develop intelligence. Teaching and learning mathematics are important aspects of promoting students’ intellectual development. The purpose of teaching mathematics in elementary and secondary schools should enable students to acquire the basic knowledge of mathematics, master skills such as calculation, graphing, and measurement, and apply this knowledge and skills to life, study, and even to nature; through mathematics teaching, students develop their operational ability, spatial imagination ability, and logical thinking ability so that the mathematics teaching task is subordinate to the overall educational level. One of our educational tasks is to promote the psychological development of the educated, especially the development of intelligence and abilities. From educational measures to the realization of intellectual and psychological development, as described in Chapter One, is intermediated by the student’s understanding or mastery of the educational content and goes through a certain process of quantitative and qualitative change. After experiencing education and teaching, it is very important for students to gradually understand and acquire knowledge and experience. Regarding the knowledge and experience that need to be mastered and acquired, in terms of content, there are ideological and moral, academic knowledge, basic activity experience, etc. In terms of form, there are basic knowledge (including basic concepts) and basic skills (some further developed into technique). Acquisition and mastery of knowledge and experience is the intermediate stage from education to intellectual to psychological development, which is a process of “quantitative change” for intellectual to psychological development and is the basis for the “qualitative change” of intellectual to psychological development, as shown in Fig. 2.1. We can see from Fig. 2.1 that intellectual development must not be limited to the understanding and mastery of knowledge and experience. The development of intelligence is inseparable from knowledge and experience; however, it is not only about the improvement of basic knowledge and skills but also about the development

Education

Repetitive implementation

Acquisition Knowledge experience

Constantly internalization

Mastery

Fig. 2.1 A diagram of the relation between education and development

Development

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of observation, memory, thinking, imagination, language abilities, and manipulation abilities (including experimental abilities and practical abilities), especially thinking quality, by guiding students to experience, understand, think, and explore. Therefore, the purpose of education and teaching is not only to make students understand and acquire knowledge and experience but also, more importantly, to develop their intelligence. Only when intellectual and even psychological changes are evoked does the process of qualitative changes from education to psychological development emerge. The relationship between the understanding and mastery of knowledge and experience and intellectual and even psychological development is a relationship from quantitative change to qualitative change. This is a requirement in traditional Chinese culture, prevalent from over 2000 years ago, and called “XunKuang: Quanxue,” elaborated by the metaphor, “There is never an end to learning. The dye extracted from the indigo is bluer than the plant, so is the ice colder than the water.” This means that education could make an individual’s intellectual nature undergo a process of quantitative accumulation and produce a qualitative change, such as the transformation of green into blue and water into ice. The relationship between knowledge, skills, and intelligence is one of the most discussed topics in the education and psychology community. Knowledge and skills are not equal to intelligence, and the level of knowledge and skills does not necessarily mean the level of intelligence of a student; these aspects are complementary, and the development of intelligence is accomplished in the process of mastering and applying knowledge and skills. Try to think about how a person who does nothing, who does not study, who does not train, could have developed intelligence. Knowledge and skills are the basis for the development of intelligence. That is, the level of intelligence depends on the knowledge and skills that the student has acquired. At the same time, intelligence, to a certain extent, restricts the possible achievement of knowledge and skills, and the development of students’ intelligence and ability can promote the improvement of “double basics.” Therefore, the basis of mathematics teaching is to impart mathematical knowledge. Teachers must simultaneously consider how to cultivate and improve students’ mathematical ability, that is, arithmetic, spatial imagination, and logical thinking, to achieve the goal of teaching mathematics in elementary and secondary schools. Specifically, as stated in the Mathematics Curriculum Standard for Compulsory Education (2011 edition) issued by the Ministry of Education: “In the mathematics curriculum, emphasis should be placed on developing students’ sense of number, symbolic consciousness, spatial ideas, geometric intuition, data analysis concepts, arithmetic skills, reasoning skills, and modeling thinking…application and innovation ideas” (Ministry of Education of the People’s Republic of China, 2012); in other words, to develop students’ abilities in mathematics.

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Heuristic Teaching in Mathematics How should we help students acquire knowledge and skills by promoting their interests and hobbies and then qualitatively develop to the basis of intellectual development? One of the most important methods is heuristic teaching. By focusing on heuristic teaching and teaching in accordance with students’ aptitudes, teachers can guide students to think and explore independently and actively so that they can understand and master basic mathematical knowledge, skills, ideas, and methods and obtain basic experience in mathematical activities, thereby providing a foundation for mathematical ability, especially in terms of abstract thought, generalization, reasoning, and model building. The most important feature of heuristic teaching is to stimulate students’ interests, guide them to think about problems, mobilize their initiative and enthusiasm in learning to the maximum extent, and make them work hard to acquire knowledge. Thus, students can gradually develop the habit of thinking diligently about problems and the ability to solve them. How can we apply heuristic teaching? The organization of the teaching mission must conform to the law of cognition so that students can understand a principle or a method and transform from being unknowing to understanding, from incapable to capable, and from asking questions to solving problems and reaching conclusions. Hence, students can become satisfied with it and become interested. The purpose of teaching should be clear so that it promotes learning; the clearer and deeper students understand the purpose and task of learning, the more interested and self-motivated they will be in learning. Applying full use of intuitiveness and experience will ensure that students can actually “see” and “feel” when they acquire more abstract things and can progress from being perceptual to being rational, from simple to complex, and from shallow to deep, which facilitates students’ interest and acceptance of knowledge and their transition from concrete image thinking to abstract logical thinking. We should emphasize the “comparison” method so students can recognize the similarities and differences between things and become interested in their essence to be good at raising questions, revealing contradictions, guiding students to think actively with great interest, solving problems step by step, and acquiring new knowledge. Heuristic teaching is not just “asking a question,” asking “what is it?” or “is it right?” Asking the right question is the key to inspiration. An experienced mathematics teacher, after teaching “The logarithm of a rational number other than an integer power of 10 cannot be an integer or a fraction,” teaches the following: Q: Then, what is the number? A: An irrational number. Q: What is an irrational number? A: Infinite acyclic decimals. Q: Based on what we discussed, could you guess what the majority of numbers in the mantissa of the logarithm are? A: They are irrational numbers.

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The teacher found an error in the figure of arithmetic series in the textbook, and instead of asking the students to correct it, he asked them a question after explaining the arithmetic series general term formula an = a1 + (n − 1)d: “If a1 and d are known numbers, which variable is a and n’s function? What is the degree of the function?” Students answered the question. The teacher then asked, “What does the linear function look like?” Students answered, “A straight line.” The teacher then asked the students to look at the image in the book, which is indeed a straight line. There is apparently no need to doubt this, but then he asked: “Can the independent variable n in this function be any number?” Students answered: “It must vary within the set of natural numbers,” The teacher asked again: “Then, the image of the function can still be a straight line?” Students realized: “No.” By asking questions and inspiring students to discover, understand, and correct errors, the teacher increases their interest in learning mathematics and improves their motivation. This not only helps students absorb knowledge and acquire problemsolving skills but also develops logical thinking abilities, which is much more rewarding than asking students to correct errors in books directly. Heuristic teaching should pay attention to the age characteristics of students’ psychological development. The way and depth of questioning should be different for different grades of students. The questions should not be too complicated for younger students, and the difficulty and depth of the questions should be increased appropriately with increasing age. In this way, by asking questions and promoting enlightenment, students’ interest in learning can also increase. As long as they have an interest in learning, the internal contradiction of their intellectual development, that is, motivation, can be stimulated.

The Relationship Between Age Characteristics and Individual Differences The issue of age and individual differences in intellectual development is very important. Many ideologists and educators in ancient China affirmed that there are both age and individual differences in human development. Confucius said in The Analects of Confucius—Weizheng: “At fifteen, I had my mind bent on learning. At thirty, I stood firm. At forty, I had no doubts. At fifty, I knew the decrees of Heaven. At sixty, my ear was an obedient organ for the reception of truth. At seventy, I could follow what my heart desired, without transgressing what was right” (Confucius, 2011, p. 17). This is the most classic discourse of the age characteristics in human development. At the same time, he emphasized the need for “teaching in accordance with aptitude” because of individual differences. In the Ming Dynasty, Tingxiang Wang1 supported the idea of individual differences, and in his “Shen Yan—Asking about nature,” he 1

Tingxiang Wang: A Chinese philosopher in the Ming Dynasty (1474–1544).

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elaborated on the concept of individual differences: “The nature and gifted talents of people are different since birth.” Age characteristics include physiological and psychological characteristics. These are closely related and influence each other. Psychological age characteristics refer to the qualitative psychological characteristics formed by children and adolescents at different stages of mental development under certain social and educational conditions.

The Phase of Intellectual Development in Psychological Development Intellectual and even psychological age characteristics are based on the phase of intelligence and even psychological development. From birth to maturity, children go through six major stages: infancy (0–1 year old, or early infancy), toddlerhood (1–3 years old, or late infancy), preschool years (3–6 years old), early school years (equivalent to elementary school, or childhood), adolescence (about middle school), and young adulthood (about high school). These stages are interrelated and distinct from each other and are consecutive; new stages replace old ones and cannot be surpassed or regressed. From the perspective of developmental trends, the order and time interval of various intellectual and even psychological phenomena in each age phase or stage are generally constant.

The Representativeness of Age Characteristics of Intellectual and even Psychological Development The age characteristics of intellectual and even psychological development are those general, typical, and essential characteristics expressed by children and adolescents in intellectual and psychological development to a certain age. By “general,” we mean “non-personal”; by “typical,” we mean representative; by “essence,” we do not mean “phenomenon.” Students’ general, typical, and essential characteristics when learning mathematics are reflected in the development changing of abstract generalization abilities of quantities and quantitative relationships, shapes, and shape relationships. When studying the laws of a particular matter, all sciences generalize the essential things from their specific and diverse manifestations. Concrete matter is the most abundant, but the essence of matter is the most concentrated. The psychological age characteristics of children and adolescents are summarized from many specific, personal facts about the psychological development of children and adolescents, which are general, typical, and essential. The development of thinking also shows such stages with stability.

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The thinking form from birth to three years old is mainly intuitive-action thinking. The form of thinking for a toddler or preschool-year child is mainly concrete image thinking. The form of thinking for early school-aged children is mainly abstract image thinking, which is in the transition from concrete image thinking to abstract logic thinking. The form of thinking in adolescence is mainly abstract logic thinking based on experience. The form of thinking in early youth is mainly abstract logic thinking based on theory. Considering early school age or childhood, for example, the thinking of elementary school children is image-abstract thinking, which is in terms of the most general, typical, and essential characteristics. The fact is that first-grade children’s thinking is still primarily in the form of concrete image thinking. Fifth-grade children are basically abstract logical thinkers, although they still possess the ability for concrete image thinking. The general trend of elementary school children’s thinking is from concrete image thinking to abstract logic thinking. Thus, the psychological age characteristics of children and adolescents refer to the general, typical, and essential characteristics of a certain stage, while at the beginning of this stage, many age characteristics of the previous stage may be retained. At the end of this stage, there may be many age characteristics from the next stage.

The “Critical Age” in Intellectual and even Psychological Development The age characteristics of intellectual and even psychological development are also characterized by the “critical period” that appears at each age. There is a process of quantitative to qualitative change in intellectual and even psychological development and a process of small qualitative changes to large qualitative changes. Thus, each intellectual process or individual characteristic must go through several leaps or qualitative changes and exhibit a certain age characteristic. The form of this age characteristic is called the critical period. Research has shown that at the ages of 2–3 years (mainly 2.5–3 years) and 5.5– 6 years, the preschool stage is a critical period in developing children’s intelligence (thinking). The former is a turning point in the development of intuitive thinking to concrete image thinking. The latter is the beginning of producing abstract logical thinking based on concrete image thinking. In elementary school, fourth grade is a critical period in the development of students’ thinking, which is reflected more clearly in their mathematics learning; that is, fourth grade is a turning point in the development of concrete image thinking to abstract logical thinking. In secondary school, the eighth grade is a critical period. The eighth grade is a turning point in secondary school students’ thinking development. Mr. Hong Chen, the famous mathematics educator of Beijing Fifth Secondary school, said to me when talking about students’ mathematics learning: “The eighth-grade students truly step

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into the stage of abstract logical thinking.” In other words, the eighth grade is an important period in the development of students’ thinking. Seventh-grade students are still not very different from upper elementary grade students regarding their type of thinking, while eighth grade is a new starting point for abstract logical thinking. From this time, abstract logical thinking starts to develop gradually from empirical to theoretical. Therefore, the abstract logical thinking of eighth graders is in a qualitative “leap” period. In addition, eighth grade is an important time in the development of students’ moral qualities. The “chaotic classes” at the secondary school level often emerge in the eighth grade, the “differentiation” of academic performance occurs in the eighth grade, and the progress or regression of moral qualities of secondary school students often appears in the eighth grade. The qualitative change in thinking is related to physiology and learning, while the leap in moral development is more closely related to the function and role of education. At the same time, it is unnecessary to consider the critical period in the extreme, as it often comes from education. Therefore, it is also unnecessary to exaggerate the role of the critical period by thinking “now or never.”

The Maturation Period of Elementary and Secondary School Students in Intellectual and even Psychological Development In the physiological and even psychological development process in elementary and secondary school, there is a maturation period, usually at the end of the first year and the beginning of the second year of high school. When they reach the maturation period, individuals’ mental processes and personality characteristics are basically shaped and remain relatively stable. Several secondary schools in Beijing have conducted longitudinal studies and found that the intelligence, learning ability, and academic performance of the “top students” who graduated from the ninth grade and enrolled in high school changed greatly after one year. However, from the end of the first year of high school to graduation, they remain relatively stable in both morality and academics. After entering college, the majority of those students who had good moral qualities and academic performance in high school remained so in college. The findings suggest that the difference between premature and post-mature psychological development is their plasticity. Before maturity, students have great plasticity, so they should train and cultivate. It is not impossible to develop after maturity, but it is difficult to train and cultivate because of poor plasticity. Therefore, professors must pay close attention to the shaping of students when they are in the premature stage.

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The Stability and Variability of Age Characteristics of Intellectual and even Psychological Development Considering age characteristics in intellectual and even psychological development, are these characteristics identical in students of the same age at varied times, in different regions and different individuals? No, they are not. On the one hand, some intellectual and psychological development factors, such as the sequential and systematic movement of the stages and the process, range, amplitude, and speed of change of each stage, are largely stable and common. On the other hand, because social and educational conditions work differently in each student, there can be certain differences in the process and speed of intellectual and psychological development, which is also known as variability. This variability shows the personality or individual differences between students and shows that under different socioeconomic statuses or educational conditions, the degree and speed of psychological development of social groups produce certain changes. The stability and variability are complementary, and their existence is relative. Their relationship is the unity of generality and individuality and the unity of typicality and diversity. In education, why do we advocate “individuality learning,” and why do we emphasize “teaching in accordance with aptitude?” We do this because there are individual differences in human talents and intellectual abilities. 1. From the perspective of the difference in development degree, this can be expressed as a difference in supernormal, normal, and subnormal levels, highlighted by the quality of an individual’s thinking or intelligence. For example, in calculations, some people are fast, some are slow, some are flexible, and some are dull; some have innovative solutions, and some apply the formula rigidly. 2. From the perspective of the difference in development styles, there are differences in cognitive styles, particularly in field independence and dependence on cognitive styles. 3. From the perspective of the difference in types of components, the differences are in the combination and use of various psychological or academic abilities. Due to the complexity of intelligence composition, there are differences in intelligence types: some people have an advantage in abstract logical thinking, some people have an advantage in concrete image thinking, and some people are more in the middle. Some students are good at arts, and some students are good at science, which shows this difference. In particular, the same performance of students in the same activity can be determined by a combination of different intellectual abilities. Therefore, students of the same age will have various ways to solve the same problem. 4. From the perspective of the difference in the range of manifestation, it can be represented as the distinction between learning and non-learning, performance and nonperformance, and academic and non-academic. To a certain extent, this difference is also a reflection of special abilities. 5. From the perspective of the relationship between special intelligence and general intelligence, we can also see the difference between the early and late stages of

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the development of intelligence. There have been records of “young genius” and “late bloomer.” The individual differences in students’ intelligence and ability development make it necessary for teaching to emphasize students’ aptitudes. By paying attention to such differences, we can better deal with the relationship between general education and “individual (personality) education” to “open one lock with one key.” Therefore, mathematics teaching should be oriented to all students and meet the requirements of their individual development. In other words, it is necessary not only to make every student receive the quality mathematics education they deserve but also to enable those students who have expertise in mathematics to have more development opportunities.

References Confucius. (2011). The discourses and sayings of Confucius (H. M. Gu, Trans.). Yunnan People’s Publishing House. (in Chinese and English) (Original work published 540 BC - 400 BC) Ceci, S. J., Rosenblum, T., Bruyn, E. D., & Lee, D. Y. (1996). A bio-ecological model of intellectual development: Moving beyond h2 . In R. Sternberg & E. Grigorenko (Eds.), Intelligence, heredity and environment (pp. 303–322). Cambridge University Press. Dekaban, A. S., & Sadowsky, D. (1978). Changes in brain weights during the span of human life: Relation of brain weights to body heights and body weights. Annals of Neurology, 4(4), 345–356. https://doi.org/10.1002/ana.410040410 Fancher, R. E. (2004). The concept of race in the life and thought of Francis Galton. In A. S. Winston (Ed.), Defining difference: Race and racism in the history of psychology (pp. 49–75). American Psychological Association. https://doi.org/10.1037/10625-003 Gesell, A. (1929). Maturation and infant behavior pattern. Psychological Review, 36(4), 307–319. https://doi.org/10.1037/h0075379 Gesell, A., & Thompson, H. (1929). Learning and growth in identical infant twins. Genetic Psychology Monographs, 6, 1–123. Gesell, A., & Thompson, H. (1934). Mental growth and maturation. In Infant behavior: Its genesis and growth (pp. 292–325). McGraw-Hill. https://doi.org/10.1037/11333-005 Jensen, A. R. (1970). Race and the genetics of intelligence: A reply to Lewontin. Bulletin of the Atomic Scientists, 26(5), 17–23. https://doi.org/10.1080/00963402.1970.11457807 Krantz, D. L., & Allen, D. (1967). The rise and fall of McDougall’s instinct doctrine. Journal of the History of the Behavioral Sciences, 3(4), 326–338. https://doi.org/10.1002/1520-6696(196 710)3:4%3c326::AID-JHBS2300030403%3e3.0.CO;2-6 Lin, C. D. (1980). A study on the development of number concept and computational ability of preschool children. Journal of Beijing Normal University (Social Sciences), 2, 67–77. (in Chinese). Lin, C. D. (1981). A study on the development of number concept and computational ability of primary school children. Acta Psychologica Sinica, 3, 289–298. (in Chinese). Lin, C. D. (2006). Creativity training and experiment on intervention. Journal of Beijing Normal University (Social Sciences), 01, 41–47. (in Chinese). Lin, C. D. (2013). Psychology of secondary schoolers (pp. 40–41). China Light Industry Press. (in Chinese). Liu, S. Z. (1962). Age characteristics of brain development of Chinese children. in Chinese psychological society educational psychology committee. In Selected papers on educational psychology:

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Selected papers of conference on educational psychology of the Chinese psychological society. People’s Education Press. (in Chinese). Livingstone, D. (2011). Environmental determinism. SAGE Publications Ltd. https://doi.org/10. 4135/9781446201091 Ministry of Education of the People’s Republic of China. (2012). Mathematics curriculum standard for compulsory education (2011th ed.). Beijing Normal University Press. (in Chinese). Stich, S. P. (Ed.). (1975). Innate ideas (Vol. 10). University of California Press. Weisberg, R. W. (2015). Toward an integrated theory of insight in problem solving. Thinking & Reasoning, 21(1), 5–39. https://doi.org/10.1080/13546783.2014.886625 Ye, G. S. (1980). General rules of growth and development. In G. Ye (Ed.), Child and adolescent hygiene. People’s Medical Press. (in Chinese). Zhou, X., Chen, C., Chen, L., & Dong, Q. (2008). Holistic or compositional representation of twodigit numbers? evidence from the distance, magnitude, and SNARC effects in a number-matching task. Cognition, 106(3), 1525–1536. Zhou, X., Chen, C., Dong, Q., Zhang, H., & Chen, C. (2006a). Numerical distance effect in the n240 component in a number-matching task. Neuro Report, 17(10), 991–994. Zhou, X., Chen, C., Dong, Q., Zhang, H., & Zhou, R. (2006b). Event-related potentials of single-digit addition, subtraction, and multiplication. Neuropsychologia, 44(12), 2500–2507. Zhou, X., Chen, C., Zang, Y., Dong, Q., & Chen, C. (2007a). Dissociated brain organizations for single-digit addition and multiplication. Neuro Image, 35(2), 871–880. Zhou, X., Chen, C., Zhang, H., Chen, C., Zhou, R., & Dong, Q. (2007b). The operand-order effect in single-digit multiplication: An ERP study of Chinese adults. Neuroscience Letters, 414(1), 41–44. Zhou, X., Chen, Y., Chen, C., Jiang, T., Zhang, H., & Dong, Q. (2007c). Chinese kindergartners’ automatic processing of numerical magnitude in Stroop-like tasks. Memory & Cognition, 35, 464–470. Zhu, Z. X. (1981). Several problems about children’s intellectual development. Journal of Beijing Normal University., 01, 39–46. (in Chinese). Zhu, Z. X. (2018). Child psychology. People’s Education Press. (in Chinese).

Chapter 3

Intelligence and Creativity

In psychology, creativity and innovation are often considered to be synonymous. However, the relationship between creativity and intelligence remains controversial in this field. Intelligence is a necessary condition for creativity; that is, to solve creative problems, a person must have intelligence. However, intelligence is not a sufficient condition for creativity—intellectual and non-intellectual factors are required in the process of creativity or innovation, as is an appropriate environment (Karwowski et al., 2016). In other words, there is a proper correlation between intelligence and creativity. However, the correlation between a high level of intelligence and creativity is weak. Thus, although intelligence may stimulate creativity to a certain extent, it cannot guarantee the existence of creativity in itself. People with a high level of intelligence may have the potential to develop a higher level of creativity than those with lower levels of intelligence. However, their level of creativity depends on other conditions, including the external environment, atmosphere, and other natural and social factors. China is highly creative. The author of The Genius of China: 3000 Years of Science, Discovery, and Invention, Robert Temple, highlights that the Chinese have 100 “word firsts” (Temple, 2004). For example, in mathematics, the Chinese were the first to propose “decimal notation,” “the placement of zero,” “negative numbers,” “extraction of higher-order roots and solutions of higher-order numerical equations,” “decimal fractions,” “algebraic applications in geometry,” “the Yang Hui triangle,” and “the exact value of π (Pi).” As early as the third century A. D., Hui Liu started from the standard 192 inscribed polygons of a circle to calculate the exact value of Pi until it was closer to 3072 inscribed polygons of a circle, finally obtaining the approximate value of Pi as 3.14159. By this time, the Chinese had surpassed the Greeks in mathematics. By the fifth century, mathematicians Chongzhi Zu and his son Geng Zu had “refined” Pi to 35 decimal places. These innovations illustrate the innovative spirit of the Chinese people and reveal that mathematics can facilitate innovation or creation.

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Characteristics of Creative Talents and Growth Environment Today, we need all kinds of talent, the most prominent being creative talent, which explains why “education for all-round development” is emphasized in education. Quality education has the spirit of innovation at its core, advocating students’ learning, practical, and innovative abilities (creativity). What is creativity, specifically (Walia, 2019)? However, this remains controversial. At present, three tendencies have emerged in international psychological research: the first considers creativity as one or more psychological processes; the second considers creativity as a product; and the third considers creativity as a personality, wherein different people possess different kinds of creativity. I maintain that creativity combines all three: a psychological process, complex and novel product, and personality characteristic or quality. Professor Zhixian Zhu (founder of Chinese Child Psychology) and I define creativity as the intellectual quality of using all known information to generate a novel, unique, socially, or personally valuable product with a specific purpose. Product here refers to the outcome of thinking in a particular form, which can be a new concept, idea, theory, technology, process, or work (Lin et al., 2003; Zhu & Lin, 1986). Three criteria were employed to judge whether a product fits this definition of creativity: (a) novel, (b) unique, and (c) socially or personally valuable. “Novel” mainly means not adhering to conventional rules—daring to break the old and establish the new. This criterion is relative to history and is a vertical comparison. “Unique” mainly refers to being out of the ordinary and ingenious. This criterion is relative to other individuals and is a horizontal comparison. Finally, “socially valuable” indicates a significant contribution to the progress of humankind, the nation, and society, such as major inventions, creations, and innovations. Simultaneously, “individual value” implies that it plays a significant role in an individual’s development. The history of human civilization is a splendid history of creativity. It is reasonable to judge an individual’s level of creativity based on a product, as the concept is generally reflected through creative activities and outcomes. Additionally, a product is visible, tangible, and easy to understand. However, the nature and structure of an individual’s psychological processes and personality characteristics are not always clear. Therefore, it is more credible to consider a product as a criterion than to use the mental process or personality traits of the creator as an indicator. This finding is consistent with the operational principles of psychological research. Given that no other better method exists, it can be concluded that judging creativity based on a product or outcome is the most practical approach. The main reason for emphasizing creativity as an intellectual quality is that it is regarded as a thinking quality. Creativity is an intellectual quality that values individual differences in thinking ability and emphasizes individual differences (Lin, 1999; Zhu & Lin, 1986). Notably, creativity is an intellectual quality that generates a socially (or personally) valuable novelty element for a certain purpose. In summary, when considering the relationship between intelligence and creativity, I presume that creative talent consists of creative thinking (intellectual

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factors) and creative personalities (non-intellectual factors). In addition, a creative environment should be considered. There are many mathematics scholars in China, including Mr. Luogeng Hua. One day, when Hua was in elementary school, the teacher asked the class a question: “There are certain things whose number is unknown. If we count them by three, we have two leftovers; by five, we have three leftovers; and by sevens, we have two leftovers. How many things are there?” Hua immediately gave the correct answer: “23.” The teacher was delighted and praised him for his arithmetic abilities. Later, Hua was forced to drop out of school to work in sales because of his family’s economic status, but he continued to study while working. In 1930, Hua published a paper claiming that the quintic was fundamentally flawed (Hua, 1931). Hua’s analysis caught the attention of Prof. Qinglai Xiong, the Dean of the Department of Mathematics at Tsinghua University. At that time, Prof. Qinglai Xiong thought that Hua was a returning professor, so Xiong was surprised when he discovered that Hua was only 19 years old and working in sales. In 1931, Hua attended Tsinghua University. During his time at Tsinghua, Hua published a dozen papers, completed a study abroad program in England, obtained a Ph.D., became a famous mathematician, and proposed Hua’s theorem. Hua’s story helps us to understand the meaning of creative talent.

Creative Thinking (Creative Intellectual Factors) Creative thinking, that is, creative intellectual factors, comprises five characteristics and manifestations. Creative Thinking is a Novel, Unique, and Meaningful Thinking Activity. Innovation is the primary characteristic of creativity. However, “novel” and “unique” do not equate with “good.” Furthermore, we should emphasize that “meaningful” refers to aspects that are valuable to society or individuals. It is imperative to solve problems creatively and adequately in teaching experiments. For example, our research group asked elementary school students to compose their word problems to breakdown the difficulties so that they would further understand the interdependence of numbers. This approach not only improves their ability to solve word problems but also promotes the development of intelligence and creativity. The Content of Creative Thinking is Thinking Plus Imagination. Through imagination, conceiving helps us solve problems that others cannot. The learning outcomes that students receive in any subject are closely related to their imaginations. Therefore, the recommended teaching practices are as follows: enriching students’ relevant representations, using vivid and affective language to describe what students need to imagine, training students to formulate correct and realistic imagined outcomes, and guiding students to read literary and science fiction works. Developing students’ spatial imagination is one of the main goals of mathematics teaching.

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Inspiration Occurs in the Creative Thinking Process. In the process of creative thinking, there sometimes appears to be a sudden emergence of new images and hypotheses, often referred to as “inspiration.” However, the reality is that most of the time, inspiration results from long-term thinking and hard work—the outcome of an individual’s powerful and positive spiritual energy. Inspiration is closely connected to creative motivation and a constant search for new ways of thinking. Many creative mathematicians have similar views in this regard. One’s inspirational status is characterized by the fact that a person’s attention is intensely focused on the object of creation, such that under the inspired status, creative thinking is highly productive. Elementary school students lack inspiration. Even when they reach secondary school age, their ability to be inspired is only in the initial stages and is insignificant. However, after 18 years of age, inspiration rapidly develops. Nevertheless, both children and adolescents have the capacity for conscious attention, which is one of the foundations of inspiration; therefore, our group attaches great importance to cultivating conscious attention in elementary and secondary school students. In efforts to cultivate students’ conscious attention and non-intellectual factors, we also reformed teaching content and didactics. The supplementary textbooks for secondary school mathematics, experimental textbooks for secondary school and elementary school mathematics, and thinking quality training textbooks compiled by our research group are conducive to motivating students to learn, enhancing conscious attention, and providing a foundation for the embryonic and formation of inspiration (Lin, 1983a, 1983b). Creative Thinking Combines Analytical and Intuitive Thinking. Analytic thinking is logical thinking conducted step-by-step, whereas intuitive thinking directly apprehends a concept or situation. There are two different approaches to thinking: the first is analytical, requiring one to follow strict logic or laws, deriving step-by-step until a logical answer or reasonable conclusion is reached; the other involves a fast, direct, leap to a conclusion, that is, intuitive thinking, in which the derivation process cannot be identified. For example, a teacher presents a difficult factorization problem on the class board. Almost immediately after the instructor finishes writing, a student rushes up and solves the problem by applying cross multiplication. The teacher asks him, “Can you explain the reasoning behind the solution?” The student then shook his head. The teacher asks again, “How did you get the answer?” “It’s hard to explain,” he says. “Then, why did you use cross multiplication?” “I do not know. I just knew it was the right thing to do,” the student responded: This is a typical example of intuitive thinking, from which we can see six characteristics: (a) fast, (b) direct, (c) no derivation process, (d) arising from a sense of conviction, (e) personalized, and (f) probability. At face value, the process of intuitive thinking is the result of highly concentrated “assimilation” or “knowledge transfer” without the manifestation of “indirectness,” “linguization,” or “internalization” of thinking. Not surprisingly, Einstein regarded intuitive thinking as the budding of creative thinking. Therefore, during lessons, teachers should protect and guide students’ intuitive thinking

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skills. Especially after the eighth grade, students should be gradually guided to apply the mantra: “I do not seek the answers; I seek to understand the question.” Intellectual Creativity is the Unity between Convergent and Divergent Thinking. Convergent and divergent thinking are complementary and dialectically integrated; that is, they are two forms of intellectual activity that seek common ground while reserving their unique differences. Convergent thinking emphasizes the participant finding the “correct answer” to a question and the role of memory in intellectual activity. Divergent thinking emphasizes that the participant actively seeks multiple solutions other than “an answer” to the problem, thus encouraging flexibility in intellectual activity and knowledge transfer. Convergent thinking is the basis of divergent thinking, and divergent thinking is the development of convergent thinking. In a complete intellectual activity, the absence of divergent thinking and the lack of training and cultivating flexible thought among students will be dull in mind; even if they have acquired knowledge, they cannot be divergent and creative, which in turn affects the acquisition of further knowledge and the development of convergent thinking. Therefore, when developing intellectual flexibility, attention should be given to the “correct answer” and “multiple solutions.” Moreover, by combining the two, we can reasonably call it reasonable and flexible intellectual quality. For over 40 years, we have used these five characteristics as indicators of creative thinking and measures to foster creative thinking in experimental schools. Although what we have obtained thus far is merely rough theories, they still influence experimental schools and even the educational community. Here, I would like to clarify the relationship between intelligence and creativity: intelligence is a necessary but insufficient condition for creativity. People with a high level of intelligence certainly have greater potential for creative development than people with a low level of intelligence; however, whether they can generate creativity is most often determined by personality and environmental factors.

Creative Personality (Creative Non-intellectual Factors) Creative talent needs a more creative personality, which is composed of creative non-intellectual factors. An American psychologist collected Nobel laureates from their adolescent IQ material. The results indicated that most laureates did not have an extremely high IQ but were at an average level or above. Moreover, their nonintellectual factors differed greatly from those of other people in general (Eubanks et al., 2016). Luogeng Hua, as mentioned above, was born poor and only had a middle school education. Apart from his talent, he became a world-class mathematician because non-intellectual factors also played a large role in his success, particularly his personal qualities of interest, diligence, perseverance, and confidence. Two researchers are famous worldwide for studying creative personality traits. Guilford (1966) identified eight characteristics of creative personality: (a) a high degree of self-consciousness and independence; (b) thirst for knowledge; (c) a strong

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sense of curiosity and motivation to deeply investigate the mechanisms and roles of the movement of matter; (d) versatility and adeptness at observation; (e) ability to work in an organized, rigorous, and clear manner; (f) a rich imagination, keen intuition, a preference for abstract thinking, and a broad interest in intellectual activities and games; (g) a good sense of humor and excellent literary talent; and (h) excellent mental qualities, able to eliminate external distractions and focus on an issue of interest for long periods. Sternberg (2018) proposed the triangular theory of creativity. The third dimension comprises personality characteristics among the intelligence and cognitive dimensions, consisting of seven elements: (a) tolerance of ambiguity, (b) overcoming obstacles, (c) creative self-concept, (d) risk-taking, (e) intrinsic motivation, (f) expectation to be recognized, and (g) willingness to fight to be recognized again. After decades of research, Prof. Zhu and I have summarized the personality of creative people in terms of the following five aspects: (a) healthy emotions, including the degree and nature of emotions and rational feelings; (b) a strong will, that is, purposefulness, persistence (perseverance), decisiveness, and self-control; (c) positive personality consciousness trends, especially in terms of interest, motivation, and ideals; (d) a resolute character, especially one with attitudinal (e.g., diligence) and dynamic features; and (e) good habits (Lin, 1992, 1999; Zhu & Lin, 1986). Over the past 40 years, I have used these five aspects as indicators for research on the characteristics of creative personalities and as a measure to develop students’ creative personalities in experimental schools. This experience has allowed me to obtain valuable perspectives in basic education. Most importantly, there are indications that to nurture and foster creative talent; attention should be given not only to creative thinking but also to creative personality; creativity should not simply be regarded as a gift but also a consequence of nurturing. Moreover, education related to creativity should not be limited to intellectual education but rather seen as an essential principle to be encouraged holistically throughout the entire education system, including moral, intellectual, physical, and aesthetic education.

Creative or Innovative Environment Creative talent development depends on various settings, including family, school (education), cultural, social, workplace, and resource environments. It should be emphasized that developing creative talent requires a democratic and harmonious environment for creativity or innovation. Harmony refers to the well-managed and coordinated relationships of all types. Moreover, psychological harmony is consistent with social harmony. Therefore, it is necessary to establish an environment that encourages innovation. All types of creative talent can compete with each other, from which superior creative talent can emerge, thus fostering and producing world-class scientists and leading experts in science and technology.

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The three types of relationships that should be addressed in a harmonious society are the relationships between the individual and the self, between the individual and others, and between the individual and society. From the perspective of psychological harmony, six primary relationships must be considered when educating individuals on these three types of relationships. 1. The relationship between individual and self. Confidence is the primary condition. Innovation requires the innovator to have self-confidence, self-esteem, self-reliance, and self-improvement. 2. The relationship between individuals and others. Team building should be the first in the interpersonal relationship of a creative team since with the spirit of teamwork, each team member can perform better and be more creative. 3. The relationship between individuals and society. Patriotism is the core motivation for innovation. We should seek to understand national conditions, recognize national dignity, build national prestige, know national humiliation, and promote national spirit. 4. The relationship between individual and nature. This is the requirement of “man-nature harmony.” Innovation requires establishing healthy environmental values. Creative talent must have the qualities of caring for life, caring for the environment, and caring for nature. 5. The relationship between software and hardware. Hardware is a necessary condition for innovation. Moreover, we need to adhere to the principle of being peopleoriented to fully activate people’s motivation. People are the foremost element of innovation. 6. The relationship between China and the world. “Making the Foreign Advantages Serve China” is the guideline for this relationship. The innovation process requires us to deal with the relationship between internationalization and nationalization. The more national the innovation is, the more internationally it will appear. In summary, harmony builds power and accomplishes greatness. The development of creative talent cannot be achieved without a healthy democracy or harmonious environment.

Implementation of Creative Education and the Training Model of “T”-Type Talents In 2003, I hosted a key grant project for philosophy and social science research sponsored by the Ministry of Education of China, entitled “Research, not creative talents and creative education” (Jin et al., 2010), for which Prof. Shenghua Jin conducted a subproject called “Research on the group of creative talents.” The research participants were 34 science and technology academics and 38 national art experts in China. Their growth shares one commonality, and their success is strongly correlated

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with education and teachers, which is related to the international issue of “creative education.” How then should creative education be understood? The premise consists of the three elements mentioned in the previous chapter: the relationship between intelligence and creativity, individuals’ intellectual and non-intellectual factors, and the environment. Therefore, creative education as an educational concept, regardless of different understandings, remains a common perception; that is, its essence is to develop individuals’ creative qualities. Creative qualities include creative consciousness, spirit, thinking, personality, creativity, and practical abilities, which encompass the dynamic, cognitive, personality, and behavioral systems of creativity that influence and interact with each other, forming an organic whole of creative qualities. In other words, creative education is the process of cultivating creative qualities through the creative teaching methods provided by teachers in a creative school context.

Creative Education Comprises Three Groups in Schools that Produce Five Components of Effectiveness The three groups refer to the management team (headed by the principal), the school faculty, and students. The five components of effectiveness that they aim to achieve are as follows: (a) the creative principal produces creative management, (b) creative management produces a creative environment, (c) the principal drives teams constructed of creative teachers, (d) creative teachers provide creative lessons, and (e) creative students are cultivated through creative teaching. Specifically, creative education does not require a specific curriculum or form but must be achieved by reforming existing educational ideas, contents, and methods, thus permeating all educational activities. We should also consider the presentational, discovery, discussion, and creative teaching methods; the teaching effect of convergent and divergent thinking; the relationship between creative education and students’ physical and mental development; and the role of subject teaching, didactics, and extracurricular activities. Having identified the necessary aspects, how can these elements be used for effective creative education? Promote Creativity in School. First, it is important to promote creativity in the school context, which includes the principal’s personality qualities, guiding ideology, school management, working methods, environmental arrangements, assessment systems, learning atmosphere, and other school-related factors. It should be stated that a democratic atmosphere is one of the key factors for success in school, as its presence determines whether creative education will thrive. Schools are supposed to be places where creative talent is fostered, but the reality is that most schools are overly focused on students’ academic performance to exclude other aspects, thus suppressing both teachers’ and students’ creative abilities. Accordingly, optimizing creativity in the school context is necessary to promote students’ creativity.

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Construct Teams of Creative Teachers. It is necessary to construct a team of creative teachers to develop creative education because teachers play a major role in fostering students’ creative qualities. Creative teachers are good at absorbing the latest educational and scientific achievements and including them in their lessons. In addition, they have unique insights and can explore effective teaching methods. This includes teachers’ creative perspectives on education, knowledge structure, personality characteristics, and didactic and management skills. The teaching method is key to developing creative talent. While imparting knowledge, experience, and culture, teachers should also focus on nurturing, shaping students’ minds, and improving their spiritual world. Simultaneously, teachers should clarify that knowledge comes from creation and should be applied in practice. Therefore, a good teacher is someone who is not just a mouthpiece but also the implementer of educational purposes, the organizer of teaching activities, the explorer of teaching methods, and the creator of educational activities. One thing to consider is that the current teaching practices of Chinese college and university teachers are more rigid than those of elementary and secondary school teachers. Only by considering creative education as the basic instrument and training direction can colleges and universities adapt to society’s needs. Otherwise, they weaken their social status and special role, perhaps even to the point of being eliminated by society. Torrance (1962) found a positive correlation between teachers’ scores on a creative motivation test and students’ creative writing ability, suggesting that teachers’ creativity level is critical to the development of students’ creativity. In addition, teachers prefer high-IQ students to high-creativity students. Therefore, the study of teachers’ creative educational perspectives, personality characteristics, knowledge structures, didactics, and management skills provide practical guidance for developing and enhancing teachers’ creativity. Jingrun Chen mentioned in the previous chapter that his interest in mathematics was sparked by the encouragement of his high school mathematics teachers. Talent is everywhere. However, these are seldom recognized. Mathematicians are seldom recognized, except for Bo Le.1 When Chinese scientists discuss the contributions of Luogeng Hua, his teacher, Qinglai Xiong, is often mentioned. Qinglai Xiong primarily engaged in the study of function theory and defined an infinite series named “Xiong’s infinite number.” At the same time, Qinglai Xiong contributed outstandingly to nurturing mathematical talent in China. As mentioned above, in 1930, Qinglai Xiong, Dean of the Department of Mathematics at Tsinghua University, read a paper published by Luogeng Hua in an academic journal. After discovering Hua’s educational experience and talent in mathematics, he invited Luogeng Hua, who had only a middle school degree, to Tsinghua University and eventually made Hua a world-renowned mathematician. During his lifetime, Qinglai Xiong discovered many young mathematicians, including academicians at the Chinese Academy of Sciences, like Le Yang and Guangzou Zhang. 1

Bo Le: Herein, it refers to someone who is good at spotting, recommending, developing, and using talent—originally referring to a horse tamer in Spring and Autumn period who is good at judging and picking horse named Sun Yang.

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Develop Students’ Creative Personality or Individuality in Learning. Students are the object of creative education; therefore, it is necessary to develop their creative personality or individuality. Any creative activity is greatly constrained by personality and individuality, as it requires one to expand their thoughts beyond the existing ideas, methods, and theories they have learned. To encourage creativity, teachers should encourage students to be meticulous, diligent, studious, tireless, and perseverant when exploring the world. And about the relationship between students’ creativity, intellectual, and non-intellectual factors showed that the correlation coefficient between creativity and other intellectual thinking qualities was around above 0.4, which is neither high nor low. Therefore, in education and teaching, it is necessary to consider the role of both intellectual and non-intellectual factors on students’ development of creativity, particularly the role of “creative” achievement motivation.

Everyone Has Creativity Throughout our research, we stated that everyone possesses some degree of creativity and that creative education should be open to all students. On September 15, 2016, Guangming Daily published a front-page article, “Shanghai’s Secrets in Mathematics Teaching,” which introduced the success of mathematics education reform in Shanghai’s elementary and secondary schools. For 30 years, the mathematics education community in Shanghai has remained committed to developing students’ mathematical quality and lifelong learning ability, taking students’ development as the primary focus, and changing from exam-oriented education and compulsory courses-oriented education to quality-oriented education and updating the required courses’ teaching models. Their reforms were meant to benefit the development of all students and have yielded fruitful results. Shanghai students ranked first worldwide in mathematics in two consecutive terms in the Program for International Student Assessment (PISA) conducted by the Organization for Economic Cooperation and Development (OECD). Teachers enthusiastically guide students, fill the classroom with fun, and allow students to discover their creative sides. In this way, students become increasingly eager and confident in experimenting and exploring. Examining past psychological research, the research object of creativity has been limited to a few outstanding inventors and artists. However, it is important to note that creativity is a continuum rather than an all-or-none quality. Everyone, including children, has a certain degree of creative thinking and creativity. The creative qualities of individuals and their development differ only in type and level; therefore, the same model cannot be used to develop students’ creativity. It is also essential that creative education be popularized, especially in schools, so that every student can have the opportunity to develop their creativity through creative education. We must have a big-picture understanding of the problem at hand, consider the future, analyze development, and never make arbitrary statements to the students, such as, “I know you inside out.” It seems to ask, “What kind of creativity is there in a person like you?” contrary to the idea that “everyone has creativity.” As the educator Xingzhi Tao

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once said, “You have Watt in your taunt, Newton in your cold eyes, Edison in your sneer (Tao, 2018).” All people (including great men) have weaknesses and strengths in the way they develop creativity. Thus, creative education should be based on the principle of “teaching in accordance with aptitude” so that the educated can “exploit strengths and avoid weaknesses.” In creative education, the first step is to establish the correct educational concept, especially the correct talent concept (referring to diversity, generalization, and level of talent), and to consider all those who can contribute to society as talent. At the same time, we should note that although society needs talent with a high level of education and advanced degrees, it also needs talent with a low level of education and limited degrees; it needs talent who receives extensive systematic education and talent who are self-taught and have certain specialties or expertise in practice. Both types are all creative in varying aspects, and some of them are outstanding, that is, one way or another, all roads to Rome—they are all essential talents for society. Modern educational concepts have also proposed new requirements for the cultivation of future talent in schools. Specifically, importance is attached to developing students’ innovative spirit and creative ability, growing their capacity to independently obtain knowledge and apply it to solve practical problems, teaching them to respect students’ different personalities, and developing their sense of individuality. With this approach to education, we should be future-oriented, aim at nurturing “T”type talents, innovate teaching contents, actively and steadily reform the curriculum and teaching materials, and aggressively improve teaching methods. We should also boldly revise the content and methods of the examinations and emphasize the investigation of innovation and creativity. Only in this way can education and teaching meet the new needs of future talent.

Integrating Eastern and Western Education Models and Cultivating the “T”-Type Talents Since the late 1970s, creative education has been one of the main research focuses on China. Psychology is increasingly concerned with studying human resources such as human abilities, intelligence, knowledge, skills, activeness, initiative, creativity, and the distinction between the breadth and depth of knowledge structures. Researchers of talent studies in China also value the type of talent according to the structure of knowledge, figuratively using “—” to indicate the breadth of knowledge and “|” to indicate the depth of knowledge. In addition, the “T”-type talent concept has been proposed, i.e., a talented person with a wide range of knowledge and deep knowledge of expertise. Here, we provide a new interpretation of talent or human resources: to nurture “T”-type talent by integrating Eastern and Western education mode and establish this as a goal of creative education in China.

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Western mode

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Range of knowledge, creativity, adaptability, independence, practical abilities depth of knowledge logical thinking

Eastern mode

comprehension ability uniformity collectivism

Fig. 3.1 “T” type talents model

In a certain sense, East and Southeast Asian countries are more typical of Eastern countries, whereas the West mainly refers to European and American countries. The “T”-type talent cultivated by integrating Eastern and Western education mode are world citizens representing the best qualities of both regions, which is the fundamental goal of creative education in China in the twenty-first century. The successful development of “T”-type human resources would bring about a global educational revolution. However, doing so is quite challenging, requiring the reform of the old educational content, concepts, methods, and approaches. As shown in Fig. 3.1, “—” represents Western educational concepts, teaching methods, and teaching mode, while “|” represents Eastern educational concepts, didactic, and teaching mode. What are the characteristics of each of the Eastern and Western education models and the talent they produce? Western education emphasizes the development of a broad range of knowledge, creativity, adaptability, independence, and practical skills. Adaptability serves as the basis of this educational model, practical training as the instrument, the growth of creative ability as the fundamental goal, and the development of individuality as the goal. Western education places great importance on students’ adaptability or social adjustment, regarding adaptability as a component of both intelligence and mental health. Western education emphasizes practical activities for students, from elementary school to graduate school, craft (or hands-on) courses, and teachers who oppose “empty talk” but instead encourage students to practice while guiding them to solve practical problems. Western education also follows the principle of creativity cultivation and therefore advocates for “creative education” and “creative learning” so that students can fundamentally improve their creativity. One of the important goals of Western education is to fully develop an individual’s personality. In other

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words, the purpose of education is to make full use of macro-social correlations, form micro-interpersonal relations through group interactions, promote personality development in various ways, become a lively social individual, mobilize individual agency, and serve society. Eastern education emphasizes the development of in-depth knowledge, logical thinking, comprehension ability, uniformity, and collectivism. This kind of educational model is characterized by knowledge understanding as its base, formal studying as the instrument, the growth of logical thinking as the fundamental goal, and the pursuit of unified norms as the goal. Eastern education focuses on students’ knowledge and emphasizes the depth of knowledge and the degree of understanding, which can be seen as “I do not seek the answer; I seek to understand the question.” In the East, since ancient times, regardless of country, the main test of all subjects has been the examination of knowledge. Eastern education places a special emphasis on reading. There are sayings in China: “In the book, there is a house of gold; in the book, there is a shade of jade” and “Nothing is more important than education.” Eastern education focuses on the development of logical thinking. Human thinking is logical, which means that the thinking process has a form and manner and is carried out according to laws. Among Chinese international students in the West, a considerable number study mathematics and computer science, and their grades are quite good. Why is that? This is because Chinese students have accepted logical thinking since their childhood. Eastern education values the deep understanding of knowledge, emphasizing rational knowledge and that students “see through the appearance to perceive the essence.” Eastern education also underlines the concept of collective collaboration, rules, and standardization: “Nothing can be accomplished without norms or standards.” Therefore, the pursuit of unified norms is a goal of education. Which one is better? Each method has its own advantages. The two models were different and compatible. More than a hundred years ago, Chinese education began to advocate “conversant in both Chinese and Western.” This principle should continue to be advocated in conjunction with “exploiting strengths and avoiding weaknesses” to cultivate several innovative or creative talents.

Approaches to Developing Students’ Creativity Through Creative Learning Students should be encouraged to engage consistently in creative learning. In general, learning refers to the process of experience acquisition and behavioral changes. Creative learning is the product of the theory of discovery learning proposed by Dacey and Madaus (1969). Bruner and Guilford proposed the theory of creative thinking. Learning can be divided into two styles: reception and discovery learning.

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Reception learning refers to the situation in which learners convert others’ experiences or knowledge into their own; the content learned is delivered in a definite form by educators without any independent discovery on the part of students. The corresponding teaching method is the expository method, in which the learner internalizes and organizes the experiences and knowledge of educators so that they can be reproduced and used when necessary. Discovery learning, or the discovery method, encourages learners to discover and solve problems themselves. It aims to develop learners’ independent thinking and uses basic textbooks as content to enable them to learn through self-discovery. Discovery learning can be classified into independent and guided learning methods. The former is the same as scientific research and is less common in the school context. Simultaneously, the latter occurs in the classroom context, where students are asked relevant questions, guided to study, gather relevant information, and then discover concepts and principles by actively thinking and experiencing. According to Bruner (of discovery learning, discovery learning has four advantages) (Bruner, 1981): first, it is conducive to mastering the knowledge system and learning methods; second, it helps arouse students’ motivation and enhance their confidence; third, it is beneficial to develop students’ discovery and creativity attitudes and their inquiring mindset; and fourth, it facilitates the consolidation and transfer of knowledge and skills. Guilford’s (1957) creative thinking mainly refers to divergent thinking, which consists of three characteristics: flexibility, the capacity to consider a variety of approaches to a problem simultaneously; originality, that the ideas differ from others; and fluency. Creative learning was developed based on research on discovery and creative thinking. The term “creative learning” comes from “innovative learning.” The concept of innovative learning was first introduced in the work of Botkin et al. (1979) : No Limits to Learning, in response to global environmental issues, energy crises, and so on. Innovative learning is a type of learning that is opposed to traditional learning methods such as maintenance learning. Maintenance learning obtains fixed insights, methods, and rules to address known and ongoing situations that are necessary for closure and fixed situations. In contrast, innovative learning causes renewal, reorganization, and the formation of a series of changes. Its main characteristic is its comprehensiveness, meaning that it can be applied to a broad range of open environments and systems. Anticipation and participation form the conceptual framework of the innovative learning process that requires creative work. Another difference between maintenance learning and innovative learning is that the problem needs to be solved through maintenance learning derived from scientific authorities or administrative leaders. Its solution is easily understood and accepted by the public. By the early 1980s, the international psychology community realized the great importance of “creative learning.” Thus, it was at this time I proposed the concepts of “repetitive learning” and “creative learning.”

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Characteristics of Creative Learning The purpose of learning activities is to apply all experiences, knowledge, and cultural achievements obtained by humankind to arm the minds of the new generation to find ways to positively change individuals’ behavior, thus benefitting civilization and social development. Education is the foundation of learning activities. Today, we further emphasize creative learning, a form of creative education. Creative education is developed through creative training that is driven by creative theory. This includes two aspects. First, psychologists recommend creativity training programs to develop creative abilities. It is important to emphasize here that cultivating such abilities is not limited to processes such as “creative problem solving.” Incredibly, it is inseparable from the entire process through which one develops a multifaceted and complete personality. As the formation of personality and intrinsic motivation is essential for the development of creativity, education must positively influence personality formation. Psychologists also promote “problem-solving” exercises and other methods, such as encouraging students to ask questions and understanding teachers the ways on how to formulate questions in such a way as to elicit creative abilities. Second, apart from being crucial for the sustainability and success of creativity, the important role of educational measures can be attributed to organizational factors. The purpose of educational measures is to ensure the high efficiency of the participant and maintain a highly creative mental state. In recent years, many proposals have been made to apply various organizational procedures to stimulate creativity. Creative education is based on creative training, which takes special account of ways of thinking about presentation, discovery, divergence, and creation. In creative education, it is necessary to promote schools’ context creativity, employ creative teachers, develop creative learning behaviors, and adapt to the characteristics of creative education so that students form affective and voluntary learning activities focusing on presentational, discovery, divergent, and creative questions, which is creative learning. Creative learning is a form of creative education. Creative learning has the following main characteristics: 1. Emphasizes learners’ subjectivity. Subjectivity is the basic characteristic of the learner as the subject of practical and cognitive activities, and its essence lies in the learner’s self-consciousness in monitoring learning activities. Studies have shown a high correlation between creative thinking and self-consciousness and that individuals with high levels of self-acceptance, independence, autonomy, and emotional frankness have higher levels of creativity (Silvia & Phillips, 2004). The subjectivity of the learner is reflected in the fact that the student is the embodiment of the educational purpose—the student is the master, active explorer, and reflector of the learning activity. 2. Encourages learning research and focuses on learning strategies. The most essential tool for all students was to know how to study; the most effective knowledge was self-controlled knowledge. To learn how to study, learners must first acquire effective learning strategies—what, when, where why, and how to study.

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Learning strategies refer to the rules, methods, and techniques required to achieve the stated goals of a learning activity. Creative learning involves the application of learning strategies. Students learn to study, establish creative learning environments, find unique approaches, and grasp opportunities to identify and solve problems. All students were required to use the appropriate learning strategies. 3. Requires learners to develop a novel, flexible, and efficient learning method. Creative learners are motivated to organize the time spent learning, have more systematic and appropriate learning methods, and develop good study habits. 4. Motivates students to learn from creative activities and pursue creative learning goals. Creative learners have a unique motivation for learning: a strong curiosity, a desire to learn, an outstanding quality of will to eliminate distractions and focus on an interesting issue, the motivation to investigate the changing mechanisms of matters, the desire to find answers to difficult questions, a diligent attitude toward learning, a broad range of independent thinking that is not influenced by teachers, admiration for the learning ideals of celebrities and great minds, and a goal to strive to spur themselves on. Thus, the goals, content, and approaches to learning should extend beyond unusual ideas and actions.

Behavioral Characteristics of Creative Students In addition to having a unique personality, creative students also exhibit distinctive behaviors. The American psychologist Torrance (1974) surveyed 87 educators and asked each to list five behavioral characteristics of creative students. The results are presented in Table 3.1. Table 3.1 shows the behavioral characteristics of creative students—curiosity, flexible thinking, independence, inquisitiveness, desire to explore, and so on. This result was consistent with the actual situation. Innovation is valuable for learning. Some argue that learning is simply a process of accepting knowledge from predecessors and books and that there is no innovation or creation. However, learning differs from scientific research because it requires students to break old rules and explore what they have learned by applying multiple ways of thinking. Although most students in schools employ repetitive thinking, the development of students’ creative thinking is an important goal in education and teaching. Students’ thinking is unique, divergent, and novel in the learning process and is a manifestation of creative thinking. A new and important research topic in the psychology of thinking and learning is to study the development of students’ creative thinking and the characteristics of their creative learning and conduct scientific analyses to promote the effects of students’ creative learning, which is a fundamental task for educators in the information era.

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Table 3.1 Behavioral characteristics of creative students listed by 87 educators and mentioned proportions Behavior characteristics of creative students

Percentage of mentions (%)

Curiosity, constantly asking questions

38

Originality of thought and action

38

Independence of thought and action, individualism

38

Imaginative, likes to narrate

35

Does not follow the crowd, not overly dependent on the collective 28 will Likes to explore various relationships

17

Has a lot of ideas (fluency of thought)

14

Likes to perform experiments

14

Highly flexible

12

Tenacious, persevering

12

Likes fiction

12

Interested in the intricacies of things and likes to explore complex 12 things with multiple ways of thinking Delight in fantasy

10

Seven Approaches to Developing Students’ Creativity in Creative Learning Seven approaches to enhancing student creativity have emerged through research on theories of creative learning, experiments, and studies on the development of students’ creativity. 1. Improve the campus atmosphere. Establish a campus culture for creative learning, including recognizing and internalizing creativity and creating a widespread, strong sense of cultivating innovation. Furthermore, building a supportive, creative campus environment and conducting creative teaching activities will improve the creativity of students and faculty. 2. Improve students’ creativity in teaching each subject. Develop a series of requirements related to a specific competency in a specific subject to inspire students’ creative learning and enhance their creativity by meeting these creative learning requirements. 3. Inspire students’ creative learning and cultivate creativity in classroom teaching. By motivating students to be creative, asking flexible questions, and assigning homework, teachers master and apply creative teaching strategies such as discovery teaching, problem-solving teaching, discussion teaching, and openapproach teaching, through which they creatively guide students’ learning and cultivate their creativity by solving problems. 4. Promote new interpersonal relationships on campus. Based on the views mentioned earlier on the relationship between creativity and intelligence and

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the composition of creative talent, the formation of creative interpersonal relationships should be promoted. To achieve this, school administrators must strive to establish a democratic leadership style and improve the relationship between managers and teachers. This can be accomplished by building a “youwith-me” type of teacher–student relationship and improving the relationship between teachers and students. In addition, independent, cooperative, and inquiry learning should be conducted to improve the development of non-intellectual and intellectual factors. 5. Support innovation in school organization and management systems. Based on the requirements mentioned above regarding creative education, it is necessary to establish a creative campus that includes sufficient funding for schools in the areas of teaching and student management. Split management should be actively implemented to eliminate the negative impact of a “one-size-fits-all” management approach on student creativity. An innovative evaluation system to remove restraints on the current implementation of innovative educational concepts should be developed. 6. Teach students about special techniques for creativity training. Our research group introduced teenagers to methods such as “creating an appropriate environment” proposed by Torrance for creative training and taught them how to effectively engage in divergent questioning. Students can improve their creative abilities through creative training. 7. Cultivate students’ scientific creativity in science and technology activities. Among the various extracurricular options, science and technology activities are the most closely related to the development of students’ creativity. Students should participate in scientific and technological activities to stimulate creative learning. Science and technology activities can spark students’ desire to explore new knowledge and enhance their self-learning, research, operational, organizational, and creative abilities.

Approaches to Developing Students’ Creativity in Mathematics Teaching In elementary and secondary school mathematics classes, teachers should consciously apply the principles of creative development and actively carry out creative education and creative activities to cultivate students’ creative consciousness, as well as creative and practical abilities.

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Examples of Mathematicians from Ancient and Modern Times and Creativity Development Learning stories about mathematics, including examples from mathematicians, can stimulate students’ interest, enhance their willingness to create or innovate, engage in creative mathematical activities, and gradually improve their creativity in learning mathematics. One of the top ten best-selling books in China, the Encyclopedia of Chinese Children, is divided into four volumes (Zhejiang Education Press, 2017), including the “Science and Technology” volume with special chapters such as “Science Elite” and “Mathematics Treasury.” Many elementary and secondary schools use this resource for mathematics teaching or suggest it as further guidance for students to read after class. “Giants in Mathematics,” a section of “Science Elite,” presents stories of the world’s most notable mathematicians, including Pythagoras, who believed that “everything is a number,” Euclid, the “father of geometry,” Eratosthenes, who was the first to calculate the circumference of the Earth, Chongzhi Zu, the Chinese “father of Pi,” Luogeng Hua, who worked with extraordinary enthusiasm, and Jiangong Chen, the first foreign scientist to receive a doctorate in science from Japan. The history of human evolution is closely related to the history of individual civilization development. Accordingly, “Childhood of Mathematics,” a section of the “Mathematics Treasury,” tells the origins of mathematics, the story of clay tablets, the pyramids, the papyrus, the “pearl” on the palm of Buddha, the “bridge” of mathematics, the mathematics practices of the Babylonians and ancient Egyptians, the hometown of decimal and binary, Chinese contributions to the origin of mathematics, etc. This section motivates elementary and secondary school students to study mathematics and helps them develop creativity. The section “Interesting Number” also tells numerous stories, such as the meaning of zero, the use of fractions, the experience of decimals, the introduction of negative numbers, the storm of irrational numbers, real and imaginary numbers, the infinite, Goldbach’s conjecture, and the unsolved Fermat numbers. These stories pose a creative question to school students from a positive perspective and inspire them to think creatively. In the section “The Paradox,” tales of Russell’s paradox, the liar paradox, the pirate game, the Richard paradox, and the fallacy of the isosceles triangle, etc., inform students from the opposite side that if an argument is correct, then no matter what analysis and reasoning are employed, it will never come to a wrong conclusion; similarly, if an argument is wrong, then no matter what analysis and reasoning are used, it will never come to a right conclusion. The innovation process requires a solution to the paradoxes. The section “Various and Variety” describes algebraic equations, factorization, techniques of solving equations, Diophantus of Alexandria, the hundred birds problem, Vieta’s formula, Newton’s identities for a quadratic polynomial, the question of a chicken with a rabbit cage, and so on. The section “Images” discusses the great works of mathematics, Stoicheia, the Delian problem, Euclidean tilings created by convex regular polygons, the Golden ratio, the Pythagorean theorem, the asteroid,

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the Mobius brand, and other content. The latter two sections introduce mathematical problems through stories, inspiring students to learn mathematics and helping them recognize the breadth and diversity of the range of innovation, which in turn motivates them to engage in creative activities in daily life (Lin & Jiang, 1991). In 2008, I attended the 30th-anniversary celebration of a special class of gifted youth at the University of Science and Technology of China and reported at the conference. When I stepped off the podium, a student from the special class who had graduated many years ago said to me, “Mr. Lin, I have known about you from the Encyclopedia of Chinese Children for a long time.” As we can see from the growth of this special class of gifted youth students, learning about mathematical stories can help cultivate an innovative spirit and creativity.

Development and Training of Mathematics Creativity Creativity can be developed through mathematical training. Mathematics teachers can provide various conditions for the development of students’ creativity in class. Discovery Method. Unlike the general teaching method, the discovery method emphasizes that students independently discover new knowledge and find solutions through their thinking and exploration without relying on teachers’ explanations. Certainly, a certain degree of teacher guidance is required in the discovery process. Discovery teaching typically consists of the following six steps: (a) solving a problem, (b) proposing, (c) verifying, (d) summarizing, (e) applying, and (f) improving. In mathematics teaching, with the teacher’s assistance, students create corresponding mathematical propositions through experiments or models, observe figures or tables, and obtain methods and approaches to solving mathematical problems based on the principles of comparison, analysis, induction, and generalization. An example is as follows: The distance between stations A and B was 480 km. A slow train departed from station A, traveling at 65 km/h, and a fast train departed from station B, traveling at 95 km/h. Q1: The two trains departed at the same time and met each other halfway. How many hours do they meet? Q2: The slow train departs 30 min earlier than the fast train. How many hours did the two trains meet after a fast train departed?

After presenting the problem, the teacher demonstrated to the students how to compile it. For example, the stems of the two example problems remain the same: the two trains still meet each other at some point halfway, while the second question only alters the situation by having the slow train depart 30 min earlier than the fast train. How many hours after the slow train departs do the two trains meet? After that, the students were encouraged to try to create a problem with the stem as a known condition. Next, the teacher guides the students to exchange known conditions with the initial problem. Finally, according to the questions compiled by the students, the classroom selects representatives who compete to solve the problem, thus livening up the atmosphere.

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Divergent Method. Divergent the core of creative thinking. In mathematics teaching, student creativity can be developed by drawing inferences about other cases from one instance. Here is an example: A toy factory produces a batch of children’s toys. The original plan was to produce 60 pieces per day and complete all the work within seven days; however, it only took six days to complete all the work. How many more toys per day than originally planned?

The usual solution is to determine how many pieces of work there are in total, how many pieces are produced each day, and how many more pieces are produced each day than originally planned to use the formula 60 × 7 ÷ 6 – 60 = 10 (pieces). However, one student said, “You only need to calculate 60 ÷ 6 because this day’s task had to be completed in six days, so ten more items had to be made.” From his answer, we can see that he thinks of employing a leap of faith, skipping many analytical steps. He thought in this way: seven days of work were completed in six days, which is one day ahead of schedule, so the work for that day must be assigned to be completed in six days, that is, 60 ÷ 6 = 10, which is the actual number of pieces produced per day more than planned. This unusual and innovative solution is a valuable expression of individual creativity, which should be encouraged and praised by teachers. Creative Problem Solving. The process of creative problem solving requires the individual to overcome stereotypes, think from a new perspective, and obtain a new understanding of the problem to solve it. For example, in the “four trees problem,” students are asked to plant four trees on the land so that the distance between each tree is equal. Many students try to solve problems on a flat surface regardless of how they draw squares, rhombuses, trapezoids, and parallelograms. None of these worked. To solve this problem, students need to overcome the limitations of two-dimensional thinking and construct a regular tetrahedron in three-dimensional space (see Fig. 3.2). Fig. 3.2 A regular tetrahedron in three-dimensional space

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Brainstorming. Brainstorming is a creative group technique. Applying this method to mathematics teaching can broaden students’ thinking, enrich their imagination, transform passive learning into active learning, and provide space for students to develop their creative thinking. Using the brainstorming method to design instruction can be roughly divided into the following four stages: setting up a good environment for exploration or discussion; asking questions; self-construction in the discussion; regression; and conclusion. For example, in the teaching of cos(α + β) = cos α cos β − sin α sin β, teachers can first ask students to guess the formula and then ask them to prove that it is true (how can we prove cos(α + β) = cos α cos β − sin α sin β? How can we connect cosα, cosβ, sinα, and sinβ?) Ultimately, students suggest different ways in which teachers can summarize.

Mathematics Learning Motivation and Creativity Development The Russian writer Leo Tolstoy said, “Successful teaching is not required to be mandatory, but could stimulate the student’s interest” (Moulin, 2008). Strong motivation to learn mathematics is a prerequisite for students to develop creative mathematics activities. Therefore, teachers should not only teach knowledge but also cultivate students’ interest in learning mathematics by stimulating their intrinsic motivation to learn mathematics and protecting their curiosity. In recent years, China’s secondary school students have received numerous awards at the International Mathematical Olympiad. However, a survey of 21 countries conducted by the International Assessment of Educational Progress showed that while Chinese children are ranked first in the world in calculation, they are only fifth in creativity. This contrast makes educators deeply reflect on why outstanding Chinese computing ability cannot be transformed into creative ability. One reason is the lack of motivation and interest in mathematics learning among Chinese students. Although our students have built a firm foundation for knowledge through long-term mechanical and test-oriented mathematics training, their curiosity and creativity have been suppressed. A Comparative Study of International Mathematics Learning in the U.S. in 2006 The Education Report made an interesting finding: eighth-grade students in Asian countries scored high on the Trends in International Mathematics and Science Study (IMSTS) mathematics test, while students scored low in confidence: 6% in Korea and 4% in Japan. American students did not score very high on the mathematics test, with only 39% demonstrating confidence in it. Therefore, in addition to teaching basic knowledge, educators must consider how to foster students’ interest in learning mathematics, which is a prerequisite for students to develop their creativity in mathematics. A student’s creativity can only be stimulated if they are self-motivated and deeply love mathematics. Teachers can motivate students in many ways. A good example comes from a lesson plan that teaches students about the “angle of circumference.” The teacher

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first allowed the students to try to draw a circle, giving the following instruction: “Attach one end of a thin string to a cardboard board with pins and tie a pen to the other end. Then, pull the string straight and draw a circle.” After that, the teacher asked the students to replace the string with a rubber band and draw it again. Using this method, it is impossible to draw a circle. After giving students some time to realize this, the teacher asked two questions: “Why can we draw a circle with a rubber band?” and “What are the conditions required to form a circle?” Through this kind of operation, mathematics is no longer a boring, abstract concept but a useful subject closely related to students’ daily lives. This type of activity not only facilitates students’ enjoyment of learning but also helps teachers develop their creativity. Another example comes from a lesson about “surface area.” The teacher designs a problem, such as how many different ways are there to wrap two rectangularshaped chocolates with a length, width, and height of 3, 4, and 5 into one? What is the most economical way to address this question? Real-life examples inspire students’ interest in learning, provide students with a wealth of opportunities to use their imaginations, and help them develop creativity.

Mathematical Knowledge Teaching and Creativity Development Mathematics, which originates in human production and practice, embodies innovative thinking and contains an infinite number of fascinating ideas. Mathematics, such as algebra and geometry in elementary and secondary schools, have developed step-by-step from simple to complex problems in the process of human practice over a long period, fully reflecting the wisdom of humanity. For example, in mathematics, we are taught to use letters to express numbers, which may seem simple after understanding; however, such a simple thing represents a significant leap forward in human cognition that has shifted the perspective of human cognition from the field of numbers to the field of algebra, from arithmetic to algebra, thus enabling humans to change from static to dynamic thinking when solving practical problems. Understanding this knowledge also means that children and adolescents have progressed from concrete to formal thinking, a leap in their thinking level. Another example also represents the transition of human rationality—the introduction of negative numbers—whereby we no longer stop after a positive number reaches zero. By understanding their meaning, children and adolescents expand their understanding of numbers, construct the concept of rational numbers, and thus complete highly abstract intellectual sublimation. These are rational breakthroughs made through a leap in mathematical knowledge. As mentioned above, regardless of whether they receive mathematical knowledge or use it to solve problems, mathematics is a form of mental gymnastics and a creative process for students. On the one hand, when they receive mathematical

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knowledge, especially new mathematical knowledge, they achieve knowledge growth and competence enhancement, a form of innovation based on old knowledge. For example, when learning basic knowledge of solid geometry, there is a gradual change from second- to third-dimensional thinking. This transformation process represents transcendence and innovation, a shift from an old problem-solving model to a new problem-solving model. On the other hand, after absorbing new knowledge and forming new cognitions, students unconsciously apply this knowledge to try to solve problems and discover new ones. For example, after learning the parallelogram area formula, students can try to generalize it to the rhombus area formula by comparing the similarities and differences between the two to further understand the meaning of the figure and improve their ability to absorb new knowledge based on a firm understanding of the relevant former knowledge.

Mathematics Practice and Creativity Development Mathematics is a highly practical subject, and its vitality is based on its practicality. Given the limitations of the classroom teaching model, students can only solve practical problems by solving applied problems through “simulation” after learning mathematical knowledge and solving related problems. Even so, their creativity can be enhanced. For example, when solving problems regarding time, speed, and distance, although the students are not able to complete the relevant procedures in real life, the context provided by the problem and the interest in solving the practical problems that arise from it enables them to gain new knowledge and improve their abilities. Their interest in applying mathematical knowledge is further enhanced via problem solving, which provides a background for solving real-life problems in the future. In addition to this relatively passive mode of learning, mathematics teaching in elementary and secondary schools also involves the development of compiling word problems, which requires students to be creative, as they must not only understand the relevant mathematical knowledge but also possess a strong ability to express themselves logically. This process from knowledge storage to knowledge release is the same as innovation—understanding the knowledge, conceptualizing the topic, forming images, and then manipulating specific thinking until the topic is compiled, which is the same as a new invention. Psychological research has shown that such a learning model can effectively improve students’ mathematics performance and ability to solve word problems. With the implementation of the new curriculum and the introduction of advanced teaching methods, new teaching models have been introduced in the mathematics classroom, of which research learning and problem-based learning are representative models. For example, after learning statistics, teachers can ask students to investigate the height and weight of students in the class. After collecting, organizing, and analyzing data, students can report the results to the whole class so that they can truly apply the

References

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knowledge. This helps stimulate their interest in learning mathematics, cultivating their ability to transfer knowledge, and improving their creative abilities. Another example is that when students are learning about decimals, the teacher can require them to visit supermarkets or grocery stores before class to learn about the prices of goods. During class, while listening to the student reports, the teacher intentionally records some goods with decimals and then inspires them to discover the changes in prices by constantly shifting the position of the decimal point until students can eventually set a price and compare the differences between prices. This is a lively example of how to improve students’ mathematical ability using practical problem content.

References Botkin, J. W., Elmandjra, M., & Malitza, M. (1979). No limits to learning: Bridging the human gap: A report to the club of Rome. Pergamon Press. Bruner, J. S. (1981). Some elements of discovery. Thinking: The Journal of Philosophy for Children, 3(1), 26–31. Dacey, J. S., & Madaus, G. F. (1969). Creativity: Definitions, explanations and facilitation. The Irish Journal of Education/Iris Eireannach an Oideachais, 55–69. Eubanks, D. L., Palanski, M. E., Swart, J., Hammond, M. M., & Oguntebi, J. (2016). Creativity in early and established career: Insights into multi-level drivers from Nobel prize winners. The Journal of Creative Behavior, 50(4), 229–251. Guilford, J. P. (1957). Creative abilities in the arts. Psychological Review, 64(2), 110. Guilford, J. P. (1966). Measurement and creativity. Theory into Practice, 5(4), 185–189. Hua, L. G. (1931). Contribution: Reasons why Su Jiaju’s algebraic solution of fifth degree equation cannot be established. Science, 2, 307–309. (in Chinese). Jin, S. H., Zhang, J. H., & Wang, J. (2010). On the characteristics of top-level creative talents and the enlightenment gained. Journal of the Chinese Society of Education, 6, 5–10. (in Chinese). Karwowski, M., Dul, J., Gralewski, J., Jauk, E., Jankowska, D. M., Gajda, A., Chruszczewski, M., & Benedek, M. (2016). Is creativity without intelligence possible? A necessary condition analysis. Intelligence, 57, 105–117. Lin, C. D. (1983a). An experimental study on the cultivation of elementary school students’ computational thinking qualities. Educational Research, 10, 36–41. (in Chinese). Lin, C. D. (1983b). Psychology of middle school students. Beijing Publishing House. (in Chinese). Lin, C. D. (1992). Learning and development—The development and cultivation of psychological ability among primary and middle school students. Beijing Education Press. (in Chinese). Lin, C. D. (1999). Wisdom of education—For primary and middle school teachers. Kaiming Press. (in Chinese). Lin, C. D., & Jiang, L. (1991). Children’s illustrated Encyclopedia (Science·Technology). Zhejiang Education Press. (in Chinese). Lin, C. D., Yang, Z. L., & Huang, X. T. (2003). Dictionary of psychology (p. 74). Shanghai Education Press. (in Chinese). Moulin, D. (2008). Leo Tolstoy: The spiritual educator. International Journal of Children’s Spirituality, 13(4), 345–353. Silvia, P. J., & Phillips, A. G. (2004). Self-awareness, self-evaluation, and creativity. Personality and Social Psychology Bulletin, 30(8), 1009–1017. Sternberg, R. J. (2018). A triangular theory of creativity. Psychology of Aesthetics, Creativity, and the Arts, 12(1), 50.

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Tao, X. (2018). The second change of normal university students—Becoming a child. Teaching Management and Educational Research, 16, 5–6. (in Chinese). Temple, R. K. (2004). The genius of China: 3000 years of science, discovery, and invention (Y. Z. Chen, Trans.). People’s Education Press. Torrance, E. P. (1962). Guiding creative talent. Prentice Hall. Torrance, E. P. (1974). Norms-technical manual: Torrance tests of creative thinking. Ginn and Company. Walia, C. (2019). A dynamic definition of creativity. Creativity Research Journal, 31(3), 237–247. Zhejiang Education Press. (2017). The 2017 classic is back with a vengeance, children’s illustrated encyclopedia—Chinese children’s own classic encyclopedia. Chinese Book Review, 04, 129. (in Chinese). Zhu, Z. X., & Lin, C. D. (1986). Psychology of mental development. Beijing Normal University Press. (in Chinese).

Part II

Mathematics: The Gymnastics of Human Thinking

Thinking involves many different aspects, from material to spiritual, macroto micro-, and from theory to application. Notably, the power of thought enables the subject to profoundly recognize objective reality and produce ideological products that can actively change the objective world. In recognition of its importance, people have paid much attention to the development of thinking abilities since ancient times. Mathematics is one of the most important ways to develop human thinking ability. Mathematics involves the study of quantitative and spatial relationships in the real world. These are closely related to human beings’ survival and societal progress; thus, mathematics remains embedded in and intertwined with the pursuit of science and technology (including humanity, society, and science) and every aspect of our lives. Mathematics has been called the foundation of exploration and invention, a bold claim that is entirely true, because all forms of scientific exploration and invention are impossible without mental calculations marked by mathematical thinking. Therefore, mathematics serves as the most basic lesson for developing thinking abilities. Just as people stress the importance of exercise for “good health,” people value mathematics as an essential approach to “good thinking.” As Confucius pointed out, “study without thinking is labor lost. Thinking without study is perilous” (Confucius, 2011, p. 20). Indeed, mathematics and thinking first construct the basic patterns of “learning” and “thinking,” after which mathematics continuously promotes the development of thinking ability. The three chapters of this part discuss the specific manifestations of this approach.

Reference Confucius (2011). The Discourses and Sayings of Confucius (H. M. Gu, Trans.). Yunnan: Yunnan People’s Publishing House, Yunan. (in Chinese and English)(Original work published 540 BC– 400 BC).

Chapter 4

The Complete Structure of Mathematical Thinking

Thinking and intelligence are psychological phenomena, i.e., cognition of objective facts generated by the human brain. The world of objective reality is a unified material world in which all matter and phenomena are constantly moving, changing, and developing according to the laws of material itself. Indeed, it is the complexity, integrity, and unity of the objectively perceived real world that determines the complete structure of human thinking. Because mathematics is a science that reveals the essence of matters in terms of quantitative relationships and spatial forms, learning this subject requires students’ thinking to have an overall structure. The more complete the structure is, the more students can efficiently understand the integrity and complexity of quantitative relationships and spatial forms, and the stronger their mathematical ability becomes. It can be assumed, therefore, that learning mathematics is based on the integrity of thinking and can promote the development of its overall structure as these two elements are closely related.

The Triangular Pyramid Structure of Thinking Thinking is a complete construct consisting of six elements of thinking: purpose, process, material, quality, monitoring, and non-cognitive factors (see Fig. 4.1). The concept of the structure of thinking was initially proposed in 1965. From 1965 to 1978, I was a teacher and education manager in a middle school. When I went to physics class, the teacher often spoke about physical composition, and when I went to chemistry class, the teacher and I frequently discussed the structure of chemical substances. This caused me to ponder, “if material has a structure, does not the mind have a structure?” Thus, it was the structural view of natural science that aroused my strong interest in the structure of thinking or intelligence. Later, at the first annual academic conference of the Chinese Psychological Society, recommended by my mentor Zhixian Zhu, I presented the topic of “The Number Concept and Arithmetic © China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_4

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Fig. 4.1 Model of the structure of thinking

Capability in Children and Adolescents,” in which I preliminarily demonstrated my model of the structure of thinking. Later, after discussion with my mentor and five other graduate students, I further refined the model. In 1982, at the first meeting of the Professional Committee of Developmental and Educational Psychology of the Chinese Psychological Society, I took the opportunity to select 10 questions that reflected the thinking or intellectual structure the most in the open questionnaire, asked 50 conference members (approximately half of the participants) to “check the box” in the form of a structured questionnaire, and then analyzed the results. On this basis, I took this opportunity to conduct research in elementary and secondary schools and selected 25 reputable teachers from both elementary and secondary schools to “check the box” on the same questionnaire. The results showed that the rate of agreement with six factors was above or close to the third quartile for approximately 75% of the conference members and elementary and secondary school teachers who completed the questionnaire. Subsequently, the refined concept of the structure of thinking was published in the psychological journal Acta Psychologica Sinica and in the book entitled “Psychology of Thinking Development,” which was co-written by my mentor Zhixian Zhu. As a result of subsequent efforts to improve the model, it now exists in a more mature form.

Purpose of Thinking How did the idea of the purpose of thinking originate? In March 1961, during the second semester of my freshman year, I enrolled in a basic course entitled “General Psychology.” During a class discussion after a lecture on “The Characteristics of the Human Mind,” I voiced the following opinion: “the substantial differences in mind and behavior between humans and animals arise from the purposefulness of the human mind. The human mind is characterized by conscious thought, which is referred to as the problem proposition.” The discussion facilitator found my opinion to be insightful and creative, and such “praise” made me immensely happy and more confident about expressing my beliefs. I think the primary purpose of human thought

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is to understand and adapt to the environment. Problem identification and solving are some of the most advanced intellectual activities that illustrate the determination of human intellectual activity. This purposefulness is based on the structure of thinking and is especially obvious for schemas and strategies. Continuous development and improvement of purposefulness are significant to ensure the directivity, focalization, and specialization of the thinking activity. Since returning to school to rejoin the psychology community in 1978, I have conducted a great deal of research and found that the development or improvement of the purposefulness of thinking is reflected in five indicators: orientation, adaptation, decision-making, schema, and foresight. Will this point mentioned early stand further? Since the 1990s, I have intentionally assigned this task to some of my graduate students, instructing them to conduct relevant research. Two of my doctoral students, Ziqiang Xin and Wu Kang, conducted empirical studies to explore the purpose of thinking in their doctoral dissertations. Ziqiang Xin and Wu Kang’s doctoral dissertations were “The Acquisition of Schema and Strategy in Mathematical Problem Solving of Children” (Xin, 2002) and “Middle School Students’ Mathematical Problem-posing Capability” (Kang, 2003), respectively. These two papers suggest that humans can only understand and adapt to the environment by establishing a better and more complex cognitive structure that enables the subject to generate rational cognition based on perceptual cognition. This kind of rational cognition is based on the premise of self-orientation, proactive foreseeability, plan-making, and consciously transforming nature, changing society, and adjusting oneself. Moreover, it is an implicit and complex cognitive processing process that requires active planning, hypothesizing, testing, regulation, and reflection on mathematical situations or problem solving. Therefore, while the purposefulness of intellectual activities is restricted by the schema of the subject, it remains one of the primary traits of intelligence, reflecting the consciousness, initiative, direction, and intentionality of human intelligence. The structure of problem types can guide intellectual operations in the right direction and improve the correctness of problem solving. These two studies led to three conclusions: first, there exist certain trends in the developmental changes in thinking and purposefulness of intelligence; second, thinking and purposefulness of intelligence are indicators of the level of different subjects; and third, the primary purpose of human thinking and even intellectual activity is to adapt to and understand the environment.

Process of Thinking Thinking itself has a process. I remember that when studying the psychology of thinking at university, I was particularly fascinated by Soviet psychology, especially the Rubinstein theory. There is an important viewpoint in Rubinstein’s theory, that is, thinking is not only an activity of analysis and synthesis but also a process of abstraction, generalization, comparison, systematization, and substantiation of forms (Holowinsky, 1985). Despite finding it overly simplistic to summarize thinking in this manner, I could not come up with a more convincing argument.

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In the early 1980s, I was introduced to cognitive psychology. Cognitive psychology is a science that studies human intelligence and how people think. It emphasizes that cognition is a form of information processing within a certain psychological structure and for a certain purpose. The process of information processing includes serial, parallel, and mixed processing. I believe this cognition is actually thinking or intelligence, an activity that can be regarded as a form of information processing in a certain psychological structure for a certain purpose, which is also shown as a serial, parallel, and mixed processing. Therefore, I combined both Soviet and Western theories to develop a new opinion that there exists a process in thinking that functions as an important aspect of the structure of thinking. The procedure of the thinking process can be specifically defined as follows: determining the goal → receiving information → processing and coding → abstraction and generalization → operation and application → achieving success. Will this point mentioned earlier stand further? Based on my point of view, Qi Zhang, one of my doctoral students, conducted an analysis in his doctoral dissertation concerning cognitive ability, arithmetic ability of equality quantity operations, and the anticipatory image of geometrical figures of elementary school students (Zhang, 2002). His dissertation revealed the following points. (a) The process of thinking is the main process for developing intelligence (cognition), whether it is the cognitive process of equal quantity relations or the process of the predictive image of geometric figures; (b) The development of the thinking process leads to the improvement of the intellectual process, which includes the improvement of specific thinking processes as well as better coordination and integration of the entire thinking process: (c) two trends in the development of the thinking process behave as the cognitive process improves. First, the analytical synthesis process of each specific cognition in the whole cognitive process improves. Second, coordination, unification, and perfection of the various cognitions are achieved under the purpose, task (receive information, information processing), and process (serial, parallel, or mixed) requirements of the thinking activity throughout the cognitive process. Therefore, the development or perfection of the thinking process determines the operational use and success of the entire cognition process; and (c) The development of intellectual processes is expressed in the refinement of the abstraction process of thinking.

Material of Thinking There is a popular Chinese saying that “even a clever housewife cannot cook a meal without rice,” the Western version being “one cannot make bricks without straw.” Similarly, a prerequisite of thinking is material. If we agree that the basic process of thinking is information processing, the material of thinking is logically the content of thinking, which is the internal representation of external objects or their attributes. There exist various internal types and forms of representations of external information results; however, these boil down to only two kinds: one is perceptual material, including sensation, perception, and image, and the other is rational material, which

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mainly refers to concepts, i.e., generalizations of matters’ forms, combinations, and characteristics expressed via language. The specific indicators that demonstrate the development and improvement of intellectual content are the comprehensiveness and selectivity of perceptual (cognition) material, the profundity, and generality of rational (cognition) material and the accuracy and flexibility in the process of transforming perceptual material into rational material. In the early 1990s, my doctoral student, Yinghe Chen, analyzed the material of thinking in her doctoral dissertation “The Development of Children and Adolescents’ Cognitive Operation to Acquire Geometric Concepts” (1992). In her research, she discovered that understanding the concept of plane geometry entails progressing through four distinct levels for children and adolescents. The first is the concrete level at which they are able to recognize the previously perceived figure after a certain time interval. The second is the identity level, where they are able to recognize the previously perceived figure from different visual angles. The third is the classification level. At this level, they are able to recognize that two or more different examples from a geometric concept (figure) are the same; the core competency to achieve this level is the ability of abstraction. The fourth and final level is the form level. Here, children and adolescents are able to process the connotation of the concept in essence. This process reflects that their understanding of the matter develops from perceptual to rational. Moreover, the progression of their thinking ability is not straight and linear but instead spirals upward. Chen’s (1992) dissertation can be summarized as follows. (a) The development of thinking and intellectual material (content) is a transformation from a concrete image to a logical abstraction; (b) one of the most important characteristics of the development of children and adolescents’ thinking or cognitive abilities is the continuous abstraction of intellectual material (content) and the continuous generalization of cognitive representations, which marks the increasing level of simplification and generalization in their thinking processes. This also means that they are developing abstract thinking and cognitive rationality; (c) contemporary cognitive psychology observes that the representation and generalization of matters, rational cognition, and abstract thinking should be the focus of cognitive psychology; and (d) three main types of material (content) are employed for rational cognition or abstract logical thinking: linguistic (semantic, concept, proposition, etc.), number (symbolic, operation, code), and shape (geometric figure, design chart, sketch, curve, schematic diagram, etc.). Just as the content of thinking differs, so does the process of thinking. Therefore, linguistic, numerical arithmetic, and graphic representation ability are the three basic components of intelligence in thinking psychology.

Quality of Thinking The exploration of thinking quality helps me construct the structure of thinking. Here, there is a problem between better and worse, that is, which thinking quality is better. I regard thinking quality not only as a personalized characteristic of thinking

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but also as an evaluation reference for thinking outcomes. The five thinking qualities (profundity, flexibility, creativity, criticalness, and agility) should be considered the main indicators of the “qualitative” development of intelligence and competence. Then, what about the internal relationship between the level of thinking quality and the thinking quality? This can be answered by the data constructed by my student, Chunmi Li, in his doctoral dissertation, “Research on the Development of Physics Experimental Operation Ability of High school Students” (Li, 2002), which concerns the process of changing and improving students’ thinking quality. His research results are as follows. First, a comparative study of the different thinking qualities revealed that students scored highest in the quality of profundity, indicating that this quality is the basic foundation of all other thinking qualities. This is logical, given that it is an inevitable tendency in the development of abstract logic thinking. In contrast, students scored lowest in the quality of creativity, suggesting that this quality appears later and is more difficult and time-consuming to develop than others. Second, regarding the relationships among the various thinking qualities, the quality of agility has the highest correlation coefficient with other qualities, thus indicating that agility is primarily derived from or determined by other qualities. The qualities of flexibility, criticalness, and creativity are highly connected, proving that divergent thinking is a prerequisite for or manifestation of creative thinking, the degree of which is closely related to the degree of criticalness. The lowest correlation coefficient was between the qualities of profundity and creativity, which suggests that abstract logical thinking may not necessarily produce creative thinking. It also shows that abstract logical thinking may not be the sole source of creative thinking, as imaginary logical thinking may also generate creativity. The main findings in Chunmi Li’s dissertation are as follows: (a) the improvement of intellectual quality improves the overall development and maturity of intellectual thinking quality; (b) since intelligence serves as a personalized psychological characteristic, there are naturally various levels reflecting individual differences. The levels of intelligence (above normal, normal, and below normal) are mainly represented in the level of thinking, i.e., the thinking quality; and (c) a change or improvement in intelligence is further reflected in the function of each thinking quality, the role and effect of each thinking quality in intellectual activities, the time and order of development and change, and the influence on and function qualities have with others. The perfection of these factors implies the perfection of thinking quality and is considered an important indicator of intellectual development.

Monitoring of Thinking Monitoring one’s thoughts refers to conscious self-monitoring of thinking, which I prefer to call “reflection” or “introspection.” For those naturally curious about selfreflection, the question arises: why do people need to “go over the movie of the day?” I first asked this question when I was an undergraduate. At that time, I was

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very intrigued by reflection and introspection. I also wondered, what is the “monitoring of thinking?” My conclusion was that it is a kind of self-consciousness that is the expression of self-consciousness in thought. I came up with this idea around the 1970s. Almost at the same time, the American psychologist J. H. Flavell proposed the idea of “meta-cognition,” which, in a certain sense, is the self-monitoring of thinking or criticalness of thinking. I was reluctant to call this component “monitoring” because this concept had already been adopted by Flavell. While I called it “the thinking rule,” annotated as “the process of thinking follows certain rules due to the effect of the reflection or the effect of self-consciousness in thinking.” Later, my mentor, Prof. Zhixian Zhu, suggested that it is better to use the term “thinking monitoring.” Thereafter, my students and I discussed its indicators and gradually formed the development and improvement indicators of thinking reflection or selfmonitoring, including the five aspects of planning, inspection, adjustment, management, and evaluation. There are two students who have performed research in this field: Tao Xin, who specializes in the self-monitoring of teachers, and Jianyue Zhang, who specializes in the self-monitoring of students’ learning in mathematics. Tao Xin’s doctoral dissertation is entitled “Teachers’ Teaching–Monitoring Capability: The Construct, Impact Factors, and its Relationship with the Development of Students” (Xin, 1997). He believes that teaching–monitoring capability can be divided into three aspects regarding teaching activities: first, teachers plan and arrange their teaching activities; second, teachers consciously monitor, evaluate, and obtain feedback on their actual teaching activities; third, teachers adjust, calibrate, and consciously control their teaching activities. Using a combination of correlation research and intervention studies, Tao Xin explored the structure of teachers’ teaching–monitoring ability, its influencing factors, and its impact on teachers’ behavior and students’ development from three aspects. Jianyue Zhang’s doctoral dissertation, entitled “Middle School Students’ SelfMonitoring Abilities in Mathematics—Structure, Development and Influencing Factors” (Zhang, 1999), shows how students’ reflection or self-monitoring of thinking develops and changes. Under normal school education conditions, the development of secondary school students’ mathematics self-monitoring capability has its age stage but remains relatively flat except for the period from elementary school to middle school. There was no significant difference in the development of the inspection throughout the secondary school stage, while there were different changes in regulation, inspection, and management from middle school to high school. These two dissertations identify five functions of self-monitoring, which are as follows: (a) to determine the purpose of thought; (b) to manage and control nonintellectual factors; (c) to collect and select appropriate thinking material and thinking strategies; (d) to implement and supervise the thinking process; and (e) to evaluate thinking outcomes.

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Non-cognitive Factors of Thinking The original view of the structure of thinking did not include the “non-cognitive factors of thinking.” In the fall of 1982, my mentor Zhixian Zhu organized a discussion among graduate students, during which most students suggested that non-cognitive or non-intellectual factors should be added. At the end of the same year, in the questionnaire survey, most respondents checked “non-intellectual (noncognitive) factors.” I accepted this suggestion and took it as another focus of the research. What is a non-intellectual or non-cognitive factor? It refers to a psychological factor that is not directly involved in the intellectual process but plays a direct role in it. Non-cognitive factors of thinking mainly include ideals, motivation, interests, emotions, will, temperament, and personality, all of which are related to intellection. The attributes of non-cognitive factors often depend on the relationship between the thinking material or thinking outcomes and the individual purposes. The No. 6 Secondary School at Tongzhou District in Beijing (the original No. 6 Secondary school at Tong County in Beijing, hereinafter Tongzhou No.6 Secondary school) participated in the research study. The resulting change in the quality of Tongzhou No. 6 Secondary School education indicates that non-cognitive factors play a motivational, stereotypical, and compensatory role in the development of thinking and even intelligence. Will this point mentioned earlier stand further? In 1986, one of my students, Jiliang Shen, worked with me in a research group investigating the development and training of psychological ability among elementary and secondary school students. He made great efforts in experimental research in schools with students from poor educational backgrounds, such as Tongzhou No. 6 Secondary School. He also visited Beijing No. 5 Secondary School in 1988 to study the non-intellectual factors of middle school students and the relationship between intelligence and academic performance with the help of a teacher named Jie Liang. As a result of his research, he concluded that non-intellectual factors play a significant role in the process of students’ formation and development of intelligence; achievement of good academic performance is related to not only intellectual factors but also non-intellectual factors. For this reason, Jiliang Shen points out that intelligence cannot be separated from nonintellectual factors, but the two factors supplement each other and form as a whole; it is necessary to explore the specific role of non-intellectual factors in the development and improvement of intelligence; for all intellectual activities, intelligence and non-intellectual factors work together, and about which is more or less should not be a substantive issue. The reason I published my paper on the structure of thinking in an international journal is that Chinese educational circles have too much enthusiasm for Howard Gardner’s theory of multiple intelligences. There is even a genre of Chinese children’s books called multiple intelligences. An American psychologist once said to me, “China is doing more to promote multiple intelligences than the United States. You praise Gardner too much.” It was then that Gardner’s view of intelligence first caught

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my attention, especially when I was introduced to Gardner himself in 1987 at the 7th World Council for Gifted & Talented Children in Salt Lake City. Upon returning home from Salt Lake City and further pondering Gardner’s theory, Gardner’s views suddenly seemed familiar to me. Where had I seen these ideas before? The answer finally came to me—in the thoughts of Chinese ancestors, that is, the “Six Arts” that arose 2500 to 3500 years ago from the official school of the Western Zhou Dynasty to the private school of Confucius. The “Six Arts” consist of li (rites), yue (music), she (archery), yu (chariotry), sh¯u (calligraphy), and shù (mathematics) (Confucius, 540 BC - 400 BC/2011). Thus, in the paper co-written with my student Tsingan Li (Lin & Li, 2003), we clearly indicate the similarity between Gardner’s theory and ancient Chinese thought. These “Six Arts” are also known as the seven intelligences. Some people claim, “there is no intrapersonal intelligence in the Six Arts!” to which I respond, “the Six Arts clearly mention “rites,” which concerns interpersonal relations. Moreover, teachings such as “restraint yourself and follow social norms,” “who understands the world is learned,” and “who understands the self is enlightened” in “rites” are cues for intrapersonal intelligence or self-control intelligence!” The seven intelligences mentioned in the “Six Arts” are similar to those mentioned by Gardner. However, the “Six Arts” and Gardner’s theory are not identical. That said, the difference does not lie in the different times at which they emerged—one is ancient thought, and the other is modern theory. Nor do I wish to emphasize that we are 2 or 3 thousand years ahead of him; rather, I want to highlight the differences between them. First, Gardner believes that the seven intelligences are independent and unrelated, while the “Six Arts” emphasize the interconnection with “rites” as the core: the arts are connected internally and externally and mutually related to various relationships existing between them, such as inclusion, correlation, integration, and intersection. Second, while Gardner’s “Future School” was still in an experimental stage, ancient Chinese teachers had already included the “Six Arts” in their education systems 3500 to 2500 years earlier. Therefore, we must not forget our ancestors’ wisdom and worship everything foreign. This is the first viewpoint addressed in an article published internationally in 2003: “a discussion about the relationship between the “Six Arts” and Gardner” (Lin & Li, 2003). The second viewpoint is an opinion about the structure of thinking or intelligence rooted in the ancient “Six Arts.” The third viewpoint concerns the infinity of the components of the thinking structure that is proposed below. Gardner later revised his theory of intelligence, arguing that there are nine types of intelligence and expressing his appreciation for the concept of moral intelligence proposed by Coles (1997). How many types of intelligence are there? I think the number is infinite. This reminds me of J. P. Guilford. Guilford views intelligence as a three-dimensional structure (length × width × height). This structure comprises 120 types of intelligence consisting of the contents of intelligence (4), the operations of intelligence (5), and the product of intelligence (6) for length, width, and height, respectively; thus, 4 × 5 × 6 = 120. After Guilford passed away, his students asserted that there were “180 types” or “240 types” of intelligence based on the premise that memory is divided into long- and short-term memory, and perception

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is divided into visual and auditory types. Then, there are 180 or 240 types of intelligence. That said, the substantive issue is not whether there are 120, 180, or 240 types of intelligence but that it is inexhaustible. How can it be inexhaustible? There are four additional aspects that should be considered: “nature and nurture,” “cognition and social cognition,” “contents and forms,” and “surface and depth.” I also question the inconsistency in highlighting some forms of intelligence while disregarding others: “Gardner emphasized logical–mathematical intelligence and linguistic-verbal intelligence, so why did he not mention scientific intelligence? Why did he not mention painting intelligence when he mentioned musical intelligence and bodily-kinesthetic intelligence?”. The remarkable thing about Gardner is that he drew attention to individual differences in intelligence. In proposing the theory of multiple intelligences, he emphasized that the education of individuals should be in accordance with their particular aptitudes, which Chinese ancestors had already proposed long ago. The triangular pyramid structure of intelligence proposed here demonstrates the plurality of the structure of thinking and even intelligence and shows that intelligence is primarily a critical thinking ability that people need to acquire to identify, analyze, and solve problems and achieve certain purposes in a specific physical and social–historical cultural environment. Thus, the theoretical cornerstone of true thinking psychology is the structural view of thinking, resulting in the structural view of thinking becoming an important research topic in contemporary psychology.

Accomplishment of Mathematical Entirety Mathematics teaching must equip students to deal with problems and to comprehensively understand the mathematical system in its entirety. A comprehensive understanding of mathematics helps students acquire philosophical perspectives and mathematical knowledge and develops the totality of students’ thinking structures. When students analyze quantitative and spatial relations from a self-cultivating, structural view of thinking, their ability to analyze and solve problems improves; at the same time, students’ thinking matures, and their intelligence grows. Therefore, it is crucial to strengthen their understanding of mathematics in its entirety. Experienced mathematics teachers all encourage this viewpoint. The following are some relevant examples.

Unity The law of contradiction is the law of the unity of opposites, which is the most fundamental law of materialistic dialectics.

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The quantitative and spatial relations reflected in mathematics are replete with contradictions and “the unity of opposites.” Addition and subtraction, multiplication and division, divisors and multiples, numerator and denominator, positive and negative, rational and irrational, real number and imaginary number—all relations between these quantities reflect the harmony of opposites. Spatial relations are also uniform. For example, as shown in Fig. 4.2, the intersection chords AB and CD with intersection point P stretch from the inside to the outside of the circle, and PA and PC stretch from the secant line to the tangent line. The unity that emerges in this process of transformation is a good example of the application of the unity of opposites in thinking and problem solving, which is beneficial for developing students’ dialectical thinking abilities. Moreover, the structures formed are easy to remember. For another example, see Fig. 4.3, which illustrates the unity of the opposite volume formula of a circular cylinder, frustum, sphere, circular cone, square pyramid, etc.

Fig. 4.2 Power of a point

Fig. 4.3 Example of volume formula

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Connection The study of any science requires due attention to its internal organic connection. Indeed, the inner connection of the spatial relations and quantitative relations of the real world reflected by mathematics is critical for mathematics learners. The above ideas are present from the very beginning of elementary school mathematics: addend + another addend = sum; sum − addend = another addend; factor × another factor = product; and product ÷ factor = another factor. Here, addition and subtraction and multiplication and division are interconnected and mutually transformative. Therefore, it is more effective to explain these concepts through internal connections than to explain each one independently, as students will understand more deeply and learn more flexibly. Another example is the conversion between fractions, decimal numbers, and percentages, which also reflects the inner connection of the quantitative relationships. These concepts are easier for students to understand if the questions are linked together. If the two examples used in an explanation are not connected and there is a “conversion of fraction and percent” in between, the difficulty of understanding increases. Such an approach is not conducive to flexible use and wastes time. There are numerous transformations of quantitative relations (e.g., addition and subtraction, multiplication and division) in all types of word problems; thus, understanding these relationships requires a perspective of connection. From different perspectives, one can obtain different types of word problems, and these are closely related to each other. Secondary school mathematics teaching should continue to address this problem and guide students to reveal the inner connections between objects. For example, quadratic equations, inequalities, and functions are important topics in the secondary school mathematics curriculum. The key to solving these three “quadratic” problems is the technique of completing the square. If a student has already completely understood the second degree trinomial ax 2 + bx + c(a /= 0), then the quadratic formula for the root of quadratic equation ax 2 + bx + c = 0 can be deduced, after which the vertex coordinate, axis of symmetry, and maximum and minimum of quadratic function y = ax 2 + bx + c can be obtained, and finally, the quadratic inequalities ax 2 + bx + c < 0 (or > 0) can be solved. Students should first recognize the inner connections among the three quadratics. Then, teachers can utilize specific problems to provide students with appropriate training in the technique of completing the square so that they can flexibly solve relevant problems. In mathematics teaching, examples with inner connections explained in isolation result in students taking a “leap.” However, suppose students are guided to reveal the inner connection of things. In that case, the necessary conditions for students to “view the problem comprehensively” are provided to promote the organization and systematization of students’ thinking, relieve the burden of memory, and facilitate the development of the rational component of students’ intellectual activities.

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Comparison Both differences and similarities can be identified in the quantitative and spatial relations between things; the former need to be compared while the latter need to be analogized. Comparison is an important way of developing science. The idea of “identification always with comparison” has been frequently used in mathematics. For example, in elementary mathematics, “increased by several times” and “increased to several times” include division and equal division; volume and capacity; length, area, and volume units; and differences and similarities between parallelogram and rectangles, i.e., greater than, less than, equal to and approximately equal to, etc. As long as some of the similar properties, such as the basic properties of division, fractions, and ratios, appear simultaneously, as long as they appear simultaneously and are compared to connect them organically, we can make students truly understand their meanings. The timing of the comparison will be determined by the conditions and needs. The same principle applies in secondary school mathematics teaching. For example, in plane geometry, there are theorems: “the locus of points equidistant to both sides of an angle is the bisector of the angle” and “every point in a plane which bisects a dihedral angle is equidistant from the faces of the angle.” The former is a theorem of plane geometry, and the latter is a theorem of solid geometry, which are different but similar. Likewise, “the sum of the distances from any interior point to the sides of an equilateral triangle equals a constant (the length of the triangle’s altitude)” and “the sum of the distances from a point P to the faces of the tetrahedron with faces of equal area is constant (the height of tetrahedron)” differ but similarly contain the principle, “the sum of distances equals a constant.” Comparison and analogy enable us to easily understand and remember concepts. More importantly, we can distinguish the specificity of the connotation of a concept, derive a new concept from another, and extend the solution of one problem to another problem.

Symmetries There are many objective things with symmetry, such as white snowflakes, sparkling crystals, majestic Tiananmen Square, and so on. Symmetrical quantitative and spatial relations are also very common in mathematics. In elementary school mathematics textbooks, addition and multiplication tables between 10 and 20 have symmetrical forms. The symmetrical polynomials taught in secondary schools are also an expression of symmetry, such as: Solve this simultaneous equation with integers: {

x+y+z =0 x 3 + y 3 + z 3 = −18

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Fig. 4.4 Quadrilateral CEDF

The left side of the equations is symmetrical polynomials, and when we find a set of integer solutions. ⎧ ⎨ x = 1, y = 2, ⎩ z = −3 According to symmetries, we can find the remaining sets of solutions. ⎧ ⎧ ⎧ ⎧ ⎧ ⎨ x = 2, ⎨ x = 1, ⎨ x = 2, ⎨ x = −3, ⎨ x = −3, y = 1, or y = −3, or y = −3, or y = 1, or y = 2, ⎩ ⎩ ⎩ ⎩ ⎩ z = −3 z=2 z=1 z=2 z=1 There is symmetry in quantitative relations, but the regularity in spatial relations is often more obvious, such as central, axial, plane, and spatial symmetry. The following example can illustrate this. Let quadrilateral CEDF be an interior rectangle of a circle, where a tangent line to the circle made at point D intersects the extension of CE at A and the extension 3 of CF at B (see Fig. 4.4). Please prove: BAEF = BADD3 . For this question, CD creates an identical structure (symmetry) on both sides. By drawing a tangent line and a secant line from a point outside the circle, the same number of right triangles are created. Therefore, the problem can be solved by looking at only one side. This illustrates how symmetry can simplify the solution and lead to more effective problem solving.

Conservation Everything is constantly in motion and changing, but there is a relatively “static” state or “invariant” situation between motion and changing. Thus, it is important to identify the conservation of things. Conservation is not a new concept—the law of

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conservation of energy, the law of conservation of mass, etc., have been known to us for a long time. There are also many “invariance” problems in mathematics. In kindergarten mathematics classes or under their parents’ tutelage, preschool children come to understand word-pair opposites such as “big and small,” “light and heavy,” “more and less,” and “thick and thin.” However, in fact, the objects reflected by these words are in a state of flux. By comparing two things, preschool children are able to realize that things are relative, thus obtaining the most preliminary concepts from which the concept of conservation will eventually emerge. Many conservation concepts in elementary school mathematics reflect the “invariance” of quantitative and spatial relations. For example, one of the properties of division and fractions is that multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. For another example, no matter the size of a square, its area is always equal to side multiple side. There are also many relevant problems in secondary school mathematics. For example, in the ellipses, the parameters a, b, and c are certainly related; the set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant (eccentricity). These invariants are the inherent reflections of the properties of the ellipses.

Communication With the rapid development of science and technology and the emergence of fringe science, modern subjects develop by interpenetrating and influencing each other. This is inevitable given that the different subjects reveal the internal connections and communication between objective things. Therefore, in mathematics teaching, teachers should guide students to employ a broad range of other subjects that will enhance their understanding. For example, when elementary students learn geography, they need to familiarize themselves with maps. The calculation of the map scale requires the use of the arithmetic ratio. When imagining actual geographic locations and their sizes, students must communicate with their arithmetic knowledge. Likewise, when teachers are teaching “ratio and proportion,” they should utilize maps as teaching materials. Secondary school mathematics teaching should guide students to communicate and connect with other subjects, especially the cooperation between mathematics and physics, chemistry, and other subjects. There are some students who tend to distinguish constants, variables, and functional relationships formally but do not analyze them with the essential principles of physics. For example, the power formula 2 P = UR illustrates that when U is fixed, P is the inversely proportional function of R, which provides evidence for the choice of different resistance filaments for various power bulbs. When R is fixed (for each specific bulb, P is a constant), P is a quadratic function of U.

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Fig. 4.5 Tetrahedron of methane CH4

Another example is the tetrahedron of methane CH4 ; according to the geometric position, the angle between C-H could be calculated as 109° 28, (see Fig. 4.5). In mathematics teaching, linking practices and communicating with other subjects will enrich students’ imagination, exert creativity of intelligence, improve the structural integrity of thinking, increase interest in learning, and contribute to talent development. In short, nurturing mathematical entirety involves many aspects. However, as long as students work diligently, they can continue to improve in this aspect. Our emphasis on cultivating mathematical entirety does not mean it is equivalent to the structural integrity of thinking. These concepts are different but connected. Mathematical entirety summarizes the experience accumulated by the previous studies of mathematical disciplines, which reveals the inherent and necessary essential connection between objective quantitative and spatial relations. The structural integrity of thinking and mathematical entirety is internal and external qualities for students, respectively. Each has its own characteristics and cannot be equated. Emphasizing the accomplishment of the mathematical entirety allows students to reflect on the mathematical entirety objectively in their minds so that the objective things can gradually become subjective. Mathematical entirety is a series of specific content arising from quantitative and spatial relations with different levels of complexity and layers. In accordance with a logical order and relationships composed of a rigorous system of knowledge, each part is different, and no part can be ignored. For example, there are different “primary” and “secondary” conditions for the same function. Moreover, once the structural integrity of thinking in mathematics is formed, it can be applied to the mastery of other similar specific mathematical content. Cultivation of the mathematical entirety is a key foundation for learning mathematics in arithmetic thinking. The phases of thinking material, the direction of thinking, the complexity and organization of thinking, the degree of application of rules of thinking, and so on are based on the degree of mathematics students which have acquired.

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For the same mathematics content, learning will employ different components of thinking structure in arithmetic. Mastering a particular skill also reflects a different level of thinking structure in arithmetic. Therefore, we cannot simply make the correspondence between mathematical entirety and thinking structure in arithmetic one to one. The purpose of improving the cultivation of the mathematical entirety is to inspire students to discover the components of the arithmetic thinking structure, to use and adjust them consciously, to apply them as tools to expand their mathematical knowledge smoothly, to explore new mathematical knowledge through creativity, and to enrich the mathematical entirety. Thus, improving students’ mathematical entirety will enhance their mathematical abilities, teach them how to “learn to deal problems comprehensively,” and improve the thinking structure in arithmetic, which is beneficial to their overall thinking process and intellectual development.

Students’ Mathematical Abilities Are the Structural Integrity of Thinking Since 1978, I have been conducting applied research under the theoretical system of thinking structures I constructed, with students’ disciplinary competence being one of the focuses. What is disciplinary competence? It has rarely been discussed in the basic education community. I understand disciplinary competence as a product of the organic combination of subject teaching and students’ intellectual development and define disciplinary competence as possessing three aspects: first, the specific ability to master a subject; second, the intellectual activity in learning a subject and its related intellectual components; and third, the individual differences in students’ ability to learn a subject. The components of disciplinary competence, such as mathematical competence, should be analyzed by taking into consideration the following four aspects.

The Special Abilities of Subjects Reflect This Disciplinary Competence Most Directly Special abilities related to mathematical disciplines are, as aforementioned, arithmetic (number) ability and spatial (shape) visualization ability. Mathematics is a form of intellectual gymnastics for humans, and the ability to think logically in mathematics is clearly expressed as a mathematical discipline competence. Arithmetic refers to the deduction and induction process of deforming specific equations under the guidance of an algorithm. Moreover, it only refers to numerical arithmetic but also includes the deformations and statistics of a variety of equations and the arithmetic of limit, calculus, and Boolean algebra. Spatial visualization refers to the

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observation, analysis, and abstract generalization process of the spatial form of objective things, which includes the understanding of spatial concepts and recognition of the movement, transformation, and positional relations of two- and three-dimensional geometric figures, as well as the number shape union, geometric interpretation of algebraic problems, etc. The core and foundation of these two abilities are the logical thinking ability of mathematics, which includes mathematical concepts, judgments (propositions), reasoning and other thinking forms, comparison, classification, generalization, analogy, induction and deduction, analysis and synthesis, and other thinking methods, as well as the mathematical modeling of real-world problems. Mathematical modeling is a high-level thinking activity and an essential part of mathematical competence. Arithmetic spatial imagination and abstract logical thinking constitute a special ability system of mathematics. Consistent with the three traditional competencies, contemporary core competencies in mathematics are designated mathematical abstraction, logical reasoning, mathematical modeling, intuitive imagination, mathematical arithmetic, and data analysis, thus demonstrating that the process of truth exploration is becoming more detailed. A discussion on this topic can be found in Chapter Five.

All Disciplinary Competences Are Based on Generalization Ability The same is true for mathematical competence. Good acquisition of mathematical skills such as “combining like terms” is the most graphic illustration of mathematical ability. Mathematics teaching centers on the clarification of basic concepts, and the acquisition of mathematical concepts requires generalization ability as a basis; it promotes the development of the generalization ability at the same time. Thus, the teaching of mathematical concepts and the development of students’ generalization ability are organically linked. Generalization of mathematical concepts develops from concrete to abstract, from low level to high level. For example, the learning progression from “natural numbers” to “integral numbers” to “real numbers” to “complex numbers” reflects the process of abstract generalization, which also mirrors the level of intellectual development and thinking ability as children mature. Accordingly, the first chapter emphasized that mathematical ability must be based on abstract generalization ability. That is, in certain cases, mathematical competence is the same as mathematical abstract generalization ability. We should attach importance to the cultivation of students’ abstract generalization ability in mathematics. For this reason, Mr. Guanbo Li (a former professor at Beijing Normal University before 1950 and an authority on the teaching of secondary school mathematics) highlights the following three main points when teaching basic concepts in class: (a) important mathematical concepts should appear repeatedly; (b)

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simple, brief, and easy-to-understand language should be used to guide students to generalize step by step; and (c) teachers should guide students to be good at reading as the process of reading improves their abstract generalization ability in mathematics.

The Structure of Disciplinary Competence Should Involve Thinking Qualities Competence in any discipline is developed via students’ thinking activities. Without such activities, there is no disciplinary competence. Therefore, the structure of a student’s disciplinary competence should most definitely include individual differences that reflect individual thinking, i.e., individual thinking qualities. As mentioned above, in a certain sense, thinking qualities are the manifestation of intelligence and ability. The hierarchy of intelligence and ability is inseparable from thinking qualities, namely profundity, flexibility, creativity, criticality, and agility. These thinking qualities allow us to identify and evaluate the level and characteristics that make up each individual’s certain disciplinary competence. Therefore, these five thinking qualities should be taken into account when studying the structure of disciplinary competence. Tables 4.1 and 4.2 present the models of students’ mathematics discipline competence constructed by mathematical language. Based on the above three considerations, I define mathematical competence as an open, dynamic system with 15 nodes (12 nodes for elementary school) formed by the intersection of three mathematical abilities with five thinking qualities based on mathematical generalization.

Student’s Disciplinary Competence Should Reflect Their Characteristics Disciplinary competence specifically refers to the specific abilities, intellectual components, and thinking qualities reflected in each student. On the one hand, we should pay attention to the components of students’ disciplinary competence, including learning ability, strategies, and methods for successful mastery of a particular subject. On the other hand, we should also pay attention to the personal preferences and specialties of each student. I do not agree with the early division of students into the soft or hard sciences and recommend that all-around development should be promoted at the elementary and secondary school levels. I have some doubts about the American psychologist Gardner’s view of multiple intelligences emphasizing musical-rhythmic and harmonic, visual-spatial, linguistic-verbal, logical–mathematical, and bodily-kinesthetic intelligence, which he proposes to constitute a student’s

The agility of thinking

1. On generalization With a few examples, students can quickly summarize general algorithms, laws, properties, and other rules or skills 2. On understanding With a few examples, students can understand the arithmetic principles and the basic process of arithmetic and are able to imitate examples for arithmetic 3. On application With a few examples, students can perform arithmetic correctly and quickly; they are good at capturing the essence of the problem, use appropriate and simple processes of arithmetic, simple steps, and are good at mental calculations 4. On time spent Students can respond more quickly with fewer pauses and require less time to complete operations (especially for difficult problems)

Arithmetic ability 1. On generalization With a few examples, students can generate the mathematical characteristics, laws, problem-solving skills of numbers, and formulas and quantitative relationships 2. On understanding With a few examples, students can understand the mathematical characteristics and laws of numbers and formulas and quantitative relationships; can quickly capture the essence of a problem; and are able to do equivalent transformations skillfully 3. On application With a few examples, students can illustrate the mathematical reasoning in practical problems and solve complex mathematical problems with clear thinking and direct and rapid in problem solving 4. On time spent Students can quickly solve and explain problems and take less time to complete the reasoning process

Logical thinking ability

(continued)

1. On generalization With a few examples, students can generalize the common features of geometric forms and the corresponding arithmetic formula (perimeter, area, volume, interior angel sum, etc.) 2. On understanding With a few examples, students can understand the definitions, properties, and theorem of geometry and quickly capture the essential connections between geometric forms 3. On application With a few examples, students can generalize the essential geometric connection in problems and select the correct method to solve the problem of geometric measurement, graphing and arithmetic; geometric images are clear and quickly reproduced and can be quickly decomposed, combined, and isometrically transformed to illustrate geometric phenomena and solve geometric problems 4. On time spent Students are consistent and finish their work quickly, and the overall learning process is less time-consuming

Spatial imagination ability

Table 4.1 Enumeration and dissection of the structure of mathematical abilities in elementary school students

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The flexibility of thinking

Logical thinking ability 1. On generalization Good at applying existing knowledge and experience to compare, summarize, hypothesize, and generalize the laws of numbers, arithmetic, and quantitative relationships from different perspectives 2. On understanding Good at applying existing knowledge, skills, and experience, and flexibly use analysis, deduction, imitation, imagination, experimentation, and other thinking methods to understand the problem (including definitions and problems needed to be solved) 3. On application Good at flexibly applying knowledge of numbers, equations, and geometry to consider and solve problems form different perspectives, directions, and contexts; good at solving the same problem with both general methods and special techniques; compatible with seeking commonality and seeking difference; possess the ability to consider problems from changing and moving viewpoints 4. On reasoning effects Have a strong awareness of goal tracking; timely conversion of direction, process, and skills when solving problems with multiple solutions

Arithmetic ability

1. On generalization Good at comparing and analyzing the results of arithmetic and generalizing the meaning, rules, theorems, and properties of arithmetic in relation to life experience; apply mathematical skills flexibly and thinking closely related to the goal 2. On understanding Good at using existing knowledge, skills, and life experience of numbers, formulas, arithmetic, etc., to understand mathematical arithmetic problems from multiple perspectives 3. On application Good at applying the meaning, rules, theorems, properties, and techniques of arithmetic, adjust the process of arithmetic flexibly according to the purpose of arithmetic, and select the appropriate method to carry out reasonable and effective arithmetic’s; use both general methods and special techniques to perform arithmetic, as well as multiple methods to solve the same question 4. On arithmetic effects Possess fluent, association-rich, idea-rich, flexible, and appropriate methods

Table 4.1 (continued)

(continued)

1. On generalization Good at drawing and experiments, flexibly and easily using knowledge and skills to generalize the basic characteristics and properties of geometry 2. On understanding Good at using existing knowledge and experience to understand the position and measurement relations and properties of geometry from different perspectives in a variety of ways (e.g., stability: in the case of a cone with a certain volume, identifying the inverse proportional nature of the height and base area in the cone, etc.) 3. On application Good at using different knowledge from different perspectives to analyze and solve problems; solving certain geometric problems by changing geometric positions and shapes while keeping conditions unchanged; associating known geometric conditions with a variety of other geometric positions, shapes, and measurements and flexibly solving various deformation problems 4. On geometric imagination effects Possess strong spatial imagination ability; students cannot only imagine transformations from one geometric state to another but also from certain equations to imagine the geometric shape with corresponding metric properties; have many thoughts on solutions, select the right method, and be good at solving combinatorial shape problems

Spatial imagination ability

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The creativity of thinking

1. On generalization Good at exploring, discovering, and generalizing arithmetic methods or techniques with a unique way of thinking 2. On understanding Good at understanding and explaining arithmetic methods and rules in a unique way 3. On application Good at using unique and innovative methods to perform arithmetic 4. On arithmetic effects Good at using unique and innovative methods to perform arithmetic

Arithmetic ability

Table 4.1 (continued)

1. On generalization Good at discovering contradictions, proposing conjectures, providing verification, and generalizing in student’s preferred way, with strong ability and awareness of analogical reasoning 2. On understanding Good at simulation and association and proposing supplementary opinions and different ideas and stating reasons or the basis for them 3. On application Think analytically using unique and innovative techniques; be good at developing mechanical imitation questions 4. On reasoning effects Possess novelty and a strong ability to reflect and reconstruct

Logical thinking ability

(continued)

1. On generalization Good at exploring and discovering mathematical properties and measurement characteristics of geometric forms with a unique way of thinking 2. On understanding Good at proposing equivalent geometric formulas and understanding the mathematical properties of geometry by using the thinking of general and motion 3. On application Good at creating geometric settings; good at making geometric models; good at analyzing and solving geometric problems in a unique and innovative way 4. On imagination effects Imaginative, original, and unique

Spatial imagination ability

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The profundity of thinking

1. On generalization Good at widely applying mathematical knowledge, carefully and responsibly analyzing problems related to arithmetic and generalizing and forming concepts related to the meaning, rules, theorems, and properties of arithmetic by focusing on the essence and inner connection 2. On understanding Good at understanding the meaning of different arithmetic from the perspective of the dialectical unity between the four fundamental arithmetic. Understand the theorems and properties of arithmetic from the perspective of the interrelationship between integers, decimals, and fractions. Good at clarifying the reasons for the formulas, theorems, and properties of arithmetic from experience and practice 3. On application Good at equivalent transformation of numbers, equations, and formulas; be good at handling calculations and solving deformed or uncommon arithmetic in a dialectical and uniform way; performing difficult arithmetic; and have good checking habits. Students can consciously supply sufficient bases for each step of the arithmetic 4. On arithmetic effects Have a correct and rigorous process, a high level of skill, and ability to solve difficult arithmetic problems

Arithmetic ability

Table 4.1 (continued) 1. On generalization Good at capturing the essence from mathematical materials and generalizing the basic concepts and formulas in numbers, equations, and quantitative relationships; generalizing basic quantitative relationships in complex word problems; generalizing the structure of knowledge and customary types; and classifying solution techniques in the process of problem solving 2. On understanding Good at understanding the meaning of mathematical terms and symbols correctly and mentally establishing various mathematical concepts and discovering the inner connection between knowledge. Students can mentally reconstruct the knowledge 3. On application Good at the equivalent transformations of quantitative relations and at mastering a variety of language skills to describe the same mathematical properties; using the meaning of the four fundamental arithmetic to explain quantitative relations in practical problems in a dialectical and unified way; explaining the four fundamental arithmetic and rules with specific quantitative relations; distinguishing similar mathematical concepts and discovering the intrinsic connections between different mathematical phenomena; combining and classifying knowledge skills to systematize and restructure them; thinking about problems comprehensively and rigorously, and explaining mathematical phenomena and the process of solving problems with sufficient reasons; and consciously using analysis, synthesis, induction, deduction, simulation, analogy, hypothesis, imagination, and other methods to solve difficult problems 4. On reasoning effects Possess comprehensive, rigorous, profound, and a high level of systematic skills

Logical thinking ability

1. On generalization Be good at forming the geometric concepts, measurement properties, and scales from different situations and perspectives 2. On understanding Be good at recognizing and discovering the proportional relationships quantities and relationships between forms; be good at recognizing new geometric forms by the preliminary experience and solutions; be good at interpreting arithmetical equations and rules of change in terms of geometric phenomena 3. On application Be good at recognizing and discovering the proportional relationships quantities and relationships between forms; be good at recognizing new geometric forms by the preliminary experience and solutions; be good at interpreting arithmetical equations and rules of change in terms of geometric phenomena 4. On geometric imagination effects Good at solving geometric problems described by texts with various forms of decomposition and combination transformations of geometric forms and sufficient reasons and have a clear and accurate concept of orientation, direction, shape, and measurement and a wide geometric exchange space in mind

Spatial imagination ability

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1. With a few examples, students can acquire a new method 2. With a few examples, students can apply theorems correctly to solve difficult problems 3. Think efficiently and grasp the essence of the problem quickly; the reasoning process is concise and clear 4. Use concise inference steps

1. With a few examples, students can generalize the general arithmetical method 2. With a few examples, students can apply formulas and theorems correctly for difficult problems 3. Be good at capturing the essence of the problem and quickly select the correct method and procedure 4. Use simple arithmetic steps

1. Good at using rules, theorems, and formulas flexibly 2. Easily shift from considering one type of arithmetic to another 3. Good at transforming formulas flexibly 4. Good at transforming the variables in the formula and the unknown quantities in the equation flexibly 5. Easily move from the arithmetic of an equation to the decomposition of an equation 6. Good at using multiple methods to solve a problem

The flexibility of thinking

1. Good at applying laws, theorems, axioms and methods and generalization and transfer 2. Good at changing the way of thinking and using multiple approaches to solve problems from different perspectives, directions, and aspects 3. Good at considering problems from the point of view of change and movement 4. Possess a flexible thinking process and good at organic linkage between analysis and deduction, special and general, and concrete and abstract 5. Easily shift from direct ideation to retroactive thinking 6. Possess a flexible and diverse thinking structure

Logical thinking ability

Arithmetic ability

The agility of thinking

Table 4.2 Enumeration and dissection of the structure of mathematical abilities in secondary school students

(continued)

1. Good at applying the properties of graphs flexibly 2. Analyze the properties of graphs from different perspectives and in various ways 3. Develop rules and laws from changes in the position and metric relationships of a figure 4. Good at transforming figures while maintaining known conditions of the figure 5. Good at solving trajectory problems 6. Good at associating multiple positions and measurements from known figures

1. With a few examples, students can generalize the general properties of a graph 2. With a few examples, students can perform difficult graphical analysis 3. Can quickly find the essential connection of figures 4. Use simple analyzing steps

Spatial imagination ability

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Logical thinking ability 1. Good at association and independently proposing new problems and be able to think independently, explore, and discover new patterns 2. Students have their own unique understanding of theorems and rules, can generalize them, and are good at proposing their own unique and novel methods for solving problems 3. Students are able to compile exercise questions 1. Good at making correct estimates of the solvability of problems and reasoning in a purposeful manner 2. Can appropriately select useful conditions and intermediate conclusions in the reasoning process 3. The reasoning is clear, the specific problems are analyzed in detail, and thinking can be adjusted in a timely manner 4. Good at identifying errors in the reasoning process and correcting them in a timely manner 5. Able to identify the “attraction” of mistakes and avoid generated illusions and good at overcoming the negative transfer in the learning process 6. Good at considering the pros and cons of arguments and making correct judgments

Arithmetic ability

1. Good at exploring and discovering new arithmetic patterns 2. Good at proposing unique and novel solutions to problems

1. Can clearly understand the requirements of a problem and consciously use reasonable steps when solving it 2. Can correctly select useful conditions and intermediate conclusions in arithmetic 3. Able to adjust the steps and methods of solving problems in time and adopt special solutions to special problems 4. Good at finding mistakes in arithmetic and correcting them in a timely manner 5. No confusion when using the algorithm 6. Able to use various methods to check the correctness of the results of arithmetic

The creativity of thinking

The criticality of thinking

Table 4.2 (continued)

(continued)

1. Highly purposeful in analyzing graphical relationships 2. Good at extracting useful basic figures from complex graphs and analyzing them and adding auxiliary lines correctly 3. Good at graphing and finding errors in graphical analysis and correcting them in a timely manner 4. Can easily get rid of the illusion generated by specific graphics 5. Good at changing specific graphs to test the correctness of the conclusions obtained from the analysis

1. Good at exploring and discovering new patterns in graphical relationships 2. Good at proposing unique and novel methods for graphical analysis 3. Students can design and create some special geometric teaching aids

Spatial imagination ability

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Arithmetic ability

The profundity of thinking 1. Able to correctly form the concepts of numbers, formulas, equations, and functions and understand the concepts of various arithmetic and deformations of equations 2. Able to generalize the types of arithmetic and deformations of equations and correctly determine which type a specific problem belongs to 3. Good at general studies of formulas, equations, and functions and solving problems with letter coefficients 4. Good at applying the basic laws of arithmetic, the exponential law, the idea of unity of addition and subtraction, unity of multiplication and division, unity of involution and evolution, and mastering other formulas and laws 5. Have a sufficient basis for each step of arithmetic or deformation 6. Able to clarify the reason for the formula and law 7. Good at solving difficult arithmetical problems

Table 4.2 (continued) 1. Able to form various concepts correctly and understand the meaning of terms and symbols correctly 2. Able to summarize the types and general methods of mathematical proofs 3. Has mastered the structure of example problems and the relationship between the four propositions 4. Good at systematizing and structuring knowledge, at grasping the connections between concepts and knowledge, at analyzing and combining them from different perspectives, and at forming a system of knowledge structure in a general way 5. Good at using analysis and synthesis, comparison and analogy, induction and deduction, and direct and indirect ways of proofing to reason and argue 6. Consciously reason in accordance with the laws of logic and justify every step of reasoning 7. Good at proving the theorem 8. Able to think comprehensively and carefully, engage in difficult reasoning and argumentation, and solve difficult integrated problems and word problems

Logical thinking ability

1. Able to correctly form concepts related to geometry, number axes, rectangular coordinate system, curves and surfaces of equations, images of functions, etc., and good at giving geometric explanations of certain algebraic problems 2. Good at generalizing and classifying geometric figures, curves of equations, and images of functions and grasping the connections between various figures 3. Good at imagining geometric figures based on words and analyzing the relevant position and measurement relations based on geometric figures and expressing them in words 4. Good at imagining the shape of curves based on equations and understanding the characteristics of equations from the shape of curves 5. Good at imagining the shape of an image based on a functional equation and grasping the characteristics of a function from the shape of an image 6. Able to analyze geometric figures, equation curves, and function images consciously and with good reasons 7. Good at analyzing difficult geometric problems

Spatial imagination ability

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disciplinary competence. Nevertheless, the strengths of the theory of multiple intelligences are its emphasis on the differences in students’ disciplinary competency and teaching in accordance with aptitude. This shows that disciplinary competence, including the disciplinary competence of mathematics, is an integrated structure mediated by subject knowledge, which is systematic, operable, and stable. Intelligence and competence are not void, one of the significant manifestations of which is the linkage with subject education, which constitutes students’ disciplinary competence.

Mathematics Teaching Should Start from the Integrity of Thinking Teaching mathematics from the point of integrity is necessary to develop the integrity of students’ thinking and to improve the quality of mathematics teaching. How should we consider integrity in mathematics teaching?

Study the Textbook Carefully; Understand the Systematization and Science of the Textbook Mathematics is a fundamental science with entirety, logic, and rigor. Mathematics teachers in elementary and secondary schools, regardless of the grade, should read through the mathematics textbook, understand its intention, clarify its purpose and basic requirements, and study in-depth the connections between the basic knowledge provided in the textbook as well as their effects and roles in each chapter and section to acquire a more comprehensive understanding of the entire textbook. In particular, when the content is complicated and confusing, teachers should carefully organize and analyze the textbook in detail to simplify its complexity and sort out the various ideas it presents. On this basis, we should also read the textbook intensively with further study of the specific content of each chapter and section, which includes examining the concepts, definitions, theorems, formulas, and rules in the textbook word by word, understanding their essence, and making the knowledge in the textbook systematic and organized. For example, experienced teachers focus on the keywords “both” and “same” when preparing for the section on “The Four Properties of Equation Deformation.” Students always confuse the common results with results; the reason is their misunderstanding of “both” and “same.” If the “same” number (or integral expression) is added to “both” sides of the original equation or “both” sides are multiplied by the “same” number, the new equation will have the same solution as that of the original solution. Likewise, if “both” sides of the equation are multiplied by the “same” integral expression or the “same” once involution, the new equation

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will have the same solution as that of the original solution. The use of “same” and “both” in this case differs. In this way, the four properties of equation deformation can be understood more deeply. When preparing for class, all examples and exercises in the textbook should be carefully calculated or justified. Teachers should study the multiple solutions of various exercises, summarize the laws of problem solving for the same type of exercises, and determine the purpose of the examples and exercises appearing in the textbook. Every unit’s teaching content, examples, and exercises should be clear, thus laying the foundation for determining the focus and difficulties of teaching.

Teach Students Systematic and Scientific Knowledge of Mathematics The purpose of teachers’ understanding the systematization of the entire textbook and studying its scientific properties is to systematically teach students scientific knowledge so that their mathematical knowledge is both systematic and organized. For example, when teaching algebra, one mathematics teacher drew a diagram of the structure of the algebraic system for her students (see Fig. 4.6). This teacher provided his students with an integrated understanding of algebra from the very beginning. In later teaching, he guided them to systematize and organize their knowledge. As a result, the mathematics scores of the whole class improved rapidly, and their knowledge became well organized and logical.

Fig. 4.6 Diagram of an algebraic system

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To better impart systematic mathematical knowledge to students while promoting the development of the structural integrity of their thinking, the mixed textbook for secondary school mathematics could be divided into algebra, geometry, and trigonometry. These mathematics subjects are typically taught as three separate courses. Geometry and algebra are taught simultaneously from the beginning, which is convenient for students to conduct comparisons and establish the internal connections between the concepts of “shape” and “number” while simultaneously promoting the entirety of students’ mathematical knowledge and arithmetic thinking. The three subjects are organized systematically, which fits the age characteristics of secondary school students and facilitates their systematic acceptance of algebra, geometry, and trigonometry. Such textbooks are much more systematic and scientific than the “leaping-type” mathematics textbook. Is this compilation of teaching materials for mathematics not conducive to the dissemination and understanding of modern mathematical ideas? No, modern mathematical ideas can be penetrated through various subjects, and this compilation of teaching materials will never affect students’ comprehension of modern mathematical ideas.

Effectively Strengthen the Teaching of Basic Mathematical Concepts Mathematics teachers should effectively strengthen the teaching of basic mathematical concepts according to students’ existing knowledge level, experience structure, thinking structure, etc. At present, all elementary and secondary schools teach mathematics according to the requirements outlined in curriculum standards, an approach that does not fully consider or match the actual education level and mental structure level of students’ thinking. To improve test scores and promotion rates, teachers work overtime to catch up with progress, not paying attention to the basic concepts but rather focusing on making students do a lot of mechanical imitation training after quickly introducing new knowledge. The emphasis on curriculum standards and uniform examinations fails to consider the structure of students’ understanding and the structure of knowledge and the subject itself. Teachers also often fail to consider the difference between students, blindly emphasizing unity and “alignment,” which is divorced from reality. Only when students master the basic concepts of mathematics and understand the interrelationship between basic concepts can they expand and deepen their knowledge on this basis, forming a “transfer” in learning. Therefore, teachers must effectively strengthen the teaching of basic mathematics concepts according to the actual situation of students. They should also follow a more gradual approach to learning, focusing on the key points, difficulties, and doubts to improve the effectiveness of teaching the basic concepts of mathematics.

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Form an Integrated Mathematical Knowledge System Through Review Reviews are of great importance for mathematics teaching—practice should never be underestimated but rather prioritized and valued at the same level as teaching new knowledge. Whether it is a unit review or a general review, emphasis should be placed on systematizing the knowledge learned through the review and establishing internal connections with old knowledge to form an integrated knowledge system. Suppose students effectively master this internal connection and knowledge system and fully comprehend the content of their integrated thinking structure. In that case, they can achieve the purpose of consolidating and improving their knowledge by applying it flexibly. Students can develop and structure the entirety of their thinking as they gradually internalize the knowledge system from a quantitative to a qualitative change. Basic knowledge should be comprehensively and systematically reviewed. Mathematician Chen (2008) places great emphasis on basic knowledge, asserting that although the simplest things are easy to accept, they are not easy to truly understand. Simple things often involve the most basic concepts. Only by mastering the simple things proficiently can you easily accept more complex and advanced things. If the basic knowledge is comprehensively systematized, an integrated knowledge structure will be formed. In the review, teachers should carefully guide students to read through the textbook. This involves making them clarify its purpose and learning requirements, asking them repetitively to study the development clues and having them identify the connections and the relationships between the previously learned related knowledge and the new knowledge in the textbook. Students should clarify the roles of this knowledge, further summarizing, generalizing, classifying, systematizing, and finally forming an integrated knowledge structure. An important step in the review is comprehensive exercises. By making connections, students can compare concepts, formulas, theorems, and laws they have learned, discover similarities and differences, and, at the same time, perform arithmetic to organize and systematize their knowledge and lay a solid foundation for developing an integrated structure of thinking.

Improve Mathematical Comprehension Ability and Promote the Development of the Structural Integrity of Students’ Thinking Understanding is to recognize and reveal the essence of things. The same is true of mathematical comprehension, which involves understanding (a) the quantitative causality relationships stated in the textbook; (b) the commonality and individuality

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of certain relationships stated in the textbook; (c) the logical basis of certain issues stated in the textbook, e.g., mathematical theorems, formulas, and the solutions to various exercises; and (d) the overall relationships quantities and between figures in the textbook. Comprehension is based on the structure of thinking. At the same time, any kind of mathematical concept, be it quantitative or physical relations, can be categorized and understood according to different criteria. As a result, students develop a variety of mathematical arithmetic thinking structures while learning mathematics. The structure of thinking is the basis for understanding; thus, the deeper the understanding is, the stronger the development of the structure of thinking. British psychologist M. A. Bell takes a branch of mathematics that is unfamiliar to elementary school and secondary school students—a figure from topology—as a test question to measure students’ comprehensive ability. The results of the three target groups differed: in the group with original knowledge of the rules, 75% of participants were able to adapt to the new task and understand the network rules; in the group given only the rules without instructions, 30% completed the task. In the group without any prior experience, only 17% of participants completed the task. This result clearly demonstrates that understanding new problems in mathematics must rely on past experience and intellectual level, and understanding is constantly developed in the learning process. What is the best way to understand mathematics correctly and successfully so that comprehension can be developed rapidly? First, teachers must teach starting from the level of students’ existing knowledge and experience and thinking structure. The knowledge that has already been understood is the basis for understanding new knowledge. For this reason, mathematics teachers can only target their teaching if they know their students. For example, at the beginning of the semester, when a teacher first encounters a new class, teachers need to familiarize themselves with the situation of their students. How well the students master the knowledge is an important aspect. Thus, it is necessary to produce a more comprehensive examination to test students’ initial knowledge levels. By analyzing students’ answers, we can obtain a general understanding of students’ basic knowledge and abilities, intellectual qualities, and thinking structure. In this way, lesson preparation, teaching, and tutoring will not be detached from students’ original knowledge, experience, and thinking structure, and all measures can be targeted. Second, mathematics must be taught in a progressive manner. This is determined by the characteristics of mathematics. If you fail to understand the previous material, it will be difficult to later understand more complicated material. If you have trouble with the previous knowledge, you should work harder on the review. As the proverb advises us, “sharpening your ax will not delay your job of chopping wood.” If you keep learning without properly understanding the previous knowledge, you will have many problems with the subsequent knowledge you encounter, which will likely lead to a vicious and stressful cycle of misunderstanding and confusion. It is necessary to increase the difficulty and abstraction of teaching, but we have to oppose the blind increase in the degree of difficulty, speed, and rationality that fails to consider students’ actual comprehension. There are conditions for increasing the

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difficulty, speed, and abstraction that are well known, that is, following the principles from known to unknown, from general understanding to exactly understanding, from concrete to abstract, from easy to difficult, from simple to complex, from near to far, etc. These are the objective laws that must be followed to guide students to understand mathematical materials. Third, language is a tool for understanding. Teachers’ language should be concise and clear. Examples should be easy to understand. For example, a senior teacher summed it up like this when he talked about plurals: When the baby is still in the mother’s arms, there is no one else to compete with for food, and he/she does not need to understand the number concepts. By the time he is one or two years old, he understands that his brother ate two apples while he only ate one, indicating that he already possesses the concept and innate need for natural numbers. When he is a little older and is given an apple to share with his brother and sister, he will be able to recognize the concept of fractions; that is, each person eats 13 ., which can be summarized as positive numbers. When he is able to spend money and keep accounts and experiences a loss resulting in debt, the concept of negative numbers is generated, which can be summarized as rational numbers. Later, to calculate the length of the diagonal of the unit square, there is a need to solve the equation x 2 = 2, which will introduce the concept of irrational numbers. In solving equations such as x 2 + 2 = 0, there is a need to expand the number system so that the explanation of the concept of imaginary numbers is generated. How concise and vivid is the language of this senior teacher! The teacher’s language activates students’ original thinking structures and existing experiences, contributing to students’ understanding of mathematics and language development. In addition, the teacher’s basic skills and the combination of teaching and exercise can develop the students’ thinking structure so that students, through continuous practice to strengthen and consolidate, improve their level of understanding ability.

References Chen, J. R. (2008). Memories of my middle school days: Mathematics and Physics for Middle School Students (Grade 8 Mathematics). China Normal University Edition, (9), 7–8. (in Chinese). Chen, Y. H. (1992). The development of children and adolescents’ cognitive operation to acquire geometric concepts [Doctoral dissertation, Beijing Normal University]. National Digital Library of China. Coles, R. (1997). The moral intelligence of children. Random House. Confucius. (2011). The discourses and sayings of Confucius (H. M. Gu, Trans.). Yunnan People’s Publishing House, Yunnan. (in Chinese and English)(Original work published 540 BC - 400 BC). Holowinsky, I. Z. (1985). Soviet psychology and its view of American behaviorism. Psychological Reports, 56(3), 803–810. Kang, W. (2003). Middle school students’ mathematical problem-posing capability [Doctoral dissertation, Beijing Normal University]. National Digital Library of China. Li, C. M. (2002). Research on the development of physics experimental operation ability of high school students [Doctoral dissertation, Beijing Normal University]. National Digital Library of China.

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Lin, C., & Li, T. (2003). Multiple intelligence and the structure of thinking. Theory & Psychology, 13(6), 829–845. Xin, T. (1997). Teachers’ teaching-monitoring ability: The construct, impact factors, and its relationship with the development of students [Doctoral dissertation, Beijing Normal University]. National Digital Library of China. Xin, Z. Q. (2002). The acquisition of schema and strategy in mathematical problem solving of children [Doctoral dissertation, Beijing Normal University]. National Digital Library of China. Zhang, J. Y. (1999). Secondary school students’ self-monitoring abilities in mathematics—Structure, development and influencing factors [Doctoral dissertation, Beijing Normal University]. National Digital Library of China. Zhang, Q. (2002). Research on the development of number and shape cognitive ability of elementary school students [Doctoral dissertation, Beijing Normal University]. National Digital Library of China.

Chapter 5

The Development of Student’s Thinking Ability in Arithmetic

Cognitive psychology emphasizes that human thinking activity is a complex process of analysis, synthesis, comparison, abstraction, and generalization that occurs in a serial, parallel, and mixed manner. Concepts, judgment (propositions), basic reasoning, and comprehension are manifestations of this process, as are problem understanding and solving. Notably, thinking ability is often not the manifestation of a particular process in thinking activity but the complete realization of these processes. In mathematics learning, generalization, spatial imagination, proposition proposal, and logical reasoning abilities, which make up learners’ arithmetic thinking, are developed rapidly, especially the development of mathematical generalization ability. As mentioned in the previous chapter, this generalization is the core of mathematical ability. Thus, in a certain sense, mathematical generalization ability is mathematical ability.

Mathematics Learning and the Development of Students’ Generalization Ability As mentioned in the previous chapter, generalization is the ability to concentrate on the common, essential properties of the same class of thing to form a general property of a class in thinking activity. The purpose of any scientific research is to generalize the results of the study. Darwin remarked, “My mind seems to have become a kind of machine for grinding general laws out of large collections of facts” (Darwin, 1994). Indeed, “grinding” facts into general laws is the most important and ultimate stage of all studies. The activity through which this “intelligence” is exercised is, in fact, the process of abstract generalization, thus illustrating the role and importance of generalization ability. © China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_5

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The significance of generalization ability lies in the fact that it is a prerequisite for acquiring concepts. Generalization is the reflection of the essential properties of things in the mind and is marked by words. Concepts, judgments, and reasoning constitute the basic form of logical thinking in which concepts make up the “cellular structure.” For students to acquire concepts, their generalization abilities should first be fostered. Concept acquisition necessitates comparing, analyzing, and synthesizing a class of things, determining the common and essential characteristics or properties, and then generalizing them. The level of generalization ability depends on the level of thinking required to grasp the common and essential characteristics or properties of things. The point of mathematics teaching is to clearly explain its fundamental concepts. These concepts are the core knowledge. Definitions, theorems, laws, and formulas in mathematics have their own specific mathematical concepts. If students have not fully understood and distinguished different basic concepts, they will be confused and make mistakes when using them. Therefore, it is important for teachers to pay careful attention to the basic concepts of teaching. When experienced teachers teach students the basic concepts, they are not merely explained once but repeatedly and continuously, thus consolidating the concepts. The mastering of mathematical concepts should be based on generalization ability, which simultaneously promotes the development of generalization ability. Therefore, the teaching of mathematical concepts and the development of students’ generalization ability are organically linked. The generalization of concepts develops from concrete to abstract and from low to high levels of mental processes. The level of mathematical generalization ability of elementary and secondary school students can be determined by the following six indicators. 1. The degree of reliance on visual objects. For example, whether arithmetic is focused on independent thinking or on “figure count” and other visual teaching aids. 2. Knowing the actual meaning of numbers. For example, “10” means ten, “ 21 ” means half of one, etc. 3. The understanding of the sequence and the size of the number. For example, 89 is before 98, 98 is after 89, 89 is less than 98, and 98 is greater than 89. 4. The ability to decompose and combine numbers and the ability to categorize. For example, 100 is made up of “50 + 50” or “10 tens” or “hundred ones,” and “combining like terms.” 5. The expansion of mathematical concept definitions and the ability to define and constantly reveal the essence of the concepts. For example, students are able to understand the definition of an “irrational number” is an “infinite non-repeating decimal” and to provide three examples of irrational numbers. 6. The expansion degree of numbers, that is, the five indicators demonstrated above. For example, in the expansion from “natural numbers” to “complex numbers,” the degree of expansion belongs to the respective level of number generalization, which is whether the expansion is a generalization within the “natural numbers”

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or within the generalization of “irrational numbers.” The higher the level of abstract numbers, the higher the level of generalization ability. The six indicators above are a concrete reflection of a student’s conceptual understanding and generalization abilities. However, some teachers are only satisfied when students are good at doing exercises and provide insufficient guidance for students to develop their generalization abilities. This may reflect an insufficient understanding of the importance of basic concepts and inadequate attention to generalization ability development. The result is that students only “follow the example”; students follow what teachers say and the set formula. The following illustrates a simple example. For first-grade arithmetic, the emphasis is on addition and subtraction up to 20. Although the majority of students scored 100 on the test, some of them relied on finger counting, and some of them had to speak the equations aloud to repeat their thinking and operation process. Their commonalities are a reliance on visual objects for calculation, undeveloped inner language, and a low level of generalization ability of thinking. Some students do not need to rely on visual objects for calculation, but they also do not understand that the essence of number concepts consists of three elements: their practical meaning, sequence and size, and composition. In this case, students know that 20 is 10 + 10 but cannot consciously convert it into 11 + 9, 12 + 8, 15 + 5, etc., thus indicating their low level of generalization ability. With such an outcome, it is possible that teachers have neglected to cultivate students’ mathematical generalization ability in teaching. First grade is an important stage for students to process routine training. Therefore, the cultivation of students’ generalization ability should be included in the content of routine training during first grade. When experienced teachers teach first-grade arithmetic, they tend to place more emphasis on the composition of numbers and the training of decomposition and combination. In the teaching of the number concepts of “0” to “10,” it is not necessary to describe them one by one in detail but rather comprehensively describe them with decomposition and combination. For example, “10” is a combination of “1” and “9” or “7” and “3,” and it is equal to “3 + 3 + 4,” or “5 + 2 + 1 +1 +1,” and so on. Teachers should then ask students to deduce the calculation. The teaching from “10” to “20,” after finishing the teaching of “20,” is immediately transferred to the teaching of “100” by repeating the above process of decomposition and combination and asking students to deduce the calculation. Based on the calculation, they can then guide students to summarize and generalize the law of addition and subtraction up to “100.” Experiments have shown that students can quickly comprehend mathematical knowledge and significantly improve arithmetic abilities with this approach (Sowder, 2020). For both elementary and secondary school, it is essential that mathematics teachers attach great importance to helping students acquire basic concepts that cultivate their mathematical generalization ability. When teaching the system of binary quadratic equations, some secondary school teachers provide an incisive analysis of the three properties of homogeneous solutions to systems of equations: (a) if any equation in a system of equations is replaced with an equation that has the same solution as this equation, the resulting new system of equations has the same solution as

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the original system of equations; (b) if an unknown in one equation in a system of equations is represented by an algebraic equation of another unknown, the system of equations obtained by replacing this unknown with this algebraic equation in another equation in the system of equations has the same solution as the original system of equations; and (c) if the two sides of an equation are added or subtracted to arrive at an equation, the system of equations formed with any equation in the original system of equations has the same solution as the original system of equations. Because these three properties are essential rules for solving systems of equations to maintain the same solution, the teacher should repeatedly explain these properties three times to give students a profound understanding of the three properties. Such an approach reflects a strong emphasis on the cultivation of students’ generalization abilities. An example follows below. For the first class, the teacher interprets the practical meaning of these three properties, that is, the practical meaning of mathematical concepts. This is followed by a set of fundamental examples of systems of binary quadratic equations. By showing the solutions of examples, students are provided a preliminary overview of the role of each property in ensuring identical solutions when solving systems of equations. For the second class, the teacher divides the system of binary quadratic equations into two categories and then explains and does practice exercises. One category is the equation combination of a binary quadratic equation and a system of linear quadratic equations. Another one is the equation combining two systems of binary quadratic equations. Two lessons are used for each category, and six typical examples are selected from the exercises for explanation and practice. By interpreting specific examples’ solutions, the three properties are illustrated as important guidelines to ensure identical solution equations. The teacher then gives a few examples of systems of fractional equations and irrational equations that, although they have solutions, have increasing or decreasing roots. The reason is that the three properties of identical solution equations are transgressed in solving them. At this point, the teacher tells students that when solving a system of fractional equations or a system of irrational equations, they must check the roots. This practice gives students a new overview of the intrinsic combinatorial decomposition of the three properties and takes their understanding of the three properties a step further. For the third class, the three properties of identical solution equations are synthetically applied, and seven special solutions of systems of quadratic equations that do not require root checking are provided. By this time, the students have improved their understanding of the nature of the three properties, and their generalization abilities have developed a step further. After repeating many times, the students not only firmly master the three attributes of equations with the same solution but also improve their operation speed and problem-solving abilities, especially generalization abilities. The following measures, proposed by our research group, can help students develop their generalization abilities continuously (Gu & Zhao, 1992).

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Clarify the Dominant Idea of Abstract Generalization, Guiding Students to Discover from Conjecture, Conjecture from Discovery The so-called conjecture is essentially an attempt to master new knowledge affected by a student’s original cognitive structure. To strengthen conjecture and discovery, teachers should first analyze the structure of teaching materials and the student’s cognitive structure and clarify the dominant ideology of abstract generalization. This will allow teachers to determine the appropriate scheme to guide students to conjecture and continuous discovery. For this, three points must be noted. 1. Assimilation and adaptation of students’ cognitive structure to acquired knowledge should be based on constantly discovering essential connections and differences between old and new knowledge. Conjecture and discovery should be closely focused on revealing the essential connections and inner laws between knowledge to promote assimilation and adaptation. Therefore, the analysis of the structure of the teaching materials should grasp the essential connections between facts and form the main line of conjecture and discovery throughout the subject. 2. Students’ cognitive structures are evaluated based on a structural analysis of the teaching materials. Teachers should clarify which kinds of knowledge are compatible and assimilable with students’ original cognitive structures and which are not to determine the main content of conjecture and discovery. 3. Students need to be able to freely conjecture and discover key issues. When teachers merely seek to impart knowledge smoothly, they often decrease the difficulty level for hard and error-prone content. While at first glance, it looks like students can easily solve the problems, it actually deprives students of the opportunity to exercise their thinking abilities. Problems related to key issues should be presented to promote conjecture and the discovery of assimilation and adaptation while skillfully guiding students to discover the defects of their original cognitive structure. Furthermore, teachers should instruct and guide them to adjust their cognitive structure to a higher level of development. Taking “solving quadratic equations” as an example, the textbook starts with the direct leveling method and gradually introduces the matching method, the formula method, and the factorization method. The main thread that runs through the whole solution is from particular to general; the development from the former to the latter solution is accomplished by transforming the latter method into the former one. Hence, the dominant ideology of the generalization process should be organized from particular to general and revolve around the essential connection and difference between the two methods. From the viewpoint of students’ cognitive structure, the development from the direct leveling method to the matching method and from the matching method to the formula method is easy for students to assimilate. However, when transitioning from the direct leveling method to the matching method, it is necessary to add and subtract terms to deform the equation. Moreover, the result of

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the formula leads to a variety of categories of equations that are not easily adaptable to students’ original cognitive structure and need to be adjusted to comply with the original cognitive structure. Based on the analysis above, the dominant idea of the generalization process is then clarified, and a scheme for guiding conjecture and discoveries is determined.

Provide Rich and Appropriate Materials for Student’s Abstract Generalization and Make Abstract Generation Concrete Students’ level of abstract generalization depends on the mathematical materials and their qualities. Typical materials should be selected based on certain quantities to illustrate abstract generalization so that students can easily understand abstract generalization. In this process, the contradictions of adaptation between students’ cognitive structures and abstract generalization problems are most easily exposed and stimulated. Therefore, teachers should consciously concretize abstract generalization and guide students to discover contradictions through repeated attempts. This teaching approach could produce a significant effect; for example, “compiling questions based on a certain condition” is a good method to concretize abstract generalization.

Actively Promote the Process of Assimilation and Adaptation by Variation, Reflection, and Systematization The stage of purely imparting knowledge is almost finished after concluding an argument. Although there is a summary after the conclusion, it focuses on the systematic categorization of knowledge itself, thus only solidifying memory without promoting further assimilation and adaptation. Mathematics instruction for cultivating thinking ability should extend beyond the completion of reasoning and demonstration. However, after obtaining the conclusion, the entire thinking process should be reviewed, and accuracy should be checked to deepen the understanding of mathematical principles and general methods. Finally, correlations with the common essence of previous knowledge to generalize universal laws should be made to promote assimilation and adaptation. Here is an example of a typical “variation”: To enable students to discover the essential characteristics from different aspects and avoid one-sidedness, the six triangles shown in Fig. 5.1 are all right triangles. However, due to the different positions, half of the students in the upper elementary school believe that the third and fourth triangles in the figure are not right triangles. This shows the important role of variation in teaching mathematics.

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Fig. 5.1 Six right triangles

Cultivate the formal abstraction ability based on assumptions to generalize through verbal descriptions Based on the trend of the development of generalization ability, I proposed the following three training measures. 1. Gradually cultivate formal abstraction ability in stages and gradually enable students to distinguish the essential features and properties between mathematic relationships. 2. When teaching using examples, emphasis should be placed on presenting or creating analogies and prediction of problem solutions to guide students to observe and compare so that they become proficient in basic problem types, ideas, and skills, which is beneficial for students to uncover the thinking process. 3. Clarify the dialectical relationship between formal abstraction, generalization based on concrete image thinking, and intuitive thinking assumptions. The specific content includes the following two points. (1) Revealing the connection between mathematical and practical problems while developing abstract components of thinking allows concrete image thinking to be constantly enriched and improved. Revealing the association and combinations of mathematical problems with practical problems is carried out in four main ways: the first is based on actual situations and practical problems to introduce the basic mathematical concepts, theorems, and methods to solve. The second method focuses on the geometric interpretations and physical connotations of mathematical concepts or processes. The third approach solves relevant

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practical problems by applying mathematical concepts, theorems, and methods. The last method is to define or demonstrate correlated mathematical concepts. (2) Conjecture and discovery come first, followed by analysis and proving accuracy. Cultivating intuitive thinking entails developing formal thinking, abstract thinking, and generalization thinking based on assumptions. For instance, a student, when calculating (x + y)2 − 2(x + y)(x − y) + (x − y)2 , suddenly answered “4y2 ” but had difficulty clearly explaining his thinking process. The teacher, believing that the student was displaying intuitive thinking, immediately affirmed the answer and then guided him through an analysis of the thinking process with the class. From this, it was found that this student regarded (x + y) and (x − y) as elements, respectively, as the result of the inverse application of the formula of “the square of the difference between two numbers.” Through the experiment, we obtained gratifying research results. At the experimental site of the former Beijing Tong County Educational Science Institute and Tong County No. 6 Secondary school, on the county-wide unified final exam of algebra, the algebra scores of the three experimental classes gradually and steadily improved and outperformed the control classes. The difference is significant, which indicates a small dispersion. This illustrates that experimental measures play a role in improving students’ academic performance. At the same time, according to the structure of mathematical ability, we measured students’ mathematical ability in the experimental and control classes. The results showed that students in the experimental class gradually outperformed the control class from grade 7 with a statistically significant difference. Thus, it can be seen that experimental measures play a role in the development of students’ mathematical ability in experimental classes. Additionally, the students’ differences in mathematical ability emerged later than the differences in academic performance.

Mathematics Learning and the Development of Student’s Spatial Imagination Ability When we talk about the existence of material objects, we must first specify where and how large each is, which refers to the spatial location of the object and its expansivity. It is necessary to understand the spatial connection with other material objects around it, such as the distance and order of arrangement, to determine the spatial location of the existence of a certain material object. Likewise, when we talk about something’s movement, we must relate it to the movement of its position and the increase or decrease of its size or volume. In short, the existence and movement of material objects involve the spatial form of their existence and movement. The spatial form is reflected by our minds and gives rise to the concept of space. The vast space of the universe is boundless. Recognizing this space, that is, the ability of spatial imagination, is an important intellectual power. All production work and technical and scientific creations must be completed in a certain space. All of these

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are inseparable from the ability of spatial imagination. Therefore, spatial imagination is the ability of people to think abstractly about the spatial form of objective things, which is an important part of creative thinking and plays an important role in creative activities (Suh & Cho, 2020). The concepts of shape and geometry can be perceived and are closely linked to the number concepts as an important part of mathematics. Thus, it is extremely useful if elementary school students are exposed to a large number of concepts of shape in mathematics textbooks, such as length, width, height, points, lines, surfaces, and bodies. The mastery of the concepts of shape and geometry in mathematics is based on and promotes spatial imagination ability. To master any kind of spatial thinking, students always need to understand the extent of the length, width, and height of an object in a certain position. Without knowing the positional relationship with other objects, spatial thinking cannot be accomplished, and spatial imagination cannot be developed. Therefore, an understanding of geometry is crucial for developing students’ spatial imagination ability. Secondary school students’ spatial imagination ability is related to the interpretation of graphical information, i.e., their understanding of visual representations and the spatial language used in geometry assignments, graphs, diagrams, and so on. Therefore, it has a strong relationship with the content of the curriculum and the organization of study content, especially the style in which the content is presented. The secondary school mathematics geometry curriculum, which is closely related to the development of spatial imagination, provides a wealth of graphic materials for developing students’ spatial imagination ability. However, in both ancient and modern times, all geometry courses are mainly organized into logical systems of axiom patterns to cultivate students’ abilities. In such an organizational form, geometry education does not aim to develop students’ spatial imagination ability but rather to develop their logical thinking abilities. The role of graphics here is to help students understand the abstract logical system and thus achieve developing logical thinking abilities. In other words, our current cultivation of students’ spatial imagination ability is closely linked to the development of their logical thinking abilities. Furthermore, spatial imagination ability is related to visual processing ability. Visual processing ability involves the conversion of abstract relations or nongraphical information into visual information, which is the operation and transformation of visual representations and visual images (Resnick et al., 2020). This is a kind of process ability that is independent of the presentation style of the learning content. Thus, from this perspective, the development of spatial imagination ability is inseparable from the development of concrete image thinking ability. Spatial imagination ability is closely linked to all thinking qualities. The development of spatial imagination ability is closely connected with completing the thinking quality of profundity. Without profundity of thinking, it is impossible to develop an advanced ability to interpret graphic information; at the same time, without flexibility and agility of thinking, it is impossible to flexibly convert and manipulate non-graphic and visual information. Furthermore, it is impossible to imagine the space of movement. Finally, without the creativity and criticality of thinking, it is impossible to

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decompose, combine, and recreate graphics effectively and fully develop students’ spatial imagination ability. Of course, the development of secondary school students’ spatial imagination ability is not completely synchronized with the growth of their geometric knowledge. The current secondary school textbooks arrange learning in the order of plane geometry and then solid geometry, which is not studied until the first year of high school. However, students have a relatively low level of imagination for three-dimensional geometric figures from eighth grade—they do not yet possess the high-level spatial imagination ability for three-dimensional geometric figures. Only based on understanding two-dimensional geometric figures can we achieve a higher level of imagination of three-dimensional geometric figures? This non-synchronization shows that students learn three-dimensional geometry by first learning spatial straight lines and planes, but in terms of students’ ability development, they reach the overall imagination of geometric shapes first and then the decomposition and combination of overall geometric figures. The following three criteria can be used to determine the objective indicators of a student’s spatial imagination ability: (a) the dependence on visual objects; (b) the scope of analysis and synthesis of various spatial positions in the plane and three dimensions; and (c) the degree of arithmetic of decomposition and combination of various spatial shapes. According to the above criteria, mathematics educational experts in the 1980s proposed four requirements to prove students’ spatial imagination ability (Thirteen Universities Coeditor, 1980). 1. Be familiar with basic geometric figures, able to draw them correctly, and analyze the measurement and positional relationship between basic elements (subordination, parallel, perpendicular, and other fundamental variation relationships). 2. Be able to reflect and think about the spatial shape and position relations of objective objects with the help of graphics. 3. Be able to reflect on and think about the spatial shape and position relations that are expressed in language or formula with the help of graphics. 4. Be able to recognize shapes that distinguish basic shapes from complex graphics and be able to analyze the fundamental relationships between the basic shapes and their basic elements. Although these aspects’ abilities are based on the ability to correctly draw, drawing ability does not simply refer to spatial imagination ability; rather, it is related to the application of drawing tools and the proficiency of drawing skills. Therefore, to improve spatial imagination ability, we need to emphasize the cultivation of drawing ability. However, the level of drawing ability cannot be regarded as the only indicator of spatial imagination ability. According to the above four requirements, students’ spatial imagination ability development can be roughly divided into five intellectual levels of arithmetic thinking for elementary and secondary school students, which are as follows: 1. Relying on visual objects and images to gradually identify the names and concepts of common shapes.

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2. Relying on graphics, mathematics is used to calculate the area of the regular plane and the area and volume of a polyhedral, which is a stage for performing quantitative arithmetic in three-dimensional space. The concrete image in arithmetic thinking ability still occupies a certain advantage. 3. Mastering straight lines and planes: starting from analyzing and synthesizing points, lines, and surfaces; gradually mastering the essence of intersecting lines, parallel lines, triangles, quadrilaterals, similar figures, and circles; and performing various combinations and decomposition operations of plane geometry. 4. Mastering polyhedral and gradually mastering polyhedral figures in threedimensional space based on plane geometry; analyzing and synthesizing spatial lines and planes; imagining the relationship between spatial positions; and performing combinations and decomposition operations. 5. Mastering the spatial position of cylinders, cones, tables, and balls; analyzing and synthesizing the position of their axes, the shape of their cross-sections, and their side development view to imagine the rotational changes in three-dimensional space; and performing combinations and decomposition operations. Undoubtedly, spatial imagination ability is not limited to these five levels. According to the requirements of modern science, we should not only guide students to think more about the relationships of translation, symmetry, and rotation but also teach them to abstractly consider how points form different curves through motion and the intersecting shapes of curved bodies (calculus). There are many contents of shapes in elementary and secondary school mathematics textbooks, which range from learning to calculating the area and volume of regular shapes to systematic learning of geometry. However, can knowledge of shapes in mathematics promote the development of students’ spatial imagination ability? Not necessarily! Experienced mathematics teachers attach great importance to developing students’ spatial imagination ability, and they believe that the development of spatial imagination is the focus of teaching mathematical shapes. That is, geometry is the main way to develop students’ spatial imagination ability. Some kindergartens and elementary schools start to expose children to straight lines, line segments, and various geometric figures that can be composed of line segments when they start teaching them how to count to ten. Having students draw and measure forces them to visually recognize some common figures. Based on the geometric concepts they have mastered, they are then guided to distinguish triangles, quadrilaterals, squares, rectangles, etc., from complex figures. In the 1980s, students’ spatial imagination ability improved in our one experimental school after a year of study. A mathematics competition was held at the end of the first semester, and one of the questions was to determine how many triangles were shown in Fig. 5.2. As a result, 21 out of the 40 students in the class could find six triangles. How can we cultivate the idea of space in students’ thinking through teaching points, lines, surfaces, and bodies and then improve their spatial imagination abilities? First, taking advantage of thought-provoking mathematical exercises is one way to stimulate students’ interest in spatial thinking, especially in the higher grades of

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Fig. 5.2 Big triangle

elementary and middle school stage. For example, when using six matchsticks to create four triangles, many students struggled to solve the problem on a flat surface. Others, through imagination, moving from a single plane to three-dimensional space, found the answer (see Fig. 5.3). Such fun mathematical exercises help students learn the concept of triangles and stimulate their interest in learning geometry, which contributes to their development of spatial imagination abilities. Second, visual tools can help students find solutions. The establishment of the concept of spatial shape requires a visual foundation and the support of visual tools. In addition to teaching aids and drawings, it is important to let students work by themselves. When teaching three-dimensional geometry, students can first use clay and small wooden sticks to build figures. This is not to say that we should focus on perceptual intuition and ignore abstraction. In contrast, this is just a way to guide students from visual to image and then to abstraction, thus promoting the development Fig. 5.3 Four triangles in three-dimensional space

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of spatial imagination ability step by step instead of being satisfied with applying formulas, memorizing, and answering questions. The third step is to strengthen the cultivation of diagramming ability. When formulating graphic problems, it is necessary to consider the graphic problems and their setting conditions. The graphical problem is a problem of making a graph with set conditions—whether the problem has results, i.e., whether it is determined or uncertain, naturally depends on whether the selection of conditions is appropriate. Therefore, setting conditions should not be boundless, and we should pay special attention to the following: the conditions should coordinate with rather than contradict each other; the conditions should be independent of each other, the number of conditions must be minimized to avoid non-independence and parallelism, and the number of conditions should be appropriate. Whether it is indeterminate or determinate position construction, graphical problems only have two parts: one is the setting (the given conditions), and the other is the seeking part (finding which figure is suitable for the conditions). There are also many methods of graphing, such as the trajectory intersection method, travel tangent method, triangle foundation method, contract transformation, and displacement method. In this regard, it is necessary to guide students to follow four steps in solving graphic problems. 1. Analyzing. Before formally graphing, clues about graphing methods should be found. 2. Graphing. Based on the clues obtained, the method of graphing is designed step by step. The order of the drawing actions must be recorded in detail. 3. Proving. After graphing, the graph should be verified step by step with the required conditions to test whether the graphics are correct. 4. Inquiring. Consider all possible scenarios of the problem to make an affirmative judgment conclusion. The ability to solve diagramming problems improves spatial imagination and logical reasoning ability. Fourth, properly designing and creating geometric teaching aids and models or taking field measurements are good ways to improve spatial imagination ability. Teachers can let students design and make visual geometry teaching aids and models or take field measurements based on the teaching requirements. Hands-on practice, using axioms, operating points, lines, surfaces, and bodies, making observations, dissecting, analyzing, exploring, and discovering the relationship between geometric figures and quantities are all ways to develop students’ spatial imagination ability.

Mathematics Learning and the Development of Student’s Proposition Ability Any sentence that has the form of a judgment is a proposition. For example, “4 is an even number,” “7 is not a common divisor of 14 and 15,” and so on. Therefore,

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propositions are judgments. People are judging things all the time, making different propositions. Concepts, judgments, and inferences constitute the basic forms of thinking. The essence of things reflected by concept is revealed in judgment. Judgment is the reflection of the connection between objects and phenomena or their related characteristics. Judgment is about what something is and the affirmation or denial of relations between certain characteristics of an object or phenomenon. When we say “Thunder follows lightning,” we affirm that the two natural phenomena, “lightning” and “thunder,” have a certain connection in time. When we reveal the essence of the concept of “oil,” we make many judgments about its main characteristics, such as the sources of various types of oil and their uses. There are positive or negative judgments as well as general, special, or specific judgments. Moreover, judgments can indicate conditions or causes—the inevitability, possibility, and contingency of things. Therefore, people’s practices are inseparable from the thinking form of judgment and from the intellectual power of judgment or proposition. Mathematically, all equations are predictions of propositions, such as “6 = 7, ” “4 is an even number,” “ABC is an isosceles right triangle,” etc., which are all judgments. Mathematics reveals the relationship and connections between quantities and forms, which must be expressed in the form of judgments. Mathematics learning must be based on propositional knowledge while also requiring students to develop higher propositional knowledge. Thus, mathematics learning promotes the development of students’ propositional knowledge. Mathematics learning requires students to make judgments independently, and this independence relies on the acquisition of judgment ability based on the habits and skills of independent judgment. It is this independence of judgment that determines the level of one’s judgmental ability. Students need to develop independent knowledge in their mathematics learning. For example, in a mathematics class, thirdgrade students struggled with “849 × 876 − 876 × 749, ” but one student used common factorization to extract “876” from it. He multiplied “876” by (849 − 749) and quickly obtained the answer “87,600.” This student demonstrated the exceptional judgmental ability that mathematics teachers must pay attention to. It must be recognized that mathematics learning demands students develop a high standard of independent judgment: from the proposition of “1 + 1 = 2” to “A grain of sand is small enough, but since it √ is a quantity, the sand is not infinitely small,” from √ “3 × 3 = 6” to “y = x + 3 −x is a functional relationship,” which is the result of the sudden change in students’ judgment knowledge. As another example, when √ √ 20 a + 4 a , he a teacher asked high school juniors to find the constant term of initially wanted the students to make a judgment via calculation: n √ n √ 20−n n a = cn a 4 · a 20 − 2 , which is n + 20−n = 0, n = 40, T = cn 4 a n+1

20

20

4

2

which is a positive integer and greater than 20, so the constant term does not exist. However, a student immediately judged by analysis without calculation and said that √ √ 20 there could not be a constant term in a + 4 a because the exponents of the two terms a in the binomial formula are positive numbers, and each term of the expansion is the product of their two multiplicities, that is, the exponent of each letter is positive

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Fig. 5.4 Relationships between four propositions

and cannot be zero, so the constant term does not exist. It is evident that this student’s judgment ability is valuable, thus confirming the necessity of developing students’ judgment (propositional) ability. In mathematics teaching, it is important to note that teachers inspire students to master the idea of the structure of “four proportions” to let students directly master the idea of positive, negative, conjunctive, and disjunctive propositions in mathematics and to guide them to independently make mathematical judgments as early as possible. In other words, the study of mathematics requires the ability to master the structure of propositions. The development of this ability is reflected in the arithmetic of the four propositions: original proposition → inverse proposition → converse proposition → contrapositive proposition (see Fig. 5.4). The structure mastery of four propositions reflects both the degree of abstraction of students’ understanding of different mathematical propositions and their mastery of the reversibility and conservation of the direction of thought in the thinking process, as well as the dialectical relationships in thinking activities. The relationship between the original proposition and other propositions reflects the law of the “inverse of inverse” in the performance of students’ arithmetic ability. Some people say that this ability does not appear until students start to learn geometry in secondary school, but this is not true. In fact, mastery of propositional structure ability begins in the first grade of elementary school. Why do first-grade students find it difficult to solve countercondition word problems? The reason is that they lack the ability to think in reverse. Some first-grade teachers encourage students to memorize various “types” of problems instead of flexibly mastering various “types.” However, not flexibly mastering the nature of reverse exercises is detrimental to the development of elementary students’ proposition ability. Mathematics learning requires students to master the form of propositions. In mathematics, positive or inverse propositions always appear first, followed by “and” and “or” propositions; for example, asking elementary school students to determine “36 is a multiple of 12 and a multiple of 9”; asking middle school students to state that “a and b are parallel” and “a and b are intersecting,” which cannot both be true at the same time. This is a conjunctive proposition “ p ∧ q” (p and q) and a

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disjunctive proposition “ p∨q” (p or q). Based on these four primary forms, compound propositions gradually appear in the textbook. For example, the possibility of the root of an equation is determined, i.e., whether it has only one solution, multiple solutions, infinite solutions, or no solution. These equations are the evolutionary relations of compound “equivalent” propositions. Ultimately, students are required to determine the deformation of the propositions according to the rules of arithmetic, such as solving equations and determining the deformation of the same solution of a system of equations, as mentioned earlier. The emergence of this content in mathematics teaching requires students to have the corresponding proposition ability as a foundation, which in turn requires teachers to consciously cultivate students’ proposition ability so that they can better determine simple propositions, master and judge compound propositions, and determine the deformation of propositions according to the laws of arithmetic. Based on the mathematical logic used, clusters and lattices, that is, 16 binary propositional arithmetic, characterize the maturation of children’s and adolescents’ thinking structures. In my research, secondary school students’ level of propositional arithmetic is constantly improving, reflecting the development trend of their logical thinking abilities. Another study on the syntactic structure of propositions showed that secondary school students’ proposition ability develops from simple to complex ideas. The measurement of a student’s level of judgmental (propositional) ability in arithmetic is based on five indicators: the level of independent judgment in mathematical arithmetic, the understanding of the mathematical proposition structure, the understanding of mathematical propositions, the degree of decomposition and combination of mathematical propositions, and the degree of propositional deformation. How can we cultivate students’ proposition ability through these indicators? The key is to guide students in determining the relationships between the conditions and conclusions of different propositions. A proposition is composed of two parts: the condition and the conclusion. How are they related? As mentioned above, “if A then B,” the presence of A guarantees the conclusion of B, and it can be assumed that condition A is sufficient for conclusion B. Likewise, if there is no condition A, there is no conclusion B, and the implication is that for now, condition A is necessary for conclusion B. In mathematics, there are three types of conditions that must be distinguished. 1. Sufficient and unnecessary conditions. In this case, the original proposition is true, and the inverse (or negative) is false. For example, if the original proposition is “two angles are opposite angles, so they must be equal” (correct), the inverse proposition would be “two equal angles must be opposite angles” (incorrect), and the negative proposition would be “two angles are not opposite angles, so they must be unequal” (incorrect). Therefore, while “opposite angles” is a sufficient condition for “equal,” it is not a necessary condition. 2. Necessary but not sufficient conditions. In this scenario, the original proposition is false, and the inverse (or negative) is true. For example, if the original proposition is “two angles are equal, so they must be opposite angles” (incorrect), the inverse proposition would be “two opposite angles must be equal” (correct), and the

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negative proposition would be “two angles are not equal, so they must be not opposite angles” (correct). Therefore, while “equal” is a necessary condition for “opposite angles,” it is not a sufficient condition. 3. Sufficient and necessary conditions. In this situation, both the original proposition and the inverse (or negative) proposition are true. For example, if the original proposition is “the two sides of a triangle are equal, the opposite angles are equal” (correct), then the logic that “if the two angles of the triangle are equal, the opposite sides are also equal” (correct) can be applied. Thus, two equal sides of an isosceles triangle and two equal opposite angles are sufficient and necessary conditions that support each other. Each proposition is valid only under certain conditions. However, students often do not pay enough attention to the conditions that are required to reach a certain conclusion. If they do not know how to consider the sufficiency and necessity of conditions, they are prone to making mistakes in judgment, which will affect the development of their proposition ability. Therefore, when teaching mathematics, it is important to guide students to consciously distinguish the relationship between conditions and conclusions. To develop students’ proposition ability, we should cultivate a habit of considering conditions precisely stated from learning arithmetic in elementary school. The following are example questions that would be beneficial in helping students make appropriate judgments. 1. Which is heavier, 1 kg of cotton or 1 kg of iron? Why? Which is longer, 2 m of leather or 2 m of cloth? Why? 2. Six trees were planted in 3 rows. Do you think it is possible? Why or why not? 3. Is a number with 0 as the last digit always divisible by 5? Must a number divisible by 5 always have 0 as its last digit? 4. To find the common multiple of two numbers, is it enough to multiply the two numbers? Is this the least common multiple? Why? 5. Must a number whose sum is divisible by 3 be divisible by 3? If a number is divisible by 3, can the sum of its digits also be divisible by 3? If the sum of the digits is not divisible by 3, can the number be divisible by 3?

Secondary school students can be required to answer similar questions and make correct conclusions based on what they have learned, based on which the following three levels of proposition ability are demonstrated. 1. At Level I, students are able to perform calculations on simple propositions with universal qualifications (when the proposition is a positive judgment, the universal quantifier is usually omitted) but still struggle to understand the meaning of logical connectives in the process of propositional calculation. In other words, students at this level cannot perform calculations on formal propositional syntactic structures without the semantic content of the propositions but can perform calculations on simple positive propositions with particular quantification and can understand the meaning of quantifiers in the propositions.

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2. At Level II, students can combine simple propositions, that is, the calculations of conjunctive ( p ∧ q) and disjunctive ( p ∨ q) propositions, and perform the calculation of a negative proposition—the key here is to correctly convert quantifiers and subject–predicate conjunctions (the conversion between positive and negative propositions). 3. At Level III, students can perform negative calculations of compound propositions. Compound propositions refer to those propositions with quantifiers and logical connectives (negation, conjunction, disjunction, implication, and equivalence). Students at this level understand the meaning of logical connectives and can perform correct calculations based on the laws of propositional calculus (e.g., the laws of exchange, union, distribution, and double negation). The development order of proposition ability outlined above reflects the development of secondary school students’ logical thinking ability, which is a process from low to high and from simple to complex. At the same time, this process reflects their development of antithetic thinking ability, which is a process of moving from cluster structure to lattice structure. Secondary school students’ logical thinking ability and structure develop and mature by performing calculations of increasingly complex forms of propositions.

Mathematics Learning and the Development of Students’ Logical Reasoning Ability When you get up in the morning, look outside, and find the yard, flowers, and other plants are all wet, you might immediately conclude that it rained the night before. This process of deducing a conclusion not by direct judgment but by “thinking” is called reasoning. Logical thinking is a general, indirect reflection of objective reality, while reasoning is the logical formation of a new judgment from two or more existing ones. Notably, any new judgment is always deduced from several other judgments; thus, logical reasoning is the highest basic form and core of thinking. As mentioned in the previous chapter, thinking ability is the core of intelligence; logical reasoning ability then is the core of the core. What ability do people rely on to solve problems? The answer is logical reasoning ability. It is no wonder then that many psychologists’ research on problem-solving thinking ability is centered on logical reasoning. Therefore, it can be said that logical reasoning ability is the ability to solve problems. As all human activities are based on solving problems, logical reasoning ability is an absolute necessity for our existence. Mathematics is characterized by not only its existence in objective reality but also its high degree of abstraction, accuracy, logic, and rigor. The process of mathematical arithmetic is the process of solving problems, which must be based on logical reasoning ability, which simultaneously promotes the development of students’

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logical reasoning ability. In arithmetical operations, this ability is expressed in abundance. The five indicators for determining a student’s logical reasoning ability are as follows. 1. Whether the steps of reasoning are direct or indirect. For example, for problems needing to be solved, some can be solved in one step, and others require multiple steps of reasoning. The level of the latter is higher than that of the former. 2. The degree of completeness of the type of logical reasoning. For example, deductive syllogism requires a major and a minor premise and a conclusion. The lack of any of these elements makes the process imperfect. 3. The range of reasoning. Reasoning in the arithmetic domain can be clearly differentiated from reasoning in the algebraic domain. Indeed, in the same field of arithmetic, reasoning within the integer range and reasoning within the fraction range are two different levels. The higher the level of reasoning in the abstract number range, the higher the level of logical reasoning ability. A student whose level of logical reasoning allows him/her to rationalize in the range of “complex numbers” would be generally higher than a student who can only reason in the range of “rational numbers.” 4. The correctness of the reasoning process. This refers to whether an appropriate conclusion can be obtained by reasonable reasoning. 5. The characteristics of reasoning, such as generality, self-consciousness, and the extent of revealing essence. In a similar arithmetic process, some students have learned to infer, generalize, transfer, and form analogies between problems at a higher level of logical reasoning ability. That is, there are those who can consciously conduct various inferences at a higher level of logical reasoning ability and those who cannot. Those who can solve problems through reasoning generally have a higher level of reasoning abilities than those who cannot.

Indirect and Direct Reasoning Both elementary and secondary school mathematics instruction require students to develop their thinking skills from direct to indirect reasoning. However, the range and degree of abstraction of direct and indirect reasoning differ. First-grade elementary school word problems mostly focus on one-step word problems, which are basically direct reasoning. For two-step word problems, where the condition and the problem are not directly connected, we add one more step. Thus, one more connection is added to the intellectual activity, which develops into indirect reasoning. After the third grade, word problems are primarily taught by gradually increasing the number of steps. Some think that more steps do not necessarily represent a higher level of logical reasoning ability. However, thinking is a reflection of the nature of the inner connections and relationships of things. In solving word problems, the more steps there are, the more connections are required in higher reasoning ability. Furthermore, as elementary students solve progressively more difficult word

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problems, their logical reasoning ability develops in line with the increasing number of steps. Both direct and indirect reasoning in secondary school are utilized within the range of abstract algebra and geometry and are roughly divided into four degrees. 1. Direct reasoning refers to applying formulas and matching the conditions that lead directly to a conclusion. 2. Indirect reasoning refers to formulas that cannot be directly applied, thus requiring changing the conditions and finding evidence, which then leads to a conclusion via multiple steps. 3. Roundabout reasoning refers to analyzing the premises and proposing a hypothesis. A conclusion is reached after repeatedly testing and verifying the hypothesis. 4. Comprehensive reasoning occurs in accordance with a certain mathematical format. Students with this degree of reasoning will gradually simplify and rationalize their reasoning process. For example, for one step of the solution, further steps can be recognized and formed.

Synthesis and Analysis Logical reasoning comprises synthesis and analysis methods according to forward thinking and reverse thinking, which are the most common methods in elementary school mathematics teaching. Synthesis. The synthesis method starts from the hypothesis and progresses through a series of stated propositions. Deductions are gradually made until a new proposition is obtained or the current problem is solved. In short, the synthesis method is to draw results from a cause, that is, to draw a conclusion by analyzing the known conditions. For example, to prove the theorem “If A, then D,” the thinking process applied in the synthesis method is shown in Fig. 5.5. The synthesis method draws results from a cause, moving from conditions to the conclusion. Thus, we can observe the downward deductions from A that form various pathways until D is achieved. However, there may be more than one result produced by A. Assume that B, B1 , and B2 are all the results of A and that each B has its own result. Suppose that B produces C and C 1 , B1 produces C 2 , and B2 produces C 3 and C 4 . Some of these resulting Cs can produce D, and some cannot. Finally, we assume C can produce D, and the inference of “A to B to C to D” is completed. Analysis. To prove the correctness of a proposition, we could track backward from the conclusion when thinking. That is, starting from the conclusion of a proposition, we investigate the reason and then consider these reasons separately to see what conditions are required for its establishment until we gradually reach the truth. Simply speaking, the analysis method seeks a cause from the results, i.e., it discovers the required conditions based on the analysis of the conclusion. For example, to prove the theorem “If A, then D,” the thinking process applied to the analysis method is shown in Fig. 5.6.

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Fig. 5.5 Thinking process of proving “If A, then D” via applied synthesis

Fig. 5.6 Thinking process of proving “If A, then D” via the applied analysis method

The analysis method seeks to determine the cause from the results, that is, to observe upward from D to find the cause. If this is the case, then what can be deduced from D? Suppose that C, C 1 , and C 2 can all produce D; then, what can produce C? Suppose that B and B1 both can produce C; then, what can produce C 1 ? Suppose that B2 can produce C 1 ; then, what can produce C 2 ? Suppose B3 and B4 can produce C 2 . All these causes can lead to D, but which one is the result of A? After the checking, it was found that B is the result of A. That is, from known D to unknown A, the inference of “D to C to B to A” is established. In fact, the synthesis and the analysis methods are closely integrated. Regardless of which method is adopted, both synthetic and analytical processes are involved. When the solving path is from the conditions to the question, one should pay attention to the requirement of the question asked, whereas when the solving path is from the

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question to the condition, one should pay attention to the unknown condition. This process entails both synthesis and analysis, and the two cannot be separated, as illustrated by the following example: In a village, 30 hectares of cotton are planted, and 4 21 times as many hectares of wheat are planted as cotton. The total number of hectares of cotton and wheat is 75% of the total number of hectares, and the total number of hectares of rice is 41 of the total number. How many hectares of rice are planted?

Synthesis method: From the known conditions, we know that there are 30 ha of cotton and that the number of hectares of wheat is 4 21 times that of cotton. Thus, the number of wheat hectares is 30 ha × 4 21 = 135 hectares. Because the number of hectares of cotton and wheat is 75% of the total number, the total amount of cotton and wheat can be accounted for (30 ha + 135 ha) ÷ 75% = 220 ha. Since the number of hectares of rice is 41 of the total number, the number of hectares of rice is 220 ha × 41 = 55 ha. Analytical method: From the requirement, there are three simple questions that can be classified after analysis: (a) the total number of hectares of rice is 41 ; thus, to determine how many hectares of rice are needed, the count of total hectares is needed; (b) the total number of hectares of cotton and wheat is 75% of the total number; thus, the number of hectares of cotton and wheat is required to find the total number of hectares; and (c) the number of hectares of wheat is 4 21 ; thus, the number of hectares of cotton is required to find the number of hectares of cotton and wheat. The known condition is 30 ha of cotton. From the known conditions, solving these questions step by step, 55 ha of rice can be obtained. The above process reflects that analysis is inherent in the synthesis method and vice versa. In the process of teaching, teachers should guide students to combine synthesis and analysis so that they can develop a well-founded, methodical, and organized manner of thinking.

Inductive and Deductive Reasoning According to the relationships among the individual, the general, and the special, logical reasoning is divided into inductive and deductive reasoning. These two methods of reasoning are more commonly used in mathematics teaching at the elementary and secondary school levels. Inductive Reasoning Inferences are deduced from the validity of particular facts to validate general facts. This process, which means extrapolating from the particular to the general, is called inductive reasoning or induction. There are two kinds of inductive reasoning: ordinary and enumerative. Ordinary Induction. For example, in the section “properties of fractions” in elementary schools, two lessons are generally needed. However, when ordinary induction is applied, it only takes half a lesson to make students understand the properties of fractions. Here is an example.

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“ 41 ” is written on the blackboard. The teachers ask, “What is the value of 41 ?” The student answers, “0.25.” Then, the teacher asks students to calculate

1×2 4×2

= 0.25,

1×3 4×3

= 0.25,

1×5 4×5

= 0.25…

Finally, the teacher asks, “What do you see from this?” The students answer, “The numerator and denominator are both multiplied by the same number and the size of the fraction remains the same.” The teacher writes on the blackboard:

1×0 4×0

=?

The students immediately correct the teacher, saying, “The numerator and denominator are both multiplied by a number that is not equal to 0. The size of the fraction remains the same.” By using the same procedure, the students were able to deduce that “if the numerator and denominator are divided by a number that is not equal to 0, the size of the fraction remains the same.” The teacher then asks the students to make further ordinary inductions and concludes, based on these two properties, that “the size of a fraction remains the same if the numerator and denominator are increased or reduced by the same multiple (except 0).” Then, the teacher asks students to do exercises to consolidate their knowledge in class. This process enables most students to master the properties of fractions and, more importantly, teaches them the proper way of thinking and understanding the process of ordinary induction.

Enumerative Induction. If one is able to think through all the possible scenarios of a matter to be studied, the answer derived from this general reasoning is correct. This method of reasoning is called enumeration induction. Enumeration induction is often used in mathematics, and it allows students to develop the habit of being thoughtful and careful when considering problems. Therefore, it is crucial for students to master the method of enumerative induction in their development of intelligence. A theorem should be proved via enumeration if the nature or interrelationship of the assumption changes and the reasons for the proof differ. Specifically, by first assuming various possible cases of the matter and then proving them one by one after each situation to be proved, an assertion can be drawn: the proposition is true in general. The following serves as an example: Problem: If the sum of squares of a pair of opposite sides of a quadrilateral is equal to the sum of squares of the other pair of opposite sides, the two diagonals are perpendicular to each other. Problem: In quadrilateral ABCD, AB 2 + C D 2 = AD 2 + BC 2 . Conclusion: AC⊥B D. Proof: A quadrilateral is divided into a planar quadrilateral and a space quadrilateral, and a planar quadrilateral and a space quadrilateral are divided into a convex quadrilateral and concave quadrilateral. According to these three cases, each statement is examined separately (see Fig. 5.7).

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Fig. 5.7 Three situations for proving the proposition

(The detailed proof is omitted) All three situations are valid, so the proposition is valid in general.

Deductive Reasoning Deductive reasoning moves from general to specific concepts, usually applying the method of syllogism. Syllogism is composed of three judgments, two of which are the premises and the third of which is the conclusion. The first premise is a general affair, called a major premise; the second premise is a special affair, called a minor premise; and the final judgment based on these two premises is called a conclusion. For example, in the calculation of proportion, the product of two inner terms is equal to the product of two outer terms (the major premise). ∵ 50: 300 is proportional to 2: 12 (minor premise). ∴ 300 × 2 = 50 × 12 (conclusion).

For another example, to determine the congruence of two triangles, it is necessary to prove that two sides of the two triangles are equal to an angle (minor premise), which conforms to the theorem “side by side” for determining congruent triangles (the major premise) so that the two triangles can be proven to be congruent (conclusion). In elementary and secondary school mathematics teaching, students solve a specific problem based on the definitions, laws, properties, formulas, theorems, etc., a process referred to as deductive reasoning. However, deductive reasoning often entails more than simply applying a formula. It usually contains a number of steps, each of which is analyzed in the form of a syllogism. Therefore, the process of proving a proposition via deductive reasoning is actually a series of three coherent steps of reasoning, that is, an indirect multistep reasoning process. To simplify the description of the proof process, we often omit one of the two premises (usually the major premise) in the steps of reasoning (the three-step method) or even just write down the conclusion.

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Analogical and Comparative Reasoning Comparison is the method of mentally determining the similarities and differences between two objects. The saying that “comparison is the only way to discriminate” emphasizes its significance. The comparison process is conditional: first, objects that are related to each other can be compared, and second, the comparison should be made with essential or practical features and under the same standard. In mathematics teaching, the use of comparison as a method of logical reasoning can motivate students to actively think about the problem and consciously acquire knowledge, thus improving their thinking abilities. This method of logical reasoning can be divided into analogical and comparative reasoning according to the nature of the comparison. Analogical Reasoning. Analogical reasoning is reasoning through comparison between one particular case and another. It compares two objects with the same (similar) properties and deduces that some of their other properties are also the same (similar). Analogical reasoning is often applied in elementary and secondary school mathematics to identify connections between knowledge and build a new scientific concept system. For example, in elementary school mathematics, from the judgment that “the dividend and the divisor are enlarged or reduced by the same multiple (except 0), and the ratio remains unchanged,” the analogy derives “the former and subsequent terms of the ratio are enlarged or reduced by the same multiple (except 0).” The ratio remains unchanged, and the judgment that the “numerator and denominator of the score are expanded or reduced by the same multiple (except for 0), as well as the size of the score, remains unchanged.” Comparative Reasoning. Comparative reasoning is the comparison of two or more objects with different properties to draw a new conclusion. In terms of the nature of its judgment, the conclusion may be similar to the premise or may be more general than the premise, so comparative reasoning is a form of reasoning that exists from the particular to the particular or from the particular to the general. Comparison is often used in elementary and secondary school mathematics teaching to identify the differences between mathematical proportions and establish a scientific concept system. Comparative reasoning can be employed to analyze and explain concepts, definitions, theorems, formulas, and confusing laws. For example, teaching elementary school students the meaning and properties of positive and inverse proportions can guide students to understand the relationship between “positive” and “inverse.” “Positive” cannot be valid without “inverse” and vice versa. Another example is that the formula of polyhedrons and rotating bodies can be distinguished through comparative reasoning and understanding of each figure’s essence. The above illustrates the cultivation of the content of logical reasoning and corresponding abilities in the arithmetic thinking process. When we solve a practical problem in mathematics, we engage in complex thinking activities. The logical reasoning abilities mentioned above are related to each other and cannot be separated

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entirely; they are neither subordinate nor exist in isolation. When students acquire these reasoning abilities, they can gradually make their arithmetic thinking more concise and rationalized and progressively more abstract and “internalized,” so that the abstraction of logical thinking, precision, and rigor steadily improves. Therefore, it is essential for teachers to have a concrete plan and purposefully choose appropriate reasoning methods according to specific mathematics textbooks and teaching content. In this way, students, in the process of learning mathematics, can be guided from unconscious to conscious and gradually master reasoning abilities to develop logical thinking abilities.

References Darwin, F. (1994). The life and letters of Charles Darwin (D. Ye., & G. Meng., Trans, Vol. 1). Science Press. Gu, J. F., & Zhao, R. L. (1992). An experimental report on cultivating students’ thinking quality based on generalization in algebra teaching in middle school. In C. Lin (Ed.), Ability development and cultivation of middle school students. Beijing Education Press. (in Chinese). Resnick, I., Harris, D., Logan, T., & Lowrie, T. (2020). The relation between mathematics achievement and spatial reasoning. Mathematics Education Research Journal, 32(2), 171–174. Sowder, J. T. (2020). Making sense of numbers in school mathematics. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 1–51). Routledge. Suh, J., & Cho, J. Y. (2020). Linking spatial ability, spatial strategies, and spatial creativity: A step to clarify the fuzzy relationship between spatial ability and creativity. Thinking Skills and Creativity, 35, 100628. https://doi.org/10.1016/j.tsc.2020.100628 Thirteen Universities Coeditor. (1980). Middle school mathematics teaching material teaching method general. People’s Education Press. (in Chinese).

Chapter 6

Differences and Development of Students’ Intellectual Qualities in Arithmetic

As stated in Chap. 1, intellectual thinking quality is the performance of an individual’s intellectual characteristics represented in his or her thinking activities. It is an important indicator to distinguish between supernormal, normal, and subnormal intelligence in a certain context. In student life, especially after enrollment, teaching requires engaging students in intellectual activities structured to reflect a certain level of purpose, direction, certainty, and criticality. It also requires a degree of agility, flexibility, breadth, and depth of intellectual activities, enabling the rapid development of independent thinking abilities. Acknowledging this, what are the characteristics of the developmental trends of students’ intellectual qualities in arithmetic? First, there are clear differences in students’ intellectual qualities in arithmetic. These become increasingly pronounced with age and as students’ progress through grades. Consider agility (speed) as an example: Table 6.1 shows the statistical results of seven grades of students who were instructed to complete all calculations on a mathematical test (with an average completion time of 40–45 min). The table indicates significant differences in students’ mathematical arithmetic agility by age (grade). Notably, this difference (including the deviation within the same grade) is age-specific; that is, the higher the grade, the more significant the difference. That said, the difference in overall intellectual quality gradually stabilized after Grade 9. Second, during early youth’s mental or intellectual maturity, the intellectual component of intuitive images gradually decreases, and the intellectual component of logical abstraction steadily increases. Likewise, the self-consciousness of abstract logic develops gradually. In the process of learning arithmetic, students in the lower grades of elementary school can solve problems but are unable to explain their thinking process; however, after Grades 3 and 4, with the development of internal speech, they begin to regulate, examine, and discuss their thinking

© China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_6

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Table 6.1 Standard deviation of arithmetic speed in different grades Grade

Grade 3

Grade 5

Grade 7

Grade 8

Grade 9

Grade 10

Grade 11

SD

6.90

9.70

10.04

12.91

13.01

13.56

13.60

processes more consciously and can explain how they reached a particular conclusion. This shows those intellectual qualities of profundity and criticality of thinking gradually determine the development of all intellectual qualities. Third, the complexity of the arithmetic process reveals the diversity of intellectual qualities. Same-aged students may show different levels of thinking quality when performing calculations on different types of mathematical problems. Consider the profundity (abstraction level) as an example. Fourth graders can think abstractly about integer operations without concrete objects, but their understanding of fractions still needs the support of visible teaching approaches. Fourth, the level of intellectual quality in arithmetic depends on education. Mathematics learning must be based on and promote the development of various intellectual qualities. Mathematics teachers must apply effective methods and teaching approaches to cultivate students’ intellectual qualities. The following three examples illustrate this. Example 1: In the 1980s, an experimental class teacher applied a teaching measure to cultivate the intellectual qualities of elementary school students. As a result, the four-year teaching task was completed in only three years with excellence and quality, and the students’ intellectual qualities improved significantly. Example 2: In the early 1990s, a group of secondary school experimental mathematical sites in Beijing and Shanghai strengthened the quality of mathematics training for students in experimental classes. All students’ mathematical ability indicators in the experimental class exceeded those in the control class. Example 3: At the beginning of the 21st century, in an article published in the journal Cognitive Development, Zhou et al. of St. John’s University in the United States noted that they used the intelligence (cognitive) development scale to conduct a comparative study of an experimental site that insisted on training students to improve their thinking quality (Zhou et al., 2000). The study was conducted across three sites: (a) Jinghai County, Tianjin (Today’s Jinghai District), a remote rural elementary school, (b) a normal elementary school in Beijing, and (c) an urban elementary school in the United States. The comparison showed that the participants in the rural elementary school in Jinghai County, Tianjin, had the highest academic performance. From this, they concluded that thinking quality training is paramount in children’s and adolescents’ intellectual development and that the longer the training period is, the more pronounced the effect. Many intellectual qualities appear in student arithmetic. Thus, as illustrated in Chap. 1, the focus of cultivation should be on developing the five intellectual qualities of thinking profundity, flexibility, creativity, criticality, and thinking agility.

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Profundity of Students’ Thinking and Its Cultivation The profundity of thinking, also known as logic, is the degree of abstract generalization of the thinking process or intellectual activity, which reflects that intelligence entails being good at abstraction and generalization and proficient at grasping the laws and essence of things to carry out systematic, rational activities. Mathematics requires a high degree of logic; thus, mathematics teaching first requires the profundity of students’ intelligence, which is based on logic, followed by active efforts to promote the development of the same, meaning the development of their logic. In Chap. 5, we discuss arithmetic thinking abilities. From this understanding, it follows that individual differences in thinking abilities can be characterized as variations in the degree of abstraction of students’ intellectual qualities in mathematical arithmetic. Students with subnormal intelligence cannot calculate without intuition or a visual image; they rely on their fingers to perform the calculation. When the fingers are forbidden, the calculations are terminated. Therefore, their abstraction in arithmetic is low; likewise, their logic (i.e., profundity) is poor. In contrast, students with supernormal intelligence are often good at seeing through appearances to perceive the essence of the problem. They diligently study difficulties, solve abstract exercises, and enthusiastically reason logically, from which the profundity of their intellectual qualities becomes obvious. Here, I introduce the story of “Einstein’s algebraic problems.” When Einstein was in secondary school, he got sick and was hospitalized. A friend gave him a mathematical problem to amuse himself during his hospital stay. The problem is as follows: A clock’s minute and hour hands can possibly accurately indicate an exact moment in time if the minute hand is changed to the hour hand and vice versa. However, in general, such a change would no longer indicate the exact moment. For example, at 3:30, the hourly hand points to the middle of 3 and 4, and the minute hand points to 6. If the minute and hour hands are switched, the hour hand points to 6, and the minute hand points to the middle of 3 and 4, which is not an accurate indication of the hour. If it is 6:00, the minute hand should point to 12; if it is 6:17:30, the hour hand should point somewhere between 6 and 7. However, as mentioned above, the hour and minute hands can be interchanged and still indicate the right moment, for example, when the hour and minute hands overlap. Except when the two needles overlap, how many cases exist where the hour and minute hands are interchangeable and still indicate the right moment and at what times?

This was a logical algebraic problem that required complex reasoning. However, as soon as his friend finished speaking, Einstein finished thinking, took a pencil and paper to create a formula, and immediately worked out the answer. His answer was this: The clock dial was divided into 60 grids (i.e., the dial per minute was one grid). Starting the observation at 0:00, when the hour and minute hands overlap, it can be seen that when the hour hand moves one grid, the minute hand moves 12 grids; when the hour hand moves five grids (i.e., 1 h), and the minute hand moves 60 grids (i.e., 1 lap). Let x be the number of grids the hourly hand moves, let a be the number of complete turns the minute hand moves,

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and y be the number of grids the minute hand moves from 0 after deducting the number of complete turns, which gives us:

12x = 60a + y. (Obviously, y < 60)

(6.1)

Let the minute hand be in front of the hour hand; then, the hour hand continues to move to the position of the minute hand. At this time, the minute hand moves b laps to the position of the hourly hand, which can be expressed as follows:

12y = 60b + x. (Obviously, b > a)

(6.2)

From Eq. 6.1, we can obtain y = 12x − 60a, and then substituting it into Eq. 6.2, we obtain (12x − 60a) = 60b + x. Consequently,

x=

720 60 a+ b 143 143

(6.3)

where a and b are the number of laps in the minute hand, which must be zero or a positive integer. Furthermore, because the hour hand returns to point 0 when a = 12, we derive the following:

12 > b > a

(6.4)

As you can see, the solution to this problem is not unique but rather a set of solutions. What is this set of solutions? According to Equation 6.4, we can list the determined values (see Table 6.2).

Table 6.2 Values of A and B Value of B

Corresponding value of A

0 1

0

2

0

1

3

0

1

2

4

0

1

2

3

5

0

1

2

3

4

6

0

1

2

3

4

5

7

0

1

2

3

4

5

6

8

0

1

2

3

4

5

6

7

9

0

1

2

3

4

5

6

7

8

10

0

1

2

3

4

5

6

7

8

9

11

0

1

2

3

4

5

6

7

8

9

10

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This gives 66 answers: (11 + 1) × 11 ÷ 2. Each time a pair of corresponding values of a and b is taken and substituted into Equation 6.3 to find x, the two moments represented before and after the two hands are interchanged can be found. For example, when b = 6 and a = 3, 60 720 60 the hour hand is at x = 720 143 a + 143 b = 143 · 3 + 143 · 6 ≈ 17.622377 (grids). Here, x refers to the number of grids in which the hourly hand moves. To obtain the corresponding time, the above results are divided by 5; the resulting integer quotient part is the number of hours, and the remainder is multiplied by 12 which is the number of minutes. The resulting time sought was approximately 31 min and 28 s, respectively, past 3. When the hour and minute hands are interchanged, the hour hand is at y = 12x − 60a = 12 × 17.622377 − 60 × 3 = 31.468524(grids). According to the previous method for calculating the number of hours and minutes, it is known that the time at this moment is approximately 17 min and 37 s 6. That is, when the hour and minute hands were interchanged at 31 min and 28 s past 3, the time resulted at 17 min and 37 s past 6. The above set of answers was obtained for b > a. If b = a, we can find x = y, , which means that the hour and minute hands overlap. Therefore, in answering the question, “how many times do the two hands overlap during the course of the hour hand turning around the clock face?” We cannot simply respond with “12 times” but must consider exactly how many times and why. Furthermore, what is the case when b < a? A set of values was also obtained.

For such complex algebraic exercises, Einstein was able to provide the solution quickly and answer it in an organized way, which shows that Einstein had excellent generalization and logical reasoning ability and profundity in his logical abstraction. The development of students’ thinking profundity in arithmetic enables the cultivation of mathematical generalizations and reasoning. As mentioned earlier, the profundity of mathematical thinking is a concentrated reflection of a student’s ability to generalize specific mathematical material, abstract specific quantitative relationships, and spatial forms and consider breadth, depth, difficulty, and level of rigor in the reasoning process. In other words, students with a high level of profundity in mathematical thinking can think more comprehensively, deeply, accurately, and carefully about mathematical activities. They excel at grasping the essence, laws, and inner connections of things; are gifted at abstract generalization, classification, and reasoning; possess a high level of systematization of knowledge and skills; and can competently answer questions. These factors provide measures to cultivate students to think profoundly in mathematics. How does the profundity of students’ thinking in arithmetic develop? Apart from basic knowledge, this mainly arises from training.

Cultivation of Profundity in Elementary School Students’ Mathematical Thinking In elementary schools, teachers should cultivate students’ thinking from direct to indirect reasoning in the early and middle grades to improve the level of abstract generalization and logic.

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An experiment showed that teachers could inspire students to infer arithmetic problems by “assumption,” that is, to learn to find the “standard” or central contradiction in every mathematical system. Accordingly, teachers should guide students to gradually expand the number of “steps” in solving word problems to learn to infer and solve complex word problems in a multistep relationship. The profundity of students’ thinking in the experimental class improved significantly after three years of training. Students’ mathematical knowledge and logical thinking abilities were assessed using a district-range teaching quality test with high-difficulty, high-logical questions that required indirect reasoning. The results showed that the experimental class ranked first out of the 49 classes at the same-grade level in the district range, with an average score of 89.8. Furthermore, the class ranked second among all education nationwide or in the province, with an average score of 79 (the significance test of difference showed p < 0.01, a significant difference). The overall average of the district was only 61.3 (which also showed a significant difference from the experimental class). There were 953 third-grade students in the district, of which only 42 received perfect scores. The experimental class accounted for 41 of the students. This shows the need to cultivate profundity in thinking at the elementary school level. With reasonable measures, the proportion of logical abstraction in thinking can be effectively enhanced, and the students’ level of abstraction can be refined more rapidly.

Cultivation of Profundity in Secondary School Students’ Mathematical Thinking With the increased difficulty of secondary school mathematics, textbooks have been structured to incorporate more complex logical reasons. How can we cultivate profundity in mathematical thinking among secondary school students? I studied the experiences of Mr. Wanli Jia, a special-class mathematics teacher in Jilin, China, in the 1980s. When clarifying the definition of a mathematical term, Mr. Jia focuses on the profundity of the explanation and allows students to participate in exploratory activities to conclude. In the beginning, the teacher and students first reached a common understanding. The teacher then gradually transitions to allow the students to draw their conclusions. For example, when defining “ellipse,” the teacher first teaches students to draw several ellipses and then asks them to compare these with the definition of a circle to determine the most crucial part of the concept of an ellipse, that is, “the distance from a moving point to two fixed points is a constant.” After this, the teacher had the students summarize the definition according to the drawing process. Having built a foundation, the students can independently define a hyperbola. This training on different definitions effectively improves students’ abstract generalization abilities.

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Moreover, when explaining formulas, Mr. Jia believed it was insufficient for students to understand, memorize, and apply them. Rather, he emphasized that students should be allowed to participate in the entire reasoning process: analyzing conditions, performing derivations until the formula is derived, and analyzing its internal structure, thus improving their abstract generalization abilities. To improve the level of the profundity of students’ thinking and consider students’ acceptability, Mr. Jia trains students in three stages. First, the teacher points out the direction of the proof, clarifies the difficulties, returns to highlight the important thoughts of the proof after the certification is completed, and then asks the students questions about the constraints between various quantities. Second, only the conditions and conclusions are provided: students complete the proof process independently. Third, only the conditions were given, and students were allowed to guess the conclusion and then prove and test it. Thus, the abstractness of students’ intellectual activities gradually improves, and the profundity of their intellectual qualities becomes progressively stronger.

Flexibility of Students’ Thinking and Its Cultivation The flexibility of thinking refers to the degree of flexibility in an intellectual activity, also called “cleverness.” Flexibility is the basis of creativity, and any creative invention is inseparable from the human attribute of “cleverness.” Mathematical arithmetic is an indispensable foundation for intellectual development. Therefore, a study of flexibility is necessary. As mentioned earlier, the flexibility of mathematical thinking involves an immediate conversion of the concentrated reflection of the student’s thinking direction, the process of thinking, and the skill of thinking in students’ mathematical thinking activities. Students with a high level of flexibility in mathematical thinking exhibit fluent thinking and rich imagination. Moreover, through correlations, they master richer mathematical thinking skills to seek common ground while shelving differences. Other characteristics of such students include their ability to track a goal positively and negatively, as well as horizontally and vertically, to transform or manipulate an object or to expand/compress it in a way that allows them to select the most appropriate and reasonable answer. Over the last half-century, the topic of flexibility in thinking has attracted much attention from psychologists. This interest has led to the development of theories and experiments on the nature of creativity. The initiator of this study was an American psychologist, Guilford (1956). In a lecture he gave at Stanford University in 1969, he divided thinking into convergent and divergent. Divergent thinking, which increases the mind’s tendency toward flexibility, similar to the meaning of this word, that is, tending to be different or developing in different directions, can be regarded as a process of speculation, imagination, and creativity. Notably, this thinking process arises from the assumption that there are several correct ways of dealing with a problem. In particular, creativity may be more associated with divergent thinking; therefore, teachers should strengthen guidance to engage in divergent thinking and

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encourage students to seek more answers. Guilford also pointed out that most current classroom teachings are related to convergent thinking. Instead of employing divergent thinking to achieve higher grades flexibly, rote memorization and repetitive thinking are emphasized to achieve higher grades. The “following procedure,” “rote memorization,” and “exercise and exercise” all come from this type of convergent thinking. Even students with supernormal intelligence who receive intensive or focused convergent instruction tend to lag behind students with normal intelligence who receive divergent instruction regarding final learning outcomes, especially creativity. I have a different perspective. As mentioned earlier, convergent thinking mainly emphasizes one solution, whereas divergent thinking highlights multiple solutions. For an integrated thinking activity, one solution is the basis for multiple solutions, which represent the development of one solution. The best solution can be found using multiple solutions, which returns to “convergent” thinking. Notably, divergent and convergent thinking are important forms of human thinking; they are both indispensable prerequisites for creative thinking and the base of mathematical ability, neither of which can be ignored. Chapter 3 assessed Guilford’s ideas. This section focuses on the flexibility of thinking and the multiple solutions in mathematics. Thus, while the cultivation of convergent thinking is necessary, the development of divergent thinking plays a more important role in improving academic performance and developing intelligence. Many answers can be obtained through divergent thinking. One of the critical reasons divergent thinking can develop intelligence is that it improves the flexibility of thinking quality. For example, for the question “1 =?” 1 + 0 = 1 (thinking about addition) 100 − 99 = 1 (thinking about subtraction) 1 × 1 = 1 (thinking about multiplication) 20 ÷ 20 = 1 (thinking about division) 2 + 13 = 1 (thinking about the whole 1) 3 n 1 = 1 (thinking about multiplication) sin2 x + cos2 x = 1 (thinking about sine and cosine) tan α · cot α = 1 (thinking about tangent and cosecant) loga a = 1 (thinking about the operation of logarithms) 0! = 1, 1! = 1 (considering the factorial definition) …… From the divergent process of “1 =?” several characteristics of divergent thinking have emerged. 1. Multitermine. One problem can have various starting points, thus generating various associations and leading to various conclusions. A simple question such as “1 = ?” could lead to ten more answers, which broadens the aspects of thinking. The development of divergent thinking requires a wealth of knowledge as the basis; only with a large amount of knowledge can we consider problems from different aspects and associations to avoid one-sidedness and narrowness.

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2. Flexibility. Views on one problem can change according to the objective circumstances; that is, the original idea can be modified in time according to the new facts discovered. At first glance, seeing the question “1 =?” the first thing that comes to mind is that “1” is the first natural number. If you are sticking to that definition (“1” is “1”), what other answer can there be? However, through divergent thinking and flexibility emerges: addition, subtraction, multiplication, division, and logarithmic, among others 3. Refinement. The “1” is valuable in mathematics. To think “1” in a divergent way, you have to consider the problem in a comprehensive and detailed manner, not only to consider the problem as a whole but also to consider the more minute details of the problem, i.e., not only considering the problem itself but also other conditions related to the problem. 4. Novelty. If a group of students is asked to divergently think about “1 = ?” simultaneously, each student, given their particular characteristics, will offer different, novel, and unique responses. It is no wonder psychologists in international academic communities often associate divergent thinking with creativity! The above demonstrates that the more comprehensive the range of divergence in thinking, the more flexible the performance. The focus of cultivation on the divergence of students’ thinking in arithmetic helps to improve their thinking flexibility. In arithmetic, flexibility is manifested as a flexible starting point for calculating all types of mathematical problems from different perspectives by applying various methods. These include a flexible operation process that easily employs all kinds of mathematical concepts, theorems, and laws, learning and comprehension by analogy, and high efficiency in the calculation. Students with supernormal intelligence, or those who outperform their peers in mathematics, are generally more flexible in their intellectual activity. We ran a mathematical competition in a school, and the average score of the winners who completed the multisolution and divergent questions was 85. In contrast, normal students scored 35 or below. There are 15 students with subnormal intelligence who had rigid thinking and limited flexible thinking in their arithmetic (Lin, 1981). Mathematics learning requires the ability to utilize multiple solutions, equivalent transformations, and identical deformations in arithmetic, which are manifestations of the flexibility of thinking in arithmetic, which is also a prerequisite for flexibility. The cultivation of flexibility in mathematics teaching often starts with cultivating one’s ability to solve multiple problems. Finding multiple solutions to one problem, a traditional method of teaching mathematics in China, is a good method for cultivating thinking flexibility. Its requirements correspond with the requirements proposed by Guilford; that is, “multisolution” means “divergent.” Therefore, a multiple-solution approach should be promoted for mathematics teaching. Such an approach is also beneficial in helping students learn from each other while simultaneously maximizing the effects of problem-solving training, which are considerably better than merely completing exercises and helping reduce students’ learning burden.

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Fig. 6.1 Example in the 1979 National College Entrance Examination in Science and Mathematics

Multiple solutions are an essential feature of mathematical exercises. For example, in the 1979 National College Entrance Examination in Science and Mathematics, there were ten questions, the most difficult of which is illustrated below: Let the top angle of the isosceles triangle be ∠AO B = 2θ , as shown in Fig. 6.1. The x-axis bisects ∠AO B, and the height on the bottom side AB is h. (1) There is a moving point P inside ΔO AB, and the distances from this point to the three sides OA, OB, and AB are |PD|, |PF|, and |PE|, respectively, and satisfies the relationship |P D| · |P F| = |P E|2 . Based on this information, the trajectory of point P. (2) Determine the coordinates of point P in the above trajectory such that |P D| + |P E| = |P F|. Given that |PD|, |PF|, and |PE| are the distances from point P to OA, OB, and AB, some students easily think of applying the law of the “formula of distance from a point to the line” in analytic geometry. This idea is correct. However, there are other solutions. Yes, there are. Notably, more than 10 solutions have been explored (Gu, 1980). If high school graduates can solve similar mathematical problems, they achieve a high level of flexibility in thinking. This problem requires multiple solutions, which are common in mathematics. For example, consider the following common algebraic problem, a normal algebra problem: “a train is delayed by 6 min and travels at a speed of 16 km/h faster than the train’s regular schedule for a 20-km segment so that the train can remain on schedule. What is the speed of the train over this distance according to the train schedule?” There were more than 15 and 16 solutions, respectively. As another example, consider the common plane geometry problem: “prove that the distances from the midpoints of the hypotenuse in a right triangle to the three vertices are equivalent.” There are at least ten ways to prove this. A simple trigonometric equation, sin x + cos x = 1, also has nearly ten solutions. Thus, if we often guide students to find laws from multiple solutions and learn by analogy, will their flexibility in thinking not be greatly improved?

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As mentioned earlier, our multiple-solution approaches correspond with divergent thinking in international psychology, except that we emphasize the “multiple solutions” while paying attention to the premise of “one solution.” At the same time, we must point out that strengthening the study of multiple solutions to a problem is meaningful for improving the quality of elementary and secondary school mathematics teaching and has critical value for discovering the theory and practice of flexibility of thinking. As a multiple-solution approach is an excellent way to cultivate students’ flexibility of thinking, teachers need to develop this ability from an early age. Research has shown that from second or third grade onward, experimental classes that inspire students to compare and distinguish similarities and differences and to identify regularities through multiple solution techniques from the very beginning of their schooling result in classes exhibiting increasingly different skills from their peers in regular classes in the same grade in terms of their ability to reach multiple solutions. We tested the students’ ability in a particular grade to solve problems using multiple solutions. One of the questions was as follows: “there are 30 young pioneers in class A and 15 in class B. Please add questions based on these two conditions: the greater, the better, and the more difficult, the better.” The results of the test showed that the average score of the experimental class was 92.5, the average score of the control class was 78.3, the experimental class added, on average, 9.4 problems, and some students added up to 23. In contrast, the control class added, on average, 6.5 questions, and few people added more than ten questions (the significance test of the difference between classes shows p < 0.01, a significant difference) (Lin, 1984). This indicates that as long as the important role of training students’ ability of multiple solutions from an early age in mathematics teaching is emphasized, students can learn to adapt to variable exercises and be motivated to transfer knowledge when thinking about problems, thus promoting a significant improvement in their flexibility of thinking.

Creativity of Students’ Thinking and Its Cultivation The creativity of thinking, also called originality, can be regarded as synonymous with “creativity,” “creative thinking,” or even “innovation” for convenience. Because the topic of “intelligence and creativity” was already discussed in Chap. 3, the related issues will not be demonstrated here. As mentioned in the previous chapter, one of the most important reasons for scientific discovery and invention is the promotion of creative thinking. The creative problem in mathematics teaching is primarily concerned with the manifestation of creativity in students. In other words, a person’s ability to solve problems depends on their degree of creativity. Several psychologists regard the intellectual quality of creativity as an essential psychological factor or condition for learning. Learning can be repetitive or creative in terms of the degree of creativity. Repetitive learning refers to relying on books, limited adaptation, or following others. In stark contrast, creative learning is unconventional, non-conforming, non-routine, and innovative. Whether a person’s learning style is

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repetitive or creative is often directly related to their intelligence level. Innovation is one of the most valuable aspects of learning. Some people consider learning to be just absorbing existing knowledge rather than inventing and creating it and find that it has no relevance to innovation. I think that learning differs from scientific or academic research and requires people to get rid of the old and create new ideas, especially in mathematics. There is a story about the German mathematician Gauss (Tent, 2006). When he was a child calculating 1 + 2 + 3 + . . . + 100, he ruled out the “stupid” method of adding mechanically in a step-by-step manner and used 1 + 100 = 101, 2 + 99 = 101 . . . , 49 + 52 = 101, 50 + 51 = 101, which transforms the problem of adding different numbers into the problem of 50 multiple 101, the results of which can be quickly calculated as 5050. This was due to the emphasis on creative learning that Gauss discovered some mathematical formulas in secondary school and later became a world-renowned mathematician. How can we cultivate the intellectual quality of creativity in elementary school students’ thinking in mathematics teaching? The teachers in our experimental classes employed the following methods. First, they cultivated the habit of independent thinking as part of the learning “routine” for lower grade students; second, they guided students to compile exercise questions, especially word problems, to inspire them to break through the difficulties of independent work and to understand the interrelationship between quantities further; third, teachers encouraged students to be “creative” and to explore various methods in arithmetic and to name them after the students, such as the “× × algorithm” and “× × × × × × method” for recognition and encouragement. By the third grade, many students in the experimental class displayed increasingly significant differences in their independent thinking abilities compared to their peers of the same grade. We tested the students’ ability to formulate questions. The results showed that the average score of the experimental class was 86, while that of the other classes was 62.4 (p < 0.01, a significant difference). The students in the experimental class were able to compose multistep word problems, identify any problems, ask questions, combine multistep equations into one-step synthesis equations, and one-step synthesis equations into five-, six-, or even sevenstep word problems. Most importantly, they could think independently and solve the problems novelty, which was less common in the other classes (Lin, 1981). Creativity in middle school mathematics teaching is also valuable. Consider the word problems as an example, which reveals two common weaknesses among secondary school students. First, they formulated equations without clearly understanding the meaning of the questions. Such equations often “cannot match the theme” or even miss some provided conditions, leading to ridiculous answers. The other is understanding the meaning of the question (provided the conditions are not missing). Nonetheless, students only follow the steps of the teacher’s example for the solution and do not try to come up with other possible solutions. Although their equations’ answers were correct, they failed to meet the requirements for the intellectual qualities of creativity. The key to improving problem-solving abilities is thinking independently, analyzing carefully, and being creative. Thus, not only were the questions answered

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x liters of mixture

A liquid x liters

Cup A liquid

Cup B liquid

Fig. 6.2 Two cups of 10 L of liquid

correctly and quickly, but more importantly, the intellectual qualities of creativity were cultivated. Therefore, experienced mathematics teachers actively compile independent thinking questions for their students. An example is as follows: There are two cups, A and B. Cup A contains 10 liters of liquid a, and cup B contains 10 liters of liquid b. Now, take a certain amount of liquid from cup A, fill cup B, and stir it. Then, the same amount of mixture was removed from cup B, poured into cup A, and stirred thoroughly. The ratio of liquid a to liquid b in cup A was 5:1 (Fig. 6.2). The amount of liquid removed from cup A was determined for the first time in this study.

In general, students make an equation according to the question: let x liters of liquid a be taken out of cup A and injected into cup B, so the ratio of liquid a to the 10 x , and the ratio of liquid b to the mixture is 10+x . Then, x mixture in cup B is 10+x x liters of the mixture were removed from cup B, which contained 10+x · x liters of 10 · x liters of liquid b. Therefore, the ratio of liquid a to liquid b in liquid a and 10+x (10−x)+

x

·x

10+x cup A should now be = 51 . Solving this equation yields x = 2. That is, 10 10+x ·x the first time that the liquid was removed from cup A was 2 L. However, after comprehensively understanding the problem, some students noticed two things:

1. When liquid a from cup A was poured into cup B and then the mixture from cup B was poured into cup A, the two cups still contained 10 L of liquid. That is, the same amount of liquid b is in cup A as that of liquid a in cup B. When the ratio of liquid a to liquid b in cup A was 5:1, the ratio of liquid b to liquid a in cup B was 5:1. 2. The composition of the mixture in cup B was determined when liquid A was removed from cup A to cup B. Removing the mixture from cup B did not affect the liquid-to-liquid ratio b in the mixture. Therefore, they understood the problem as follows: how much liquid a is removed from cups A to B so that the ratio of liquid a to liquid b in cup B is 1:5.

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If x liters of liquid a are removed from cup A, the equation can be expressed as = 15 . This can be easily solved with x = 2. Professor Caihan Cao, a mathematics educator, points out that creative thinking in secondary schools is more often manifested in discovering contradictions and then integrating knowledge to break through the contradictions in an offensive posture and eventually solve problems (Cao, 1990). Consider the following example. Find the proof that x 10

n−1 Σ k=0

Σ 2kπ 2kπ = =0 sin n n k=0 n−1

cos

To answer this question, it is more complicated think purely in terms of trigonom−−→ etry. If we consider the point Ak , which corresponds to the vector O Ak , then prove that the proposition holds. We only need to prove that n−1 Σ −−→ O Ak = 0 k=0

−−→ Additionally, the vector O Ak can be considered as the force f k , and then the force system with the same size and terminal distribution on the n vertices of the regular n-sided polygon has a combined force of 0. Therefore, n−1 n−1 Σ −−→ Σ O Ak = fk = 0 k=0

k=0

is proven. The proof is omitted. This indicates that students can simplify the calculation process in various ways by employing careful analysis and thinking. The process of simplifying a calculation is a manifestation of creative intellectual quality. As long as the teacher is reasonable and actively guides and inspires students to use their existing knowledge to flexibly employ new ideas and find ways to solve problems on their own, students cannot only deeply and actively learn mathematical knowledge but also constantly improve the intellectual quality of creativity in the process of arithmetic. The development of creativity in children’s and adolescents’ mathematics is influenced by numerous factors, such as natural conditions and the nurturing environment. Therefore, different characteristics and developmental trends appear at different ages. Individual differences in creative development are clearly recognizable for different individuals. Therefore, studying creative development is a prerequisite for cultivating and nurturing creative talent. Creativity is expressed in the development of children’s movements, speech, perception, imagination, thinking, and personality characteristics. The development

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of children’s curiosity and creative imagination is the two most important manifestations of the formation and development of creativity. Generally, children can express their creativity through various activities, such as mathematics (arithmetic), language, painting, music, dance, handcrafting, and games. Among them, games, as the dominant activity for young children, have the dual function of (a) satisfying their need to participate in adult social life and practical activities and (b) enabling them to combine imagination and real life uniquely, thus leading to an important impact on their mental behavior and creative development. After entering school, the children’s thinking and imagination were further developed. In particular, voluntary imagination gradually develops into the domain position, and the imagination’s purpose, generalization, and logic become more advanced. The creativity of imagination also improves, including both reproductive imagination and creative imagination, which are characterized by originality. Over time, the originality of arithmetic thinking in elementary school students matures. Our research (Lin, 1983; Zhu & Lin, 1986) on the development and cultivation of children’s creativity in elementary school students’ mathematics learning revealed that in learning mathematics concepts, it is essential to change narrative expressions, conduct multidirectional comparisons, and use image associations. In learning calculation, the key elements are multiple solutions, step and calculation simplification, visualization of the calculation process, and estimation ability. The following are required to acquire preliminary geometric knowledge: attention to observation, hands-on operation, use of association, and knowledge. When learning how to deal with word problems, comprehensive perception, intuitive thinking, the discovery of conditions and keys, use of comparison, overcoming stereotypes, supplementary practice, segmentation and reassembly practice, expansion and contraction practice, and self-programming word problems, among others, are extremely beneficial for mastering mathematical knowledge and improving mathematical ability and important manifestations of elementary school students’ creativity. Our research found that the creativity of elementary school students’ arithmetic thinking was mainly manifested in independence, divergence, and valuable novelty. Its developmental trend is expressed in two aspects: first, in the content, from the processing of concrete image materials to the processing of abstract verbal materials; and second, from gaining independence, moving from easy to difficult, from imitation to a transitional period of semiindependence, until finally, students can engage in fully independent creative thinking. The characteristics of the physical and mental development of secondary school students during adolescence indicate that their creativity differs from that of young children, elementary school students, and adults. We (Lin, 1983; Zhu & Lin, 1986) found that compared with the creativity of preschool and elementary school children, secondary school students’ creativity has the following characteristics. 1. It is no longer illusory and detached from reality but more realistic and inspired by real problems and difficult situations. 2. It is more active and intentional, and they are able to use their creativity to solve new problems.

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3. It gradually matures. We (Zhu & Lin, 1986) observed that, in the process of mathematics learning, secondary school students’ creativity was manifested in both the flexibility and diversity of approaches in thinking about mathematical problems and in the reversibility of the reasoning process. In addition, it manifested in the ability to ask questions, propose guesses and hypotheses, and prove them when solving mathematical problems. The study of physics and chemistry, as well as science and technological activities, requires secondary school students to experiment, think about, and explore experimental phenomena, try to reveal and discover the inner laws of things, deepen their understanding of the laws by using methods of comparison and induction, and use these laws to explain phenomena and solve problems. All these are important to motivate secondary school students to explore the mysteries of nature, improve their practical operational skills, and promote the development of creativity that involves arithmetic skills.

Criticality of Students’ Thinking and Its Cultivation Criticality of thinking refers to intellectual quality in thinking activities that rigorously estimate thinking materials and meticulously examine the thinking process. As early as the 1950s, there emerged among the international psychology community a concept corresponding to critical thinking quality, now commonly known as “critical thinking.” Metacognition, self-monitoring, reflection, and other similar concepts belong to the category of criticality, which is the manifestation of selfconsciousness in thinking. In other words, critical thinking is rigorous, comprehensive, and self-reflective thinking. When employing this type of thinking to solve a problem, all available conditions should be considered, and the formulated hypothesis should be constantly verified to obtain a unique solution to the problem. Therefore, critical thinking should be considered an integral part of problem solving and creative thinking (Russell, 1956). I elaborate on critical thinking in terms of individual differences in thinking, which I call the “critical thinking quality.” It has the following five characteristics: 1. Analytics, which is in the process of thinking, constantly analyzes the conditions under which the problem is solved and repeatedly verifies the hypotheses, plans, and programs that have already been formulated. 2. Strategy, which precedes the thinking question. According to an individual’s original level of thinking, knowledge, and experience, the corresponding strategies or means of solving the problem emerge. These are then applied to solve the problem of thinking effectively. 3. Comprehensiveness means being good at objectively considering both positive and negative arguments in thinking activities, carefully grasping the progress of the subject, always adhering to the correct plan, and revising the wrong solution.

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4. Independence, which is not to be swayed by contextual hints, not following others, and not following blindly. 5. Correctness is a rigorous and organized thinking process, correct thinking results, and reasonable conclusions. As mentioned earlier, criticality in thinking results from self-consciousness in the thinking process. Self-consciousness is the highest form of human consciousness, and its maturity is an essential feature of its maturity. Self-consciousness takes the individual as the object of consciousness and is the monitoring system of the thinking structure. By monitoring the self-consciousness system, the human brain can manipulate an automatic control system that regulates the input, processing, storage, and output of the information. In this way, people can regulate their thinking and behavior by adjusting their consciousness. The self-regulation of thinking activities is manifested in individuals regulating the thinking process, revising thinking issues, and changing the approach to solving issues based on the requirements of thinking activities. Notably, a process of active self-feedback exists. As a result, the efficiency of the thinking activity was improved, the analysis of the thinking activity was developed, the initiative of thinking was enhanced, blindness and randomness were reduced, erroneous outcomes of thinking were corrected, and narrowness and inaccuracy decreased. The ancient Chinese thinker Lao Tzu said, “who understands the world is learned; who understands the self is enlightened.” The level of criticality in monitoring selfconsciousness in human thinking activities reflects an individual’s level of thinking activity. Soviet psychologists have shown that self-evaluations of children with subnormal intelligence tend to be non-critical (PybinxteЙn, 1984). Studies by American psychologists have demonstrated a strong correlation between creative thinking and self-concept. Datta conducted a test of creative thinking on a group of children who were divided into three groups according to their scores—high creativity, low creativity, and no creativity—and then measured their basic characteristics in terms of self-concept and found that subjects with high levels of selfassertion, independence, autonomy, and emotional openness possessed a high level of creativity (Yawkey, 1980). These facts show that the criticality of thinking arises from the adjustment of all aspects and links of the thinking activity. Corrected selfconsciousness, an indispensable element in creative activities and creative thinking processes, is an extremely significant thinking quality. Therefore, it is necessary to conduct research on the criticality of thinking. Professor Zhixian Zhu pointed out that while thinking psychology is mainly concerned with the process of thinking, that is, the ability to manipulate thinking, it is also related to the products and outcomes of thinking and the strategy of thinking (i.e., the control of one’s own thinking process, especially self-conscious self-control) and the criticality of thinking (Zhu, 1984). Criticality in mathematics learning reflects students’ discovery, exploration, and variation in the mathematics learning process. This quality of self-monitoring is essential for secondary school students to assume the mantra of “I do not seek the answers; I seek to understand the question.” Criticality is often manifested in the

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process of systematizing learned knowledge. Nonetheless, it also focuses on examining and regulating the thinking activities that are part of the learning process. Students should introspect how they discovered and solved problems, what basic thinking methods, skills, and techniques they have used, what roundabout way they took, what mistakes they made and why, what lessons should be learned, and so on. For example, in the section “square roots,” based on systematic knowledge, teachers should guide students to reflect on the following three points: (a) by introducing the concept of square roots and analyzing the relationship between reciprocal operations; (b) by discussing the features of square roots and summarizing how to classify them in the process of generalization; and (c) by calculating square roots and comprehending related mathematical thinking methods (e.g., the approximate approximation method). Jianyue Zhang’s doctoral dissertation (1999), “Middle School Students’ Self-monitoring Abilities in Mathematics—Structure, Development, and Influencing Factors,” centered on the progression from being controlled by others to establishing self-control, from unconsciousness and automation to consciousness and independence. It identified the gradual increase in transferability and sensitivity and the transition from mathematical examples with external monitoring to monitoring the entire process, which is sufficient to illustrate these points. When cultivating students’ critical thinking, it is vital not to turn a lesson into logical preaching in the textbook. Teachers should pay attention to the accumulation of materials that reveal students’ inspiration to enhance their mental abilities and overcome obstacles in thinking. Reflective questions should be designed in a targeted manner, and students should be encouraged to speak and participate actively in the discussion. In our experiments, to cultivate this criticality, I suggested that teachers should not only be concerned with the “reflection” section in classroom teaching but also assist students in developing the habit of monitoring their mathematical thinking at all times. To this end, teachers can require students to make reflective notes on their homework, including (a) the concepts, theorems, and laws underlying each step of derivation and calculation; (b) a brief analysis and correction of errors; (c) a summary of question types or ideas; (d) notes on problem solving; and (e) notes on other experiences. Teachers should ensure that homework is viewed on the same day and categorized and reinforced promptly, which is beneficial to developing students’ critical thinking in mathematics.

Agility of Students’ Thinking and Its Cultivation Thinking about agility is a matter of speed in intellectual activity. Daily, we often think of people who react quickly and assume they are “smart.” There is also a matter of speed training in mathematical arithmetic, which involves developing students’ ability to solve problems correctly and quickly. The difference between students with super, normal, and subnormal intelligence often manifests in the speed of mathematical arithmetic.

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In 1978, a district in Beijing selected 38 students who were thought to have supernormal intelligence for the National College Entrance Examination that year. The subjects of these examinations were mathematics and physics. The 38 students scored above 80 in mathematics and took less than two hours to complete the mathematics test. Students in the experimental classes in the first grade of secondary school were asked to complete the same examination. The average score of these students was only 35, and the average time to finish was three hours. Among these 38 students, the youngest was a 15-year-old ninth-grade student. He was asked to take a mathematical test for the selection of the district’s extracurricular mathematics activity group. He took only 50 min to complete the test and received a score of 100. A total of 20 mathematics enthusiasts in the first grade of secondary school were required to complete the same test for comparison; the results showed that they took an average of 150 min to complete only 60% of the test questions, with an average score of only 31. What about elementary school students? We counted the time spent completing a test in a mathematical competition in a district in which the top ten winners took, on average, less than half the allotted time to complete the test. An elementary student took only 22 min to complete a 90-min test. As a result, top mathematics students in elementary schools generally manifest the characteristics of fast calculation. According to the calculation time records and statistical results of supernormal intelligent individuals aged 5–18 years, their arithmetic speed is generally 13 to 21 times that of normal students. Therefore, arithmetic speed should be regarded as a difference in the level of understanding of mathematical knowledge and as a difference in arithmetic habits and generalization ability thinking. Such differences can be expressed in four situations: correct-quick, correct-slow, incorrect-quick, and incorrect-slow. We should cultivate the first type, which is a quick arithmetic ability. There are two common ways to do so: one is to always have strict speed requirements in mathematics teaching, and the other is to teach students how to calculate quickly. One of my students, Siqing Lian, conducted a wonderful experiment on the role of working memory in simple algebraic operations, in which he made a significant discovery, even though the judgment of very simple integer equations and exponential equations is related to visuospatial templates and phonological loops, meaning that the perception of visuospatial patterns and phonological loops necessitates agility (Lian et al., 2007). Speed is required in mathematics teaching (including assigning homework to students). For example, a middle school mathematics teacher brought a small blackboard to every class, and the first thing he did was to show the mathematical questions on the board and required the students to calculate them while checking their accuracy and speed in a time-limited task. In this way, he took advantage of students’ desire to “win” by conducting a competition. After more than two years of training, students’ mathematical performance and intellectual agility improved significantly. Teachers typically do not emphasize the speed in their teaching, especially in lower elementary school routines. This affects not only students’ mathematical performance but also their intellectual development. In my research, I found many cases in which children

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with a high level of thinking agility when they entered school became progressively more “sluggish” in their thinking process due to the lack of speed training, which is a lesson worth noting for teachers. To improve students’ ability to calculate quickly and accurately, teachers should teach them techniques and speed calculation methods. For example, in elementary school arithmetic, when doing addition, if among these numbers, there is a pair of complementary numbers, it can be added first. The addition of consecutive numbers can be summarized as the first term plus the last term and then multiplied by half the number of terms; multiplying or dividing by 5, 25, 125, 625 … can be calculated using 1 divided by 5 as 0.2. There is a “practice that makes perfect” process in mastering and applying quick calculation methods. Accomplishing not only improves mathematical knowledge but also promotes the development of intellectual quality. Our study showed that the teachers in the experimental class not only taught students various types of fast calculation methods but also encouraged them to create their own methods, actively sharing the reasonable fast calculation methods they found. Consequently, they gradually developed the habit of calculating quickly. After two or three years, the difference between students in the experimental class and those in other classes at the same-grade level in terms of correct and quick arithmetic ability became more obvious. In a third-grade quick calculation competition, students in the experimental class had an average calculation time of 8 min and 37 s and an average score of 98.9, while students in other classes had an average calculation time of 14 min and an average score of approximately 90. (The significance test of the difference between the two classes in calculation time was p < 0.01, indicating a significant difference.) This showed that the students in the experimental class, due to proper teaching methods, were more accurate in their calculations than their same-grade peers, with a statistically significant difference in the speed of calculation. It can be seen that the cultivation of the agility of intelligence in arithmetic is achievable; as long as certain reasonable measures are taken and students are required to practice consistently, their thinking process in arithmetic will gradually become more agile more responsive and faster when performing calculations. In particular, I would like to point out that while agility of thinking can be improved through the training of “correct and quick arithmetic ability,” it should be noted that agility is not a form of independent thinking or intellectual activity but is derived from or determined by other thinking qualities. Improvement in the level of profundity, flexibility, creativity, and criticality of arithmetic thinking is also an important measure for promoting the development of correct and rapid arithmetic ability.

Importance of Studying Students’ Thinking Qualities I learned in the course of studying the thinking quality or intellectual quality in students’ arithmetic that the study of the development and cultivation of thinking quality is an important aspect of thinking development, psychological research, and educational research. This led to the following findings:

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1. Current research on the development of students’ intelligence and abilities involves a wide range, but the focus should be on the development of students’ thinking abilities. Thinking quality is the manifestation of thinking ability, and different thinking qualities manifest as different thinking abilities. Because of differences in intelligence and ability, thinking quality is the most critical difference. All research on the development of intelligence and thinking ability is based on the individual, seeking to study the improvement of individual thinking ability and changes in interindividual differences. Studies on the development of elementary and secondary school students in terms of mathematical concepts, reasoning, problem solving, and understanding and studies that seek to distinguish their levels of thinking ability and intelligence are inseparable from the depth, breadth, speed, degree of flexibility, degree of abstraction, degree of criticism, and degree of creativity of thinking, that is, the five thinking qualities. The thinking quality manifests as thinking ability, and the study of the thinking quality of elementary and secondary school students and young children reveals the development of their thinking ability. 2. In studies on thinking or the development of thinking, it is always a difficult issue to formulate and find objective indicators. Piaget’s experimental research method for thinking and thinking development is a major breakthrough worth learning from. At the same time, I have seen in my research that objective indicators of thinking quality for elementary and secondary school students and young children in teaching environments or in daily life are easy to determine. The differential performance of agility, flexibility, profundity, creativity, and criticality can be documented using objective methods, which can serve as indicators of thinking quality. Therefore, it is possible to explore some characteristics of the development of thinking in elementary and secondary school students and young children based on a study of the development and cultivation of thinking qualities. 3. The study of the thinking development of elementary and secondary school students and young children is inseparable from the Discussion of the Issue of “Education and Development.” Traditional teaching has many weaknesses, such as the “one step at a time” and “one size fits all” approaches. The differences in thinking, intelligence, and abilities between elementary and secondary school students and young children are difficult to identify, often making students passive learners. Many international psychologists, such as Dewey, Zemkov, and Bruner, have worked on the development of thinking, intelligence, and ability while also committing to reforming traditional teaching. My research found that the development and cultivation of thinking qualities are conducive to further exploring the potential of thinking and intelligence in elementary and secondary school students and young children. 4. The level of development of thinking quality is an indication that distinguishes elementary and secondary school students and young children as having normal, subnormal, or supernormal intelligence. My research has shown that elementary, middle school, and high school students and young children with supernormal intelligence are quick to react, flexible in thinking, profound in understanding,

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able to grasp the essence of things, and creative in solving problems. Elementary and secondary school students with subnormal intelligence and young children are slow and dull in their thinking, can only recognize the superficial phenomena of things, and lack unique and novel viewpoints with social meanings. The study of thinking quality is of great significance for discovering the supernormal and subnormal intelligence of elementary and secondary school students and young children, studying their thinking and intelligence, and formulating targeted training. It should be pointed out that the teachers of the experimental classes at our sites in 26 provinces, autonomous regions, and municipalities directly under the central government take “improving the quality of teaching and reducing the excessive burden of students” as a starting point, and they never work overtime or give students extra homework. Most students, except those with extremely poor grades, can complete their mathematics homework at school. Our practice indicates that suitable and reasonable educational measures can foster the quality of arithmetic thinking among elementary school students and promote their learning performance in mathematics. In this way, they can learn faster and more flexibly; in other words, it can improve the quality of elementary school mathematics teaching. In my opinion, the ways and results of developing thinking quality in mathematical arithmetic described above are an example of thinking quality development, but thinking quality is by no means a special ability that can be developed only in mathematical arithmetic. The development of thinking quality has the feature of universality, meaning that it can be cultivated in any subject, such as language, physics, chemistry, biology, or foreign language. Our study confirms this finding. Therefore, our experiments in basic education in China are called experimental research on “thinking quality development.” I firmly believe that developing thinking quality is the breakthrough point for developing thinking ability and the best way to improve teaching quality and reduce student burden.

References Cao, C. H. (1990). Introduction to middle school mathematics teaching. Beijing Normal University Press. (in Chinese). Gu, Y. S. (1980). How to think? Science Enthusiast, (2). (in Chinese). Guilford, J. P. (1956). The structure of intellect. Psychological Bulletin, 53(4), 267. Lian, S. Q., Zhang, H. S., & Lin, C. D. (2007). The role of working memory in the verification of simple monomial addition equations. Psychological Science, 2, 281–284. (in Chinese). Lin, C. D. (1981). The study on the development of the number concept and operational ability in schoolchildren. Acta Psychologica Sinica, 3, 289–298. (in Chinese). Lin, C. D. (1983). Psychology of middle school students. Beijing Publishing House. (in Chinese). Lin, C. D. (1984). Education and child psychological development-experimental summary on the cultivation of operational thinking quality in primary school students. Journal of Beijing Normal University, (01). (in Chinese). PybinxteЙn, C. X. (1984). Psychology of mentally retarded students (Y. Piao., Trans.). People’s Education Press.

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Russell, D. H. (1956). Children’s thinking. Ginn & Company. Tent, M. B. (2006). The prince of mathematics: Carl Friedrich Gauss. AK Peters/CRC Press. Yawkey, T. D. (1980). The self-concept of the young child (pp. 151–159). Brigham Young University Press. Zhang, J. Y. (1999). Secondary school students’ self-monitoring abilities in mathematics—Structure, development and influencing factors [Doctoral dissertation, Beijing Normal University]. National Digital Library of China. Zhu, Z. X. (1984). Some basic problems about the study of thinking psychology. Journal of Beijing Normal University, 1, 1–7. (in Chinese). Zhou, Z., Peverly, S. T., Boehm, A. E., & Chongde, L. (2000). American and Chinese children’s understanding of distance, time, and speed interrelations. Cognitive Development, 15(2), 215– 240. Zhu, Z. X., & Lin, C. D. (1986). Psychology of mental development. Beijing Normal University Press. (in Chinese).

Part III

Development of Mathematical Abilities in Children and Adolescents

All of us have mathematical abilities. However, when does mathematical ability emerge, and how can it be developed? This part discusses these questions and seeks to answer them. In infancy, the emergence of “mathematical ability” follows the appearance of intelligence. For example, at eight or nine months of age, children tend to choose the larger one or the one with more; this “mathematical ability” starts to develop gradually after the age of 2 years. This “mathematical ability” lays the foundation for developing mathematical intelligence in early childhood “mathematics teaching.” In elementary school, students’ mathematical abilities start to improve rapidly, maintaining the same trend as the intellectual and thinking development that characterizes elementary school years. Students develop a new level of mathematical knowledge, skills, and culture at the secondary school level in line with their maturing intellectual and logical abstraction. Thus, secondary school students (especially high school students) already exhibit some mathematical abilities characteristic of adults. Children and adults exhibit age-specific characteristics in their development of mathematical talent, which is primarily governed by a combination of the quality of mathematics education and significant individual differences. Therefore, teaching that matches an individual’s aptitude should be the most fundamental principle in teaching mathematics.

Chapter 7

Preschool Children’s Arithmetic Thinking Ability and Early Education of Mathematics

The psychological development of preschool children is one of the most important research topics in child psychology. Revealing the age characteristics of intellectual development at this stage is a starting point for determining early education. Experiments have shown that if children do not receive sufficient appropriate learning opportunities, some of their learning abilities will decline with age (Kohlberg, 1968). In contrast, the effectiveness of learning for children who are provided proper training immediately after birth nearly doubles after three months compared to that of untrained children of the same age. This suggests that early education positively affects later learning and that learning abilities are impacted by use and disuse. Some studies have also shown that it can improve intelligence so that over time, children with subnormal and normal intelligence can develop near supernormal intelligence (Lazar et al., 1982; Protzko, 2015). How preschool children’s intelligence develops, how to assess their level of development of number concepts and arithmetic abilities, how to promote their intellectual development through early mathematics “teaching,” and so on are all new topics in the field of child and adolescent psychology. I conducted a study of 1005 preschoolers from eight weeks of birth to seven years of age (0–7 years) and obtained a relatively objective result (Lin, 1981). The analysis of preschool children’s arithmetic thinking abilities must be situated in the framework of preschool children’s thinking or intellectual development because arithmetic thinking abilities should be considered an integral part or component of children’s thinking and intelligence.

© China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_7

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The Characteristics of Thinking and the Developmental Profile of Arithmetic Thinking Abilities Children aged 0–6 years, distinguished by developmental stages, include the nursing stage (0–1 year), infancy stage (1–3 years), and early childhood (preschool, 3– 6 years), the first two of which can also be referred to as infancy. Children in infancy are dominated by intuitive-action thinking, whereas children in early childhood are dominated by concrete image thinking as the main form of intellectual expression. We regard infancy as the stage of occurrence or emergence of children’s thinking. Generally, infants under the age of one have only perceptions of things and basically possess no thinking abilities; after the age of one, in the process of infants’ activities, based on their image and verbal development and due to the accumulation of experience, infants begin to emerge with a certain amount of “generalized” thinking activity. Before the age of three, children’s thinking can basically be categorized as “intuitive action,” that is, the coordination of “perception” and “action,” as emphasized by Piaget (1977). The emergence of this kind of thinking at this point in human development is highly significant. It implies the real beginning of the intellectual activity, which entails both intelligence and the foundation of mental freedom (or autonomy). Based on the level of infantile thinking under the new life status, the child’s thinking gradually develops under the premise of verbal development, and concrete image thinking emerges. Thinking involving concrete images indicates that children’s thought processes mainly materialize via the manifestation of concrete images or representations of things, that is, by the associations of concrete images, but not by the understanding of the inner essence of things and relationships, that is, concepts, judgments, and reasoning. For example, a child can correctly answer the question “If you have six apples, how can you share them equally among two people?” but does not know how to calculate 3 + 3. When a child notices an alarm clock ticking every day, he or she will guess that there may be a little person inside who is pushing the hands around and will even open it up to see what it has inside. Young children generally like fairy tale books and cartoons, which is related to the fact that they have to rely on vivid and concrete images to understand the stories. The nature of the concrete images in young children’s thinking also derives from the nature of experience, superficiality, and anthropomorphism in young children’s thinking. These characteristics of young children’s thinking are inseparable from their poor knowledge and experience and the intuitive image activity system. However, in early childhood, the abstract logic of thinking begins to emerge, and the role of language in the development of young children’s thinking increases. Abstract generality of thought and the conscious regulation of action are two basic features of human consciousness. Although those in the early childhood stage are still quite far from acquiring these two features, these features start to become more prominent. The arithmetic thinking ability of children aged 0–6 years, especially the generation and development of the number concept, is predicated on the above characteristics and manifests as a form of preliminary concept acquisition.

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Preschool Children’s Acquisition of Preliminary Concepts Concept formation and concept acquisition are two different processes. The former refers to the historical evolution of a concept from nothing to something—concepts are formed and developed gradually as human societies develop. In contrast, the latter refers to an individual’s acquisition of pre-existing concepts, that is, those notions that have already been formed in society. Adults use language instruments to impart concepts to children through a combination of verbal communication and educational approaches. A child’s acquisition of a concept is not always accomplished at its first introduction; rather, a concept is continually enriched and modified according to the child’s increased knowledge and experience, which is in line with the development of his or her level of thinking. Therefore, the depth and breadth of the same concept are different for each child. Moreover, the same is true for the same child at different stages of his/her personal development. As noted, this process of concept acquisition is also the process of developing from concrete image thinking to abstract logic thinking. Children’s level of generalization ability directly constrains the characteristics of their acquisition of concepts. Specifically, the features of young children’s generalizations are manifested in the following three ways: 1. The content of generalization is relatively poor; a word initially represents only the characteristics of a specific thing(s) instead of the common characteristics of a class of things. The content generalized by a concept is gradually enriched in late childhood. 2. Many of the generalized features are external and non-essential, such as the concept that young children mostly characterize things by their functionality. 3. The connotations of generalizations are often imprecise; sometimes too broad and sometimes too narrow. It is because of these characteristics that children in early childhood’s understanding of the breadth and depth of the concept are very poor—they can generally only master the concept of concrete objects and find it difficult to grasp some of the more complex abstract concepts concerning nature, relationships, and morals. It is only in late childhood that children may acquire some more abstract concepts, such as beasts, animals, furniture, and bravery. The general developmental process for children’s mastery of concepts related to physical matter is as follows. 1. At the infantile stage, the content of physical concepts basically represents one or several of the things with which children are familiar. 2. As toddlers, children are able to point out the more distinguishing features of certain objects at the generalization level, especially the functional features. 3. Preschool-age children are able to identify the sum of several features of an object, but they are limited to certain external and internal features of familiar things and cannot clearly distinguish between essential and non-essential features.

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With proper education, it is possible for preschool children to master the essential features of an object concept, such as “a horse is an animal,” but this depends on whether the objects are familiar to the children and whether they have acquired a broad enough lexicon to express abstract generalizations. When learning concepts, classification, which is based on the essential properties of things, can be made once the meaning of a thing or phenomenon is fully understood. Through categorization, children can gradually grasp the conceptual system. Categorization is also one of the common methods used in psychology to study children’s conceptual levels. Materials for studying children’s conceptual categories usually employ objects or figurative materials (Zhou et al., 2003). The Soviet psychologist Vygotsky (2012) conducted an exercise using some “experimental blocks” (geometric objects) of different sizes, colors, and shapes. These blocks were presented to children, who were then instructed to group them. Vygotsky’s (2012) results revealed that children over six years of age considered only single properties (e.g., color or shape) as necessary and sufficient for classification, while children under six years of age constantly changed the criteria, using shape at one moment and color or size at another as the basis for classification. Vygotsky called this phenomenon the “chain concept.” In later research, Piaget and others further explored children’s object classification and noted that young children do not use taxonomic methods but rather a thematic form of classification (Inhelder & Piaget, 2013). For example, a toy cat and a chair were grouped together on the grounds that the cat liked to sit on the chair. Researchers thus suggested that children’s concept development goes through three stages: (a) a thematic concept; (b) a chain concept; and (c) a true concept based on stable, salient criteria. Chinese psychologists such as Liu et al. (1963) used picture materials to study the development of category concepts in children aged 4–9. The content of each picture could be divided into first-level and second-level concepts, and children were asked to classify them, as shown in Fig. 7.1. The results show that the developmental process of children’s categorization occurs in the following order: unable to categorize → categorize according to perceptual characteristics → categorize according to life contextual → categorize according to function → categorize according to the concept (Liu et al., 1963). Approximately 82.3% of children under the age of four could not categorize the image at all, while

Fig. 7.1 Concept map example

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children aged 6–7 could gradually categorize it according to function and essential characteristics. This indicates that children’s abstract generalization ability starts to develop around the age of six (Liu et al., 1963; Wang et al., 1964). This result is basically consistent with the results of some domestic and international studies. The number concept is more abstract compared with the object concept; thus, it emerges later and is more difficult to master in the course of children’s development. Accordingly, parents and kindergarten teachers are concerned about the age at which children can learn to count and how difficult it is to teach them arithmetic.

Development of Preschool Children’s Number Concepts Preschoolers’ acquisition of preliminary number concepts is a complex process. Being able to count is not the same as mastering the number concepts, and counting orally is only the first step. The second step is to “count objects and tell.” The adult asks the child to “count,” and the child begins to count the objects. After finishing counting, the child tells the total he/she has come up with. The third step is “taking objects by number,” in which the child takes objects after hearing a certain number. The fourth step is the acquisition of the number concept. Acquisition of number concepts includes progressively understanding the following: (a) the actual meaning of a number (e.g., “3” means 3 objects); (b) the order of a number (e.g., 2 comes before 3, 3 is after 2, 2 is smaller than 3, 3 is larger than 2); and (c) the composition of number (e.g., “3” is made up of “1 + 1 + 1” or “1 + 2”). In summary, children’s formation of number concepts goes through four developmental stages: counting orally → counting objects and telling → taking the object by the number → acquisition of the number concept. Table 7.1 shows the data of the study on this developmental trend (Lin, 1980). As presented in Table 7.1, the following observations can be made: 1. The formation and development of children’s number concept have obvious age characteristics: from 1.5 to 2 years old, children begin to use the number phrase “1 and 1” to count numbers, e.g., “1,” “2,” “3,” and gradually enter the level of oral counting. From 2 to 2.5 years old, children can not only count up to “5” but also name 2 or 3 by counting objects or taking 2 or 3 objects when instructed to remove a given number. At this stage, they begin to grasp the concept of counting within 2. On this basis, children’s counting ability develops gradually, and by the age of 5–6, most children can count up to “100” orally (the average number in Table 7.1 is “88”) as well as say the number of objects and take the correct amount of objects according to the number. The ability to count and tell and name objects by number is also close to “100” (84 and 80, respectively, in Table 7.1). Children’s level of acquisition of number concepts before the age of 7 is as follows: children aged 2–3 years can master “2”; children aged 3–4 years can master “5”; children aged 4–5 years can master “11”; children aged 5–6 years

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Table 7.1 Development distribution of number concept table Age

Achieved number Count orally

Count objects and tell

Take the object by number

Acquisition of the number concept

8 weeks–0.5 yr 0.5–1 yr 1–1.5 yr 1.5–2 yr

1

0

0

0

2–2.5 yr

4

2

2

1

2.5–3 yr

9

5

4

2

3–4 yr

19

15

9

5

4–5 yr

50

39

34

11+

5–6 yr

88

84

80

23+

6–7 yr

97

92

87

29+

Note The test of significance, p < 0.01 (no significant difference)

2.

3.

4.

5.

can master “23”; and children aged 6–7 years can master “29” (in fact, more than half of the children in this age group can master the number concepts up to “50”). Ages 2–3 years and 5–6 years are two critical periods in the formation and development of number concepts in children. The former is a leap from perceiving things to the budding of number concepts, that is, from “blank” to the emergence of counting ability, and the latter is a leap in the formation and development of number concepts in preschool children. The emergence of counting ability is not equal to the formation of number concept, and the degree of counting is not equal to the degree of acquisition of number concept because the concept reflects the essential properties of things and is more abstract. Therefore, moving from counting to the formation of number concepts requires the subject to go through a complex learning process. In the process of developing number concepts, children do not take the same amount of time to acquire each number concept. For example, children quickly acquire the number concepts of “1” and “2,” while the transition from “2” to “3” takes almost twice as long (on average, more than half a year). Mastering the numbers “11” to “23” (13 numbers) may take one year, while the transition time from “23” to “29” (only 7 numbers) may also take one year. Thus, it can be seen that children acquire the number concept of “10” to “20” much faster than “20” to “30.” There are individual differences in the formation and development of number concepts, and these differences increase with age.

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Development of Preschool Children’s Arithmetic Abilities As preschoolers develop their number concepts, children’s preliminary arithmetic abilities are actively developing. In general, the formation and development of children’s arithmetic abilities are characterized by distinct age characteristics. Preliminary arithmetic begins after age 2, but children aged 2–2.5 can only do simple operations (add and subtract within 3–4), and the vast majority of children rely on real objects to do so. Most children aged 5–6 years can perform verbal or written operations up to “20” without objects. A few children are beginning to master addition and subtraction operations within “50,” and very few can use numbers above “100” to perform addition and subtraction exercises, as well as simple multiplication exercises. In the preschool years, ages of 2–3 and 5–6 are two critical ages and significant turning points in the development of children’s arithmetic abilities. In the development of preschool children’s arithmetic abilities, there is a clear difference between relying on real objects and not relying on real objects for operations from “1 to 20.” For arithmetic operations with numbers above “20,” children often have no real objects to rely on and can essentially only perform direct operations without relying on real objects. In the development of arithmetic abilities, different children show individual differences that increase with age. The relationship between the formation of number concepts and the development of arithmetic abilities in children aged 0–6 years shows that the formation and development of children’s arithmetic abilities are consistent with the formation and development of number concepts at this stage. Children move from learning to recognize numbers to learning to use numbers, which is the universal development trend. From this trend, it can be assumed that the development of children’s thinking is a form of progressive learning in which the student moves from recognition to application. Of course, there are a few children who can operate add, subtract, and even multiply and divide numbers above “100,” which often exceeds the number concepts they have mastered. In this case, they can perform arithmetic, but they cannot understand the actual meaning of the numbers. The reason for this is twofold: first, children’s number concepts and arithmetic abilities develop to a certain degree and then show differences, and second, children’s acquisition of number concepts is based on the images of numbers.

Development of Preschool Children’s Ability to Recognize Shapes To better study the formation and development of children’s number concepts, I investigated children’s knowledge of different geometric shapes from 0 to 7 years old, and Table 7.2 showed the results of the study (Lin, 1978).

20%

18.7%

40.1%

49.4%

5.7%

40.1%

50.6%

92.5%

100%

3–4 yr

4–5 yr

5–6 yr

6–7 yr

Able to tell “square”

2.5–3 yr

Able to tell “four-sided”

2–2.5 yr

1.5–2 yr

1–1.5 yr

0.5–1 yr

8 weeks to 0.5 yr

Age

Table 7.2 Survey results

100%

95%

75%

55%

15%

Able to tell “round”

37%

25%

40.1%

18.8%

21.3%

Able to tell “circle”

100%

88.8%

52.2%

50%

15%

Able to tell “three angles”

11.2%

47.8%

34%

16.2%

21.3%

Able to tell “triangle”

61.4%

49%

9.5%

Able to tell “trapezoidal”

10%

25%

30%

Able to compile name

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Children’s understanding of the names of plane geometric figures develops from prescientific concepts in their daily lives to scientific concepts: from “four-sided” to “square,” from “round” to “circle (shape),” from “three angles” to “triangle,” and from making up their own names (according to common sense) to “trapezoidal.” These changes reflect the development process of children’s thinking from a visual image to a linguistic abstraction under certain educational conditions. There are differences in the degree of children’s awareness of different shapes; for example, they recognize “circles” at an early stage but have greater difficulty comprehending “trapezoids.” This reflects the closeness and frequency at which children experience these shapes in their daily lives and their own level of abstraction. Importantly, it shows that the formation of children’s number concepts must be based on their perceptual life experiences. Children’s knowledge of plane geometric figures has a process: children before the age of 2 can identify the size of objects, and 2–3-year-olds begin to call out the prescientific concepts (names) of individual figures, thus reflecting their ability to differentiate these figures from other objects through language. In normal educational settings, more than half of children aged 4–5 years can name mathematical shapes (scientific concepts) such as squares, triangles, and circles. After the age of 5, they basically recognize the concept of these shapes. A comparison of children’s acquisition of number concepts and the development of their arithmetic abilities reveals a consistency in the formation and development of number and shape concepts in children aged 0–7 years. Although there are some differences between them, the overall consistency of the development trends of the two number concepts and the stability of their age characteristics, as expressed at the level of thinking activities, is clear.

The Development of Level of Thinking Activity in Acquisition of Number Concepts After exploring the characteristics of 0–6-year-old children’s number concept acquisition and arithmetic abilities, the characteristics of their deeper thinking or cognitive activities should be discussed.

Related International Academic Communities and Domestic Perspectives I have categorized the problem of children’s acquisition of number concepts for ages 0–6 as the field of children’s thinking or cognitive development centers on this period of human development. As demonstrated in authoritative international studies on children’s thinking, Chinese research on the development of number concepts in

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preschool children, and the Guidelines for Kindergarten Education (for trial implementation) promulgated by the Ministry of Education in 2001 (hereinafter referred to as “the 2001 Guidelines”),1 we can derive the following: Piaget’s Idea of Conservation. In the international psychological community or academic communities, when talking about the level of thinking activity of children aged 0–7 years (or 0–8 years), it is often necessary to refer to Piaget’s “conservation” study. The term “conservation” was introduced by Piaget and Szeminska (1952) to denote the understanding that the content of matter remains constant when it changes from one form to another. According to Piaget, children’s thinking in the preoperational stage can only focus on one dimension of the problem, and children pay attention to the superficial and apparent features of things, which are characterized by centrality. To prove this, he designed a series of conservation experiments (see Fig. 7.2). For example, in the liquid conservation experiment (see Fig. 7.2), children are presented with two identical glasses containing equal amounts of liquid. After children are assured that the amount of liquid in the two glasses is equal, the experimenter pours one of the glasses into a taller, thinner glass next to it, and the liquid level naturally rises. The child is then asked, “Is the liquid in the new cup more or less than the liquid in the original cup, or is it the same amount?” Most 3–4-year-olds will answer “more” because they only notice the height of the new cup. In contrast, 5–6-yearolds are at the transition stage of conservation and seem to realize that they must consider both the height and the thickness of the cup; however, when comparing, it is difficult for them to consider both dimensions at the same time. According to Piaget, children generally comprehend the concept of conservation at around age 8. At this point, children are able to realize that a change in one dimension is always accompanied by a change in the other dimension, which they demonstrate through their understanding of the logical relationships in terms of identity, compensation, or reversibility. Some children still make the same mistakes in experiments of conservation of quantity. However, some recent studies have argued that Piaget underestimated children’s abilities (Moll & Meltzoff, 2011; Viarouge et al., 2019). For example, in a repeated experiment on the conservation of quantity, only a minority (16%) of 4–6year-olds understood the conservation of quantity. However, shortly afterward, the experimenter borrowed a bear toy and had it come out of its box to arrange buttons in a line, and 63% of the children said the number of objects did not change. Gelman and Baillargeon (1983) tested 3–4-year-olds with smaller numbers and found that they were aware of the one-to-one correspondence of numbers and the conservation of quantity. However, if the number was increased, 6–7-year-olds could not achieve the state of conservation (Cowan, 1987).

1

Guidelines for Kindergarten Education (for trial implementation): This guideline was already applied officially when this book was published.

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Children are presented with two identical glasses containing equal amounts of liquid.

The experimenter pours one of the glasses into a taller, thinner glass next to it.

Children are presented with two identical mud balls.

The experimenter flattens one of the mud balls.

Children are presented with two arrange of buttons.

The experimenter creates a new arrangement of buttons.

Fig. 7.2 Illustration of three conservation problems that measure children’s concrete image thinking

The Spirit of the Ministry of Education’s 2001 Guidelines. The 2001 Guidelines were the curriculum standard for kindergarten. It contains only rough descriptions and regulations of the targets and contents of each learning area for preschool children, unlike the “Kindergarten Education Outline (Trial Draft)” promulgated by the Ministry of Education in 1981, in which detailed requirements were created for each subject and each age class (infant, toddler, and preschool classes). The reason for this change may be the concern that overly detailed regulations would cause a tendency to turn kindergarten education into a format resembling elementary school. In this government document, mathematics is not presented as a separate subject but rather as part of the science domain, thus emphasizing the “cognitive development field” and “mathematical thinking” and highlighting thinking activities in the area of number concepts and operations that children can acquire (Ministry of Education of the People’s Republic of China, 2001).

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There are seven criteria for this type of thinking activity. Standard 1: children have a preliminary understanding of numbers and the relationships between numbers. Standard 2: children have a preliminary understanding of the meaning of addition and subtraction operations. Standard 3: children are able to distinguish differences in quantities. Standard 4: children have a preliminary understanding of spatial relationships and geometric concepts. Standard 5: children are able to correspond, order, and classify. Standard 6: children are able to discover, imitate, and create patterns through observation and reasoning. Standard 7: children are able to apply mathematical methods to collect, organize, and express (a previous section does not diverge much from these seven standards).

What I greatly appreciate about these seven standards is that they have rules or specific content, which is helpful in guiding kindergarten education and teaching. Although the 2001 Framework does not list mathematics as a separate subject, it still emphasizes mathematical thinking and its own rigorous disciplinary system.

My Research The level of thinking activity demonstrated by preschool children in the acquisition of number concepts and the development of arithmetic abilities are a reflection of their thinking ability and intelligence level, which can be divided into four levels. Intuitive-action Perceptual Generalization. When children observe objects, they are able to distinguish size and quantity. For example, during one of the experiments, children were observed to ask for a large number of sugar cubes and larger apples, and they were happy if given them and unhappy if not. Children at this level are not yet able to speak, and their level of intuitive action “thinking” is fairly low and very general. We can only determine the emergence of children’s ability to perceive size and magnitude by their emotional responses. Intuitive-image General Generalization. In the study, we set up different numbers of sugar cubes. When asked how many there were, children could use their fingers, nod their heads, or say yes. Children at this level already possess the basic number concepts, but it is inseparable from concrete things. For example, while they begin to understand “one apple” and “two pieces of candy” and can say phrases such as “a lot of candy,” what these words represent is very general and undifferentiated. The number of objects is not yet differentiated from the perception of the set of objects (Lin, 1980). Intuitive-speech Number Generalization. Children at this level are able to count orally and take objects by number. However, they have four distinctive features in their thinking when forming preliminary number concepts.

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1. They must be based on the true objects; removing the objects in arithmetic often leads to interruptions in thinking. 2. Number words are often followed by “measure words,” especially in arithmetic, such as “1 (cat),” “2 (cars),” “3 (apples), etc.” 3. The actual meaning represented by mathematical words often cannot exceed the reality of children’s lives, such as “10” fingers or “20” toys, and number words that exceed reality are often meaningless. 4. It is not possible to produce the image of a number group (the simplest combination of decomposition). Image-speech Number Generalization. From this level, children begin to understand the actual meaning, order, and composition of numbers, but these are still inseparable from concrete images and from life experiences in arithmetic; they can successfully perform operations up to “20” without relying on objects, and some can calculate numbers of “50,” “100,” or more, or even thousands of thousands of numbers. Children at this level can also remember and apply the steps of addition, subtraction, and even multiplication and division, but they cannot understand the actual meaning of these numbers. I plotted the data obtained in the study into four curves (see Fig. 7.3) as a way to illustrate the trends in children’s thinking activities from 0 to 7 years old (Lin, 1980). From Fig. 7.3, we can derive the following observations: 1. There are significant age characteristics in children’s thinking activity and development. The stage of perception is 0–2 years old, while 2–7 years old is the stage where the main focus is on image or concrete image thinking, which gradually develops from image to preliminary abstraction. In the development of number concepts and operations, the 5-year period from 2–7 years can be divided into three levels of thinking activity: (a) 2–3 years is the intuitive-image generalization stage; (b) 3–5 years is the intuitive-speech number generalization stage;

Age (y)

Fig. 7.3 Curves of the developmental trend of children’s thinking activities between the ages of 0 and 7

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and (c) the age of 5 is the stage of image-speech number generalization. At this time, although children’s thinking is mainly based on image, they can already generalize the actual meaning of numbers, understand the order of numbers, and especially understand the composition of numbers—number groups (decomposition and combination of numbers)—and perform operations without relying on intuition. Undoubtedly, children’s thinking at this stage has begun to abstract and initially generalize, taking a significant step toward the level of logical abstraction. 2. From birth, children progressively form number concepts and perform operations. At 8 months of age, children begin to perceive the size and number of things. At approximately the age of 2, children begin to make intuitive-image generalizations about the number of things. With the development of children’s daily practical activities and language, the complex connection between two signaling systems, intuition and language, begins to communicate and form. By the age of 3–5 years, children can differentiate based on a concrete collection of objects to abstract the number of objects, that is, the ability to count, which includes counting orally, counting by objects, saying the total number, and taking objects by numbers, gradually develops. At the age of 5, their development progresses from forming images of numbers to forming concepts of numbers, but children’s concepts of numbers are still concrete and figurative until the age of 7. As mentioned above, children gradually move from recognizing and mastering numbers to using numbers and performing number operations. 3. Although there are individual differences in thinking activities in the development of number concepts and operations, in general, children’s first seven years of life are when they are in the process of forming and developing number concepts. Three critical ages for the development of activity levels are (a) from eight or nine months; (b) from age two to three (mainly); and (c) from age five to six. Focusing on critical ages, setting requirements according to children’s existing levels, and providing early education in a timely manner can assist in promoting the formation of number concepts and the improvement of arithmetic abilities, as well as the development of children’s intelligence.

Recent Research on the Development of Mathematical Abilities in Children Aged 0–6 Years Recent international research on the development of mathematical abilities of children aged 0–6 years has increased. Likewise, our team has invested in relevant research and achieved some results.

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Development of Quantitative Representations Quantitative representations contain both symbolic and non-symbolic representation systems. Among them, non-symbolic quantitative representation refers to the process of expressing and operating without relying on symbolic knowledge of physical quantitative stimuli presented by sensory channels such as sight and sound; its development does not depend on language and has some innate properties (Mao et al., 2021). Non-symbolic quantitative representations are the most important element in the field of mathematical cognitive research in infants and young children. Research from comparative psychology has shown that all forms of life—from insects to primates—are capable of non-symbolic number representation (Brannon & Terrace, 2002). Moreover, research from cognitive developmental psychology has shown that humans possess the ability to represent non-symbolic numbers from birth with precise discrete representations of small numbers (up to 3 or 4) and approximate continuous representations of large numbers (Carey, 2009). Regarding discrete representations of small numbers, my student, Prof. Yinghe Chen, and several colleagues conducted research focused on the role of the influence of perceptual cues (Lai et al., 2012). The study used a number comparison task in which children aged 3–6 years were presented with a pair of patterns containing anywhere between one and four elements and were asked to determine which pattern contained more elements (see Fig. 7.4). The visual elements used by Lai et al. (2012) varied in size and shape dimensions. The findings suggest that the cumulative area of these visual elements affects the correct rate of children’s number comparisons, and when it creates a conflict with the actual number (e.g., the total area of three elements is larger than that of four elements), children’s correct rate of comparison decreases. This suggests that children’s representations of small numbers are not always accurate and that visual perceptual cues affect their representations of discrete amounts of small numbers.

(a) Fig. 7.4 Questions used in small-quantity comparative studies of children

(b)

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In the earliest stages of mathematical development, children’s representations of large numbers (> 4) are approximate. In the preschool period, this representation gradually becomes more precise. The transition from approximate to precise representations of large numbers, that is, the formation of linear representations of quantities, is a critical step in the development of children’s mathematical abilities. Using a variation of the number line task, Xu and Chen (2012) examined the developmental pattern of linear number representations in a Chinese cultural context and found that Chinese children form linear number representations between the ages of 4 and 4.5. However, counting ability and number concept level affect the pattern and accuracy of children’s quantity representations—the higher the counting ability, the more accurate and closer to linear the children’s representations are for large numbers. This suggests that the acquisition of number language has a facilitating effect on the formation of linear number representations. In addition, correct numerical feedback appears to facilitate the conversion of children’s quantitative representations to linearity and improve their representational accuracy.

Understanding and Applying Number Labels Understanding number terms and knowing the meaning of number labels are the foundations for the further development of mathematical knowledge and abilities. From the age of two or three, children’s level of understanding of the meaning of number words gradually improves. Han et al. (2013) used the “comprehend or level theory” (Carey, 2001, 2004) to analyze children’s performance in a task involving taking things by number. The study classified children aged 2–5 years into various levels. 1. Pre-number-knower Knowledge Level. At this level, there is no relationship between the number of objects taken by the child and the target number word of the task. It can be assumed that the child has not yet attached any meaning to the number word. 2. Subset Level. Children comprehend the number terms “one,” “two,” “three,” and “four” (i.e., children understand only the meanings of 1, 2, 3, and 4). These children’s number word comprehension levels are within their counting range. 3. Base Principle Comprehension Level. This group of children can extend the principle of correspondence between number words and quantities to “five” and beyond; that is, they have acquired the meaning of all number words within their own number range. Reaching the level of comprehension of the cardinality principle also means that children understand the successor function (Gelman & Gallistel, 1978); that is, if the cardinal value referred to by a number n is n, and if the number p comes immediately after n in the number sequence, then the cardinal value corresponding to p is n + 1. The results of the study are shown in Table 7.3. It can be seen that the distribution of children aged 2–5 years varies significantly across levels and that age 3–4 is

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a period of rapid development in children’s number word comprehension. Most children at ages 2 and 3 years are still at the subset level, whereas children at ages 4 and 5 years have largely reached the base principle comprehension level. It was also found that children who reached the base principle comprehension level possessed a better understanding of subsequent functions. Another study by Han et al. (2013) also found that children who were good at using number labels performed better in tasks such as equivalence matching and quantity comparison.

Fraction and Proportional Cognition Fractions and proportions are important contents of children’s mathematical cognition. Generally speaking, elementary school mathematics textbooks are designed so that knowledge of fractions is mostly learned from the third grade. However, the emergence of the concept of fractions and proportions occurs in the preschool stage. Although preschool children cannot understand fractions represented in symbolic form, they already have a certain degree of simple understanding of the concepts of fractions and proportions in non-symbolic form. My student, Prof. Ziqiang Xin and Yulei Han, has pointed out that children’s initial knowledge of fractions is often related to life experiences in the sharing, division, and distribution of food (Xin & Han, 2014). Although preschoolers initially acquired simple concepts of fractions and proportions, they still have difficulty understanding words such as “fraction” and “proportion.” Therefore, it is difficult to accurately test children’s mastery of proportions or fractions through direct questioning. In particular, question-setting and instructional language design have long been critical challenges in this area. In one study, our team used a Stroop-like task that subtly avoided direct questioning and examined children’s representation and manipulation of proportional information at an automated level. The study asked children to compare the area of the presented sectors but used proportional information as an interference dimension, thereby examining children’s automatic activation and processing of proportional information in area comparisons. It was found that proportional information as an interference dimension exhibited a greater degree of interference from at least 5 years of age, and the magnitude of the effect was similar to that of adults. This finding suggests that children show automatic activation and processing of proportional information early in development, so proportional cognition is likely to be a highly intuitive cognitive process (Yang et al., 2015). Xin and Liu (2011) also explored children’s fraction calculation abilities and found that although children’s non-symbolic fraction calculation abilities were slightly weaker than their whole number calculation abilities, both showed similar developmental patterns. Again, from around age 5, they were able to complete graphical fraction calculation problems in ¼ units, and both number recognition and number recall abilities significantly predicted their performance in fraction calculation tasks.

0

0

10

0

0

4 yr

5 yr

In total 3

3 (19%)

7 (17%)

1 (6%)

2 (5%)

2 yr

3 yr

18

0

1(4%)

10 (24%)

7 (44%)

16

0

0

12 (28%)

4 (25%)

1

0

0

0

1 (6%)

52

15 (100%)

26 (96%)

11 (26%)

0

100

15

27

42

16

Pre-number-knower Subset level of “one” Subset level of “two” Subset level of Subset level of “four” Base principle In total knowledge level comprehension level comprehension level “three” comprehension level comprehension level comprehension level

Age

Table 7.3 Distribution of children in different age groups in terms of number word comprehension levels (number and percentage)

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Early Teaching of Mathematics

183

Early Teaching of Mathematics In the first two sections, we discussed the individual differences among children. The causes of such differences are, as stated in Chap. 2, firstly, genetic factors and secondly, environmental education. My experimental study found that a major factor is the difference in educational instruction; such a gap not only causes differences in preschoolers’ levels of number concepts and operations, but, more importantly, affects the level of thinking activity or intellectual activity in the development of their number concept formation and arithmetic abilities. In my study, the subjects came from two types of kindergartens: those that teach according to the unified mathematics syllabus of the Beijing Municipal Education Commission and those that run their own kindergartens and focus on “cultivating” rather than “teaching” (Lin, 1980). The current situation is even more complicated, as many private kindergartens are also adept at providing high-quality education. Among the many kindergartens, the quality of education varies, resulting in both environmental and educational differences. I analyze the differences in the impact of these two types of kindergartens on the development of children’s minds using children aged 6–7 years as an example. The reason for selecting this particular age group is twofold. First, the differences are more significant due to the older age of the children. Second, and most importantly, 6–7-year-olds have been in kindergarten for a longer time period; thus, they have been influenced more deeply by this educational environment and are more representative of the target group. I compared four kindergartens/nurseries that used a uniform mathematics syllabus. A total of four kindergartens/nurseries that had a unified curriculum but with a different scope and nature of parental work (an urban kindergarten, a factory nursery, a rural nursery, and a military nursery) were selected. I observed the similarities and differences between these kindergartens/nurseries in terms of the children’s levels of mastery of number concepts and thinking activities, the results of which are shown in Table 7.4. Table 7.4 Comparison of children’s mastery of number concepts in the four kindergartens/nurseries Unit

Up to 5 (%)

Up to 10 (%)

Up to 20 (%)

Up to 50 (%)

Up to 100 (%)

More than 100

Urban kindergarten

96.7

96.7

96.7

93.3

33.3

6.7

Factory nursery

100

100

96

87

40

N/A

Rural nursery

95.5

95.5

95.5

82

27.3

N/A

Military nursery

100

100

100

86.7

36.7

10

Note The test of significance, p < 0.01 (no significant difference)

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Table 7.4 shows that although there is a difference in the acquisition of number concepts among the 6–7-year-old children under the requirements of the unified mathematics syllabus (a slightly higher level of acquisition of number concepts for children in the urban kindergarten than those in the rural nursery), this difference is not significant (p > 0.1). As seen in Table 7.5, despite some slight differences in the level of thinking activity of 6–7-year-old children in the four kindergartens in terms of the mastery of number concepts and the development of arithmetic abilities (marginally higher in urban children than in rural children), this difference was not significant. In the case of early education under the unified mathematics curriculum, the level of thinking activity of 6–7-year-olds was generally similar, despite individual differences caused by genetics and environment, and more than 82% of the children had entered the “image-speech number generalization” stage. At the same time, I compared two kindergartens in rural areas of the same region, which had different teaching contents but children of the same age. The rural kindergarten mentioned above, where early education is conducted according to a uniform mathematics syllabus, is referred to as “A.” A factory kindergarten, where mathematics teaching is not required and children’s attendance is more casual, resulting in a significant difference in the level of children’s acquisition of number concepts and thinking activities in the two kindergartens (see Table 7.6), is referred to as “B.” In Table 7.6, although we can see that there are certain age-specific characteristics in the acquisition of number concepts for 6–7-year-olds, it is also clear that different educational conditions significantly impact children’s acquisition of number Table 7.5 Comparison of children’s thinking activity levels in the four kindergartens Level

Urban kindergarten (%)

Factory nursery (%)

Rural nursery (%)

Military nursery (%)

Intuitive-speech number generalization

13.3

13

18

13.3

Image-speech number generalization

86.7

87

82

86.7

Note The test of significance, p < 0.01 (no significant difference)

Table 7.6 Comparison of children’s acquisition of number concepts in the two rural childcare centers Unit

Up to 5 (%)

Up to 10 (%)

Up to 20 (%)

Up to 50 (%)

Up to 100 (%) More than 100

A

95.5

95.5

95.5

82

27.3

N/A

B

84

84

52

10

6

N/A

Note The test of significance, p < 0.01 (no significant difference)

Early Teaching of Mathematics

185

Table 7.7 Comparison of thinking activity levels of 6–7-year-old children in two rural childcare centers Level

A (%)

B (%)

Intuitive-speech number generalization

18

42

Image-speech number generalization

82

58

Note The test of significance, p < 0.01 (no significant difference)

concepts. This indicates that early mathematics education plays a dominant role in children’s number concept development. As seen from Table 7.7, different educational conditions not only affect children’s mastery of number concepts and their comprehension of mathematical knowledge but also directly affect children’s level of thinking, which is, in part, reflective of their level of intelligence. Children with better early education have a strong thinking ability, whereas children lacking reasonable early education rely more on visual objects, which seriously hinders the development of their abstract logical thinking. Through surveys, I found that among children starting school, in general, rural children tend to have a lower level of intelligence than urban children. This was not innate. At that time, the conditions of early education for rural children were poor, and few of them had attended kindergartens. In addition, many had not been taught with teaching materials that adopted a uniform syllabus, which later emerged as a key factor affecting the intellectual development of rural children. At that time, preschool education in China was far from universal, and there was a lack of institutions specializing in the psychology and education of infants and children. Domestic and foreign psychological experiments have proven that early instruction in literacy, reading, and mathematics is not only appropriate for preschool children’s level of physical and mental development but also significantly develops the corresponding intellectual abilities (Breit et al., 2021; Thurstone & Thurstone, 1941). According to many studies of factor analytics, the first and second factors of intelligence are language (mainly vocabulary) and mathematical ability, respectively. It is well known that language is a tool for receiving knowledge and learning all sciences, as well as for thinking, expressing ideas, and recording the results of thinking. Likewise, mathematics is a tool for learning all natural sciences and most social sciences, inspiring thinking, carrying out thinking, and expressing the results of thinking. According to the modern information processing (processing) theory of thinking and learning principles, thinking starts with learning, inputting, and storing information (knowledge). Learn more, memory storage, thinking is easier, and intellectual development is difficult. Confucius also said, “Thinking without learning is perilous.” Thus, early education in language and mathematics from the preschool age is the fundamental way to develop children’s thinking and intelligence. Some people are concerned that early education may inhibit children’s physical health and may even lead to brain damage and shorter lifespans. However, this has not been found to be the case in existing examples of early learning (e.g., several junior classes at the University of Science and Technology of China and other children

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engaged in early learning). There has never been a clinical case of brain damage and death caused by brain overuse. There are also many examples throughout the Chinese history of early learning and high achievement, such as Bai Li,2 who “recited the six aces at the age of five and observed a hundred schools of thought at the age of ten” and died at the age of 61. If he had not been a lifelong alcoholic and suffered persecution in his later years, he might have lived longer. Juyi Bai3 was also an early learner who passed away at the age of 74. Although the “Four Masters of the Early Tang Dynasty” and He Li4 did not live long, the causes of their deaths were not related to early learning, such as Bo Wang’s5 death by drowning and He Li’s suicide due to anger and depression from extensive persecution. Some researchers have conducted a series of experiments that examine the brain and its physiological changes in animals exposed to early learning (Rosenzweig et al., 1972). The experimental facts were contrary to the general conjecture, and the rich environmental stimuli and learning opportunities in the early years did not harm but rather promoted the development of the animals’ brains. As the saying goes, “Use it to advance, waste it to retreat.” The brain develops through movement and practice. The supposition that learning must take place only after the nervous system or brain is fully mature—or else it will either damage development or be difficult to collect results—is also unfounded. The nervous system or brain develops earlier and faster than other systems and organs in the body, and people are constantly engaged in a process of learning after birth. Examples include getting to know people and things in the surrounding environment, becoming familiar with life and habits, listening, speaking, dressing, eating, walking, singing, dancing, and playing the piano. There are complexities and logical differences between learning activities in life and learning activities in school, such as reading, writing, and arithmetic. Since these activities do not have to wait for the body or brain to fully mature before learning is possible, nor do they hinder the development of the body or brain, how can they hurt us mentally or physically? In early education, language and mathematics are two of the principal factors that contribute to the development of children’s thinking and intelligence. Below, I offer some suggestions specifically for the early education of mathematics.

From Early Education to Early Mathematics Teaching Early mathematics teaching is a part of early education. The United Nations Educational, Scientific, and Cultural Organization (UNESCO) has long emphasized that early childhood education should be one of the main goals of future education. 2

Bai Li (李白): A Chinese renowned poet in the Tang Dynasty (701–762). Juyi Bai(白居易): A Chinese renowned poet and government official in the Tang Dynasty (772– 846). 4 He Li(李贺): A Chinese renowned poet in the mid-Tang Dynasty (790–816). 5 Bo Wang (王勃): A Chinese renowned poet in the Tang Dynasty (650–676). 3

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In 2010, China released the “National Medium and Long-term Education Reform and Development Plan Outline (2010–2020),” which likewise stated that “preschool education is important for the physical and mental health, habit formation, and intellectual development of young children significance.” In the following, I start from the view of early childhood education in various countries and make some further suggestions about early mathematics teaching.

Stages of Early Education and National Perspectives on Early Childhood Education In the mid-1980s, when my experiments in educational reform and attempts to develop students’ thinking and intelligence were developing vigorously, several newspapers published materials from Bloom’s (1964) study on intellectual development. The study concluded that if we take the intellectual level reached at the age of 17 or 18 as 100%, approximately 50% of development takes place at the age of 4, 30% occurs at the ages of 4–8, and the remaining 20% is developed at from ages of 8 to 18. I find this to be extremely perplexing. If this were indeed the case, many scientific questions would lead to a dead end. First, if this were true, the relationship between intelligence and knowledge and experience would not be very important because the knowledge and experience gained from elementary school to secondary school after the age of 8 would be presumed to have very little effect on intelligence. Moreover, at the age of 17 or 18, if intelligence has reached its limit, the knowledge gained in university would be meaningless to the development of intelligence. Second, it is not easy to explain the lifelong developmental perspective of intelligence, especially the theory of fluid and crystallized intelligence proposed by Horn and Cattell (1966) after Bloom (see Fig. 7.5). Third, if this theory were true, it can be declared that experimental studies of intellectual development are of little significance, and school education, including the role of teachers in the intellectual development of students, need not be investigated.

Maximum effect

crystallized intelligence fluid intelligence

Infant Children Youth Middle age Old age Fig. 7.5 Trend of crystallized intelligence and fluid intelligence

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Consequently, my own experiments on thinking and intellectual development would be called into question. At the time, I did not put much stock in Bloom’s data. I believed that the purpose of promoting Bloom’s view of intellectual development was to overvalue “early education.” To be clear, I am not opposed to early education. I believe having quality teachers is a prerequisite for the development of students’ minds and intellect from kindergarten through elementary schools, secondary school, high school, and even college. However, I cannot ignore the importance of elementary, middle school, and high school and university education in the development of human thinking and intellect just because I want to explain the importance of early education. For this reason, I have been experimenting more intensively with intellectual development to reveal the relationship between the quality of teachers and the intellectual development of students, as well as the underlying mechanisms of influence. On May 30, 2011, Reference News published “Early Childhood Education in Different Countries” in the Global Survey section, which greatly inspired me. The USA is a country that values early education; as early as the 1980s, the American Educational Development Institute claimed in its book Early Education in America that “we are on the verge of fundamental developments that will soon involve the early education of American children.” The US Department of Education also states on its website, “The period before kindergarten is one of the most important periods in a child’s life that affects cognition.” The US federal government urges states to invest more public funds into early childhood education and to provide more early childhood education opportunities. Thus, it follows that parents should place more emphasis on early childhood education. With the assistance of professional organizations such as the National Association for the Education of Young Children and the American Montessori Society, all US states and local governments have basically provided preschool children with considerable access to early education. Japan also has a tradition of welcoming early education. In the 1980s, Masaru (2001) noted in his book Kindergarten Education is Late that the greatest period of human intellectual potential is not in college, nor in elementary or secondary school, but in the preschool period and that proper training must be provided from birth. In the 1990s, early education began to increase in Japan. Currently, early education has become a huge industry in Japan, and many specialized companies have emerged. For example, the “Shichida Children’s Academy,” which boasts of its capacity to develop children’s right brains (I have a different opinion on this), has 460 classrooms for young children across the country. In addition to large education companies, various small parent–child classrooms have sprung up nationwide. Many publishers are working on publishing various early education textbooks, some of which are so detailed that they are based on months rather than years in terms of age. However, there are also critical voices in Japan about early education, arguing that it is not only useless but harmful for infants and young children and pupils in the early grades of elementary school. The UK has a long history of theory and practice in early education. Early education for children in the UK is a long-lived tradition that has existed for nearly a

From Early Education to Early Mathematics Teaching

189

century. As early as 1923, the Early Childhood Education Association was established to assist parents in providing quality head start for children from newborns to the age of 8. One of the most principal elements is to allow children to be carefree and enjoy their childhood while simultaneously exploring and discovering their interests early to provide more references and options for further education in the school system. Most early education activities in the UK are universal and public serviceoriented, aiming to promote social equity. For example, early education activities are organized at community libraries three times a week and are free for all. Most Korean families also plan for their children’s development from an early age. Many Korean children are sent to various extracurricular classes from kindergarten onward in order to gain an advantage over their peers in the highly competitive society when they grow up (Li, 2007). The so-called English boom has been around for a long time in Korea, and fluency in English is considered an “asset” for young people in the workforce. Thus, many parents believe that the earlier their children learn a foreign language, the better. Some even go so far as to start their children’s English learning in the fetal stage, with “English fetus education” becoming very popular in Korea in recent years. In addition to English, there are also taekwondo, swimming, basketball, piano, rope skiing, roller skating, painting, science, and many other hobby classes available at private institutes. However, Korean parents are not keen on the certification and examination of various talents. What are the benefits other than adding to the burden of the child? No one has conducted a tracking study. I do not want to analyze the various early education philosophies in China, nor do I wish to evaluate the various early education methods in China. I believe that early education should be viewed as a “two-sided” concept that should be valued but not absolutized. I cannot quite understand the meaning of “the starting line6 in the competition” when society and parents are quick to express concerns about not letting the children “lose at the starting line.”

Some Suggestions for Early Mathematics Education Based on my research, I would like to make the following educational suggestions for teachers of childcare centers and kindergartens to consider when teaching mathematics. 1. The teaching of numbers and calculation in nurseries and kindergartens should be based on children’s physical and mental development, especially their age characteristics, and should not follow the order of “objects → image → abstraction” in order to ensure children’s happy and healthy growth. (1) Before the age of 2, children should be allowed to play with various toys so that they can develop intuitive-action thinking and gradually develop the perception of space, size, and quantity from intuition. 6

The starting line: Here is a metaphor for the fierce competition in study and life.

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(2) Between the ages of 2 and 3, children should be taught the number concepts mainly through objects and “intuitive → speech.” When teaching children to count, teachers can make use of the fact that this is a critical age for children’s verbal development and make up some counting songs so that children can enjoy counting. (3) For 3–4-year-olds, “objects →image → number concept” approach to teaching should be employed. Children should be given children objects to count, asked to count them, and then be instructed to count the total number of objects. (4) At 4–5 years of age, children should again be given objects to count, emphasizing the role of words in guiding visual objects. At 2 years of age, mathematics instruction focuses on the development of counting abilities. (5) After the age of 6, children should be taught to understand the actual meaning of numbers, the sequence of numbers, and their composition by using visual aids and moving away from objects as much as possible to “image → speech” instruction. 2. Throughout the teaching of numbers and calculation in nursery and kindergarten, it is important to cultivate children’s interest in things and events around them, such as numbers and shapes, and to stimulate their curiosity and desire for knowledge. For this, children should be guided to use their senses, hands, and brains to investigate problems. In particular, children must be taught how to write numbers early and allowed to use more language to express and communicate the process and results of mathematical operations appropriately. Especially at the ages of 4 or 5, which form a critical period for children’s written language development, it is more important to provide children more opportunities to speak and write so that they can use various analyses to communicate the connection between the two systems of image and language, thus facilitating the formation of number concepts. 3. Teachers should adopt a variety of interesting teaching activities, combine the teaching of numbers and calculation with games, and carry out some colorful “counting” and “arithmetic” competitions to cultivate children’s interest in numbers and calculation so as to impart such knowledge through an enjoyable activity. For example, a kindergarten held a “mathematical competition” for children in the middle class (4–5-year-old students) that consisted of assigning numbers to objects and taking objects by numbers. At the beginning of the competition, the children took their positions, each with more than 100 small stones in front of him/her, and the teacher beat the drum. The teacher then said a number, and the children immediately started counting. They were instructed prior to starting the contest that as soon as they finished counting, they should raise their hands. The teacher first counted the time it took for each child to complete the task and then checked the objects taken out by the children to see which children were accurate and fast. Similar “games” and “competitions” can not only make children gain a sense of the number of things from life and games but also improve their ability to count. What is more, they can help develop their interest in mathematics and contribute to improving their thinking abilities in counting and other operations.

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4. Children’s counting and arithmetic should be reinforced over time to confirm their accuracy in arithmetic and that they are employing appropriate learning strategies. Reinforcement is provided not only via verbal affirmation but also by appropriate material rewards, which improve memory and lay the foundation for further arithmetic. The State Key Laboratory of Brain Cognition and Learning at Beijing Normal University has published several articles in Science Citation Index (SCI) journals, which propose and test through a series of experiments the hypothesis of a “dual representation-based arithmetic brain” shaped by early learning experiences (Zhou et al., 2006, 2007, 2011). This hypothesis refers to the influence of learning experiences where visual-Arabic numerical representations of addition and subtraction formulas (e.g., 9 + 7 = 16, 8 − 3 = 5) and auditory-verbal representations of multiplication formulas (e.g., 3 × 7 = 21) predominate. Using early arithmetic learning experiences as the study variable, we used behavioral experiments and functional brain imaging techniques to reveal a more comprehensive picture of the enduring role of early learning experiences in shaping the brain’s mathematical cognitive functions. Our results indicate that brain activity patterns for addition, subtraction, multiplication, and division diverge depending on learning strategies and that learning content directly influences the way arithmetic knowledge is organized in long-term memory. 5. After age 5, increasing levels of abstraction should be taught under acceptable conditions. For example, the ability to decompose and combine numbers up to “20” can be developed by mastering clusters of numbers without the use of physical arithmetic. In the older classes, verbal word problems up to “10” can be taught by simply stating the conditions and letting the children ask questions, which is a semi-finished verbal word problem. For instance, the teacher points to the four pears on the left side of the board and asks the children, “How many are there?” to which they respond “four pears.” Then, the teacher points to the three apples on the right side of the board and elicits “three apples” from the children. The teacher can also let the children act as the teacher, meaning that the children can ask the questions and then everyone can answer them. This not only promotes the rapid development of children’s arithmetic and independent thinking abilities but also promotes the improvement of children’s oral language abilities. It is on the basis of the development of one’s “mathematical” ability and the improvement of one’s “verbal” level that children’s logical abstraction and intellectual qualities are constantly developing. 6. To avoid the contradiction between the kindergarten and elementary school “textbook mixture,” the focus of kindergarten “mathematics teaching” can be on the development of children’s thinking activity level and intellectual quality. For example, children should be guided earlier on to develop from “intuitiveverbal number generalization” to “image-verbal number generalization.” For instance, to develop the structure of children’s arithmetic thinking, for example, conservation, and children’s arithmetic thinking structures (e.g., conservation and reversibility), a piece of clay can be shaped into a “pancake” and then into a “sausage” to have children judge the “size” and see that it is constant

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regardless of the shape. Similar activities allow children to determine different quantities, thicknesses, weights, and the relationships between them to develop their conservation skills. Another example is to ask children to determine how many squares and rectangles there are (see Fig. 7.6) and how many squares there are (see Fig. 7.7) from different angles to develop children’s judgment, observation, and spatial imagination. Another example is to give children six stones and ask them to arrange them into a triangle so that there are an equal number of stones on each side (see Fig. 7.8), which not only develops children’s spatial imagination but also their logical reasoning ability.

Fig. 7.6 How many rectangles are there in this image?

Fig. 7.7 How many squares are there in this image?

Fig. 7.8 Triangle

References

193

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Ministry of Education of the People’s Republic of China. (2001). Guidelines for kindergarten education (for trial implementation). Beijing Normal University Press. (in Chinese). Moll, H., & Meltzoff, A. N. (2011). How does it look? Level 2 perspective-taking at 36 months of age. Child Development, 82(2), 661–673. https://doi.org/10.1111/j.1467-8624.2010.01571.x Piaget, J. (1977). The role of action in the development of thinking. Knowledge and development (pp. 17–42). Springer. Piaget, J., & Szeminska, A. (1952). The child’s conception of number. Basic Books. Protzko, J. (2015). The environment in raising early intelligence: A meta-analysis of the fadeout effect. Intelligence, 53, 202–210. https://doi.org/10.1016/j.intell.2015.10.006 Rosenzweig, M. R., Bennett, E. L., & Diamond, M. C. (1972). Brain changes in response to experience. Scientific American. Scientific American, 226(2), 22–29. https://doi.org/10.1038/scientifi camerican0272-22 Thurstone, L. L., & Thurstone, T. G. (1941). Factorial studies of intelligence. Psychometric Monographs, 2, 94. Viarouge, A., Houdé, O., & Borst, G. (2019). The progressive 6-year-old conserver: Numerical saliency and sensitivity as core mechanisms of numerical abstraction in a Piaget-like estimation task. Cognition, 190, 137–142. https://doi.org/10.1016/j.cognition.2019.05.005 Vygotsky, L. S. (2012). Thought and language. MIT Press. Wang, X. D., Liu, J. H., & Fan, C. R. (1964). An investigation in the development of concepts in children of 4–9 years—ii. generalization development as reflected in children’s ability to classify. Acta Psychologica Sinica, 4, 352–360. (in Chinese). Xin, Z. Q., & Han, Y. L. (2014). The developmental and interventional research on lower graders’ concept of equivalent fraction. Acta Psychologica Sinica, 46(6), 791–806. (in Chinese). Xin, Z. Q., & Liu, G. F. (2011). The development of children’s nonsymbolic calculation ability of whole-number and fraction and its relationship with number memory. Psychology Science, 34(3), 520–526. (in Chinese). Xu, H., & Chen, Y. H. (2012). Review and prospect on the research of children’s number line estimation. Psychological Research Psychologische Forschung, 5(5), 46–50. (in Chinese). Yang, Y., Hu, Q. F., Wu, D., & Yang, S. Q. (2015). Children’s and adults’ automatic processing of proportion in a Stroop-like task. International Journal of Behavioral Development, 39(2), 97–104. https://doi.org/10.1177/0165025414556520 Zhou, R., Zhang, H., & Lin, C. (2003). Studies on the acquisition of “zero” concept in children. Psychological Exploration, 1, 29–32. (in Chinese). Zhou, X., Booth, J. R., Lu, J., Zhao, H., Butterworth, B., Chen, C., & Dong, Q. (2011). Ageindependent and age-dependent neural substrate for single-digit multiplication and addition arithmetic problems. Developmental Neuropsychology, 36(3), 338–352. https://doi.org/10.1080/875 65641.2010.549873 Zhou, X., Chen, C., Dong, Q., Zhang, H., Zhou, R., Zhao, H., Chen, C., Qiao, S., Jiang, T., & Guo, Y. (2006). Event-related potentials of single-digit addition, subtraction, and multiplication. Neuropsychologia, 44(12), 2500–2507. https://doi.org/10.1016/j.neuropsychologia.2006.04.003 Zhou, X., Chen, C., Zang, Y., Dong, Q., Chen, C., Qiao, S., & Gong, Q. (2007). Dissociated brain organization for single-digit addition and multiplication. NeuroImage, 35(2), 871–880. https:// doi.org/10.1016/j.neuroimage.2006.12.017

Chapter 8

Mathematics Learning and Intellectual Development of Elementary School Students

Elementary school students (6–12 years old) enter a structured academic environment with learning-oriented activities. Elementary schools set the stage for future learning and are the basis of national education wherein mathematics is one of the core subjects comprising foundational concepts elementary school students require to master other subjects. Promoting the basic concepts and arithmetic abilities of mathematics is also necessary for the development of elementary school students’ intelligence.

Development of Mathematical Intelligence in Elementary School Students Based on their development during the preschool period, elementary school students take learning as the primary focus when entering school. Learning is the purposeful and systematic acquisition of knowledge, skills, and behavioral norms, as well as a social obligation. In the process of completing learning tasks, children’s relationships with the world around them also change, thus continuously and comprehensively developing their intellectual and mental activities. The development of all facets of intellectual activity in elementary school children is characterized by the transition from oral to written language and from concrete image thinking to abstract logical thinking, which accordingly constrains the development of all aspects of their intelligence. In the area of perception, purposeful and conscious perception and observation ability and spatial and time perception are continuously developing, thus providing the basic conditions for children to analyze and synthesize more accurately when learning. In terms of memory, although unconscious, mechanical, and concrete image memory still play an important role, other types of memory—namely conscious, comprehension, and abstract logic—develop rapidly under teaching requirements. In terms of thinking, elementary school students gradually transition from analyzing concrete images to utilizing abstract logic, which © China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_8

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dominates a large component of perceptual experience. In terms of imagination, due to the development of abstract logical thinking, the intentional, creative, and realistic components of imagination are steadily cultivated. Elementary school mathematics instruction (mainly arithmetic instruction) requires students possess a certain intellectual foundation, which, in turn, facilitates their intellectual development. Elementary school students begin to learn mathematics systematically and gradually acquire arithmetic rules. They not only have to think and solve various problems (especially word problems) but also pay attention to how to think and discover the essential connections of things. In addition to remembering various formulas, definitions, and properties, they must also focus on how to recognize and memorize them so that they can remember them consistently for future use. These needs motivate elementary school students to develop the consciousness and intentionality of mental processes in the process of mastering number concepts and operations so that their arithmetic abilities can gradually improve. The arithmetic abilities of elementary school students are consistent with the general trend of intellectual and thinking ability development.

Development of Elementary School Students’ Number Generalization Ability As mentioned earlier, mastery of concepts is mainly linked to children’s and adolescents’ accumulation of knowledge and intellectual development, and the level of generalization is a direct prerequisite for mastery of concepts. In my own research, I analyzed the level of development of elementary school students’ number concepts in terms of the trends in their number generalization capacity. The indicators used in the study to determine the developmental level of elementary school students’ number generalization ability were as follows: (a) the degree of reliance on intuition; (b) the understanding of the actual meaning of numbers (range of number representations); (c) the awareness of the order and size of numbers (cognition); (d) the composition of numbers (decomposition and combination); and (e) the expansion of number concepts and the development of definitions. Our group analyzed the results of the study based on these five indicators and identified five levels of elementary school students’ number generalization ability. The first level (I) is intuitive generalization. Number concepts up to ten are learned by relying on objects, teaching aids, or finger-counting. The second level (II) is the intuitive generalization of arithmetic. Students who belong to this level enter the “integer propositional arithmetic” stage. There are three indicators of achievement: mastery of the actual meaning of certain integers, the order and size of numbers, and the composition of numbers. This level can be subdivided into a number of different substages, such as the number concepts up to “20,” the number concepts up to “100,” the number concepts up to “10,000,” and the concept of the four operations on integers. At this stage, due to the limitations of experience,

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although some operations can be performed in the range of numbers outside their life experience, students cannot truly understand the actual meaning of the numbers of all operations due to the lack of number images. The third level (III) is figurative abstract arithmetic. Students at this level are in the process of developing from figurative generalization to abstract generalization, and the indicators they achieve are as follows. 1. Number images and the actual meaning of numbers are combined in students’ thinking, forming the characteristics of new generalizations. Students understand the actual meaning of a large number of numbers, mastering not only the concepts of multidigit integers but also fractions. They comprehend the magnitude, order, and composition of simple positive and negative numbers and are able to synthesize the concept of genera and form the concept of species. 2. Spatial images are developed so that students can generalize geometric concepts from a large collection of geometric figures and master some formulas and definitions of geometry, which are important indicators of the generalization ability at this level. Therefore, this level can be called “elementary geometric propositional operations.” The fourth level (IV) is the initial level of essential abstract generalization, that is, the level of generalization of initial algebra. At this level, students achieve the following indicators. 1. Students master not only “subsets” in arithmetic operations but also “intersecting sets” and “concatenating sets” in the scope of arithmetic; for example, finding common multiples and common divisors, which is actually mastering the relationship between “intersecting” and “concatenating.” 2. Students use abstract representations of letters instead of abstract representations of numbers; for example, this is the level at which they are initially exposed to solving word problems with equations. 3. Students are able to use analytical and synthesis methods in the thinking process to answer “typical word problems,” and they are able to analyze the operations of combinations and master the various relationships between quantities. The fifth level (V) is the generalization of algebraic propositions. Students at this level are able to generalize based on assumptions and completely leave arithmetic block diagrams aside; that said, very few students reach this level of generalization in elementary school. These five levels provide an overview of the development of elementary school students’ generalization levels. Elementary school students make the transition from primarily concrete image generalization to primarily abstract logical generalization in arithmetic. However, the latter still contains a large component of concrete imagebased generalization. How does the level of generalization develop in elementary school students? Generally, the first grade develops on the foundation acquired from preschool thinking activities, which is basically intuitive image generalization. The second and third grades teach students to transition from intuitive image generalization to

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abstract generalization, which most students achieve at some point in the third grade. The fourth grade produces a sudden change wherein most students in the fourth or fifth grades can enter the preliminary level of essential abstract generalization of arithmetic. There are a few students who begin to progress to the preliminary algebraic propositional generalization arithmetic level under the influence of a well-organized education. That said, there are, of course, a few students who do not have strong generalization abilities and are limited to a low level of generalization, which would be an individual difference.

The Development of Elementary School Students’ Proposition Ability and Arithmetic Rules Characteristics of Elementary School Students’ Proposition Ability. In my research, I found that there are four trends in the form of propositional arithmetic for elementary school students (Lin, 1981). 1. Positive (p). For example, p = 150 = 70 + 80, and elementary school students quickly respond in the affirmative. 2. Negative (p). For example, p = “31 is an even number”; negating this proposition leads to another proposition; i.e., q = “31 is not an even number.” Proposition q is called the negation of p. Students are prone to make mistakes in propositional arithmetic, especially when checking the answer negating the wrong proposition or checking for falsely negating the correct answer. 3. Conjunctive, p ∧ q (p and q). For example, elementary school students decide that a = 36 is a multiple of both 12 and 9. 4. Disjunctive, p ∨ q (p or q). For example, elementary students determine that a = 15 is either a common multiple of 21 and 14 or a common multiple of 3 and 5 (only one conclusion is correct). The same is true for the concept of probability, where students determine that the probability of rolling two dice is “either 2 points, 3 points, 4 points … or 12 points,” with 11 possibilities. In elementary school, the ability (level) of students to master these four forms of propositions varies from grade to grade (age). The tendency of students to master the different forms of propositions (in order) is as follows: positive → negative → conjunctive → disjunctive. Characteristics of Elementary School Students’ Arithmetic Laws. The algorithm of laws is associated with mathematical propositions in elementary schools. The process of thinking follows certain laws, and the laws of thinking are a reflection of the objective laws of things. There are many laws that elementary school students should follow to master number concepts and arithmetic thinking. There are four main laws of arithmetic, namely the law of exchange {x ∧ y = y ∧ x}, the law of distribution {x ∧ (y ∧ z) = (x ∧ y) ∨ (x ∧ z)}, the law of union {(x ∧ y) ∧ z = x ∧ (y ∧ z)}, and the law of double negation {¬(¬x) = x}.

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Table 8.1 Development of elementary school students’ ability to use the laws of operations at different grade levels Grade The law of exchange

The law of union

The law of distribution

The law of double negation

Numerical Word Numerical Word Numerical Word Numerical Word operation operation operation operation operation operation operation operation (%) (%) (%) (%) (%) (%) (%) (%) 1

83.3

–

80

–

80

–

–

–

2

90

83.3

86.7

40

83.3

40

–

–

3

100

90

100

70

96.7

73.3

–

–

4

100

100

100

100

100

100

23.3

13.3

5

100

100

100

100

100

100

86.7

76.7

In terms of the range and correctness of the algorithm law, there are three levels of mastery in elementary schools: (a) using the algorithm in numerical operations; (b) using the algorithm in simple word operations; and (c) using the algorithm in algebraic and geometric operations. Only the results of the first two levels are analyzed here. Table 8.1 shows that more than 80% of the first graders surveyed in our study were able to use the laws of exchange, union, and distribution in simple numerical operations from the second semester of their schooling. After experiencing the transition to grade 2, most elementary students in grade 3 can use the laws of exchange, union, and distribution in simple word operations. After grade 4, they gradually master the law of double negation in arithmetic operations. The mastery of the double negation law is a turning point (a leap) in the ability of elementary school students to use the laws of arithmetic.

The Development of Elementary School Students’ Reasoning Ability The development of elementary school students’ reasoning ability is mainly expressed in two kinds of ability: inductive and deductive reasoning. The Development of Inductive Reasoning Ability. In arithmetic, elementary school students’ inductive reasoning can be divided into four levels. The first level (I) is direct inductive reasoning in arithmetic operations, which is the induction of the principle of arithmetic by direct observation of the equations provided in simple number operations in one step. For example, the exercise asks, “6 + 0 = 6, 8 + 0 = 8, 19 + 0 = 19. What does this mean?” Elementary students can correctly answer, “Any number plus 0 equals the original number.” The second level (II) is direct inductive reasoning in simple word operations; that is, in simple operations in the abstract of letters, a step is taken to introduce the principle of arithmetic. For example, when students are faced with a set of equations

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“x = y, x + a = y + a, x + b = y + b, x + c = y + c,” they can correctly summarize the conclusion that “the equation still holds when the same number is added to both sides of the equation.” The third level (III) is indirect inductive reasoning in arithmetic operations via complex operations and complex word problems to summarize the conclusion or principle and calculation formula. For example, students identify fraction properties through multistep fraction operations. The fourth level (IV) is indirect inductive reasoning from preliminary algebraic equations or multistep inductive reasoning. For example, students correctly reason inductively about how one number follows another from multiple preliminary algebraic equations. This logical reasoning actually shows that they have generalized the initial functional relationship of y = f (x). Research has shown that students in different grades (ages) have different inductive reasoning abilities in arithmetic (Lin, 1981). Most first graders can reason directly and inductively in simple arithmetic operations; more than half of the students in grades 2 and 3 can reason directly and inductively in simple word operations; and most students in grades 4 and 5 can reason indirectly and inductively in complex arithmetic in multiple steps. Moreover, a few students move to the level of indirect inductive reasoning in preliminary algebraic formulas. Development of Deductive Reasoning Abilities. In arithmetic, elementary school students’ deductive reasoning can also be divided into four levels corresponding to inductive reasoning. The first level (I) is the level of arithmetic in which the principles and laws of simple arithmetic are directly specified. For example, after mastering the types of simple word problems, students in the first and second grades of elementary school can perform deductive operations correctly when they encounter various word problems. The second level (II) is the level of operations in which simple arithmetic principles and laws are directly concretized by letters. For example, after learning the law of exchange, second graders can express it as “a + b + c = c + b + a = a + c + b…,” and when they encounter problems that use the law of exchange, they can reason deductively according to the law correctly. The third level (III) is based on arithmetic principles, laws, and formulas as the main premise and requires logical multistep and concrete reasoning to draw conclusions correctly and complete arithmetic exercises. The fourth level (IV) is based on elementary algebraic or plane geometry principles as the major premise and requires multistep deductive reasoning to reach correct conclusions and complete algebraic or geometric exercises. Research has shown that students’ ability to reason deductively in arithmetic varies by grade (age) level (Lin, 1981). Most students in grade 1 can apply simple arithmetic principles and laws as major premises for deductive operations; more than 70% of students in grades 2 and 3 can represent simple formulas, principles, and laws with letters and make them concrete; and most students in grades 4 and 5 can perform multistep deductive and concrete operations on principles and formulas within arithmetic and begin to master some elementary algebra or plane geometry principles for deductive operations.

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inductive reasoning deductive reasoning

level Fig. 8.1 Comparison of the development of inductive and deductive reasoning abilities in elementary school students

We statistically calculated the scores or data of elementary school students’ acquisition of inductive and deductive reasoning in mathematical operations and obtained a high correlation coefficient of r = 0.89 between these two reasoning abilities. A comparison of elementary school students’ acquisition of the two forms of reasoning is shown in Fig. 8.1. The study results indicate that the trends and levels of mastery of the two forms of reasoning (inductive and deductive) are similar in terms of the development of elementary school students’ arithmetic abilities (Lin, 1981).

Basing Elementary School Mathematics Teaching on Age Characteristics of Arithmetic Thinking Ability The development trend of intelligence and the comprehensive thinking ability of elementary school students with age and grade is expressed in the enhancement of various abilities, abstraction, concision of the steps of intelligence, the correctness of the thinking process, and rational, logical, and self-awareness. The content and methods of teaching mathematics should follow this change to target and gradually increase the difficulty based on elementary students’ actual ability levels. This is conducive not only to the improvement of students’ knowledge but also to their intellectual development. In terms of the development of the various aforementioned abilities, the fourth grade is a critical period for developing students’ generalization ability—it is at this juncture that they begin to make propositions, apply relevant laws, and employ deductive reasoning. Students at this age/elementary school stage undergo a significant turning point in transitioning from concrete image thinking to abstract logical thinking as the new main form of elementary school thinking. Therefore, elementary

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school mathematics teaching should pay due attention to this critical age. Reasonable educational measures should be taken to promote the early realization of sudden changes in the arithmetic thinking ability of elementary school students and the early realization of the turnaround, which is one of the important tasks of elementary school mathematics teaching.

Recent Studies on the Development of Elementary School Students’ Mathematical Ability Research on the development of elementary school students’ mathematical ability is also a distinctive topic of recent research by our team.

Further Development of Elementary School Students’ Quantitative Representational Ability In the early elementary school grades, children’s numeric representations are further refined, thereby achieving a shift from logarithmic to linear representations. Logarithmic representations and linear representations are typical expressions used by researchers in recent years to refer to the degree of precision of children’s numeric representations. At younger ages, children’s numeric representations are imprecise, and there is a logarithmic compression of objective quantities. That is, children widen the gap between small numbers and narrow the gap between large numbers in their mental representations. Thus, the larger the quantity is, the more difficult it is for them to distinguish. As they grow older, their numeric representations gradually tend to be linear; that is, the distance between two adjacent numbers is equal regardless of whether they are larger or smaller numbers (Dehaene, 1997). For the examination of children’s ability to make such numerical representations, a number line task is generally used. That is, children are given a line segment of a specific length and asked to mark the position where a specific number should be or the number corresponding to a specific position. In a study by my student, Professor Yinghe Chen, who used the number line task to examine changes in children’s numeric representation ability, it was found that there was also a shift from a logarithmic model (judging the location task according to whether the number fits the logarithmic function model) or an exponential model (judging the number task according to whether the location fits the exponential function model) to a linear model in Chinese children’s number line representations, which occurred mainly in the early elementary grades (Zhang et al., 2015).

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Elementary School Students’ Computational Strategies and Their Relationship with the Development of Basic Cognitive Functions Computation is one of the most important aspects of mathematical development for elementary school students. The strategies that can be used in calculations of various difficulties are quite diverse. The choice of calculation strategies is a complex cognitive processing activity and a typical expression of children’s mathematical ability. It is also closely related to the development of children’s basic cognitive functions. Zhao and Chen (2010) conducted a systematic study on the characteristics of elementary school children’s computational strategies and their relationship with basic cognitive functions. They found that fourth-grade elementary school children used as many as 12 strategies in multidigit multiplication calculations, such as extracting, listing vertical formulas, and repeating sums. Most of the children used two or more strategies. Among them, children with higher working memory breadth were more flexible and effective in their choice of strategies, and they were able to choose the best strategy according to the difficulty and characteristics of different problems, thus showing greater adaptability. In addition to working memory, children’s executive functions are also related to the ability to select and use calculation strategies (Zhao & Chen, 2010). Children with strong inhibition ability have higher rates of correct strategy execution and are able to select more extraction strategies in response to interference. In contrast, children with weak inhibition ability are less likely to use exchange and decomposition strategies to solve addition problems, and they also have lower rates of correct strategy execution.

Development of Fraction Concepts in Elementary School Students Fractions are a major focus of mathematics instruction in the middle grades of elementary schools. The learning of fractions, especially the understanding of fraction concepts, is highly demanding and challenging given the thinking level of elementary school children. Although a formal introduction to fractions does not occur until the middle grades of elementary schools, children’s initial understanding of fraction concepts begins very early on in their lives. First, from only a few months of age, infants can distinguish between sets of objects with different proportions and possess an innate sensitivity to proportions (McCrink & Wynn, 2007). Second, fractions or proportions are widely present in everyday life and therefore accumulate in children’s personal experiences. Children show the emergence of fraction cognition in everyday problems such as equal fractions and proportional comparisons. Upon entering the middle grades of elementary school, children begin to learn about fractions in the formal mathematical notation system, and related knowledge and

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abilities grow rapidly in the years that follow. From at least third grade onward, they can represent fractions on a mental number line from left to right based on their values and in order from low to high (Xin & Li, 2013). For the initial stage of fraction symbol comprehension, Zhang et al. (2014) used a number line task to examine the characteristics of fraction number representations of children in grades 3 to 6. They found that while children in grades 3 and 4 lack a clear tendency to use linear or logarithmic representations of fraction number lines, those in grades 5 and 6 predominantly apply linear representations. This implies that it may only be after fifth and sixth grades that children can truly understand the meaning of symbolic fractions and more accurately represent the real values of fractions. It has been suggested that students’ understanding of fractions progresses through three levels from low to high, which are as follows: (a) characterizing fractions as two natural numbers that are independent of each other; that is, they cannot understand the meaning of fraction symbols; (b) characterizing fractions as a “part-whole” relationship; and (c) characterizing fractions as a ratio of two numbers (Stafylidou & Vosniadou, 2004). Based on this framework, Liu and Xin (2010) conducted a systematic study of Chinese students’ fraction concept development and analyzed the fraction concept development and misconception types of 199 students in grades 5–8. The results revealed that Chinese students’ understanding of fraction concepts was well-developed. In grade 5, 19.2% and 34.6% of the students were at levels 1 and 3, respectively. In grade 8, only 3.7% of the students were at level 1, while 74.1% had reached level 3. At the same time, however, the errors students showed in the experiment continued to reflect some of the common misconceptions about fractions, which were particularly typical in that they had difficulty getting rid of the familiar integer representations and often misapplied them to their understanding of fractions, that is, integer bias. In the early stages of learning fractions, the most prominent problem children exhibit is integer bias; that is, they use the previously familiar integer schema to understand fractions and tend to use separate unit counts to understand fractions, thus often mistaking fractions with large denominators for larger ones. At this point, using discrete objects to provide them with concrete figurative fraction materials for practice can promote their understanding of fraction symbols and reduce the integer bias; moreover, its intervention effect is better than using number lines (Xin & Liu, 2013). In addition, our team conducted research on other issues in elementary school children’s fraction perceptions and obtained some valuable results. For example, it was found that children in grades 1 to 3 were equally disturbed by integer equivalence thinking on the equivalent fraction task, and their relative quantity concepts were not sufficiently developed (Xin & Han, 2014). Moreover, children in grades 4 to 6 were able to employ more than two strategies for proportional reasoning (Lai et al., 2016a, 2016b).

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The Characteristics and Development of Elementary School Students’ Level of Representation in Word Problems Mathematical word problems are problem situations that can be solved only by stating them in natural language and by performing mathematical operations. They are an important form of problems and teaching content in elementary school mathematics. The most critical step in solving mathematical word problems is to correctly represent the numeric relationships in the problem, which contain all the conditions needed to solve the problem. Different numeric relationships vary in their difficulty of representation; for example, both semantic meaning and hierarchical complexity affect the difficulty of representation in fraction multiplication word problems. Among them, the representation of comparative meanings is more difficult than combinations and conversions (Zhang et al., 2016). This may be because elementary school children understand multiplication operations more as the accumulation of a certain quantity, and the comparative relations in multiplication word problems are far from the accumulative meaning. Based on these findings, Xin and Zhang (2009) developed the measurement of students’ representation level on arithmetic word problems based on the relational representational complexity model for measuring children’s representation level. In addition, when solving word problems, individuals construct problem representations either in the form of a problem model, which extracts only information from the problem that is relevant to the solution of the problem (e.g., objects and quantities), or in the form of a contextual model, which extracts information from the problem that is easy to understand, including contextual information such as actions and events. It was found that students in grades 4 to 6 apply more contextual models to characterize the problem during the problem comprehension stage, whereas they drop much of the contextual information in favor of the problem model during the implementation stage. Student representations of problems indicating high performance were more focused on information directly related to problem solving, and this ability increased according to the grade level. In contrast, worse-performing students often had difficulty identifying redundant conditions, and this did not improve as they progressed through the grades (Zhong et al., 2014). In addition, as the difficulty of the questions increased, the gap between students’ different representation levels in terms of correct solution rate also widened. The more difficult the topic is, the more obvious the role of representation level becomes (Zhong et al., 2009).

Elementary School Students’ Emotional Experience of Mathematics and Mathematics Achievement There are significant individual differences in children’s mathematical abilities, some of which are related to the development of the basic cognitive functions described earlier, whereas others are related to noncognitive factors. Among the non-cognitive

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factors that influence the development of mathematical ability and academic performance in mathematics, the emotional experience of mathematics has received the most attention. The emotional experience of mathematics refers to various emotions directly related to the process of learning mathematics and academic performance in mathematics, such as enjoyment of learning mathematics, pride in successfully mastering mathematics concepts, or test anxiety related to mathematics. Our children’s emotional experiences of mathematics have been manifested since at least kindergarten (Lai et al., 2016a, 2016b). When faced with nonsymbolic mathematical tasks that they are more familiar with and better at completing, older children in kindergarten have more positive emotional experiences compared to those in grades 2 and 3 in elementary schools. However, this positive emotion disappears around third grade. At the same time, children’s negative experiences with symbolic mathematical tasks increase with grade level. When presented with pictures of lecture and problem-solving situations, third-grade children exhibited significant negative emotions. Indeed, the emotional experience of mathematical tasks and lecture situations in elementary school children predicts their mathematical performance; the more negative the emotion, the worse the mathematical performance. Among the various emotional experiences that affect the development of mathematical ability, mathematics anxiety has received the most attention. Mathematics anxiety refers to negative emotional experiences such as uneasiness, nervousness, fear, and worry when confronted with a mathematics-related situation. Our study found that mathematics anxiety affects mathematical performance in fifth-grade children, wherein the higher the level of anxiety is, the worse the mathematical performance and that mathematics anxiety affects children’s performance on mathematical tasks by negatively affecting working memory (Cui et al., 2011). Moreover, mathematics anxiety affects children’s level of metacognition about mathematics, which subsequently affects their mathematics performance (Zhou et al., 2014).

Analysis of the Characteristics of Elementary School Students with Mathematics Learning Difficulties Learning difficulties in mathematics reflect a group of children who have a normal level of intelligence, without significant sensory deficits or emotional disorders. These children receive normal mathematics education suitable for their age, but their mathematical ability and performance significantly lag behind their peers. Professor Chen’s research team conducted a long-term study of children with learning difficulties in mathematics in China and obtained several valuable results. In their study, the detection rates of children with mathematics learning difficulties in grades 2 to 6 were 8.03%, 2.26%, 3.00%, 2.75%, and 3.32%, respectively (Huang & Chen, 2016). It is evident that mathematics learning difficulties emerge in the early grades of elementary school. The mathematical cognitive abilities of children with learning

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difficulties in mathematics appear to increase slowly with grade level, consistently lagging behind average children overall. Children with mathematics learning difficulties struggle with mathematical cognition in several ways: they are less accurate than average children on number line estimation tasks, and they also use exponential representational forms in second grade (Han et al., 2010). In addition, they have developmental lags in symbolic and non-symbolic number conversion (Zhang et al., 2018) and spatial ability (Lai, et al., 2014) and are less competent than average children. They also lack metacognitive skills, such as monitoring the process and checking the results during word problem solving, and are less effective in executing a plan (Hao et al., 2011). Some researchers who studied the basic cognitive functions of children with learning difficulties in mathematics found that they were significantly less able to extract information from long-term memory activation and direct access areas, but their ability to switch the focus of their attention was not significantly different from that of the average child (Chen et al., 2011).

Improving Elementary School Students’ Abilities to Solve Word Problems Word problems in elementary school mathematics make up a great proportion of the problems distributed across all grades, with various kinds of tasks and levels of difficulty. Considerable class time is reserved for teaching word problems each semester. The study of elementary school students’ abilities to solve word problems is an important part of research on elementary school students’ arithmetic thinking abilities and intelligence. Notably, improving elementary school students’ abilities to solve word problems is not only necessary to improve the quality of elementary school teaching but also an essential way to develop elementary school students’ intelligence and thinking abilities. Common elementary school mathematics word problems include integer and fraction word problems. Elementary school students attempting to master the full application of the general trend of thinking activities generally go through four stages: mastering one-step word problems, mastering two-step word problems, mastering two or more steps of typical word problems, and mastering the various types of structure of word problems for integrated operations. In the teaching of elementary school word problems, it is necessary to effectively grasp the teaching of one- and two-step word problems because one-step word problems are the foundation and two-step word problems are the key that opens the door to solving a variety of more complex word problems. The ability of elementary school students to solve word problems should also include an understanding of the regularity of word problems and mastery of mental representation strategies and other mental abilities. The former refers not only to students’ ability to understand and solve rule-based word problems but also to their

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ability to think deeply about the relationship between real situations and mathematical operations (application of life knowledge) in the process of solving problems. The latter refers mainly to how to build teaching mental representations and to better master direct conversion strategies and problem-modeling strategies.

One-Step Word Problems: The Basis for Solving Compound Word Problems A one-step word problem is the first step to transforming a specific column calculation problem into a word problem. To solve a word problem, the calculation methods of addition, subtraction, multiplication, and division cannot be clarified as in the case of columnar calculation problems. Instead, students need to judge for themselves what method to use (independent thinking), which is an indicator of complete mastery of one-step word problems. Students are required to not only answer one-step word problems correctly; they must thoroughly understand the relationship between conditions and problems, that is, under which condition the problem can be solved and which numbers can be applied to create a new condition. Therefore, word problems can test students’ thinking ability to uncover the relationship and essential connections between things. There is a process of the development of elementary students’ ability to answer one-step word problems. An in-depth analysis of the formation process and its conditions reveals the following features. 1. The structure and nature of different types of one-step word problems require different ways of thinking, and the problem-solving process shows different levels of thinking abilities. To fully master one-step word problems, an individual first needs to know the relationship between two quantities. If you find the sum of two numbers, you must know what each of the two addends is; if you ask for the product of two numbers, you must know what each of the two factors is. However, this knowledge is not enough; students must be further guided to initially understand the relationship between the three quantities based on the concepts of addition, subtraction, multiplication, and division equations, that is, the relationship between addition and subtraction and the relationship between multiplication and division. For example, students are told that “Class A has 30 young pioneers and Class B has 15 young pioneers.” If you ask, “How many young pioneers are there in the two classes?” The answer is 30 + 15 = 45 (students). According to this equation, two subtraction questions can be asked: ➀ “There are 45 young pioneers in total in the two classes. It is already known that there are 30 in Class A, so how many are there in Class B?” and ➁ “There are 45 young pioneers total in the two classes. We already know that there are 15 in Class B, so how many are there in Class A?” In this way, the inverse relationship of the three quantities between addition and subtraction and sum and difference is clarified.

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Similarly, if you ask, “How many times more young pioneers are there in Class A than there are in Class B?” the answer is 30÷15 = 2. According to this equation, two new multiplication and division problems can also be asked: ➀ “There are 30 young pioneers in Class A. The number of young pioneers in Class A is twice as many as that in Class B. How many are there in Class B?” and ➁ “Class B has 15 young pioneers, and the number of young pioneers in Class A is twice as many as that in Class B. How many are there in Class A?” In this way, the inverse relationship of the three quantities between multiplication and division and product and quotient is also understood. Thus, addition, subtraction, multiplication, and division one-step word problems have a connection with each other, which is learning to look at the problem comprehensively and applying knowledge flexibly. If you can master one-step word problems, you can then learn two-step, three-step, and even more complex word problems. Thus, a one-step word problem is considered to be a reliable foundation. 2. In students’ arithmetic operation, the correct rate of positive condition questions (i.e., cis-conditional questions) is higher than that of negative condition questions (i.e., inverse questions). Therefore, fully grasping the one-step word problem requires a different thinking approach to different types of word problems, especially to guide students to be good at analyzing the conditions of the inverse topics. Here are two examples: (a) “Elder brother and younger brother went to cut grass together. The elder brother cut 20 kg, and the younger brother cut 10 kg. How much grass did they cut altogether?” and (b) “There are two ropes. The first one is 25 m long. It is 15 m shorter than the second one. How much longer is the second rope?” These two questions are one-step addition word problems, but I found in my experimental study that while first-grade students successfully answered the first question, nearly half of the students had difficulty answering the second question. It can be seen that in answering “more than” and “less than” word problems, it is easy to master the normal topic but difficult to master the inverse topic. In the past, some people thought that first-grade students could not easily understand word problems expressed in the form of questions and advocated teaching them in the third grade, while others advocated not teaching them at all. I believe that the development from the forward (one-way) to the reverse is the order of the intellectual development of the direction of thinking, not to mention the reason why elementary school students experience difficulty solving reverse word problems. Additionally, because they are accustomed to answering only the direct narrative “more than” and “less than” word problems, the formation of thinking hinders their ability to solve indirect word problems later. Therefore, to develop children’s intellectual flexibility from an early age and improve their ability to solve word problems, we should let them do a variety of word problems from the first grade onward, including both addition and subtraction problems of various structures as well as direct and indirect narrative topics. This diversity of integrated topics is extremely helpful to develop the flexibility and independence of the thinking activities of elementary school students. 3. The ability to propose questions lags behind the general arithmetic ability of word problems.

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For example, returning to the word problem mentioned earlier, “Class A has 30 young pioneers, Class B has 15 young pioneers…,” if the ellipsis is changed to a question, such as “How many more young pioneers are there in Class A than in Class B?,” students can calculate the correct answer without any difficulty. However, asking students to propose their own questions leaves a good majority of first graders dumbfounded and helpless. As we can see, elementary students’ thinking moves from passive to active in regard to answering one-step word problems. Therefore, experienced mathematics teachers inspire students to ask questions from the very beginning of their schooling. For elementary school students to fully master onestep word problems, they should not only learn to calculate based on two numbers but also be able to pose questions in many ways and apply their knowledge flexibly. This also lays a solid foundation for them to learn compound word problems and self-programming word problems in the future.

Two-Step Word Problems: The Key to Solving Compound Word Problems Since the two numbers required to solve two-step word problems often include a directly stated number and an indirectly stated number, the problem must have a hidden number, which requires students’ thinking activities to consider an additional intermediate link. The number cannot rely on direct judgment but depends on the form of reasoning, thus leaving the direct question and using internal speech at the heart of the “mental calculation.” This is the only way to further the calculation task. The aforementioned thought process is an indicator of mastery of two-step word problems. Still using the previous question as an example, we changed the question to “Class A has 30 young pioneers, and Class B has 15 fewer pioneers than Class A. How many young pioneers are there total in Classes A and B?” This problem has the same characteristics as the one-step word problem mentioned earlier—both belong to the exercise of “compare how many” and “find the sum,” i.e., add to find the total number of young pioneers in the two classes—the difference is that this problem lacks the number of young pioneers in Class B. How many young pioneers are there in Class B? The question does not directly state this, but gives a clue to find this number: Class B has 15 fewer young pioneers than Class A. The number of young pioneers in Class A is already known, so 30 − 15 = the number of young pioneers in Class B. This makes the sum of the two numbers complete: 30 + 15 = 45. A composite equation can be formed: 30 + (30 − 15) = 30 + 15 = 45(students). This is how the two-step word problem is formed. The important approach in two-step word problem calculation is to move from step-by-step presentation to integrated presentation. This highlights the characteristics of the two-step word problem: (a) it is composed of two one-step word problems, and (b) it is important to analyze the interdependence of the quantities possessed by

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each one-step word problem. It is difficult to make elementary school students understand the dependencies between the quantities of two-step word problems and what numbers are needed, what conditions must be present, and with what conditions, as well as what numbers must be or could be sought. An experienced teacher in an elementary school analyzed this problem. The following example provides insight into a similar problem. “The factory plans to produce 1200 sets of uniforms and has completed 38 of production.” [According to these two ] conditions, either how many sets have been completed 1200 × 38 = 450(sets) or how many sets remain to be completed [ ( ) ] 1200 × 1 − 38 = 750(sets) is needed; one of the two must be present. Which condition is needed for the development of the problem must depend on what number is required at the end of the problem. If it is said that there remain 10 days to complete production, then the remaining number of sets is the first detail to determine. Alternatively, if it is said, “The factory took only 9 days to complete the task of 38 production, the remaining uniforms need to be completed in 10 days.” Based on the average daily production, how many sets can be completed? This requires not only the condition that 450 sets have been completed but also the condition that 750 sets have not been completed. Above, we start from the known condition, and we know what number we can find when the condition appears. In turn, starting from the problem and then based on what problem arises, we can determine what condition must be found. As another example, “How many sets need to be produced on average per day for the remaining 10 days to complete production…?” To answer this question, you must find the hidden condition of “how many sets are left” because the condition of 10 days is stated in the question. To determine how many sets are left, you must know how many sets are planned to be produced and how many sets have already been completed. It is stated in the question that the planned production is 1200 sets. Next, the hidden number of completed sets must also be found. The question says that 38 have been completed, 1200 × 38 = 450 (sets). How many sets are left? 1200 − 450 = 750 (sets). How many sets are produced on average per day? 750 ÷ 10 = 75 (sets). In this way, the problem is solved smoothly. Teaching two-step word problems is not only about making students know and understand what conditions must be present to solve a problem but also about developing their analytical reasoning skills. If a condition is missing in a problem, you have to analyze and reason based on the quantitative relationship to find it. The teaching of two-step word problems plays an important role in developing students’ analytical reasoning ability. For this reason, I think two-step word problems are the key to analyzing and solving word problems. Some students find it difficult to solve word problems with more than two steps, and they do not even know where to start because teachers do not make students master the method of analytical reasoning when teaching two-step word problems. When such students are encountered, teachers should start by training them in how to answer two-step word problems.

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Mastering the Various Types of Structures of Word Problems for Composite Operations Students must master the various structures of compound word problems and perform integrated operations to systematize their ability to solve word problems and “simplify” their intellectual activities. The mastery of typical word problems with two or more steps is a significant manifestation of integrated arithmetic. The index of mastering typical word problems requires students to use certain methods to answer compound word problems with certain characteristics that require further analysis of the quantitative relationships between known and unknown conditions in the problem. A typical word problem has its own characteristics in form and its own rules for calculation. Students must recognize this feature, master this law, think about the problem more comprehensively, and apply their knowledge and skills flexibly. When teaching typical word problems, teachers should clarify various knowledge structures and ensure students understand several common typical word problems and their characteristics, such as the average problem, the normalization problem, the multiplication problem, the proportional distribution problem, the sum-fold problem, the difference-fold problem, the sumdifference problem, and the travel problem. This will allow students to understand and master the laws of arithmetic early on, thus enhancing their ability to answer word problems. When elementary school students perform various types of word problems, new indicators of the thinking process emerge, namely, “generalization-systematization” and “abstraction-concretization.” For example, students are able to breakdown a problem into two, three, or more steps and combine them into a single step; i.e., they master the structure of various types of word problems and perform comprehensive operations. The various steps or reasoning processes of this thinking activity are gradually simplified or certain “links” are omitted to perform operations faster. The ability to analyze problems is an important part of the thinking process. Guiding students to create their own word problems is not only an effective way to guide them to break through the difficulties of word problems and further understand the interdependence of quantities but also an essential method to improve the creativity of students’ thinking. In the process of brainstorming problems, students can further experience how a two-step, three-step, and multistep word problem is composed of simple one-step word problems. In this way, when they encounter a two-step, three-step, or multistep word problem in the future, they will be able to break it down into several one-step word problems or solve it step by step. At the same time, students can solve multistep word problems with integrated formulas and synthesize all of the problems in one step. By doing this repeatedly, students’ analytical and synthesizing abilities are continuously improved through self-programming word problems. It is in this process that elementary students’ ability to solve word problems improves. The structure of their thinking is gradually systematized and organized on

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this basis—the more tedious steps are omitted and tend to be “internalized,” and the “quantity” of their intellectual activity is gradually “simplified.”

Research on the Regularity of Mathematical Word Problems In recent years, my student, Professor Yinghe Chen, has been conducting in-depth research on the regularity of students’ word problems. She noted that written word problems, that is, general word problems, are a kind of practice task that connect mathematical knowledge with real situations. However, through an analysis of relevant domestic and international studies, Chen et al. (2003) illustrated that students perform better in solving textbook regular word problems than irregular word problems that are closer to real-life contexts. The form of regular word problems, students’ mental representations, teachers’ attitudes, and teacher–student relationships all contribute to students’ inability to analyze irregular word problems in context. Chen et al. also highlighted the limitations of regular word problems and made some suggestions for future teaching reform. A word problem is a task that is linked to a real-life situation and requires the subject to choose an arithmetic form and perform calculations to solve the problem; it is an extension and expansion of numerical problems. Topics simulating reallife occurrences not only exercise students’ arithmetic ability but also, and more importantly, their mathematical judgment and reasoning abilities in a context close to reality, thus achieving the purpose of cultivating students’ mathematical thinking and improving their real-life problem-solving abilities. However, most of the word problems in textbooks currently ignore the in-depth consideration of the realistic situations in the topics and evolve into a kind of practice problem that simply exercises students’ calculation ability, which somewhat deviates from the original purpose of word problem design. According to the different understandings of the function of word problems, Chen et al. divided them into regular and irregular word problems. The so-called regular word problems are those word problems that appear frequently in traditional mathematics classrooms—those are stereotyped, standardized, and must have a solution—and students can obtain the correct answer by performing operations on all the numbers appearing in the problem. Irregular word problems are those that are closer to real life and may or may not have a solution; the conditions in the problem may be sufficient or missing, and some conditions may be necessary or unnecessary. Students have to use their lived experiences and reason with mathematical thinking to solve these problems. Teachers should first analyze the reasons for the difference in performance in solving the two types of word problems and then teach them in a targeted manner. Students are unable to analyze irregular word problems in context because of cognitive deficits. That is, the real reason is the stereotypical form of regular word problems, students’ stereotypical mental representations of word problems, teachers’ negative

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attitudes toward irregular word problems, and hierarchical relationships in teacher– student interactions that often cause students to fail to solve problems correctly. Therefore, mathematics teaching should emphasize the establishment of relationships among conceptual, declarative, and procedural knowledge. Moreover, students should focus on both mathematical computation and understanding the nature of mathematical knowledge so that they can engage in broader mathematical reasoning that allows them to solve problems in different contexts. Rule word problems and their current teaching model precisely neglect to combine textbook knowledge with practical problem-solving abilities, so they are not conducive to the establishment of relationships between the three types of knowledge, thus hindering the development of students’ mathematical ability and limiting their creative thinking. Meanwhile, in addition to resolving the above issues, there are other ways to improve students’ effectiveness in solving irregular word problems. One example is encouraging students to have active discussions in class so that they can work together to find the right way to solve problems in a cooperative learning atmosphere. This can, to some extent, eliminate the single solution idea taught by teachers. Instead, teachers can use multiple methods, such as diagrams and objects, to represent problems and break the stereotypical representations in students’ minds. Another option is utilizing modern teaching technology to present problems via multimedia. The multimedia teaching environment can make the presentation of problems more concrete and vivid, thus helping students consider problems from both visual and auditory perspectives in a holistic manner and promoting the development of their mathematical thinking. In short, teachers should use flexible and varied forms of instruction to help students better understand problems and solve them successfully.

Strategies for Valuing Mental Representations of Mathematical Word Problems The issue of strategies for the mental representation of mathematical word problems is also a recent result of Professor Chen’s research. The term representation seems slightly complicated and abstract. An early concept in psychology is image, which refers to the perceptual image formed in the mind based on perception, including both memory and imagination images; these are more specific than the term representation. In the early development of psychology, specific aspects of image were mainly evaluated quantitatively. After the rise of modern cognitive psychology, the study of image developed rapidly, and most studies focused on the representation of information (where representation seems to mean expression), such as the studies of mental rotation and scanning, which are quite novel. The dual-code theory of cognitive psychology (Paivio, 2006) asserts that information is stored in our long-term memory in the form of both visual image and verbal

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representation. Thus, representations or mental representations are defined as information symbols that represent coded external things or events in the information processing system. (Lin et al., 2003). Mathematical mental representations are, of course, coded information symbols representing mathematical objects in the mathematical information processing system. Examples include mathematical symbolic constructions, the confirmation of the meaning of mathematical concepts, spatial schema and strategy-inspired processes, and even emotional factors associated with mathematics. There are two strategies in mathematical mental representation: the direct translation strategy and the problem-modeling strategy. The direct translation strategy means that when the subject is confronted with a mathematical word problem, he or she first selects numbers from the problem and then processes them. This first strategy thus emphasizes quantitative reasoning, that is, the arithmetic process. In contrast, the problem-modeling strategy means that when the subject is confronted with a mathematical word problem, he or she first tries to understand the situation of the problem and then makes a plan based on the contextual representation. Accordingly, the latter strategy emphasizes qualitative reasoning, that is, understanding the relationship between conditions in the problem. Chen et al. (2004) reviewed relevant studies from abroad and identified four basic stages in mathematical problem solving: translation, integration, planning, and execution. Translation is the construction of a mental representation of the conditions in a problem. Integration is the construction of a mental representation of the relationships between conditions in a problem. Planning is the development of a solution, and execution is the implementation of the solution. Based on these four stages, the researcher concluded that there are also four corresponding stages in the mental representation of mathematical word problems. Specifically, the cognitive process of the direct translation strategy consists of updating the database, selecting numbers and keywords, generating, and then executing a plan. The cognitive process of the problem-modeling strategy differs from that of the direct translation strategy in the second stage (selection of numbers and keywords vs. the construction of a contextual model) (see Fig. 8.2). Chen’s team has performed much research on the issue of strategies for representing mathematical word problems for elementary school students, specifically on strategies for mathematical word problem representation of elementary school students in grades 2 to 4 (Chen et al., 2004). By applying experimental and clinical interview methods, they administered a mathematics word problem test to secondto fourth-grade students in a general elementary school to investigate the differences in the representational strategies of mathematics between academically gifted and struggling students in solving comparative word problems. The results indicated the following: 1. From the second- to fourth-grade students’ solutions to consistent and inconsistent word problems, it appeared that the higher-performing students used

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Direct translation strategy

Problem-modelling strategy

Database updating

Database updating

Translation stage

Numbers and keywords selecting

Contextual model construction

Integration stage

Plan generation

Plan generation

Planning stage

Plan execution

Plan execution

Execution stage

Fig. 8.2 Comparison of the cognitive processes of the two strategies

more problem-modeling strategies to represent the problems, while the worseperforming students used more direct translation strategies to represent the problems. 2. The gender differences were not significant in all areas except for the fact that worse-performing girls’ correct problem-solving rates were lower than those of worse-performing boys and that worse-performing girls used more direct translation strategies than worse-performing boys in their self-reports. 3. As the grade level increased, the academically gifted students became more sophisticated in using problem-modeling strategies. The struggling students did not learn to use more effective problem-modeling strategies and continued to utilize direct translation strategies, but they improved in terms of their knowledge about strategy use (Chen et al., 2004). Chen et al. also conducted a study on mathematical word problem representation strategies of elementary school students in grades 4 to 6. This study used an experimental method to administer a rectangular area task and the mathematical processing instrument (MPI) to 161 students in grades 4–6 in a general elementary school to examine the level of students’ use of visual-spatial representations in solving mathematical word problems and their effects on problem solving. The results showed the following: 1. Representation levels increased with grade level, and the gender differences in students’ representation levels were not significant. 2. Regardless of the grade level, in terms of the level of representation, the topperforming students are better than the average-performing students, and the average-performing students are better than the worst-performing students. That is, the higher the level of representation, the better the student’s problem-solving result. 3. As the difficulty of the questions increased, the gap in correct problemsolving rates between students at each level of representation also increased. In other words, the more difficult the problem, the more pronounced the role of representation level in problem solving (Zhong et al., 2009).

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It is evident that elementary school students’ mistakes in answering word problems are due to the failure of stored representations of problem information. Representation strategies play a crucial role in solving complex or new problems; elementary school students who build contextual models not only extract information from the questions relevant to problem solving but also construct problem representations of the information they understand, such as objects, behaviors, and relationships between events, which in turn contribute to their understanding of word problems.

From “Wormy Formula” to Thinking Training Questions In the early 1980s, I saw that other countries were using traditional four-operation word problems to train students’ thinking while simultaneously introducing more new thinking problems. One of them, called the “Wormy Formula,” caught my interest. In the first edition of “Intellectual Development and Mathematics Learning” (1982), I wrote a section on “The thinking training of Wormy Formula.” Now that “Wormy Formula” is a component, or a small part, of Olympic mathematics for elementary school students, I think it is necessary to discuss this type of problem for elementary school students.

“Wormy Formula” Thinking Training Questions In 1982, before the book “Intellectual Development and Mathematics Learning” was published, I saw the so-called Wormy Formula thinking training problems and thought it was not a novel thinking training method. I said at that time, “Wormy Formula” thinking training problems are very common in our elementary school mathematics teaching—these are the fill-in-the-blank type questions. This type of exercise requires more thinking. Since this kind of thinking exercise requires logical reasoning based on the provided conditions to find the correct answer, it develops elementary school students’ logical reasoning abilities. I also made a comparison between the so-called Wormy Formula and fill-in-the-blank problems. The first six questions below are typical examples of “worm-eating arithmetic.” Example 1 Try to fill in the “⛛” of the following equation with the appropriate number to make the equation work.

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The process is omitted. The answer is 975 × 18 = 17,550. Example 2 Try to replace the letters in the following equation with appropriate numbers to make the equation work.

The process is omitted. The answer is N = 7. In fact, such arithmetic problems were occasionally encountered in elementary school arithmetic in the past and were only largely referred to as training questions recently. The value of these training questions is that they contribute to students’ skills of observation, analysis, and synthesis, as well as reasoning ability. The following is an illustration of the process of solving the test problem. Example 3 Try to replace “×” in the following equation with an appropriate number to make the equation work.

Analyzing the feature of the equation, since the last digit in (2) is 7, we can obtain that the unit of multiplicand is 3, and the last digit in (1) is 1, so the unit of multiplier is 7. Then, according to the known digit in (2), it can be found that the multiplicand is 3413. Hence, the equation is: 3413 × 97 = 23,891 + 307,170 = 331,061.

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Example 4 Try to replace “×” in the following equation with the appropriate number to make the equation work.

Solution: From (1), the first digit of the quotient must be 2, and from (2), the third digit is 6, so the last digit in (1) must also be 8, so the unit digit of the divisor is 9. Since the divisor is 319, the quotient is 274, and it is not difficult to fill in the empty digits. From the front course, the basic process of solving such problems is to observe and analyze the characteristics of the given equation first, analyze all the possible reasons based on these characteristics, then test each of the possible reasons and find the digits of each blank space in turn. Let us illustrate the basic process of solving this problem with a more complicated example. Example 5 Try different numbers to represent different letters in the following equation so that the equation works.

Solution: From the tens digit of the answer Q = Q + T and the hundreds digit A = A + T + T , we know that T = 0. Additionally, from result (2) obtained by multiplying the tens digit D of the multiplier by multiplicand we can get D × G = D, so G = 1. Because the last digit of N × E is N, the last digit of N × D is 0, and the last digit of N × A is 0, we can get N = 5. Further, E = 3 or 7 or 9; by examining whether they meet the requirements of (1) in turn we can get E = 9. Then from the requirements of (1), we can get Q = 2, R = 3, and D = 4.

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Since the multiplicand multiplied by A is (3), we can get A = 6 or 8. The remaining L = 7 and S = 6. Therefore, the original equation is 123,425 × 67,849 = 8,374,262,825. Example 6 Try to replace the “×” sign in the following equation with an appropriate number so that the equation works.

Solution: Since (6) is obtained by dropping two digits in a row from the divisor, the ten digits of the quotient are 0. Multiplying the divisor by 7 gives (3) a three-digit number, and (2) minus (3) still gives a three-digit number. Therefore, the hundreds digit of the divisor is 1, and the tens of digits is either 1 or 2. If we analyze and compare the cases of (1), (2), …, and (7), we know that the quotient must be 97,809. If the tens digit of divisor is 1, the maximum divisor is 119. However, 119 × 8 = 952, even if (4) takes the minimum value of 1000, then (4) minus (5) is 1000 − 952 = 48, so (6) is 48 ××, which is also impossible. Because 48 × divided by 119 is 4, we cannot get the quotient of 0. Therefore, the tens digit of divisor must be 2. Further, from (5) we know that the divisor 12 × multiplied by 8 gets a three-digit number, so the divisor’s units digit can only be less than or equal to 4 with which to test we obtain 12,128,316 ÷ 124 = 97,809. From the above examples and their solutions, it can be seen that such exercises require students to constantly analyze, explore, and experiment with various solutions, but basically, they are not beyond the student’s original knowledge and intellectual level. Therefore, I think this is a good type of thinking training problem for upper elementary and lower middle grades. I also provide examples of fill-in-the-blank problems in China for comparative analysis. Example 7 Fill in the following brackets with numbers so that the equation works.

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Take (4) as an example. The unit digit of () × 7 is 1. In the “seven” multiplication table, there is only 3 × 7 = 21, so the units digit of the multiplicand should be filled with 3; the product of () × 7 in the tens position is only a single digit. In the “seven” multiplication table, there is only 1 × 7 = 7, so the answer is 13 × 7 = 91. The answers to the above fill-in-the-blank questions are (1) 68 + 13 = 81; (2) 52 – 28 = 24; (3) 104 – 95 = 9; (4) 13 × 7 = 91; (5) 92 × 9 = 828; and (6) 102 ÷ 48 = 2…6. Example 8 Find the numbers represented by a, b, c, d, or ✩, Δ (different letters or symbols represent different numbers in the same question).

Take (4) as an example. ➀ The multiplicand is a four-digit number, multiplied by 9, and the product is still a four-digit number, so a can only be 1. ➁ The units of the product are 1; i.e., the units digit of the product of d × 9 is 1, so d can only be 9.

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➂ As a = 1, b /= 1. If b is any number from 2 to 9, the product of b × 9 must be a two-digit number. In this way, the 9 in product’s thousands place plus any number not equal to 0 must be carried to five digits, which does not meet the requirement, so b can only be 0. ➃ The product of multiplicand multiplied by multiplier 9 is 81, which is carried to 8. However, the product has 0 in the tens place, so the units digit of the product of c × 9 should be 2. Therefore, c can only be 8. 5. ➄ The answer is 1089 × 9 = 9801. The answers to the whole letter-symbol problem are (1) a = 8; (2) ✩ = 5, Δ = 4; (3) a = 8, b = 1, c = 3, d = 0; and (4) a = 1, b = 0, c = 8, d = 9. Example 9 These are word problems. 1. Three 1s and two 0s make up a five-digit number. To make this five-digit number read: (1) none of the 0s; or (2) both of the 0s; or (3) only one of the 0s (more than one answer). Let us use (3) as an example to perform the arithmetic. Reasoning: There are two cases of reading with only one 0: one is a 0 is at the end of the number and another 0 is in the middle of the number, so there are two answers, i.e., 10,110 or 11,010; the other is two 0s are put together (but not at the end of the number), so there are also two answers, i.e., 10,011 or 11,001. Therefore, there are four answers to this question, i.e., 10,110/11,010/10,011/11,001. 10,011 and 11,001. The answers to this question are (1) 11,100; (2) 10,101; and (3) 10,110, 11,010, 10,011, and 11,001. 2. “2 + 3 = 5.” In this addition equation, the two addends and their sum are prime numbers. How many addition equations can you give where the sum does not exceed 40 and the two addends and the sum are prime numbers? Reasoning: ➀ All prime numbers except 2 are odd. The sum of two odd numbers must be an even number, and any even number greater than 2 is a composite number. To satisfy the requirement that the sum of two prime numbers is still a prime number, one of the addends must be 2. ➁ An analysis of two consecutive odd numbers up to 40 that are prime shows 3 and 5, 5 and 7, 11 and 13, 17 and 19, and 29 and 31. ➂ The answer to this question is 2 + 5 = 7, 2 + 11 = 13, 2 + 17 = 19, and 2 + 29 = 31, for a total of four addition equations. 3. A and B each took several saplings to plant. If A gives 1 sapling to B, the number of saplings for both of them is equal; if B gives 1 sapling to A, the number of plants for A is 2 times that of B. Q: How many saplings each did A and B originally take? Reasoning:

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➀ The total number of saplings remains the same whether A gives to B or B gives to A. ➁ If A gives 1 sapling to B, the number of saplings is equal, so we can see that A has 2 more saplings than B. ➂ If B gives 1 sapling to A, then A has 4 more saplings than B. When there are 4 more saplings, A is twice as many saplings as B, so B has 4 saplings and A has 8 saplings. ➃ Therefore, it turns out that A took 8 − 1 = 7 (plants) and B took 4 + 1 = 5 (plants). The teacher can combine mathematical training with teaching, giving students some exercises similar with “Wormy Formula” or fill-in-theblank problems. In this way, students can be appropriately taught some key methods of problem solving, which is conducive to the development of their thinking abilities and intelligence.

Thinking Training Questions and Mathematical Olympics Training Both the Wormy Formula and fill-in-the-blank questions are thinking training questions. In the use of mathematics for thinking training, China’s mathematical community has the longest history and the richest content. Famous problems such as the “chicken and rabbit cage” problem are common today in our elementary school thirdand fourth-grade training; this kind of household name in mathematical thinking training problems began as early as the ancient “Sun Tzu Arithmetic” appeared. In a cage, there are chickens and rabbits; a total of 10 heads and 28 feet can be counted. Q: How many rabbits are there?

The numbers can be flexibly changed. Sometimes the mathematics teacher turns the heads into 46 and the feet into 103 and other times the heads into 30 and the feet into 70. The “chicken and rabbit cage” problem can be used flexibly in similar mathematic examples. For example, a similar hypothetical problem would be: There are 27 bills comprising 10 yuan and 50 yuan, totaling 990 yuan. Q: how many bills of 10 yuan and 50 yuan specifically?

There is a Chinese folk saying: A team of hunters and a team of dogs, two teams walk together as one. The total number of heads is 306, and the total number of feet is 809. Q: How many hunters and how many dogs are there?

It is important to note that any mathematical topic can be used as a test question for thinking training; the key is how the teacher applies it. Experienced elementary school teachers emphasize strengthening thinking training in their daily mathematics teaching.

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In modern times, such difficult thinking training questions are used as elementary school “MO” questions. The term “MO” is short for “Mathematics Olympic,” also known as the “Mathematical Olympiad,” which is the result of people comparing mathematical competitions with athletic sports. In 1894, the Hungarian Mathematical Society decided to hold a national annual mathematics competition for secondary school students each October to commemorate a member of the society becoming the Hungarian Minister of Education. After decades of practice, several world-renowned mathematicians emerged from the Hungarian contestants, and the secondary school mathematics competition became the birthplace of mathematical masters. In 1934, the Soviet Union held a mathematics competition for secondary school students in Leningrad (now St. Petersburg), which was called the “Secondary School Mathematical Olympiad,” linking the mathematics competition with the Olympic sports competition in ancient Greece for the first time and implying that the mathematics competition would carry forward the spirit of the Olympic Movement. In these cases, the spirit of the Olympic competition is centered on human intellectual competition. In 1956, Luogeng Hua, a mathematician of the old generation, after examining the Mathematical Olympiad (MO) in the Soviet Union, thought that such an activity was exactly what was needed for the development of the new China. Consequently, he joined Buqing Su, Zehan Jiang, and other famous mathematicians in China in advocating for a Chinese version of the competition. In time, many provinces and cities joined the competition, which lasted until 1964. The event was interrupted from 1965 until 1978, when it was resumed, and in 1981, the high school mathematics competition was started in the form of a national league, which was expanded to include middle school students in 1985 and then to the elementary school level in 1991. By this time, the pattern of mathematics leagues for elementary and secondary school students in China had taken shape, and Chinese students had begun to achieve impressive results in international competitions. The original purpose of MO was to popularize mathematics, cultivate students’ interest in mathematics, and motivate mathematics enthusiasts to dare to challenge scientific problems. However, in some provinces and cities, schools have linked the results of the Olympiad to higher education, leading to an increasingly obvious trend of utilitarianism, which is contrary to the original purpose of the Olympiad. Although the current trend of MO has become tasteless, we must not ensure that MO continues. In the next chapter of this book, I have written a special section on “Secondary School Olympiads and the Intellectual Development of Secondary School Students.” For the elementary school Olympiad, I regard it as a mathematical thinking exercise of some difficulty for elementary school students. I have read many books on elementary and secondary school MO training, including a set of “100 types of MO training books” edited by Mr. Biao Xu and published by Nanjing University Press, which I greatly appreciate. There are 100 topics in each of the four grades from grades 3 to 6, and each topic is used as a unit of mathematical thinking training, showing the four characteristics of “comprehensive content, spiral direction; from the foundation, focus on improving; an example of multiple practice, to reflect on three; and with the times, follow the times” (Xu, 2006). This kind of mathematical thinking

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training is helpful to further improve the thinking ability and even the intelligence of elementary school students. It can inspire many students to learn to look at problems comprehensively, to learn how to reveal the initial laws of quantity and form and their intrinsic relationships and to master the methods of comparison (analogy and contrast), proposition, and reasoning. It should be noted that some of the questions have depth and difficulty. For example, here is a fourth-grade question. Each bus needs 8 manufacturing parts of type A and 3 parts of type B. Each minibus needs 4 manufacturing parts of type A and 10 parts of type B. To date, 52 parts of type A and 79 parts of type B have been used. How many buses have been assembled from these parts?

At first glance, I was stumped by the question, but then I made a quadratic equation and solved the problem. After that, I thought about the problem and analyzed the conditions given in the question that enables the problem solver to determine which quantity is equivalent to the large bus and which quantity is equivalent to the small bus, the number of parts needed for the large and small buses, respectively, and the total number of parts used (A and B). Then, based on the correct assumptions, the problem was solved by correct projection with substitution according to the quantity relationship given in the question. The answer is 3 buses and 7 min. From this, I concluded that a significant number of elementary school students can devote themselves to training in OM if they have the foundation and the conditions to do so. However, given that each student’s interests and hobbies differ, as do their future development characteristics, I am against the inaccurate slogans of “Olympiad knowledge for all” and “Learning Olympiad knowledge starts from elementary school.” This not only defeats the original purpose of the Olympiad but also goes against the requirement of teaching students based on their aptitude, thus placing an excessive academic burden on students. All elementary school students should learn the basic knowledge of elementary school mathematics as stipulated in the national curriculum. Instead of forcing every student to participate in Olympic training, teachers should strengthen their thinking training in the teaching of this basic knowledge. From the requirements of mass education, we should strengthen the training of students’ thinking in daily mathematics teaching. The understanding that only Olympiad questions can be used to improve thinking levels is wrong.

Key Points that Elementary School Mathematics Teaching Should Pay Attention to To develop the thinking ability and intelligence of elementary school students, it is crucial to improve the teaching of elementary school mathematics. From the various requirements of elementary school mathematics pedagogy, what particular requirements should elementary school mathematics teachers pay attention to when seeking to develop students’ logical thinking abilities and intelligence?

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Explain the Concepts, Laws, Formulas, and Methods of Problem Solving The concepts, rules, formulas, and methods of solving problems in elementary school mathematics reflect the basic knowledge, skills, ideas, and activities necessary for elementary school students to adapt to social life and further development. This is one of the general objectives of mathematics learning for elementary school students. The Mathematics Curriculum Standard for Compulsory Education (2011 edition) set out the knowledge and ability requirements for elementary school grades 1 to 3 and 4 to 6. Academic Period 1 (Grades 1 to 3) 1. Experience the process of abstracting numbers from daily life, understand the meaning of numbers up to ten thousand, and have a preliminary understanding of fractions and decimals. Understand common quantities: appreciate the meaning of the four operations, master the necessary arithmetic abilities, and be able to perform operations accurately. In concrete situations, appropriate units can be chosen for simple estimation. 2. Experience the process of abstracting simple geometry and plane figures from actual objects and understanding some simple geometry and common plane figures. Feel the phenomenon of translation, rotation, and axis symmetry. Recognize the relative position of objects. Master the preliminary skills of measurement, map reading, and drawing. 3. Experience the process of collecting, organizing, and analyzing simple data and understanding simple data processing methods. Academic Period 2 (Grades 4 to 6) 1. Experience the process of abstracting numbers from concrete situations and recognizing numbers over ten thousand. Understand the meaning of fractions, decimals, and percentages and the meaning of negative numbers. Master the necessary arithmetic abilities. Understand the meaning of estimation, be able to represent simple quantitative relationships with equations and be able to solve simple equations. 2. Explore the shape, size, and position relationships of some figures. Understand the basic features of some geometric bodies and plane figures. Experience the process of motion of simple figures; be able to draw simple figures after motion on square paper. Understand some basic methods of determining the position of objects. Master the basic methods of measurement, map reading, and drawing. 3. Experience the process of collecting, organizing, and analyzing data and master some simple data processing abilities. Experience random events and the equal likelihood of events occurring. 4. Be able to solve simple word problems with the help of a calculator. (Ministry of Education of the People’s Republic of China, 2012, pp. 10–12, cited with modifications)

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The mathematical concepts, rules, and formulas that reflect the basic knowledge and skills of elementary school mathematics do not seem difficult to understand at first glance, but they are not easy to teach in practice, especially in regard to making students understand, master, and make them their own, i.e., the process of “internalization.” Each unit or lesson in elementary school mathematics often begins with some new concepts. For example, when you start learning about fractions, regarding the words “fraction” and “number,” there are numerous different concepts, such as approximate, multiple, prime, composite, prime factor, reciprocal prime, common divisor, greatest common divisor, and least common divisor. This is followed by common fractions, approximate fractions, fractions, numerators, denominators, true fractions, false fractions, band fractions, etc. The development of thinking ability also starts from the “cells” of these basic concepts. Therefore, whether it is lesson planning or classroom teaching, we should seriously consider how to make these concepts clear—not only because it seems very simple and easy to let go but also because a seemingly simple concept is often the most basic core of mathematics. The beginning of a new semester means the start of new material for any grade level. Some talk about area, some about fractions, some about decimals, etc. The concepts established at the beginning of a book or unit are the foundation of this knowledge and must be learned solidly, without any “almost” thinking. The requirements for mastering basic mathematical concepts, rules, formulas, and problemsolving methods should be extraordinarily strict. I have already talked about this in the previous chapters, so I will not repeat it here. In conclusion, knowledge and skills are both foundational goals for student development and a vehicle for implementing the goals of mathematical thinking and problem solving, which are the foundation for developing elementary school students’ intelligence. Therefore, we must focus on students’ understanding and mastery of basic knowledge and skills. For the same reason, we lay a solid foundation before building a house, and we must ensure that students have a firm grasp of core concepts. Seeking to rework or repair a weak foundation after the walls are built high and appear crooked and distorted is too late.

Highlighting the Key Points, Solving the Difficulties, and Explaining the Doubts Elementary school teaching is very systematic and, as aforementioned, cannot be arbitrarily abridged. However, there is a distinction between major and minor in terms of textbook content. In any mathematics textbook, there is some content that is substantial, important, or major in the kind of knowledge offered, which constitutes the focus of classroom instruction. The primary content is very important for further learning and is the pillar of basic knowledge for developing intelligence, which is the focus of the textbook. Generally, in the lower and middle grades, the focus is on the four operations with integers up “20” (the addition of a single digit and its

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corresponding subtraction), while the focus of “100” is the addition and subtraction of two digits, multiplication, and division within a table. The focus of “10,000” is multidigit addition and subtraction and multidigit multiplication and division, and the focus of “1,000,000,000” (billion) is multidigit multiplication and division. When teaching, instructors should highlight the key points in the material so that students can learn the main content well. In mathematics teaching, some content is not easily understood or mastered by students, which constitutes a difficult point in classroom teaching, for example, the concept of fractions. Difficult points in the textbook should be solved by appropriate approaches according to different situations. Teachers need to identify these key points when teaching and discuss the difficult points. There are some key and difficult points in mathematics textbooks that can often play a decisive role in mastering a certain part of knowledge. These elements are key to understanding the other contents in the textbook. For example, in general, trial quotients are the key to multidigit division, and the handling of decimal points is the key to the four operations with decimals. Teachers should seize the key in the textbook and concentrate on explaining it clearly and thoroughly so that students can learn and practice it well. Of course, sometimes the focus of mathematics teaching is the difficult point or the two are quite close; for example, “the nature of fractions” is both the focus and the difficult point. At other times, the focus of teaching and the difficult point are some distance apart. All these are what we should pay attention to in teaching. In other words, to teach a mathematics textbook, we should grasp the key, the substantive content of the textbook, master the key points and difficulties, familiarize ourselves with their status in the overall textbook, and understand their interconnection with the textbook before and after the place. The so-called doubtful points are where students are easily confused or misunderstand the content of mathematics. For example, according to the properties of division, fraction, and fractions, the divisor and the dividend, the numerator and the denominator are multiplied or divided by a number, and the size of the number remains the same. These are key points we should pay attention to when teaching.

Strengthen the Practice Practice is not only a means of review conducive to the development of memory but also a kind of “abstract-concrete” thinking and a deductive reasoning process, which aids in the development of logical thinking abilities. In teaching, classroom lectures and exercises are complementary and mutually reinforcing so that students can better grasp knowledge and develop their intelligence. I would like to make three suggestions on how to guide elementary school students to practice. Pat attention to “Timely Reinforcement.” Fig. 8.3 is a curve describing the speed of forgetting created by the German psychologist Ebbinghaus (1885). It shows the relationship between the forgetting variable and the time variable. As seen, the

Percentage of material memorization after different time intervals/%

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Time/Day

Fig. 8.3 A curve describing the speed of forgetting

just-learned material is rapidly forgotten within the first few hours, but the rate of forgetting slows after two days and is even slower thereafter. Although the degree of memorization is directly related to the material memorized, it is still important to review and practice in a timely manner. How can students be guided to practice effectively? It is important to note the following: (a) practice must be based on an understanding of basic concepts, definitions, and laws; (b) practice must have clear and strict requirements and be “reinforced” in a timely manner so that students know early on what is correct and what is wrong, as well as the nature of the errors; (c) practice must stimulate creativity and flexibility; and (d) practice must be constantly demanding and well planned. The plan for speed, correctness, and the content, methods, and forms of practice should be diversified. Teachers should pay attention to the allocation of practice time and focus on teaching students within their aptitude, making certain arrangements for both outstanding students and students with difficulties, and treating them differently. Be Flexible. The amount of “practice” and the quality of “practice” assigned to students must be flexible enough to take into account the differences in students’ development. When discussing “flexibility in the design of the content of teaching materials,” the Mathematics Curriculum Standard for Compulsory Education (2011 edition) cite six areas that are appropriate for student practice. 1. Ask different levels of questions or open-ended questions about the same problem situation. 2. Provide a certain amount of reading material, including historical information, background materials, and applications of knowledge for students to choose to read. 3. The selection and arrangement of exercises highlight the hierarchy. Set consolidation problems, extension problems, and exploration problems; any exercises that do not require mastery by all students need to be clearly marked.

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4. When designing integrated and practical activities, we choose topics that enable all students to participate; different students can gain different experiences through problem-solving activities. 5. Compile some optional learning content that broadens knowledge or methods. The added content should focus on introducing important mathematical concepts and mathematical ideas and methods rather than one-sidedly pursuing the depth of content, difficulty of problems, and problem-solving abilities. The exercises in this area are more reflective of the flexibility characteristics of individuality. 6. Design some topics and reading materials to guide students to conduct exploratory learning activities with the help of tools such as abacuses, function calculators, and computers (Ministry of Education of the People’s Republic of China, 2012, p. 66, cited with modifications).

Connection with Reality The syllabus of elementary school mathematics in the 1980s pointed out that “elementary school mathematics teaching should make students not only develop knowledge but also develop wisdom, train students from a childhood love of science, science, use science… inspire students to analyze quantitative relationships and grasp the laws of problem solving.” Getting elementary school students to relate to the real world is also an important aspect of meeting the requirements of the school syllabus. Contact with practical situations should not only enable them to give examples of what they have learned and argue with intuition and experience but also lead them to nature and society. In this way, they can utilize their mathematical knowledge to observe space, measure area, measure distance, and make some simple mathematics teaching aids. This helps students obtain a large amount of perceptual materials for mastery, which provides pillars for mastering mathematical knowledge and developing rational thinking. It further helps enhance their interest and desire to learn mathematics and improves their logical reasoning and spatial imagination abilities, as well as the ability to analyze and solve problems. The Ministry of Education’s Mathematics Curriculum Standard for Compulsory Education (2011 edition) places great emphasis on relating to real-life problems and gives many examples that can be used as references. Mathematics originates from actual life, and students’ daily lives are full of mathematics. Reflecting on their life experience, students learn to think about problems, which helps them realize that mathematics is an important focus of elementary school teaching both at home and abroad. I browsed through the elementary school mathematics textbooks of developed countries and found that many mathematical problems are related to students’ daily lives. In recent years, there have also been a large number of lifelike examples in our elementary school mathematics textbooks and classroom teaching. I read in the 2011 Issues of Primary Mathematics Reference that many teachers have adjusted the original design of small-step problems to open-ended

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problems so that students not only learn life skills but also systematize and generalize their existing direct experience. I also saw that students were guided to measure length and area, leading to a series of practical activities related to geometric figures. They were also guided to use calculators and even computers to find some interesting patterns. These practice-related exercises make up a considerable proportion of the exercises. In my opinion, no matter what methods are used to relate to the real world, students can feel the truth that “there is mathematics everywhere in life.” This kind of mathematics teaching activity stimulates students’ interest in learning mathematics, guides them to think mathematically, and encourages their creative thinking. Over time, this kind of mathematics teaching activity can not only encourage students to learn to solve some practical problems but also, more importantly, motivate students to develop good mathematics learning habits and master appropriate mathematics learning methods.

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in spatial abilities in 3rd to 6th grades. Studies of Psychology and Behavior, 12(1), 36–44. (in Chinese). Lin, C. D. (1981). The study on the development of the number concept and operational ability in schoolchildren. Acta Psychologica Sinica, 26(3), 289–298. (in Chinese). Lin, C. D. (1982). Intelligence development and mathematics learning. China Science Publishing & Media Ltd. (in Chinese). Lin, C. D., Yang, Z. L., & Huang, X. T. (2003). The comprehensive dictionary of psychology. Shanghai Educational Publishing House, p. 74. (in Chinese). Liu, C. H., & Xin, Z. Q. (2010). Development of Chinese students’ understanding of the concept of fractions from grade fifth to eighth. Journal of Mathematics Education, 19(5), 59–63. (in Chinese). McCrink, K., & Wynn, K. (2007). Ratio abstraction by 6-month-old infants. Psychological Science, 18(8), 740–745. Ministry of Education of the People’s Republic of China. (2012). Mathematics curriculum standard for compulsory education (2011th ed.). Beijing Normal University Press. (in Chinese). Paivio, A. (2006). Mind and its evolution: A dual coding theoretical approach. Erlbaum. Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503–518. Xin, Z. Q., & Han, Y. L. (2014). The developmental and interventional research on lower graders’ concept of equivalent fraction. Acta Psychologica Sinica, 46(6), 791–806. (in Chinese). Xin, Z. Q., & Li, D. (2013). Primary school students’ representation of fractions in non-symbolic materials. Journal of Psychological Science, 36(2), 364–371. (in Chinese). Xin, Z. Q., & Liu, C. H. (2013). The effect of number line task and discrete object task training on children’s whole number bias. Journal of Psychological Science, 36(1), 78–85. (in Chinese). Xin, Z. Q., & Zhang, L. (2009). Measuring students’ representation level on arithmetic word problems: Based on the relational representational complexity model. Psychological Development and Education, 25(1), 34–40, 53. (in Chinese). Xu, B. (2006). Mathematical Olympiad training, 100 categories of learning by analogy: Third grade. Nanjing University Press. (in Chinese). Zhang, F., Lai, Y. H., & Chen, Y. H. (2015). Development of children’s number line estimation: The influence of mental distance. Psychological Development and Education, 31(2), 149–152. (in Chinese). Zhang, L., Jiang, H., & Zhao, L. (2018). The impaired transformation ability between symbolic and non-symbolic numerical magnitude for developmental dyscalculia. Journal of Psychological Science, 41(2), 337–343. (in Chinese). Zhang, L., Lu, C. F., & Yang, X. R. (2014). Relationship between the magnitude representation of whole numbers and fractions for third to sixth graders. Psychological Development and Education, 30(1), 1–8. (in Chinese). Zhang, W., Xin, Z. Q., Chen, Y. H., & Hu, W. P. (2016). Influence of set-relation attributes on children’s multiplicative word problem solving. Journal of Mathematics Education, 25(1), 43–46. (in Chinese). Zhao, H., & Chen, Y. H. (2010). The characteristic of 4th graders’ multidigit multiplication and their relations to working memory. Journal of Psychological Science, 33(4), 938–941. (in Chinese). Zhong, N. N., Chen, Y. H., & Wang, J. (2014). The research of elementary fourth to sixth graders’ representational model in math word problem. Psychological Exploration, 34(3), 218–222. (in Chinese). Zhong, N. N., Chen, Y. H., & Zhang, X. L. (2009). The feature of elementary fourth to sixth graders’ representational level in math word problem and its influence on problem-solving. Journal of Psychological Science, 32(2), 293–296. (in Chinese). Zhou, S. Z., Han, C. C., & Chen, Y. H. (2014). Mediating role of pupils’ metacognitive ability in math between mathematics anxiety and math academic achievement. Journal of Mathematics Education, 23(5), 14–18. (in Chinese).

Chapter 9

Mathematics Learning and Intellectual Development of Secondary School Students

Secondary school students1 who experience both juvenile and early youth stages are collectively referred to as “adolescents.” Middle school students (12–15 years old)2 are entering their first years of adolescence, which is a period of transition from childhood to adolescents. In terms of psychological development, juvenile have the characteristics of being semi-infantile and semimature; thus, the adolescent period is often referred to as the transitional age. High school students (15–18 years old)3 also undergo an important period of physical and psychological development. Adolescents are often energetic and have a strong enterprising spirit as they develop vigorously, both physically and intellectually.

Development of Mathematical Intelligence in Secondary School Students The transition from elementary school to secondary school is the most important condition for the psychological changes of students in this period. The number of required subjects in secondary school increases compared to elementary schools, and the content of each subject tends to be specialized. For example, arithmetic is replaced by algebra, geometry, and trigonometry, whereas general knowledge is replaced by chemistry, physics, history, geography, and biology. All of these subjects are part of a scientific system that requires secondary school students to begin to acquire the basics of systematic science and some of the more general laws of development 1

Secondary school: A collective period of middle school and high school, which refers to the grades 7–12. 2 Middle school students: generally, children enter school at age 6. After six years of elementary school education, they enter middle school (usually at age 12) for three years (age 12–15). 3 High school students: similar to middle school students, they enter high school at age 15 for three years (age 15–18). © China Light Industry Press Ltd. 2023 C. Lin, Intellectual Development and Mathematics Learning, https://doi.org/10.1007/978-981-19-8757-1_9

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in the real world. These factors strongly contribute to secondary school students’ psychological development, including the development of intelligence and thinking.

Characteristics of Secondary School Students’ Thinking Development Secondary school students’ thinking is more often referred to as adolescent thinking in developmental psychology research. The basic characteristics of adolescent thinking during the secondary school period are that thinking abilities develop rapidly and abstract logical thinking is in a dominant position. However, thinking in adolescence (middle school students) and early youth (high school students) differs. In adolescent thinking, although abstract logical thinking begins to dominate, it is still largely empirical, and their logical thinking needs direct support from perceptual experience. In contrast, abstract logical thinking in early adolescence is theoretical in nature—they are able to apply theory as a guide to analyze and synthesize various factual materials, thus continuously expanding their field of knowledge. At the same time, my research suggests that from adolescence onward, students may already possess a preliminary understanding of the laws of dialectical thinking, and by early adolescence, they can basically master it (Wu et al., 1984). The following will discuss three aspects of the development of adolescent thinking, that is, abstract logical thinking. Abstract Logical Thinking Occurs via Hypothetical, Formal, and Reflective Thinking. This kind of thinking is characterized by five aspects. 1. Thinking through hypothesis. The purpose of thinking is to solve problems, and problem solving relies on assumptions. The beginning of adolescence is the period where abstract logical thinking using concepts without concrete things first occurs. Thinking through hypotheses allows the thinker to follow a series of abstract logical processes to achieve the purpose of asking and clarifying the question and then formulating and testing the hypothesis. 2. Anticipatory thinking. The hypothetical nature of thinking necessarily prioritizes the subject in complex activities before the expected factors such as intentions, schemes, plans, programs, and strategies. The ancients said, “Preparedness ensures success while unpreparedness spells failure.” This “preparedness” is the anticipatory nature of thinking. From the beginning of adolescence, the individual’s thinking activity exhibits this “anticipatory” approach. Anticipatory thinking enables the individual to adopt certain ways and means of activity before solving problems. 3. Formalization of thinking. Starting from adolescence, under the influence of educational conditions, the component of individual thinking gradually develops from the predominance of concrete arithmetic thinking to the predominance of formal arithmetic thinking, which is the formalization of thinking.

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4. Self-consciousness or the ability to monitor thinking activities becomes more recognizable. Self-regulation of the process of thinking activity is an important condition for the smooth development of thinking. Starting from adolescence, reflective (or introspective) and monitoring thinking become increasingly pronounced. In general, adolescents become aware of their intellectual activity processes and learn to control them, resulting in clearer thinking and more accurate judgment. Of course, the development of reflective thinking in adolescence does not exclude the intuitive thinking that also appears in this period, the development of which is still an important element of education and teaching at this stage. 5. Thinking is expanded beyond the old rules. Any way of thinking can lead to new assumptions, understandings, and conclusions, all of which can contain new elements. From adolescence onward, due to the development of abstract logical thinking through hypothetical, formal, and reflective thinking, thinking is necessarily capable of formulating new ideas. Thus, from this stage, creative thinking or originality of thinking develops rapidly and becomes an important feature of adolescent thinking. In the process of thinking, adolescents seek novel and unique elements and personal characteristics, systematization, and structure. Abstract Logical Thinking, in a Dominant Position, is the Transition from Experiential to Theoretical Thinking. One of the main features of adolescent thinking development is that while the concrete image component of thinking continues to an important role, abstract logical thinking becomes increasingly dominant. Elementary school thinking and adolescent thinking differ; elementary school children’s thinking is transitioning from concrete image thinking to abstract logic thinking, while in adolescent thinking, the abstract logic component already has a certain degree of relative advantage. Of course, this “advantage” does not mean that only abstract logical thinking occurs in adolescence but that the concrete and abstract components of thinking are inseparable in a unified relationship in which the abstract component progressively occupies an important position. Moreover, because of the development of the abstract component, concrete image thinking is constantly enriched and transformed. During adolescence, it is carried out in close connection with abstract logical thinking. The development of thinking in early adolescence has a higher level of abstract generalization and begins to form dialectical thinking. Specifically, it manifests itself in two ways. 1. Abstract and concrete concepts acquire a higher degree of unity. Early adolescent thinking builds on the thinking of adolescence but differs from that of adolescence. The abstract generalization of adolescent thinking develops at a considerable pace, but because it needs the support of concrete images, it is mainly empirical, and theoretical thinking is quite immature. In early adolescence, theoretical abstract logical thinking begins to develop because of the frequent need to grasp the laws of the development of things and important scientific theories. This thinking process includes both the inductive process, that is, from the

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particular to the general, and the deductive process, that is, from the general to the particular. In other words, it is the process of upgrading from the concrete to the theoretical and using theory to guide the acquisition of knowledge. This process shows that the thinking of the early youth is transformed from empirical to theoretical as the abstract and concrete thinking processes are highly unified, thus allowing for significant development of abstract logical thinking. 2. The development of dialectical thinking is obvious. In the early stage of youth, it inevitably leads to the rapid development of dialectical thinking. In practice, and study, young people gradually realize the relationship between general and specific, induction and deduction, theory and practice and gradually develop dialectical thinking that recognizes, analyzes, and solves problems from a comprehensive, changing, and unified perspective. It can be seen that the development trend of adolescent thinking is to make a judgment by reasoning according to general principles or deduce possibilities based on a particular theory. The development of abstract logical thinking contains critical and mature periods. Specifically, based on research on secondary school students’ arithmetic ability, we found that the eighth grade is a key period for the development of thinking at the secondary school level. Starting from the eighth grade, secondary school students’ abstract logical thinking has been transformed from the empirical level to the theoretical level. This transformation is initially completed by the second year of secondary school, which means that their thinking tends to be mature. Out of a total of 500 students, 100 in each grade from seventh to eleventh grade were measured for their ability to generalize mathematically; to imagine spatially; to determine original, negative, inverse, and inverse negative propositions; and to reason logically. From these four indicators, the eighth grade represents a new “start” of abstract logical thinking, a qualitative change in arithmetic thinking at the secondary school level, and a critical period for the development of thinking (Lin, 1983). The 10th and 11th grades (15–17 years old) are when the development of abstract logical thinking tends to be “preliminarily formed” or mature. This so-called thinking maturity shows performance in mainly the following three aspects. 1. The various components of thinking basically tend to stabilize to the level of theoretical abstract logical thinking. 2. The level of individual differences, including the type of thinking (image-based, abstract, and intermediate), tends to conform to basic stereotypes. 3. There is a high level of plasticity regarding developmental changes in thinking before maturity; in contrast, after maturity, it is less plastic and remains basically the same throughout adulthood, although there is some progress. These three aspects were confirmed in a survey of several key secondary schools in Beijing. The survey results show that the intellectual performance and academic achievement of 10th-grade students are still highly variable, while those of 11th- and 12th-grade students are more stable; the ability base of college students basically remains the same as those observed in 11th and 12th graders, which indicates that

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their foundation is laid in the maturity period of high school (Wo et al., 2007). For example, students with average mathematics scores in their 11th- and 12th-grade years will almost never become highly talented students at the college mathematics level. Of course, some liberal arts students may also be late bloomers, but a certain maturation period does exist. This shows how important it is to seize the opportunity to develop various thinking skills and intelligence before students mature.

Contradictory Manifestations of the Quality of Secondary School Students’ Thinking As mentioned earlier, the emergence and development of thinking obey general and ordinary laws but also present individual differences. This difference is expressed in the intellectual characteristics of individual thinking activities, which is the thinking quality, also known as thinking traits. There are many components of thinking quality and its manifestations, such as independence, expansiveness, flexibility, profundity, creativity, criticality, and agility. The components and manifestations of thinking quality reflect different levels of development, which constitute the age characteristics of thinking. The most prominent feature of the thinking quality of secondary school students is the contradiction manifestation. Due to the requirements of independent thinking, new characteristics of the development of thinking quality in secondary school students emerge, the most prominent of which is the significant development of independence and criticality. However, their views on problems are often merely concerned with the part while ignoring the whole and focused on the phenomenon while overlooking the essence, that is, one-sided and superficial, which raises two questions: (a) Why do secondary school students “contradict” adults sometimes? and (b) Why do secondary school students tend to be one-sided and superficial? These are problems that arise from the contradictory development of thinking quality. As secondary school students gradually acquire systematic knowledge, they begin to understand some complex causal relationships in natural and social phenomena. Moreover, as their self-consciousness develops further, they are frequently unsatisfied with the description of things and phenomena by teachers, parents, or books. This, in turn, sparks their desire to independently seek answers or argue with others about the causes and laws of various aspects of the world around them. In this way, the ability to think independently reaches a new and unprecedented level. It has been said that from adolescence onward, children enter a period when they like to doubt and debate and no longer trust the “authoritative” opinions of adults, such as teachers, parents, and books, but instead approach almost everything independently and critically. This is indeed one of the important characteristics of the secondary school phase. Not only can secondary school students begin to hold critical opinions on others and books, but they can also begin to self-consciously manage their own thinking activities and consciously regulate, control, examine and prove their own thinking processes, which promotes greater independence and self-consciousness in their studies and lives.

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Teachers and parents should value this new thinking quality development in adolescents. The ability to think independently is a valuable psychological quality that should not be discouraged or suppressed because it grants young people the chance to offer different or skeptical opinions, assuming that they are deliberately “rebelling” against themselves. Of course, this does not mean that they are allowed to contradict their teachers, but rather teachers and parents should correctly respect the characteristics of children’s psychological development at this stage. We must foster secondary school students who are respectful, civilized, and polite and teach them to act in a consultative manner while simultaneously actively thinking about problems. We should also appropriately criticize those words and deeds that are indeed unreasonable and rude. This age group is one-sided and superficial, which is normal. Although secondary school students process greater independence and self-consciousness in their studies and lives, their independence and criticality are still immature, which can lead to various manifestations of prejudiced and shallow thinking: sometimes they argue without grounds—they doubt everything and insist on their own opinions with insufficient arguments. Sometimes they look at problems in an isolated and paranoid way; for example, interpreting modesty as restraint and bravery as rudeness or risk-taking. Sometimes they tend to criticize others rather than self-reflecting, and sometimes they go to extremes, often affirming everything or denying everything. The same is true in learning. They tend to take the rules or principles they have already mastered and apply them inappropriately to new conditions, leading to the problem of rigid adherence to dogma. These shortcomings in the development of secondary school students’ independent thinking abilities are related to their lack of knowledge, experience, and undeveloped dialectical thinking abilities. Teachers and parents should, on the one hand, vigorously develop their independent thinking abilities and guide and inspire them in a timely manner. On the other hand, they should provide patient and positive persuasion and education to students who have insufficient independent thinking. It is not right to ridicule or reprimand them for their weakness; likewise, it is not right to ignore their inexact thinking processes or to think that they will get better naturally when they are older.

Characteristics of the Development of Secondary School Students’ Arithmetic Thinking Secondary school students’ level of intelligence is manifested in the process of arithmetic thinking, and the acquisition of arithmetic abilities is fully consistent with the general trend of intellectual development. This conclusion can be found in my preliminary study (Lin, 2015). Manifestation of Mathematical Generalization Ability. From middle school to high school, generalization ability can be divided into four levels: the first level is numerical generalization; the second is figurative abstract generalization (where

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algebraic generalization commences but requires concrete experience to help them understand numerical knowledge); the third is generalization based on assumptions, where they completely abandon the practice of applying block diagrams of arithmetic to perform operations, formulas, and principles, which become the main instrument of understanding numerical concepts at this level; and the fourth is dialectical abstract generalization, which identifies the intrinsic connection between the opposites and unity of quantities. There are age (grade level) characteristics as well as individual differences in secondary school students’ mathematical generalization skills. 1. Seventh graders’ generalization ability is similar to that of upper elementary students—they are developing from mathematical generalization operations to figurative abstract generalization operations. 2. Eighth grade is the first turning point in the development of mathematical generalization ability in secondary school and the true “start point” of abstract logical generalization. This is also the time when there is a marked divergence in secondary school students’ mathematical performance. 3. Ninth grade is a transitional period. Based on the development of the generalization ability of ninth-grade students, the generalization ability of 10thgrade students is another remarkable development. Most students can generalize according to assumptions and are developing to the level of dialectical abstraction. 4. After the 11th grade, generalization ability according to assumption and dialectical abstract generalization is further developed. For example, students have acquired permutations and combinations, series and limits, all of which require the identification of intrinsic connections between quantities in opposition and unity. Many experienced mathematics teachers point out that students’ understanding of the concept of “infinity” progresses from difficulty to familiarity as their ability to generalize matures. 5. High school students’ generalization ability tends to be “preliminarily formed” from the second semester of 10th to 11th grades, after which their generalization ability basically tends to be consistent. Manifestation of spatial imagination ability. The spatial imagination ability of secondary school students can also be divided into four levels. The first is to calculate area and volume by numbers, which is the stage of three-dimensional space arithmetic operations, wherein the specific image is still dominant. The second level is the linear plane acquisition stage, which involves plane geometry operations. The third stage is the polyhedron acquisition stage. The final and fourth level is the rotation understanding stage, which is the acquisition of all the operations of threedimensional geometry. It should be noted that the above four levels of classification are only the result of the research in the 1980s. In fact, the difference between the content of polyhedral and rotations is not as significant as in the current secondary school stereometry. This is because instruments such as infinity and derivatives are not applied in the study of measurement. If I were to conduct the study again today, I would definitely merge the two levels.

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I learned from a large number of surveys that seventh graders are at the first level and eighth and ninth graders are at the second level. Additionally, although ninth graders can also acquire some of the operations of three-dimensional geometry, the true acquisition of polyhedral and rotational operations occurs after secondary school (Li & Lin, 2005). It can be seen that the eighth grade is a period of qualitative change in the development of spatial imagination and a critical age period that enables the maturity of spatial imagination ability of students when they reach the 10th and 11th grades. Manifestation of proposition ability. The proposition ability of secondary school students includes the level of comprehension and perfection of four kinds of propositions (original, inverse, negative, and inverse negative) as well as the level of mastery and deformation of various propositional forms, such as affirmation, negation, conjunction, and disjunctive. I have observed in my survey results that eighth-grade students can master the structure of propositions and various simple proposition forms well, while 10thgrade students can master the inner changes of propositional structure, master and discriminate the compound propositional forms, and determine the propositional deformation. Manifestation of Logical Reasoning Ability. Logical reasoning ability develops early, as mentioned above, emerging in early childhood and developing rapidly in the elementary period. In mathematical arithmetic, secondary school students’ logical reasoning ability includes the following four levels: (a) direct reasoning, that is, applying the formula with the conditions and directly drawing the conclusion; (b) indirect reasoning, that is, due to the inability to directly apply the formula, the conditions must be changed, followed by looking for a basis and drawing conclusions in multiple steps; (c) lateral reasoning, that is, conducting an analysis of the premises, proposing hypotheses after repeatedly verifying it, and then deriving the conclusion; and (d) comprehensive reasoning based on a certain mathematical and logical format. Students who have reached the fourth level have a progressively more concise and rationalized reasoning process. Research shows that the current low level of logic among secondary school students is a disadvantage in mathematics teaching—many seventh graders cannot apply formulas to solve problems. In fact, some high school students cannot even reason in one step according to formulas. Multistep reasoning has become a common problem for secondary school students, and abstract comprehensive reasoning (i.e., concise and rationalized according to mathematical logic), which moves from one step to the next steps, presents an even greater challenge for them. This result should attract the attention of the mathematics education community. That is, if teaching students how to improve their logical reasoning abilities is not well emphasized, raising the quality of secondary school mathematics teaching is merely paper talk. I have seen that the level of mathematical reasoning of secondary school students develops with grades and ages under normal educational conditions. Eighth graders are generally able to reason according to formulas, most ninth graders are able to reason indirectly in general, most 10th-grade students are able to reason multiple

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steps in a roundabout way, and most 11th-grade students’ reasoning abilities are preliminary developed in the 11th grade. This finding indicates that despite their current low level of reasoning, especially when the difficulty of mathematical exercises is increased, secondary school students have some basic reasoning abilities at all ages. If teaching measures are kept up to date and systematic remediation is employed to address students’ lack of mathematical knowledge, secondary school students can catch up to the target reasoning ability for their grade level.

Recent Studies on the Development of Secondary School Students’ Mathematical Abilities Our team has recently explored new aspects of the development of secondary school students’ mathematical ability, and the main findings are as follows.

Analogical Transfer in Secondary School Mathematics Learning Analogy transfer is a crucial process and an important learning strategy in mathematics learning. Analogical transfer in mathematics learning refers to the use of learned examples or solved problems to solve new problems. It can be said that the process of analogy transfer exists in most mathematical problem solving. Professor Siqing Lian’s research team has conducted extensive research on analogy transfer in secondary school students’ mathematics learning, covering quadratic inequalities (Lian et al., 2014), combining similar terms of polynomials (Zhang et al., 2016a), the laws of cosine (Zhang et al., 2016b), and many other areas. Overall, the more similar the target problem and the source problem are, the more effective the solution of the source problem is in transferring and applying preexisting knowledge to the target problem, and the more significant the effect of students’ level of transfer.

Cognitive Factors Affecting Secondary School Mathematics Performance At the secondary school level, basic cognitive abilities still have a significant impact on mathematics achievement. The study found a significant relationship between seventh graders’ working memory and their mathematics achievement and that analogical reasoning partially mediated the effect of working memory on mathematics achievement. At the same time, metacognition had a moderating effect on the second half of the pathway of this mediated process. That is, seventh graders’

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mathematics working memory influenced their mathematics achievement through mathematical analogical reasoning, and this role diminished with the improvement of mathematical metacognitive ability (Hao et al., 2017). This suggests that the mathematics learning of students with poor working memory can be improved by increasing metacognitive levels in mathematics teaching and that the adverse effects of poor working memory can be attenuated or even eliminated.

Secondary School Students’ Mathematics Anxiety Compared with elementary school students, secondary school students’ mathematics anxiety has a clearer component structure. Professor Chen’s team explored the structure of mathematics anxiety in high school students and developed the “Mathematics Anxiety Scale for High school students” (Huang et al., 2008). According to the results of factor analysis, the scale contains five factors: mathematical problem-solving anxiety, observation anxiety, pre-exam anxiety, exam anxiety, and life anxiety. It was found that the content of mathematics anxiety among high school students in China is relatively diverse and covers all aspects of mathematics learning.

Research on Secondary School Student’s Mathematics Core Literacy The new round of curriculum reform in China has proposed core literacies in various subjects (Ministry of Education of the People’s Republic of China, 2012). What are the core literacies of mathematics? There are mathematical abstraction, logical reasoning, mathematical modeling, intuitive imagination, mathematical operations, and data analysis. These are the very essence of mathematical abilities, with thinking as the core. Since 1996, I have enrolled Jianyue Zhang, Wenfang Zhu, Wu Kang, Siqing Lian, and Jiyuan Zhao as doctoral students in the direction of “mathematical ability development” and have actively carried out a number of studies aiming at the development and cultivation of core literacy in secondary school mathematics, including Zhang’s study on the structure, development and influencing factors of secondary school students’ self-monitoring ability in mathematics (1999); Zhu’s study on the development of function concepts in middle school students, which explores the development and cultivation of mathematical ability (2000); Kang’s study on secondary school students’ ability to formulate mathematical problems (2003); Lian’s study on the role of working memory in simple algebraic operations (2006); and Zhao’s study on the basic mental processes of mathematical modeling in high school students and their influencing factors (2007). These dissertation studies are intersectional research results between developmental psychology and mathematics education and

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are groundbreaking in their approach to mathematics education, especially the core literacy of secondary school students in mathematics, both in terms of theoretical profundity and guidance for mathematics teaching practice.

Emphasis on the Development of Mathematical Ability Before Intellectual Maturity As mentioned earlier, the second half of secondary school, around the second year of high school (16–17 years old), is the period of maturity of human intelligence or thinking before secondary school students’ intelligence, thinking, and mathematical ability develop rapidly. Thus, seizing the plasticity of their premature intellect and targeting the development of their mathematical ability is an important task in teaching secondary school mathematics.

Emphasis on the Premature Intellectual Development of Students in Secondary School Mathematics Teaching From the above four indicators of secondary school students’ generalization, spatial imagination, propositional, and logical reasoning abilities, the eighth grade is the qualitative change period of abstract logical thinking, which is the turning point or key period of the development of arithmetic thinking ability at the secondary school level. After the 10th grade, approximately 16–17 years old is the age when mathematical operations in logical thinking are “preliminarily formed.” Based on my research, I learned that the mathematical ability of middle school students is highly variable, malleable, and unstable; from the 9th grade to the 10th grade, there are still significant changes in students’ mathematical ability. In the 10th grade to 11th grade, the correlation between students’ mathematical ability is very strong, and the fluctuation of mathematical performance between 10th grade and 11th grade is very weak. As mentioned earlier, the foundation for college students’ mathematical arithmetic ability is essentially laid at the high school level. It naturally follows that age 16–17 is the basic maturation period for mathematical operations and for developing thinking skills and intelligence. Obviously, the plasticity and variability of arithmetic ability, thinking ability, and intelligence before maturity are greater than after maturity; the stability and “constancy” of arithmetic ability, thinking ability, and intelligence are greater after maturity than before. This highlights the importance of maximizing the opportunity before students mature to cultivate their arithmetic and thinking abilities and intelligence. How can we develop students’ intelligence before maturity in secondary school mathematics teaching? Below are some suggestions for educators.

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Take the Age Characteristics of Secondary School Students’ Arithmetic Development as a Starting Point for Compiling Mathematics Textbooks, Arranging Mathematics Teaching Materials, and Selecting Appropriate Teaching Methods. The goal of educators is to guide students’ intellectual development from this starting point, which is not only emphasized by psychologists but also proven by the experience of excellent mathematics teachers. There are two tendencies in mathematics teaching that deserve attention: one is to prematurely assign difficult and perplexing teaching materials to students without regard to their level of psychological development and to demand that they master difficult concepts that are age-inappropriate in terms of their present thinking abilities. Currently, the trend of replacing regular teaching with teaching that increases the burden on students remains prevalent. Although the original intention was to improve the quality of teaching, it deviates from matching the actual level of thinking and intelligence of students at a certain age. The other is to ignore students’ thinking and intellectual development, that is, not exploring students’ potential abilities or actively guiding students in a manner that ensures that they progress. In this case, there is a significant difference between the materials used and the actual grade level, and teachers do not attempt to help students catch up with their actual grade level in the long term, allowing them to fall behind. These results are contrary to the law of age characteristics. Grasp the True “Budding Stage” in the Critical Age of Abstract Logic Thinking. There is a critical age for each stage of human development. Eighth grade, for example, marks a qualitative change in the development of abstract logical thinking in normal mathematical arithmetic, whether it is a mathematical generalization, spatial imagination, mathematical proposition and operations, or reasoning and proof ability, and eighth grade is a turning point. Around this time, there is a clear contrast between the concrete figurative and abstract logical components of thinking, the level of judgment and reasoning, and the degree of spatial operations. Some psychologists believe that missing a critical age is the same as losing a developmental moment. I do not agree with the absolute view of “now or never.” However, I believe that recognizing critical periods and arranging mathematics teaching reasonably can promote the early realization of qualitative changes in arithmetic ability. We should take measures to prevent this qualitative change period from being pushed back, thus assuring a certain level of development of thinking ability and intelligence before maturity. Place Great Emphasis on the Key Points and Difficulties of Mathematics Teaching and Strengthen Basic Knowledge and Skills Training. Mathematics teaching is complicated, but the core is still to strengthen students’ basic arithmetic knowledge and skills. As mentioned above, this is the basis for the development of intelligence and thus what teachers must consider when preparing lessons and lectures and arranging homework. Basic knowledge is not necessarily deep and difficult, but it is important in terms of its role in intermediate mathematics. For example, if a student learns algebra and uses letters to represent numbers, it is easy to understand, but if he/she is asked “which number does a represent,” he/she may not be able to answer. Some teachers think

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that basic knowledge is relatively simple and easy for students to understand. Hence, teachers will focus on solving difficult problems when preparing the lesson, which is akin to putting the cart before the horse. The core of basic knowledge is grasping fundamental concepts. Therefore, to strengthen the basic knowledge, we must pay attention to the fundamental concepts. Lesson preparation is key to ensuring that students thoroughly understand the connotation and extension of each basic concept. Moreover, instructors must study various methods to describe basic concepts to the students clearly. Finally, all lessons should be organized and rigorous so that students learn strategically while leaving a lasting impression. This is not to say that every lesson is necessarily difficult but that each contains increasingly abstract mathematical knowledge that must be understood. To break through these difficulties, abstract content can generally be visualized, and theoretical problems can be approached with a concrete method. Teachers can start with familiar concepts or knowledge already acquired from daily life to introduce the topic, that is, moving from shallow to deep, from familiar to unfamiliar, so that students can remain engaged and comprehend novel concepts more easily. Only when students understand the difficulties and improve their comprehension can they promote the development of their thinking skills and intelligence. Basic skills training in mathematics can be achieved by assigning selected homework. Homework in mathematics should include understanding both materials and practices. Some teachers assign homework that only asks a few questions to students, leading them to develop the bad habit of just answering the questions and not reading the textbook, which should be ascribed to the teacher. In contrast, whenever an experienced teacher assigns homework, he or she first points out to the students that the first few paragraphs of the pages in the textbook are the content of the day’s lecture and should be read carefully. In addition, he/she then indicates that the next lesson will be about the first few paragraphs and instructs students to be prepared to preview the content. After fully informing students of what is expected of them, he/she finally assigns the questions. In this way, basic knowledge and basic skills are organically combined. The quantity of homework should not be too great, and the exercises should be selected to match the questions in the textbook and exercises with the general review questions. The purpose of doing exercises is threefold: first, to consolidate the basic concepts and knowledge taught in class; second, to prepare for the next lesson; and third, to develop students’ logical thinking abilities. If there are specific problems that require inspiration, teachers should be careful to only inspire and not tell students exactly what to do, which would hinder the development of students’ thinking abilities. Secondary School Mathematics Teaching Should Increase the Degree of Abstraction and be Incorporated into Regular Teaching Content. There are four points that should be observed in secondary school mathematics teaching. 1. Lessons should not be rigid; teachers should try to create more lively classes. In explaining some important or difficult knowledge, teachers should introduce this knowledge both from the positive side (most of the textbook) and from

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the negative side; otherwise, it is easy to copy mechanically and apply indis√ 2 criminately. For example, students tend to write (−3) = −3 because they √ disregard the constraints of a 2 = a (a ≥ 0) or do not ask about the specific conditions but simply apply the quadratic function of the extreme value method. While correcting these problems, teachers can make greater use of counterexam√ 72 = 7, ples. For example, teachers can let students judge the correctness of √ √ 2 2 0 = 0, (−5) = −5 and explain the reasons for their judgments, give examples of conditional extreme values of quadratic functions, and show that ignoring the conditions can lead to absurd conclusions. Through counterexamples, students can appreciate the conditions for the use of a formula or particular method to avoid rigid thinking. 2. Lessons should not be uniform. While not all contents need to be taught utilizing visual teaching aids, the degree of abstraction should be gradually increased, the inherent laws of teaching revealed, and the mathematical laws with universality indicated from a large amount of basic mathematical knowledge. For example, in geometry teaching, through the analysis of typical examples, students can point out that there are concrete and clear methods to prove the sum, difference, multiplication, and division of lines. This requirement should, of course, take into account students’ age characteristics and be adjusted for each grade level. 3. Teach not only mathematical knowledge but also arithmetic methods; teachers should not only correct mathematical homework but also help students summarize arithmetic methods and techniques that are appropriate to their individual differences. 4. Guide students to generalize so that they acquire mathematical knowledge more systematically. The Beijing Education Press has published the “Diagram Analysis of Examination Points–Thinking Ability Guide: Middle School Mathematics (Revised Edition)” (Ma & Liu, 2010), in which the authors created a middle school mathematics knowledge structure and mind map that is intended to help students perform well in mathematics examinations. While I do not advocate studying merely for exams, I find the central idea of the book, that is, the idea that to think and understand more deeply, a student has to learn to connect and relate knowledge points to systematize, regularize, and structure learning, to be correct. In short, the teacher’s responsibility is to guide students to unlock the treasure trove of knowledge and let them walk in. Teachers who do so can grasp the essence of the problem and obtain twice the results with half the effort. In contrast, teachers who cannot open the locks spend a great deal of effort, and students are shut out. The critical issue is to increase the abstraction level of teaching. How to unlock teaching content is also a critical issue related to how to best explain the material—critical content must be taught via the right method. The better the method is, the better the effect.

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Cultivating Secondary School Students’ Self-Study Ability Secondary school students should be allowed to engage in independent and inquiry learning. Cultivating secondary school students’ ability to learn mathematics independently is a major topic to be studied in contemporary mathematics teaching. Teaching practice proves that students who have better self-study ability tend to perform better in mathematics, and individual students who are at the top of their class engage in regular reading of mathematics-related books. However, some students who spend time doing mathematical exercises rather than reading textbooks do not see significant improvements in their performance. After entering college or society, students with strong self-study ability tend to adapt to college life or achieve competency in work relatively quickly; conversely, other students find it more difficult to adapt to or perform well at work. In the early 1980s, a survey was conducted on 400 people in scientific and technical professions who made great contributions at that time, and it was revealed that 46% of them had attended colleges and universities, 5% had attended junior college, and 49% had no post-high school education. What did the people who did not attend college rely on to achieve significant scientific and technological achievements? The answer is self-study. For example, Xiaoxing Wang, a young man from Hunan Province, insisted on self-study at work after graduating from secondary school. Later, he became an electrician and insisted on self-studying for 2 to 4 h a day. In November 1979, he was transferred to the National University of Defense Technology as a mathematics teacher after passing the entrance examination. This shows the importance of self-study and the urgency of cultivating secondary school students’ self-study ability. Many secondary school mathematics teachers and psychologists on the front line of teaching are exploring different methods and trying to cultivate students’ self-study ability. “Read, talk, and exercise” Method. Many experienced secondary school mathematics teachers employ the “read, talk, exercise” method in their classrooms, which is a practical way to cultivate students’ self-study ability and help them learn efficiently. “Read” refers to students who can read by themselves, “talk” refers to the teacher’s lecture and requirements, and “exercise” refers to the students’ purposeful exercise under the teacher’s guidance. The trinity of “read, talk, and exercise” is inseparable. Some of the specifics of the approach are as follows. 1. Crush one by one. Mathematics textbooks generally include axioms, definitions, theorems, formulas, rules, examples, and so on. These contents have unique characteristics, require reading in different ways, and vary from easy to difficult, but each one can be crushed. There are different requirements according to grade level, and the specific needs of each person should be taken into account. 2. Gradual progress. First, the teacher instructs and explains some specifics. Then, he/she gradually lets students discover and think more. According to the progress of the self-study process, teachers can approach from “talk first and then read” to “read first and then talk” and then finally to “read, talk and exercise.” 3. Gradual improvement. Students’ self-study ability improves gradually in line with their self-study requirements.

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The mathematics teaching and research group of Hangzhou No. 2 Secondary school in Zhejiang Province adopts the approach of “the boat floats when the river rises.” The specific approaches to learning various types of concepts are as follows. 1. To learn definitions, a complete description is required at the beginning, followed by an example. Later, it is necessary to clarify the relationship between the old and new concepts in the definition. Students should gradually analyze the definition, grasp the essential attributes of the concept, and then compare these with similar or easily confused concepts to clarify their similarities and differences. 2. To learn theorems, students must be able to distinguish the initial conditions and conclusions required to understand the basic application of the theorems so that they can later prove them independently and analyze the ideology of the proofs. Furthermore, it is necessary to compare related theorems and summarize the aspects of applications. 3. To learn formulas, students should know from the start how to express them correctly in their own language and English letters; after that, they must understand the derivation of formulas, the levels of derivation, the application of formulas, and the conditions for their application. In addition, they should be able to identify the characteristics of and memorize formulas to compare them with similar formulas and summarize the formula applications. 4. To learn examples, students must determine the first step, categorize them, and master the writing specifications required at the beginning. Next, they need to analyze the levels, ideas, and keys to problem solving, as well as whether there are other solutions. When dealing with a related set of examples, it is also necessary to compare similar items to find the solution rule. The above approaches reflect the following characteristics: 1. They enhance initiative to learn. One of the purposes of reading is to develop students’ initiative in learning. When they prestudy concepts, they can identify difficult points in advance or think and discuss according to the thinking questions given by the teacher. As a result, they can better concentrate on solving problems when engaged in the lesson. Before doing homework (exercises) after class, they should read first, mainly to review and compare and contrast concepts. It also affords them the opportunity to compare what the teacher says and what the book states. Reading should also occur after completing the unit with a focus on systematic organization. 2. They strengthen systemic knowledge. The self-study approach of “a boat floats high when the river rises” conveys that students can gradually begin to “look at the whole” so that they can recognize the importance of the partial knowledge they have learned from a holistic perspective. This allows them to deepen their understanding of basic concepts and concept systems of mathematics related to comparison and identification. 3. They are conducive to consolidating what students have learned. Using a combination of “read, talk, and exercise,” students must exercise by themselves. They should also create a comprehensive outline after careful consideration. This not

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only improves their thinking abilities but also improves memory and helps to consolidate what they have learned. 4. They better amplify the teacher’s leading role. Guiding students in self-study via the read-talk-exercise approach reflects that the teacher is standing higher and looking farther. The teacher’s lecture adds a finishing touch, but the process of cultivating students’ self-study, for example, the thinking questions proposed by the teacher, often plays a key role. It should reflect certain requirements and be inspiring so that students approach the purpose through the thinking questions and the process of discovering, understanding, summarizing (e.g., creating an outline), and familiarizing themselves with formulas in the self-study. “Heuristics, Reading, Exercise, and Knowing” Method. Professor Zhongheng Lu of the Institute of Psychology of the Chinese Academy of Sciences and other psychologists have applied various effective psychological principles, summarized the experimental experience of “program teaching” in Europe and America, absorbed some of its favorable factors, eliminated some of its weaknesses (combined with the actual situation in China and the experience of excellent teachers), and put forward the method of “Heuristics, Reading, Exercise, and Knowing” to cultivate students’ self-study ability (Lu, 1989). “Heuristics” refers to the need for students to enlighten their students in a timely manner when they experience difficulties in learning; “reading” refers to students independently reading and learning books; “exercise” refers to students doing exercise by themselves; and “knowing” refers to students being knowledgeable about the results of their exercises and able to instantly correct their work. Professor Zhongheng Lu and other psychologists compiled the self-study materials. This set of self-study materials comprises three books: one is the textbook, which differs from the textbook published by the People’s Education Press in that it contains psychological principles and teaching methods; the second is the exercise book, which differs from ordinary exercise books in that the exercises are printed in this book, leaving a blank space for students to do the exercise; and the third is the answer book, which is used for students to check their answers after they have completed the exercises so they can quickly check their learning results. This kind of teaching is also called “three books teaching.” Of course, it is also important to compile the necessary materials to create a selfstudy textbook. The content and requirements of the self-study textbook are the same as those of the unified textbook, except for the more detailed explanations and the implementation of eight psychological principles outlined below. 1. In terms of the process of self-study, appropriate stages are taken, transitioning from simple to difficult, to accommodate and facilitate the improvement of students’ learning ability. 2. In the process of doing the exercises, students are allowed to check their answers, and teachers provide timely feedback, which offers reinforcement and correction. 3. In terms of the form of statements, expressions, and arithmetic, there is a progression from development to compression and from detailed to abbreviated to train students to think concisely and develop generalization ability.

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4. The true or false exercises are designed to directly reveal the essential features of the problem, deepen the correct understanding of the concept, and develop the ability of identification. 5. In a series of exercises, teachers try to make the previous problems inspire the later ones so that students can learn to solve the problems and develop coherent thinking at the same time. 6. Variations of review are taken as much as possible to avoid mechanical repetition so that students can master the problem, apply it flexibly, and develop generalization abilities. 7. Operational principles are emphasized so that students can organize and apply a series of methods in a hierarchical and categorical manner and be trained in the organization and hierarchy of thinking. 8. Reversible associations are used as much as possible to develop flexibility in reversing mental procedures. Here are the three principles of thinking with examples to illustrate. The first is the principle of step-by-step thinking. When solving mathematical problems, students often feel that they do not know where to start. The best way to solve the problem of how to start—the problem of thinking in an organized way—is to think in steps. However, does step-by-step thinking create stereotypes that prevent flexibility? Take the example of a self-study textbook on factorization and have students think about the problem in steps. The first step, regardless of the number of terms, is to first consider whether the expression has a common factor; if present, we should first extract it. The second step is to consider how many terms there are; if it is binomial, we should consider whether it can be decomposed by using the formula of quadratic difference, the formula of cubic sum, or the formula of cubic difference. Then, when the expressions are trinomial, quadrinomial, pentanomial, and so on, we should apply the corresponding method. In the third step, if the resulting factors can still be decomposed, we should continue to decompose. The fourth step gives some additional questions, for example, practicing questions that are binomials but cannot be decomposed by the formula applied in binomials prevents students from stereotyped thinking. The researcher used some more difficult problems for the students to attempt, and they were able to solve them flexibly. Step-by-step thinking can have a positive effect on training the organization of thinking and how to start solving problems. However, whether step-by-step thinking hinders the flexibility of thinking requires further research. The second is the principle of reversible association. Reversible association is an important principle of mathematical thinking. For example, addition and subtraction, multiplication and division, multiplication and rooting, exponents and logarithms, as well as the original and inverse theorem in geometry, are all mutually inverse. However, the formation of reversible associations is relatively difficult. For example, if a student has learned to multiply and square and then forgets how to square, he/she will not use the reversible association to multiply the number from the multiplication to introduce the formula for squaring. This shows that reversible association is important in mathematical thinking. Accordingly, students should be conscious of the

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reversibility of each specific formula and theorem and gradually develop a reversible association with high generality. Therefore, the principle of reversible association should be a core principle when compiling materials for self-study textbooks. Third, attempts should be made to use variation review and avoid mechanical repetition. When students learn fractions in middle school, they learn that fractions have no meaning when the denominator is zero. Likewise, secondary school teachers 1 , emphasize that the denominator cannot be 0. For simple fractions, such as x(x+1) students know that when x = 0 or x = −1, this formula has no meaning, and significant mistakes are caused by carelessness. Therefore, this kind of question is used in textbooks to review the concept that fractions have no meaning when the denominator is 0. If you use a complex fraction to test this, such as “What is the value of x when the fractional equation 1+ 1 1 has no meaning?”, it is clear from 1 1+ x+1

solving this problem that the complex fraction has no meaning when x = −1 or x = −2 or x = − 23 . However, most students make mistakes and do not consider the whole; this kind of generalization is common among middle school students. If more variation questions such as this are needed, students can avoid mechanical repetition and improve their ability to solve problems, which is a principle that should be emphasized when compiling self-study materials for students. Around the 1980s, the above experimental research was conducted collaboratively in more than 20 schools in seven provinces and cities across China, and certain results were achieved: the results of the experimental classes on several exams were increasingly higher than those of the control classes; the self-study ability of the students in the experimental classes grew to a certain extent; time spent studying decreased and the study burden of the students was reduced; and students with lower grades were able to adopt a teaching method based on self-study (Lu, 1998). This shows that the method of “enlightenment, reading, exercise, and knowledge” can improve the quality of teaching and learning. This further demonstrates the important role of self-study in teaching and learning. Unfortunately, due to the death of Professor Lu and other reasons, this experiment did not continue at the beginning of the twenty-first century. The above two approaches to developing secondary school students’ self-study abilities have one thing in common, namely that self-study is an integrated intellectual activity, which is the “transfer” of one intellectual activity to another. The reason self-study can improve the quality of teaching and learning is that it conforms to the psychological principles of the human learning process. First, in the process of self-study, students’ thinking and other mental activity are in an active state from the beginning to the end. In this active state, people understand the changes and interrelationships of external things quickly and sensitively, so they receive knowledge faster and more conclusively. Second, students pick up new concepts based on their original knowledge level, which is in line with the principle of gradual progress. It is easier to accept knowledge when you have some foundation, which can also stimulate students’ interest in habitual learning and encourage them to develop a lifelong love of learning, which is conducive to the acquisition of knowledge and the improvement of understanding.

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Again, self-study takes the visual organ as the main means, and the stimulus will not disappear due to the change in time and space. Therefore, it is easy for students to analyze, synthesize, summarize, and compare the received knowledge; that is, they can properly “digest” the learned knowledge so that it can be solid and deepen their understanding. Finally, this form of self-study can be strengthened and self-regulated over time to consolidate the right and eliminate the wrong so that the mastered knowledge is highly accurate. Therefore, planning the development of secondary school students’ comprehensive intellectual self-study ability is an indispensable task in secondary school mathematics teaching.

Modern Mathematics Helps the Development of Abstract Thinking The history of mathematical development can be divided into three stages: elementary mathematics, higher mathematics, and modern mathematics. Akpilov in “Problems of the content and objects of modern mathematics” (Shi, 1983) points out that modern mathematics has the following characteristics that distinguish it from other stages of development. 1. Profound interpenetration of the main branches of mathematics—geometry, algebra, and analysis. 2. Creation of new general concepts—a shift to a new and higher level of abstraction. 3. Significant expansion of the objects of mathematics and their applications. 4. Constant generation of new theories and more effective mathematical methods. 5. Geometric arguments dominate, and rational methods have been extended, refined, and combined with the concept of set theory. 6. The foundations of modern mathematics have been deeply analyzed—the connection of modern mathematics concepts, the combination of its theories, the methods of mathematical proof, that is, the development of mathematical research based on a new general logic constructed by set theory and mathematical logic. In accordance with these features, Akpilov highlighted the nature of modern mathematical objects. 1. The definition of the object of mathematics proposed by Engels (mathematics is the science of “quantitative relations and spatial forms of the real world”) is still applicable to modern mathematics, provided that the concepts of spatial forms and quantitative relations are properly expanded. 2. Modern mathematics is the science of possible, generally changing, quantitative relations and interconnections between quantities.

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3. For modern mathematics, what is important is only the structure of the quantitative relations between the objects under study. 4. If, in the past, geometry and mathematics as a whole were interpreted as a science of quantitative relations, then geometry and even mathematics as a whole are no longer a science of quantitative relations because mathematics is clearly permeated by the category of measurement in the sense of the unity of quantity and quality. 5. Modern mathematics is becoming progressively more nonmetric and nonquantitative, and quantity belongs to a very small area of modern mathematics. 6. At the center of modern mathematics are the concepts of “order” and “structure.” 7. Modern mathematics is the science of models and structures. 8. Modern mathematics is the sum of abstract forms: mathematical structures. 9. The field of objects of modern mathematical research exists in the spirit of rationality, as well as in combination with other sciences to form a system of science. Professor Xun Du of Peking University and his colleague (Du & Sun, 1992) pointed out six characteristics that help us comprehend modern mathematics. 1. The research objects, contents, and methods of modern mathematics show a high degree of abstraction and unity. 2. Emphasis is placed on the establishment of the axiomatic system and the analysis of the structure. 3. Emphasis is placed on the combination of different mathematical disciplines and the constant development of new mathematical fields. 4. Research on mathematical models that are more compatible with the real world and solve more complex mathematical problems. 5. Close connection with electronic computers. 6. The penetration of mathematics into all sciences and social sectors (e.g., the emergence of marginal disciplines such as “mathematical psychology” in the field of psychology). Professor Du and his colleagues also pointed out that: The modern mathematical stage started with Cantor’s establishment of set theory (1874), “Cantor’s set theory, with its originality and foresight, established the foundation for mathematics.” The stage of modern mathematics is clearly marked. The objects of modern mathematics are general sets, various spaces and manifolds, which can be unified by the concept of sets and mappings, and it has been difficult to distinguish which belongs to the category of numbers and which belongs to the category of forms. (Du & Sun, 1992)

The above statement clarifies the characteristics and significance of modern mathematics and lays the theoretical foundation for the introduction of modern mathematics into secondary school mathematics. From the perspective of thinking psychology, the mathematics of secondary school basically retains the traditional system but introduces the preliminary knowledge of sets, mathematical logic, modern algebra (groups, rings, domains, vector spaces, matrix algebra, etc.), calculus, probability statistics, algorithmic language, among

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others. It divides the subject instead of unifying it, retains Euclidean geometry, makes the necessary deletion, and partially infiltrates some modern mathematical ideas on the basis of the tradition. The superiority of this arrangement is obvious. First, modern mathematical ideas must be introduced, which is “the inevitable trend of modernization of mathematics education.” Some of the contents in modern algebra, such as set theory, mathematical logic, groups, rings, and domains, reflect the high abstraction of mathematics and focus on its foundation. Calculus, probability statistics, algorithmic language, and so on are from the application perspective view, reflecting the broad application of mathematics. Chinese secondary school mathematics teaching should not be outside the world of “modernization of mathematics education.” Therefore, it is necessary to introduce the ideas of modern mathematics. Second, most of the content of traditional mathematics is basic mathematics. Mathematics education cannot be separated from history; however, it cannot ignore history. Secondary school mathematics faces the challenge of educating all students, not merely with regard to developing mathematicians. The more important purpose of mathematics teaching is to develop students’ intelligence. Euclidean geometry is important to develop students’ logical thinking ability, and it adopts a more natural and simple approach that has been effective for many years. Before students acquire appropriate abstract logical thinking ability or their thinking development is still immature, it is not advisable to introduce purely abstract concepts. If introduced to these too early, most students cannot accept or understand them. If mathematics teaching is not infiltrated with modern mathematical ideas but based entirely on modern mathematics, it will affect students’ academic performance and motivation and will not be conducive to their intellectual development. Therefore, in the secondary school mathematics curriculum, I think modern mathematics content should take the approach of “introduction” rather than “replacement” and “infiltration” rather than “all.” The “differentiated” rather than “unified” is appropriate. What are the benefits of introducing modern mathematical ideas in secondary school mathematics? Taking the basic ideas and methods of set theory permeating traditional mathematical knowledge as an example, it not only enables students to grasp some basic ideas and methods of modern mathematics earlier but also consolidates and deepens their understanding of traditional mathematical knowledge, which helps develop students’ abstract thinking ability. Seventh- and eighth-grade students learn about sets, often using mathematical concepts to express the object characteristics of the set, whether it is to give the characteristics of the elements in the set, such as {natural numbers}, {rational numbers}, {triangles}, {integers with absolute values less than 5}, to clarify what elements are in the set, or to give specific elements, for example, {3, 6, 9, 12…}, {10, 100, 1000…}. Asking students to abstract the characteristics of the elements in a set in mathematical language requires them to understand the concept accurately and use it to develop their logical thinking abilities.

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Set theory has a high degree of abstraction. If secondary school students are guided to use the set-theoretical perspective to deal with exercises of traditional mathematical content, teachers can promote a degree of abstraction in the development of intellectual qualities, that is, profundity and originality of intellectual qualities. Using vectors as the basic tool and adopting the set correspondence and geometric transformation perspectives to deal with some important content in elementary mathematics enables students to appreciate the basic spirit of modern mathematical thinking and revive traditional content. For example, trigonometric functions are very traditional contents because the basic properties of trigonometric functions are the analytic representations of the geometric properties of the circle (mainly symmetry). Therefore, with the help of the rotational symmetry of the circle, we can discuss the properties of trigonometric functions with the help of the rotational symmetry of the circle and the viewpoint of plane vectors and transformations. (1) The axisymmetric transformation of the x-axis T1 : θ → −θ , the point (x, y) on the circle becomes (x1 , y1 ) through T 1 ; then, we have (x1 , y1 ) = (x, −y), that is, cos(−α) = cos α, sin(−α) = − sin α. (2) The rotational transformation, which rotates the final side of α by π2 counterclockwise around the origin T 2 : α → π2 + α, the point (x, y) on the unit circle becomes (x2 , y2 ) through T 2 , then we have (x2 , y2 ) = (−y, x), that is, cos

 π  + α = − sin α, sin + α = cos α. 2 2

π

With the above two transformations, all the induced formulas can be obtained. For example, after two transformations of T 2 , we have α → π + α, and so cos(π + α) = cos

π

+

π

+α



= − sin

π

 + α = − cos α,

2 2 2  π  π π + + α = cos + α = − sin α. sin(π + α) = sin 2 2 2

After one-time transformation of T 1 and another transformation of T 2 , we have α → −α → π2 − α, so cos

π

 − α = − sin(−α) = sin α,

2  π − α = cos(−α) = cos α. sin 2

In fact, all trigonometric formulas can be recognized in this way: the trigonometric functions of angles with the same terminal side are rotational transformations rotating

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by integer multiples of 2π, and the induced formulas are the transformations of T 1 , T 2 , and their synthesis. In particular, the sum (difference) angle formula is the rotational transformation that rotates any angle. Obviously, any point on a circle is rotated arbitrarily on its circumference, and the −→ result is still on the circle. In other words, it means that the unit vector O A rotates −→ any angle α to O B, and point B is still on the circle. The unit circle is still the unit circle after rotation, which is its rotation invariance and symmetry. −→ −→ Let the unit vector O A correspond to any angle α. Rotating O A by any angle β −→ to O B, point B is still on the unit circle. This can be viewed from two perspectives. −→ One is that the angle corresponding to the terminate side of O B is α + β, so there is −→ O B = cos(α + β) · i + sin(α + β) · j.

(9.1)

−→ −→ Another is that O B is obtained by rotating O A by β. The coordinate system is −→ −→ established with O A as the horizontal axis, the unit vector on O A is i, , and the unit vector on its vertical axis is j, . Then, we have −→ O B = cos β · i , + sin β · j , .

(9.2)

Again, in the original coordinate system, i, can be expressed as i , = cos α · i + sin α · j.

(9.3)

and j, is the result of rotating i, by π2 , so the angle it corresponds to is α + π2 , so   π π · i + sin α + · j = − sin α · i + cos α · j j , = cos α + 2 2

(9.4)

Substituting Eqs. (9.3) and (9.4) into Eq. (9.2), we have −→ O B = cos β · i , + sin β · j , = cos β(cos α · i + sin α · j ) + sin β(− sin α · i + cos α · j ) = (cos α cos β − sin α sin β)i + (sin α cos β + cos α sin β) j. Using the fundamental theorem of plane vectors, that is, we have cos(α + β) = cos α cos β − sin α sin β, sin(α + β) = sin α cos β + cos α sin β. This is the sum angle formula. By replacing β with -β, the difference angle formula is obtained.

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In conclusion, using modern mathematical ideas (transformation ideas) and tools (vectors), one can not only manage the complexity with simplicity in content treatment but also effectively develop students’ abstract logical thinking skills. The introduction of preliminary mathematical logic in secondary schools is not only beneficial for students to learn and master the principles and use of computers but also important for the development of abstract logical thinking abilities. Preliminary mathematical logic mainly refers to propositional arithmetic, including propositions (or sentences), negation (not), conjunction (with), disjunction(or), implication (if… then), equivalence (when and only when), truth and truth tables, the principle of substitution, the format of reasoning, negative and inverse propositions, and direct and indirect proofs, among others. This knowledge enables students to improve their mathematical generalization abilities, propositional abilities, and self-consciousness in logical reasoning. Take, for example, the system of quadratic equations: 

(1) If (2) If (3) If

a1 a2 a1 a2 a1 a2

/= = =

a1 x + b1 y = c1 a2 x + b2 y = c2

b1 , the above system of equations has a solution. b2 b1 = cc21 , the above system of equations has infinite solutions. b2 b1 /= cc21 , the above system of equations has no solution. b2

There are three reasonings here, but elementary algebra simply uses the reasoning without studying it. Instead, the above three cases are analyzed with the knowledge of mathematical logic to study the relationship between the logical forms of their constituent premises and conclusions. When the premise is true, the conclusion must be true; when the premise is false, the conclusion must be false. Such a comprehensive analysis of the relationship between premises and conclusions helps students learn to reason logically and consciously and improves their reasoning ability. Mathematical logic emphasizes the form of reasoning, that is, the constant truth proposition commonly used in reasoning. For example, the is a conversation between two people like this. p = “If a = b, then a 2 = b2 . ” q = “If so, if a /= b, then a 2 /= b2 . ” This reasoning is wrong. Why? This reason is that if we assume that x means a = b and y means a 2 = b2 , we have p = x → y and q = ¬x → ¬y. The truth table of the above judgment (x → y) → (¬x → ¬y) is shown in Table 9.1 (“T” means true; “F” means false). It is not a constant truth proposition and therefore not an inference format. If secondary school students learn mathematical logic and master the reasoning format, they can not only make the reasoning process more conscious but also make the thinking activity of reasoning more abstract, thus improving the profundity of intellectual quality.

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Table 9.1 Truth Table x

y

T

T

(x → y) → (¬ x → ¬ y) T

T

T

T

F

T

T

F

T

T

F

T

F

F

T

F

T

T

T

F

F

T

F

T

T

F

T

F

F

F

T

F

F

F

T

F

T

T

F

T

T

F

1

2

1

4

2

1

3

2

1

Step

The Mathematical Olympiad and Intellectual Development of Secondary School Students In recent years, the negative effects of the Mathematical Olympiad have been disclosed by various media. At the same time, the convenience and speed of the Internet have afforded an increasing number of people the opportunity to express their opinions about the Mathematical Olympiad. Together with the media, the voices that criticize the Mathematical Olympiad are clearly the absolute majority. The Mathematical Olympiad has become a “street rat” that everyone hates. The voices of those who criticize the Mathematical Olympiad can be heard everywhere, and there are calls from the education administration, Administrative Department of Education, and representative of the National People’s Congress, China People’s Political Consultative Conference (CPPCC) members. Should we ban Mathematical Olympiad, or should we allow it to continue in a reasonable form? Why are many parents still willing to let their children participate in Mathematical Olympiad training despite the numerous voices that criticize it? What is the significance of the Mathematical Olympiad for students? Obviously, to answer these questions, it is necessary to analyze in depth the role and value of Mathematical Olympiad training for the development of intelligence and ability. I discuss the following three aspects in relation to the conditions of intellectual development.

Mathematical Olympiad Training Effectively Expands Secondary School Students’ Mathematical Horizons and Enriches and Improves Their Mathematical Knowledge Structure The drive for intellectual development comes from students’ internal needs and their original level of thinking or thinking structure. Thinking materials are the basic components of the thinking structure, and in the case of mathematical thinking abilities, students’ mathematical knowledge structure is the premise on which their development depends. Competition training in secondary school mathematics is,

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first, a further expansion or broadening of the content of the secondary school mathematics curriculum. The advantages are twofold: on the one hand, it allows students to deeply appreciate the basic knowledge of the mathematics curriculum and to grasp the essence of mathematical theories and the conditions for their application from a broader context, thus improving their mathematical cognitive structure. On the other hand, it affords students more opportunities to challenge complex mathematics with their intelligence, thus stimulating their internal desire to know and understand. According to the “Mathematics Curriculum Standard for Compulsory Education (2011 edition)” in the current middle school textbooks, for example, the content about factorization only involves the method of extracting common factors and the formula method, in which the formula includes the difference of squares and the perfect square formulas without introducing the cubic sum (difference and the perfect cubic formulas). The factorization of quadratic hierarchies has been added to the “Observation and Conjecture” section, and its status has been weakened. Students are only required to understand the factorization of the form x 2 + ( p + q)x + pq. As we know, similar to plane geometry, factorization requires certain skills, particularly strong observation and flexibility in choosing formulas according to the characteristics of the problem, which challenges students’ intellect and is an available platform to train and develop their mathematical thinking abilities. Although the development of mathematical software (e.g., the CAS system) has greatly weakened the role of skillful factorization content, it is still feasible to use it to train students’ problem-solving abilities and mathematical thinking in the middle school version of the Mathematical Olympiad. Example 1 Factorization. (1) (2) (3) (4)

ax 2 − bx 2 − ax + cx 2 + bx − cx 2   2 a 2 + b2 (a + b)2 − a 2 − b2 2y 2 − 5x y + 2x 2 − ax − ay − a 2 (1 + y)2 − 2x 2 1 + y 2 + x 4 (1 − y)2 Analysis:

(1) This equation has a large number of terms; therefore, we should consider group decomposition. After proper adjustment and two-by-two grouping, the common factorization can be found, that is,    The original equation = ax 2 − ax − bx 2 − bx + cx 2 − cx = ax(x − 1) − bx(x − 1) + cx(x − 1) = x(x − 1)(a − b + c). (2) It is observed that the latter term can be decomposed by applying the difference of squares formula, by which we can find a common factor. After extracting the common factor, we then further can collate and apply the perfect square formula. This question raises the level of abstraction in the application of the formula.

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 The original equation = 2 a 2 + b2 (a + b)2 − (a + b)2 (a − b)2

  = (a + b)2 2 a 2 + b2 − (a − b)2 = (a + b)4 . (3) Observing that the first three terms can be decomposed by the crossmultiplication method, and the last two terms can be extracted from the common factor so that the whole polynomial becomes a trinomial form again. The crossmultiplication method is applied again to try and determine whether it can be decomposed, that is, The original equation = (2y − x)(y − 2x) − a(x + y) − a 2 = (2y − x + a)(y − 2x − a). (4) After attempting the formula method or the cross-multiplication method, we find that neither of them can be used to decompose directly. However, if the middle term 1+ y 2 is changed to 1− y 2 , we can obtain the perfect square form, and then use the square difference formula, the expression can be decomposed. Finally, we can use the group decomposition method to obtain the ideal decomposition formula, that is:  The original equation = (1 + y)2 − 2x 2 1 − y 2 + x 4 (1 − y)2 − 4x 2 y 2

2 = (1 + y) − x 2 (1 − y) − (2x y)2

 = (1 + y) − x 2 (1 − y) + 2x y [(1 + y)  −x 2 (1 − y) − 2x y = (1 + x)(1 − x)(1 − x + y + x y)(1 + x + y − x y). Another example is that the drawer principle is a very lifelike mathematical principle, meaning that if you put n + k(k ≥ 1) balls into n drawers, then there is bound to be a drawer with at least two balls in it. Apart from being close to reality, this is a principle of common sense, which even elementary school students can understand. We know that the simpler a principle is, the closer it is to common sense and the wider its application. However, at the same time, its variations will be more flexible and diverse. The drawer principle is this type of principle. When applying this principle to solve a problem, the key is to build an appropriate “drawer”—a mathematical model—according to the characteristics and needs of the problem, and in building the “drawer,” strong abstract logical thinking abilities are required. Thus, the drawer principle has become an active part of the open system of mathematics—whether it is an elementary school mathematics competition or a secondary school mathematics competition, the proposer often uses the principle as a knowledge point to propose questions.

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Example 2 A secondary school mathematics interest group has 40 students, the oldest and youngest of whom are 15 and 13 years old, respectively. Try to prove that two of these students can always be found in the same year and month. Analysis: According to the question, the students are either 13, 14, or 15 years old, so there are only three possible years of their birth and thus a total of 36 possible years and months of birth. Imagine that these 36 years and months are 36 drawers, and 40 students are regarded as 40 balls that can be put into them. According to the drawer principle, there are at least 2 students born in the same year and month; that is, these 2 students were born in the same year and month. Another following example can also be derived. A secondary school has 1,500 students, the oldest and youngest of whom are 16 and 13 years old, respectively. Is it possible to find 2 students who were born on the same day?

Another form of expression for the drawer principle is that if nm + k(k ≥ 1) balls are put into n drawers, there must be a drawer that has at least m + 1 balls. Example 3 Proof: At any party of 6 people, there are always 3 people, either two of whom know each other or two who do not know each other. Analysis: Let one of the 6 people be A. A, and the remaining 5 people either know each other or do not know each other. Let us consider “know” and “do not know” as 2 drawers and people as balls. When we put 5 balls into 2 drawers, there must be a drawer with at least 3 balls in it. In other words, A and these 3 people either know each other or do not know each other. Let these 3 people be B, C, and D. If A knows all 3 people, then when at least 2 of the 3 people know each other; for example, B and C know each other, the proposition is already valid, because then A, B, and C know each other; otherwise B, C, and D do not know each other, in which case the proposition is also valid. If A and 3 do not know each other, the proposition is valid by the same reasoning. This question can also be analyzed by constructing graphs. When two people know each other, connect their corresponding two points with a solid line segment; if they do not know each other, connect them with a dashed line segment. This way, the proposition is transformed into a proof that there must exist a triangle in the graph whose three sides are either all solid or all imaginary lines. The proof is similar to the method above and will not be repeated. The key to solving the problem by applying the drawer principle is to construct the dresser and imagine the balls appear and are placed in each drawer. In this way, the drawer principle will provide a useful visual tool. However, any construction is a process from nothing to something, from void to solid, which requires creative imagination and keen insight on the part of the problem solver.

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Compared with Secondary School Mathematics Textbooks, the Mathematical Thoughts and Methods Applied in the Mathematical Olympiad are More Skillful and Flexible and Often Require Specialized Training Mathematical thoughts and methods are the soul of mathematics, and the creation of any mathematical theory or branch of mathematics is always dependent on breakthroughs in mathematical thought and methods. Descartes’ coordinate method led to the birth of the discipline of analytic geometry, and his idea of combining numbers and shapes brought algebra and geometry, both of which had been developed independently for more than 2000 years. People can use numbers to solve the problem of shapes, and vice versa. The abstract analysis pioneered by Euler led to the successful solution of the problem of the Seven Bridges, and it was this approach that showed that graphs could be used as a tool to solve many problems in real life, eventually giving birth to graph theory in mathematics. We cannot develop our students’ mathematical thinking without appropriate mathematical theories and methods. Compared with specific mathematical concepts and principles, mathematical thinking has more room for transfer, and it makes thinking deeper, more flexible, and more original. Due to the need to take into account the extent of basic knowledge in the current secondary school mathematics textbooks, many basic and important mathematical concepts and methods are often not introduced in depth, such as parity analysis, identical methods, and pairing methods. In addition, the ideas of generalization, correspondence, optimization, and other mathematical thoughts and methods do not appear in textbooks and are rarely introduced in classroom settings. In some cases, even though they are mentioned in the textbook, they are not introduced systematically, and students cannot learn them thoroughly, such as the construction method, inverse method, and recursion method. As there is no limitation on class time and content, Mathematical Olympiad training can expose students to more colorful mathematical ideas and methods and provide them with adequate training. Example 4 There are 5 rows of chairs in the classroom, 5 in each row, with one student sitting in each chair. If after 10 days each student must exchange places with one of his/her nearby (front, back, left, right) classmates, can they exchange successfully? Why?

Analysis: Answering such a problem does not require complex calculations, and the knowledge involved is only the basic parity property, so with this knowledge, elementary school students can solve it. However, even high school students may not be able to answer this question easily because it focuses more on the observational and analytical skills of the solver. As shown in Fig. 9.1, we may want to label the 25 seats with serial numbers. When a student wants to exchange seats with one of his/her neighboring students, the odd and even seat numbers are actually exchanged. Thus, to achieve one exchange

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Fig. 9.1 Figure for Example 4

between all students, the number of odd-numbered seats should be equal to the number of even-numbered seats. This is not possible because the odd and even seat numbers are 13 and 12, respectively. Applying the odd–even analysis, we can answer many similar questions, such as the following. (1) There are several people in a mathematics interest group in a secondary school. If two people talk on the phone once, it is assumed that both have made a phone call. Is the number of people who have made an odd number of phone calls odd or even? (2) There are three integers written on the blackboard. Erase any one of them and rewrite it as the sum of the other two numbers minus 1. Continue in this way and finally get 1997, 2007, and 2011. Can the three numbers originally written on the blackboard be 6, 6, and 6? Example 5 Let the real numbers x, y, and z satisfy x + y + z = x1 + 1y + 1z = 1. Proof: at least one of x, y, and z is 1. Analysis: This is a middle school Mathematical Olympiad question. The known conditions are clear, and the conclusion is easy to understand. However, when the solver tries to deform and derive from the conditions continuously, he/she finds that the goal is not clear. In other words, what mathematical equation is needed to obtain the conclusion that “at least one of x, y, and z is 1?” The human mind is goal-oriented, and if the goal is not clear, the mind naturally falls into chaos or even blindness. Therefore, the key to answering this question is to deform and transform the conclusion into a mathematical equation that can be easily derived or proved. We can take a step back

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and consider the simpler proposition that at least one of x, y, and z is 0. For such a proposition, middle school students can easily find that its equivalent is xyz = 0. From this, we can deduce that the equivalent of “at least one of x, y, and z is 1” is (x − 1)(y − 1)(z − 1) = 0. Thus, by combining the known conditions and using a combination of the forward and backward extrapolation methods, it is not difficult to find the idea of the proof. After investigation, I found that middle school or high school students can easily conclude that “x, y, and z in at least one is 0” is equivalent to x yz = 0. However, for “x, y, and z in at least one is 1,” it is difficult to find the above mathematical equivalence even if you are an undergraduate mathematics major or mathematics teacher. Why is this? I believe that it may have to do with the mathematical equations that students have mastered. Due to their daily training, students have more opportunities to encounter: if x y = 0 or x yz = 0, it means that at least one of x and y is 0 or at least one of x, y, and z is 0. This forms a mathematical relationship schema that is retained in the students’ minds that can be extracted and applied at any time. However, the connection between “at least one of x, y, and z is 1” and “(x − 1)(y − 1)(z − 1) = 0” rarely appears in daily training, so when students are confronted with the proposition “at least one of x, y, and z is 1,” they cannot quickly recall the appropriate equivalent. Therefore, when students are faced with the proposition “at least one of x, y, and z is 1,” the only way to obtain its equivalence is to make a creative transformation of the original formula. Creative thinking is reflected not only in the problem-solving process but also in the subject’s active transformation and reorganization of the existing mental schema, which is an important prerequisite for the formation of a good cognitive structure.

Mathematical Olympiad Training is a Cognitive Activity that Simultaneously, Challenges the Intellect and Stimulates Curiosity Compared with traditional textbooks, current elementary and secondary school textbooks, especially mathematics textbooks at the compulsory education level, have been enhanced in terms of content interest. However, as mathematics textbooks mainly emphasize the systematic and logical nature of the content and focus on developing students’ basic arithmetic, abstract thinking, and spatial imagination abilities, the interesting and lifelike components tend to decrease gradually as the grade level increases. In contrast, the Mathematical Olympiad is an open system; that is, as long as the content is within the student’s reach, it can be included in the system for explanation and training. Therefore, the Mathematical Olympiad training can expose students to both flexible and challenging mathematical problems and can always add interesting problems to make cognitive activities more engaging, thus stimulating students’ curiosity and mathematical imagination.

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Example 6 Given 10 points on the circumference of a circle, color six of them red and the remaining four white, and then divide the circumference into arcs that do not contain each other. The arcs with red ends are labeled with number 2, the arcs with white ends are labeled with fraction 21 , and the arcs with different colors at both ends are labeled with number 1. Multiply all these numbers together and find the product. Analysis: This example is a middle school competition question in a major city in China that was used in the 1990s. It is said that the reference solution provided by the competition proposing team to this question at that time was a categorical discussion, and the answer process was rather complicated. After the competition, the marker found that a middle school girl gave a very clever solution that was actually more concise and clearer than the reference solution and was general enough to answer the generalized form of the question, which truly amazed the experts of the proposing team. As I am sure you are curious to know, what was this clever solution? Before disclosing this, let us first follow the path of general candidates’ thinking: mark 6 red dots and 4 white dots on the circumference of the circle and mark the corresponding numbers on each arc segment as required by the question. Calculate the product of these numbers as 4, then change the position of some red or white dots, calculate the product of the numbers on the arc segment, and obtain 4. After several attempts, it was found that regardless of the position of the red and white dots, the product of the numbers on the arc segment remained the same, that is, 4. It is clear that such an answer is neither rigorous nor perfect because the subtlety of mathematics is to find the reason and essence of the changes that are constant. The girl’s brilliance also lies in her ability to analyze further: why is the product of the numbers on the arc segment independent of the position of the red and white dots? As a result, she found that the product of the numbers on the arc is related to the number of red and white dots, so she thought of assigning values to the red and white dots, which is acceptable as long as the assignment does not change the numbers on the corresponding arcs specified in the question. The assignment consisted of assigning √ 2 to the red dot and √12 to the white dot so that the number on the arc is the product of the values of the two endpoints of the arc. Thus, regardless of the position of the red and white dots, the product of the numbers on each arc segment is:  √ 6 1 4 2 2 =4 √ 2 Note that each point is used twice in the calculation of the numbers on the arc segment. The wonderful thing about the girl’s solution is that if the red dots are replaced by the usual m dots and the white dots are n dots. Consequently, the product of the numbers on each arc segment is:

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    √ m 1 n 2 2 = 2m−n √ 2 regardless of the position of these dots on the circumference of the circle. As you can see, the girl’s understanding of the above problem and her mathematically creative solution to it seems to have gone beyond the intent of the competition proposer who prepared this mathematical problem. She shows us that our youth have great creative potential, that their intelligence needs to be explored and stimulated, and that more opportunities and platforms should be made available for them to express their creativity. Overall, although the Mathematical Olympiad tends to deviate from its original purpose due to certain highly motivated students, parents, and teachers, we cannot deny its great value in developing the intelligence of the majority of young people and tapping their potential. Therefore, instead of dissolving the Mathematical Olympiad, we should revamp it so that it can be reasonably used and offer correct guidance to students. First, we hope that aside from reporting on the negative aspects of the Mathematical Olympiad, news media reporters will report on the cases of students who have achieved healthy development through Mathematical Olympiad training so that parents and teachers can see what kinds of students can benefit from the Mathematical Olympiad platform. Second, students’ parents should have a proper and realistic evaluation of their children’s abilities and interests. If their children are not interested in or particularly skilled in mathematics, it is not appropriate for them to participate in intensive Mathematical Olympiad training classes. In this regard, mathematics teachers and classroom teachers should provide necessary references and suggestions to parents. Third, all kinds of Mathematical Olympiad propositions and training materials should grasp the content and difficulty of the problem, minimize highly advanced skills or arithmetically complex topics, and instead focus on the examination and training of mathematical thinking and methods. In particular, we must firmly refuse to force elementary students to study at the middle school level, middle school students at the high school level, and high school students to comprehend collegelevel questions. As this practice often causes students to learn some solutions by rote and leads them into the trap of thinking stereotypes, it is not conducive to the cultivation of students’ flexible thinking quality, the scientific spirit of bold questioning, and being a brave spirit bent on exploration.

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