Exploring on the “Two Basics” of Mathematics Education in Mainland China 中国数学教育中的双基课程探析

1. Exploring on the “Two Basics” of Mathematics Education in Mainland China中国数学教育中的双基课程探析 Prof.SONG Naiqing, Dr.KUANG Kongxiu Southwest University 西南大学 宋乃庆 邝孔秀 Edge Hill university,UK March,2014 2014.3 英国 知山大学

2. Outline 内容提要 • 1. Background 背景简介 • 2. Connotation and Formation of the “Two Basics” 数学双基课程 的内涵与形成 • 3. Some Experience of the “Two Basics” in Mathematics Education数学双基课程的基本经验 • 4. Rationality of the “Two Basics” 数学双基课程的合理性 • 5. Academic Contention about the “Two Basics” 数学双基课程的 争鸣 • 6.Prospect of the “Two Basics” 数学双基课程发展的展望 NOTE: “Two Basics” is the abbreviation of basic knowledge and basic skills.

3. 1. Background背景介绍 Since the 1980s, mathematics education in Mainland China has attracted great attention gradually. Chinese students’ performance ranked top in many prominent international comparative studies. 20世纪80年代以来，中国的数学教育逐步引人注目，中国学生在多项国际数学测试中表现出色。 For example, the mathematics achievement of 15-year-old students from Shanghai rank first in PISA 2009 and 2012. (See Table 1) 例如，中国上海15岁学生参加2009年、2012年PISA测试，数学学科得分均为第一。（见下表）

4. Table 1: Students’ performance in PISA 2009 and 2012表1：2009、2012年PISA数学成绩(前十名)

5. A comparative study of Prof. Jinfa Cai showed that Chinese students in primary and secondary schools outperformed US students on computation, simple problem-solving tasks and complex problem-solving tasks which is process-constrained . (Cai,J. & Cifarelli, V. (2005). Thinking mathematically by Chinese Learners: A cross-national comparative perspective. In L. Fan, N.-Y. Wong, J. Cai and S. Li (Ed.), How Chinese learning mathematics: perspectives from insiders (pp.73-106). Singapore: World Scientific Publishing.) 蔡金法的跨国比较研究也表明，中国中小学生在计算任务、简单问题解决任务和过程受限的复杂问题解决任务上有明显优势。 （蔡金法,Cifarelli Victor.中国学习者的数学思维特征—一个跨国比较研究的视角[A].载：华人如何学习数学[M].南京:江苏教育出版社,2005：65. ） Many Chinese and international scholars believe that Chinese students in primary and secondary schools have a solid foundation in mathematics. 中国和国际上不少学者认为，中国中小学生的数学基础扎实。

6. The emphasis of “Two Basics” is actually the typical characteristic of mathematics education in Mainland China. A lot of experience about “Two Basics” has been accumulated in China and many experiments in exploring the reformation of mathematics education have been carried out, such as mathematics education experiment at Qingpu district and GX (Gao Xiao) mathematics education experiment in junior high schools. 事实上，重视基础知识和基本技能（以下简称“双基”）的教育是中国数学教育的典型特征，中国在这方面积累了许多经验，并出现了体现双基的青浦数学教育实验、GX初中数学教育实验等富有成效的数学教育改革探索。

7. 2. Connotation and Formation of the “Two Basics” 数学双基课程的内涵与形成 2.1 Connotation 数学双基课程的内涵 In the mathematics teaching syllabus (curriculum standards), “Two Basics” means basic knowledge and basic skills. “Basic knowledge” includes mathematical concepts, properties, laws, formulas, axioms, theorems and mathematical ideas and methods included in contents. “Basic skills” are the skills of operation, data processing, simple reasoning, graphing , charting and so on. 中国的数学教学大纲（数学课程标准）中，“数学双基”指数学的基础知识与基本技能。其中：“数学‘基础知识’包括数学中的概念、性质、法则、公式、公理、定理以及由其内容反映出来的数学思想和方法；数学‘基本技能’指按照一定的步骤进行运算、处理数据（包括使用计算器）、简单的推理、画图以及绘制图表等。”

8. 2. Connotation and Formation of the “Two Basics” 数学双基课程的内涵与形成 2.1 Connotation 数学双基课程的内涵 Mathematics educators in Mainland China commonly believed that the basic tasks of mathematics education in primary and secondary education are making students remember the connotation and extension of mathematical knowledge solidly and finally developing mathematical competence. With this idea, the systemic explorations on design, implement, assessment of Chinese mathematics curriculum were carried out. 中国的数学教育工作者普遍认为，在基础教育阶段，数学教育的基本任务是让儿童牢固地识记数学知识的内涵和外延、形成运用数学知识的技能，并由此发展数学能力。 围绕这一理念，中国的数学课程在设计、实施、评价等方面进行了系统的探索。

9. The emphasis of base and training in education is the basic characteristic and experience of Chinese traditional education, which also is the accumulation and reflection of Chinese traditional culture: “Practice makes perfect” “While Learning, the exhilaration” “gain new insights through reviewing old material” “重视基础，重视训练”是中国传统教育的基本特点和经验，是中国传统文化的沉淀和积累： 　“熟能生巧” “学而时习之，不亦乐乎” “温故而知新”

10. Based on the successful experience of educational practice and the introspection of history, “Two Basics” is gradually formed after People's Republic of China was established (1949). 数学双基课程是中华人民共和国成立（1949年）后在基础教育实践的成功经验的总结和历史教训的反思中逐步形成的。 In 1952, “Provisional Regulations in Secondary School (Draft)” issued by Ministry of Education of China proposed that, one of the aims in secondary school is to make students obtain “the basic knowledge and basic skills of modern science”. This is the first specific requirement about “Two Basics” teaching in the history of Chinese education. 1952年，教育部颁发的《中学暂行规程(草案)》提出，中学教育的目标之一是使学生获得“现代科学的基础知识和技能”。这是中国现代教育史上首次对双基教学提出具体的要求。

11. In 1963, “Syllabus of Mathematics Teaching in Secondary School (Draft)” proposed that, “mathematics teaching in secondary school aims to make students master the basic knowledge of algebra, plane geometry, solid geometry, trigonometry and plane analytic geometry, develop students' ability of calculation, logical reasoning, Spatial imagination”. According to this, the compilation of mathematics textbooks emphasized that, “the basic knowledge and basic skills which is necessary for further study and productive work should not be omited”. 1963年，《全日制中学数学教学大纲（草案）》规定“中学数学教学的目的是：使学生牢固地掌握代数、平面几何、立体几何、三角和平面解析几何的基础知识，培养学生正确而且迅速的计算能力、逻辑推理能力和空间想象能力。”数学试用教材编写时强调：“对于进一步学习和参加生产劳动必需的基础知识和基本技能的各个主要方面，注意不使遗漏……”

12. In 1988, “Syllabus of Mathematics Teaching in Full-time Junior High School in Nine-year Compulsory Education (preliminary draft)” defined the “Two Basics” specifically for the first time. Hence, this definition is used in almost all mathematics teaching syllabus or curriculum standards. 1988年,《九年义务教育全日制初级中学数学教学大纲（初审稿）》首次对“双基”给出了明确具体的界定。此后的数学教学大纲或数学课程标准基本沿用了这一界定。 In 2011, “Mathematics Curriculum Standards for Compulsory Education(2011)” stipulated that, mathematics teaching should make students “obtain basic knowledge, basic skills, basic ideas and basic activity experience which are necessary to adapt to the social life and further development; realize the connections between mathematical knowledge, between mathematics and other subjects, between mathematics and life; think mathematically; and improve the abilities to discover, pose, analyze and solve problems”. 2011年，《义务教育数学课程标准（2011年版）》提出，数学教学要使学生“获得适应社会生活和进一步发展所必需的数学的基础知识、基本技能、基本思想、基本活动经验。体会数学知识之间、数学与其他学科之间、数学与生活之间的联系，运用数学的思维方式进行思考，增强发现和提出问题的能力、分析和解决问题的能力。”

13. 3. Some Experience of the “Two Basics” in Mathematics Education 数学双基课程的基本经验 3.1 The specific requirements of “Two Basics” in Mathematics Teaching Syllabus (Curriculum Standards) 数学教学大纲(课程标准)对数学双基有具体要求 There are always specific requirements of “Two Basics” in Chinese Mathematics Teaching Syllabus (Curriculum Standards). For example, “Mathematics Curriculum Standards for Full-time Compulsory Education (Experimental Draft)(2001)” put forward 4 levels of the “Two Basics”: 中国的数学教学大纲（课程标准）历来对数学双基的要求有具体刻画。比如，2001年颁布的《全日制义务教育数学课程标准(实验稿)》对数学双基的要求提出了4个层次：

14. 3. Some Experience of the “Two Basics” in Mathematics Education 数学双基课程的基本经验 (1)Knowing —being able to get and illustrate the characteristics (or meaning) of objects from concrete cases; being able to identify objects in the context according their characteristics. （1）了解(认识) ——能从具体事例中知道或举例说明对象的有关特征(或意义)，能根据对象的特征从具体情景中辨认出这一对象； (2)Understanding —being able to describe the objects’ characteristics and origin; being able to explain explicitly the differences and connections between one object and its related ones. （2）理解——能描述对象的特征和由来，能明确地阐述此对象与有关对象之间的区别和联系；

15. 3. Some Experience of the “Two Basics” in Mathematics Education 数学双基课程的基本经验 (3)Mastering —being able to apply a object in new context based on understanding. （3）掌握——能在理解的基础上，把对象运用到新的情景中； (4)Flexible application —being able to use knowledge comprehensively, accomplish specific mathematical task with reasonable methods. （4）灵活运用——能综合运用知识，灵活、合理地选择与运用有关的方法完成特定的数学任务。

16. 3. Some Experience of the “Two Basics” in Mathematics Education 数学双基课程的基本经验 3.2 Emphasizing “Two Basics” in mathematics textbooks 数学教科书突出双基 3.2.1 Emphasizing on proposing the contents of “Two Basics” gradually and spirally in contents arrangement For example, the arrangement of knowledge in textbooks compilation emphasized learning step by step, and enabling students to understand and master “Two Basics” gradually. 3.2.1内容编排注重双基的循序渐进、螺旋上升 例如，教科书在知识点的编写中注重循序渐进，让学生能逐步理解、掌握双基。

17. Taking the solution of linear equation with one unknown as an example, five steps are included—canceling denominator, removing parenthesis, transposing, combining similar items, transforming the coefficient into one. It reflects five Naturalization processes. The textbooks published by Beijing Normal University arranges this section like this: 1)learning linear equation with one unknown which needs to transform the coefficient into one”; 2)learning linear equation with one unknown needs to transpose…… Finally，2)learning linear equation with one unknown needs to cancel denominator. In this way, students can learn linear equation with one unknown step by step with the guide of textbooks and develop the skills for solve this kind of equations. 以解一元一次方程为例，解一元一次方程的一般步骤包括：去分母、去括号、移项、合并同类项、系数化为1，它体现的是五个化归的过程。为此，北京师大版的七年级教科书是这样编排的：先学习 “系数化为1”的一元一次方程，再学习“移项”的一元一次方程，……最后学习 “去分母”的一元一次方程。 这样，教科书引导学生一步一个台阶地学习一元一次方程，有利于培养解一元一次方程的基本技能。

18. 3.2.2 Helping students know and understand the “Two Basics” in many ways 内容编写从多方面帮助学生认识和理解双基 Firstly, examples which connect to students’ life and can help them understand concepts are always presented when new concepts are introduced. 首先，新概念引入时常常给出一些联系生活、帮助学生认识概念的实例。

19. For example, five contextual problems are presented in mathematics textbooks (Grade Eight) which edited by People’s Education Press. 如，人民教育出版社的八年级教科书引入常量、变量等新概念时，给出了5个情境问题： （1）汽车以60千米/时的速度匀速行驶，行驶里程为s千米，行驶时间为t小时，先填下面的表，再试用含t的式子表示s： （2）每张电影票的售价为10元，如果早场售出150张票，午场售出205张票，晚场售出310张票，三场电影的票房收入各多少元？设一场电影售出x张票，票房收入为y元，怎样用含x的式子表示y？ ……

20. Secondly, unit basic knowledge structure charts and reflective questions are presented when one unit ended, which can help students have a overall understanding of basic knowledge. 其次，教科书往往在每单元结束时给出单元基础知识结构图和反思性问题，帮助学生对基础知识形成整体认识和理解。 For example, “knowledge structure” and “review and reflection” are laid out when the unit of “linear function ”in grade eight ended. (See the following picture.) 例如：在八年级 “一次函数”单元结束后，教科书编排了“知识结构图”和“回顾与反思”。

21. 3.2.3 Basic knowledge is expressed strictly, and its typesetting try to highlight its importance . 3.2.3基础知识表述严谨，排版突显其重要性 The expression of mathematical concepts is very strict in the Chinese mathematics textbooks. To the mathematical concepts which are generalized from specific examples, a rigorous definition will also be presented in textbooks. Meanwhile, the concepts, formulas, properties, laws, axioms and theorems are often emphasized by using different typeface, colour or by adding boxes. For instance, the concept of “inverse proportional function” in Grade Eight is showing as follows: 在中国的数学教科书中，数学概念的表述十分严谨。即使一些数学概念是从具体例子归纳而来的，教科书也会正式地给出较为严谨的界定。同时，对基础知识中的概念、公式、性质、法则、公理、定理等在排版中常常用醒目的字体、颜色，甚至加上方框以示重要。

22. 3.2.4 Examples and exercises are presented to stimulate the understanding and acquirement the knowledge of the “Two Basics” 3.2.4通过例、习题促进双基的理解和掌握 Numerous exercises and typical examples are presented in Chinese mathematics textbooks in order to assist students in understanding basic knowledge and acquiring basic skills. E.g. In the mathematics textbook of Grade Eight which edited by Mr. FAN lianghuo and published by Zhejiang Education Press, a large number of examples are followed the definition of graphic of function which aim to help students to acquire function and how to draw the graph. 中国的教科书往往用典型的例题、大量的练习题促进学生理解基础知识和掌握基本技能。 例如，范良火主编、浙江教育出版社出版的八年级教科书在“函数的图象”概念提出之后，给出了多个例题让学生学会分析并作图象：

23. 1.Please try to draw the graph of the functions in a Cartesian coordinate system, and write down the coordinates of axis intersection. y=3x y=-3x+2 There is a detailed statements related to methods, steps, key points and rules in the textbook which could help students to understand and master drawing the graph of functions. 例1：在直角坐标系中画出下列函数的图象，并求出它们与坐标轴交点的坐标。 （1） y=3x ；（2） y=-3x+2 （解法作图略） 这里，教科书例题对作函数图象的方法、步骤、要点、规律作了规范的表述，可以帮助学生理解和学会作函数图象

24. Many conventional exercises are prepared in the textbook in order to help students consolidate the “Two Basics”, such as the following examples in the textbook published the People’s Education Press. 教科书还会配备较多的常规练习题，帮助学生巩固双基。例如人民教育出版社的教材编制下面的练习题： 复习巩固 1.小亮为赞助“希望过程”现已存款100元，计划今后三年每月存款10元，存款总数y（单位：元）将随时间x（单位：月）的变化而改变。指出其中的常量与变量，自变量与函数，试写出函数解析式。 2.判断下列各点是否在直线y=2x+6上，这条直线与坐标轴交于何处？（ ） A.（-5.-4） B.（-7，20） C.（，1） D.（，） …… 综合应用 6.在某火车站托运物品时，不超过1千克的物品需付2元，以后每增加1千克（不足1千克按1千克计）需增加托运费5角，设托运p千克（p为整数）物品的费用为c元，写出c的计算公式。 ……

25. 3.3 Classroom teaching based the “Two Basics” 3.3.1The teaching of basic knowledge: improving memorization and understanding thorough explaining clearly and deeply. Teachers try to create a situation and learn new knowledge by reviewing learned knowledge in class. Teachers also use multiple ways to explain the connotation and extension of new knowledge, and to analyze the differences and connections between knowledge. Besides, a great number of examples and questions are given to guide students in positive thinking and finally understand the knowledge deeply. 3.3课堂教学立足双基 3.3.1基础知识教学：讲深讲透，层层夯实，促进记忆和理解 课堂上，教师积极创设情境，用实例、旧知引入新知。讲解时，教师从多角度分析新知识的内涵、外延，与其它知识的区别和联系等；用大量的例证和问题，引导学生积极思考，达到对新知识的深刻理解。

26. E.g. In introducing the concept of function in Grade Eight, teachers always firstly create a situation that there are different types of tickets in a cinema; secondly, let students fill out the table regarding the question; and then ask students to work out numerical relationship between ticket numbers and income. Teachers will then give the definition of function after students have the perceptual knowledge regarding the various relationships. Further more, in order to make students have profound understanding about function, teachers not only give a great deal of examples about Independent variable and Dependent Variable which are the key point in the definition of function, but also create situation by words, charts and some other ways to show the different positive and counter examples for leading students to gain the essence of function. 例如，八年级“函数”概念的教学中，教师常先给出某电影院不同场次的售票张数，让学生结合题目中的数据填表，进而让学生得出售票张数与售票收入之间的数量关系，在学生对题目中的两个变量的关系有了感性认识后，教师给出明确的“函数”定义。 为让学生对函数的定义有透彻理解，教师会对函数定义中的关键词语“自变量”、“变量”等用大量的例证说明，还会运用问题情境，以文字、图表等多种形式呈现正反两方面的例子，引导学生在多样的背景中感悟函数的本质。

27. After learning the definition of function, teachers will ask various questions to deepen students understanding about the concept of function. For example, the variable relationship between vertex angle and base angle in isosceles triangle, the variable relationship between weight and spring and so on. Teachers take these examples to illustrate the relationship between variables and function. 学生初步认识函数的定义后，教师为帮助学生理解函数的概念，还会提出许多问题，如：等腰三角形中，顶角与底角之间的关系，物重和弹簧伸长长度之间的关系等来进一步说明自变量和函数之间的关系。

28. Further more, teachers will give some difficult exercises in order to help students understand the concept of function. For example, • E.g. There is 50L petrol in a car’s tank. Suppose no more petrol will be added to the tank, the whole petrol (y) will decrease with the increase of the traffic mileage (x), and the average petrol consumption is 0.1L/KM. (1)Please try to give a equation to represent the function about y and x. (2)Write down the range of x. (3)How much petrol is left when the car drive 200 kilometers? • We can see that students need to write down the function, discuss the range of independent variable according to the real situation and then work out the value of y based on the value of independent variable. • 教师为了让学生透彻理解和掌握函数概念，还常给出有一定难度的题目。如： 问题：一辆汽车的油箱中现有汽油50L，如果不再加油，那么油箱中的油量y（单位：L）随行驶里程x的增加而减少，平均耗油量为0.1L\Km。（1）写出表示y与x的函数关系的式子。（2）写出自变量x的取值范围。（3）汽车行驶200公里时，油箱中还有多少汽油？ 该题不仅让学生写出函数关系，还要求学生结合实际问题讨论自变量的取值、根据自变量的值求出相应的函数值等

29. Overall, Chinese teachers focus on teaching the basic knowledge step by step. Using various methods, creating different situations, and designing different types of exercise, teachers try to consolidate students’ understanding of basic knowledge. Moreover, many difficult problems will be given to the students to make them master the basic knowledge comprehensively, accurately and profoundly. 总之，中国的数学教师非常注重基础知识的教学，在教学中强调脚踏实地，一步一个脚印往前走，对于基础知识，习惯综合运用多种手段，创设不同情境，设计不同类型的题目，做到层层夯实 在此基础上，为了把基础知识讲深讲透，还会不断加大题目难度，让学生全面、准确、深刻地把握数学基础知识。

30. 3.32 Teaching basic skill: more teaching and more exercise, advancing application Teachers not only pay attention to teaching basic knowledge, but also emphasize on training basic skills. And one of the most important points is that numerous teachers consider solution is the main way which students get basic skill. There is a common method that teachers emphasize on more exercise when teaching how to work out the mathematical problems, and they get more various exercises to training students in order to get the basic skill. And it worth to say that Chinese teachers not take repeated exercises, however they took various ways to help students get basic skill in a short time. 3.3.2基本技能教学：精讲多练，强化训练，促进形成和运用 教师在重视基础知识教学的同时，也注重强化基本技能的训练，并把解题作为学生获得基本技能的主要途径。 在解题教学时，教师强调“精讲多练”，常通过精讲典型例题让学生清楚基本技能，并用大量不同类型、不同形式、不同层次的习题反复训练，以形成基本技能。 值得说明的是，中国的基本技能教学并非总是重复训练。教师总是尽量采取不同方法和手段，让学生能熟练运用基本技能。

31. 3.4 Assessment based "Two Basics" For one thing, Chinese primary and middle school’s assessment is find and check the problems of "Two Basics", for another, is helping students the "Two Basics" and then developed ability. There are two ways regarding Chinese assessment, firstly, classroom exercise, extra-work, secondly, unit-exam, mid-exam and end-exam, entrance exam and month exam and so on. There is a majority proportion in Chinese mathematical assessment which accounted for 70-85% regarding the “Two-Basics”, and there is only a little questions which need advanced mathematical thought. To sum up, there are rare open-ended questions, exploratory questions and research questions in mathematical assessment. 3.4学生学业评价以双基为主要内容 中国中小学生的学业评价一方面是为了诊断双基教学中的问题，以便及时反馈和补救；另一方面是为了帮助学生巩固双基，发展数学能力。 中国的学业评价往往有两种形式：一种是课堂练习、课外作业；另一种是单元测验、期中和期末考试，以及升学考试，不少学校每月还有考试。 中国学生的数学学业评价主要围绕双基内容展开，常规性的双基类题目一般占70-85%，需要用到高级数学思维的开放题、探索题、研究性题占的比例不大。

32. 4. Rationality of the “Two Basics” 数学双基课程的合理性 We can`t build Skyscrapers without solid foundation. Only with the solid “Two Basics”，Students in primary and middle school can become creative and practice talents, and then get solid foundation to pursue lifelong education. 要建成高楼大厦,没有扎实的基础是不行的。中小学生只有具备扎实的双基，才能为他的终生教育和发展、成为创新型和实践型人才打下坚实的基础。 There is a believe that “ Understanding is priority and the practice is the byproduct” in Mainland China. However, this isn’t the common thinking and regulation, most of them think that are intergated andbetween between practice and understanding. 中国学者认为，“理解优先，操作附带”的教学观点并不完全是人的数学认识过程和规律，操练和理解是相互交替和交融的。

33. From the point of view of knowledge classification，mathematical double basis is a unity of declarative knowledge and procedural knowledge. 从知识分类看，数学双基是数学“陈述性知识”和“程序性知识”的统一体。 Ausubel argues that it can produce positive results and meaningful learning，if the teaching methods such as practicing and reviewing can satisfy the conditions of meaningful learning. 奥苏贝尔认为，讲授、练习、复习等教学方法只要满足了课堂意义学习的条件，它们能够产生积极和有意义的学习过程和结果。

34. 5. Academic Contention about the “Two Basics”有关数学双基课程的争鸣 "Two Basics" and the development of creativity 1) "Two Basics" may limit the development of " creativity. 2) "Two Basics" is the foundation of creativity 数学双基与创新能力发展 一种观点：数学双基束缚了学生的创新能力发展 一种观点：数学双基为学生的创新能力发展打下了必要的基础 "Two Basics" teaching and exercises 1) Students have excessive exercise which is called sea tactical issues. 2) Moderate exercise is the basic requirement of "Two Basics“ teaching. 数学双基教学与习题训练： 一种观点：双基教学中的习题训练过度，演变成“题海战术” 。 一种观点：适度加强训练是双基教学的基本要求。

35. "Two Basics" and Examination • 1) “Two Basics” are integrated with exam and result in teaching dissimilation • 2)” Two Basics” are complement each other • 数学双基教学与考试： 一种观点：双基教学与考试结合致使双基教学异化。 一种观点：双基教学与考试相辅相成。 • "Two Basics" and the New Curriculum Reform • 1) New curriculum are weaken “Two Basics” • 2) New curriculum are assisting and developing “Two Basics” • 数学双基教学与新课程改革： 一种观点：新课程淡化了数学双基教学。 一种观点：新课程坚持并发展数学双基教学。

36. 6. Prospects of Mathematical "Two Basics" Curriculum 数学双基课程发展的展望 There are many deficiencies of "Two Basics“ in Chinese mathematical education. For instance, Some teachers neglect student's subjectivity, and consider students as indoctrination objects, for the influence of examination- oriented education system. The memory of knowledge was emphasized too much on, but the students' understanding and insights was thought little of. 中国数学双基课程仍存在不足。比如： 受应试教育的影响，一些教师忽视学生的主体性，把学生当作灌输的对象； 过分强调记忆现成的知识，不重视学生的理解和见解；

37. 6. Prospects of Mathematical "Two Basics" Curriculum 数学双基课程发展的展望 Training is emphasized intensively, and is machinery and mechanical. The requirement of "Two Basics“ is improved continuously. The "Two Basics" curriculum aims at examination, then "Two Basics "become the “Two Basics of Examination". It will cause dissimilation of "Two Basics" . 过度进行强化训练，双基要求不断拔高，训练方式机械、呆板； 双基课程瞄准升学考试，双基成了“考试的双基”，导致双基课程“异化”。

38. 基础知识 Basic knowledge 基本技能 Basic skills 基本思想 Basic thoughts 基本活动经验 Basic activity experience 基础知识 Basic knowledge 基本技能 Basic skills Properly handling the relationship between the "Two Basics" and innovation, developing emotion, and values, and other aspects. 要处理好双基与其它方面的关系，如“打好双基”与“力求创新”，“打好双基”与发展情感、价值观的关系。 Extending "double base" to "four base“ 把“双基”扩展到“四基”：

39. Basic thoughts: abstract, model , inference Basic activity experience:the experience of mathematics learning and research which is accumulated in the process of “practicing mathematics” and “thinking mathematically” 基本思想：抽象、模型、推理 数学活动经验是在“做数学”和“数学地思考”的过程中逐步积累的数学学习和研究的经验。

40. Thank you! 谢 谢 ！