> {}za R7wjbjbYQYQ 83333R%8D`(B"ddd?[g^^^^^^^,bRd^o??oo^33ddZ`333333odd^332To^3333+\0{^Mu/[]${^p`0`]e33eH{^33DWhat Challenges Are We Confronted with in High School Mathematics?
Wang Linquan
College of Mathematics science
South China Normal University
Guangzhou, 510631, China
ABSTRACT
National Curriculum Standards of High School Mathematics were published in 2003. Three Provinces of Guangdong, Shandong , Hainan and Autonomy Region of Ningxia Hui Nationarity Hui Nationarity were part of the experimental group where the new Mathematics Curriculum was implemented. The experimental region increases each year. In 2008, more than 10 Provinces entered into the group who are implementing the new curriculum standards. We have achieved some progress in the first round, although teachers and students are confronted with great challenges.
In China, not only are teachers and students of high school mathematics faced with challenges, but the new curriculum itself contains many challenges. In this paper, we share some of thes challenges of implementing the new curriculum of upper secondary school mathematics. What problems have we experienced? What difficulties are we trying to overcome? I would like to share our experiences with other colleagues from overseas.
1. Fundamental Structure of National Curriculum Standards of High School Mathematics
Based on National Curriculum Standards of High School Mathematics, the high school mathematics consists of 5 required modules and 4 optional series. The required modules of high school mathematics are identified below.
Required module 1 Concept of function and fundamental elementary functions Include set theory4 teaching hours
Fundamental elementary functions32 teaching hours including concept and character of function, expression of function, even function and odd function, fundamental elementary exponential function, logarithmic function and power function, application of function, equation and function, model of function.
Required module 2 Introduction of solid geometry and plane analytic geometry
Include introduction of solid geometry 18 teaching hours including recognition of simple 3D figures, cylinderprism and circular cylinder , cone cylinder cone and pyramid , characters of solid figures, calculation of surface area and volume of solid figures, projection and view of simple geometric solids.
Introduction of plane analytic geometry18 teaching hours including equations of a straight line, common form, point and slope form, standard equation of a circle, relationship between straight line and circle, relationship between two circles, coordinate system in 3D space.
Required module 3 Algorithm, statistics and probability
Introduction of Algorithm12 teaching hours including meaning of algorithm, programmed chart, basic logical structures and basic algorithm sentences, classical Chinese cases of algorithm, application of algorithm.
Introduction of Statistics16 teaching hours including random sampling, population and sample, estimation characters of a population from its sample, method of least squares, linear regression equation.
Introduction of Probability8 teaching hours including random event, certain event and impossible event, classical probability model, exclusive events, complementary events, geometrical probability model.
Required module 4 Trigonometric functions and plane vector
Trigonometric functions16 teaching hours including arbitrary angle, radian measure, unit circle, definitions of trigonometric functions of general angles, sine, cosine and tangent, relations of trigonometric functions of a angle, periodic function, graphic of sine function EMBED Equation.DSMT4 and EMBED Equation.DSMT4 .
Plane vector12 teaching hours including practical background of plane vector, linear operations of vectors, collinear of vectors, included angle between two angles, coordinate representation of vector, trigonometric identity transformation.
Required module 5 Solving triangle, sequence and inequality
Solving the triangle8 teaching hours including sine theorem, cosine theorem, applying the sine theorem and cosine theorem to solve a triangle and other practical problems.
Sequence12 teaching hours including the concept of sequence, the formula of general term, arithmetic progression, geometric progression, solving problems using sequence.
Inequality16 teaching hours including relation of inequality in real life, properties of inequality, linear inequality in one variable, quadratic linear inequality in one variable, fundamental inequality, binary linear programming.
The limited optional series are divided into two series.
Limited optional series 1for the students who are preparing for further study of social science consists of 2 groups:
Group 1: Common logical language8 teaching hours including propositions and their relations, original proposition, its inverse proposition, negative proposition, inverse and negative proposition, their logical relations, sufficient condition, necessary condition, simple logical conjuctions, universal quantifier, existential quantifier.
Conics and their equations 12 teaching hours including ellipse and its standard equation, hyperbola and its standard equation, parabola and its standard equation, geometric properties of conic.
Derivatives and its application16 teaching hours including average rate of change, derivatives and its geometric meaningcalculations of derivatives, using derivatives to solve problems.
Group 2: Statistical cases14 teaching hours including main idea of progression analysis and its application, main idea of independence test and its application.
Reasoning and proof 10 teaching hours including plausible reasoning, inductive reasoning, deductive reasoning, mathematics conjecture, synthetic method of proof, analytic method, proof by contradiction.
Number system and its extension 4 teaching hours including expanding number system, real number imaginary number and complex number, number system, algebraic form of complex number and its operation.
Programmed chart6 teaching hours including flow chart and its application, structure chart and its application.
Limited optional series 2for the students who prepare for further study of science and technology consists of 3 groups.
Group 1: Common logical language8 teaching hours including propositions and their relations, original proposition, its inverse proposition, negative proposition, inverse and negative proposition, their logical relations, sufficient condition, necessary condition, simple logical conjuctions, universal quantifier, existential quantifier.
Conic and their equations16 teaching hours including background of conic, ellipse and its standard equation, hyperbola and its standard equation, parabola and its standard equation, geometric properties of conic, to solve geometrical problem and some practical problems with coordinate method.
Vector in 3D space and solid geometry12 teaching hours including space vector and its calculation of addition and subtraction, inner product of space vector, orthogonal decomposition of space vector, application of space vector in solid geometry.
Group 2: Derivatives and its application24 teaching hours including average rate of change, derivatives and geometric meaning calculations of derivatives, using derivatives to solve problems, optimization problem in real life, definite integral, fundamental theorem of calculus, simple application of definite integral.
Reasoning and proof 8 teaching hours including plausible reasoning, inductive reasoning, deductive reasoning, mathematics conjecture, synthetic method of proof, analytic method, proof by contradiction, mathematical induction.
Number system and its extension4 teaching hours including expanding number system, real number imaginary number and complex number, number system, algebraic form of complex number and its operations.
Group 3: Principle of counting number14 teaching hours including classified enumeration principle, fractional step enumeration principle, permutation and its formula, combination and its formula, binomial theorem, triangle number.
Cases of statistics and probability22 teaching hours including discrete random variable, probability distribution series, hypergeometric distribution, conditional probability, mutually independent events, independent and repeated trials, binomial distribution.
Freely optional series 3. This series includes 6 topics of introduction of modern mathematics ideas : 1) Selection of history of mathematics; 2) Information safety and code; 3) Spherics; 4) Symmetry and group; 5) Eulers formula and classification of closed surfaces; and 6) One third of an angle and expanding of number system. Each of the above topics requires 18 teaching hours for study.
Freely optional series 4. This series includes 10 topics of basic methodologies of applied mathematics:1) Selection of geometrical proof; 2) Matrix and transformation; 3) Number sequence and difference; 4) Coordinate system and parametric equation 5) Selection of inequality; 6) Introduction of number theory; 7) Introduction of optimum seeking method and experimental design; 8) Scheduling method and introduction of graph theory; 9) Risk and decisionmaking; 10) Open and closed circuits and Boolean algebra. Each of the above topics also requires 18 teaching hours for study.
2. Characteristics of High School Mathematics
Compared to the Traditional High School Mathematics, there are some characteristics in Chinese High School Mathematics that are described below:
2.1 The content of High School Mathematics has greatly increased
A large group of new series and topics has been embedded in the new course of high school mathematics, and the new course has become more abstract. Students will struggle to learn all the required content material. Consider module 1. There are a lot of new topics included in the course that students should study. Some of these new topics include power functions, approximating solutions of an equation, mathematical modeling, etc. However, students do not know anything about logarithms in primary middle school, but they need to study logarithm and logarithmic function in the first semester of high, and in a very short time!
2.2 Electivity has been showed in the curriculum
It is possible for the school to choose different optional series and topics for the students. For example, the student can choose between limited optional series 1 or limited optional series 2 according to their own needs of their future. Different test papers for the national examination can also be designed for the candidate who selected different series of mathematics.
2.3 New philosophic principles have been put forward for the new course
There are 10 principles for high school mathematics. These principles guide the implementation of the Curriculum. The 10 principls are as follows: 1) To construct a common foundation of mathematics; 2) To offer various courses to meet different students needs; 3) To encourage new style of learning mathematics; 4) To explore and discover some rules during the process of mathematics activities and to help students enhance their ability of problem solving and mathematics thinking; 5) To cultivate students application sense of using mathematics; 6) To recognize our own excellent traditions of teaching mathematics to form our new characters step by step; 7) To help students to recognize the nature of mathematics and pay suitable attention to formalization of mathematics; 8) To learn from the cultural value of mathematics; 9) To pay attention to the appropriate use of technology for the process of teaching and learning mathematics; 10) To construct a reasonable and scientific assessment system of teaching and learning mathematics.
2.4 Four main mathematical content areas have been identified.
The four main content areas in high school mathematics are Algebra and Function, Space and Geometry, Calculation and Algorithm, and Statistics and Probability. The relative knowledge of these content areas is unfolded from the beginning to the end of the high school stage. They are developed and enhanced spirally together with students mathematics experience.
3. Progress has been achieved in the first around
From the year of 2004 to 2007, we completed a first around of implementing the National Curriculum of High School Mathematics. We have experienced progress in implementing the new course in high schools.
The experimental area is becoming larger
More and more provinces, cities and nationality autonomy regions implemented the new curriculum. At beginning of academic year of 2004, there were only 4 provinces and autonomy regions implementing the new curriculum. But since academic year of 2007, there are 10 provinces, cities and autonomy regions in the experimental area of new curriculum, including our capital Beijing.
Inservice mathematics teachers training has been held
Both central and local governments paid close attention to the professional development of inservice mathematics teachers. A large scale of teachers training has been held. The focus of teachers training is to help teachers master the new course, including content, as well as instruction. As a professor of mathematics education, and also a standing member of the group of National Curriculum Standards of High School Mathematics, I attended over 30 different level mathematics teachers training. As a teachers educator, I also had numerous discussions with teachers. For example, mathematics teachers training has been held 6 times at South China Normal University with more than 1500 math teachers participating in the training.
The initial implementation of the new curriculum was completed.
The first round of implementation was completed, and the results of the national uniform examination indicates success. For example, according to systematic sampling from 224 084 mathematics test papers of candidates in 2007 national uniform entrance examination of higher education, the average score is about 84 from the full of 150. After the examination, both teachers and students had more confidence in the new high school mathematics curriculum.0
Strong emphasis in the process of learning mathematics.
Close attention has been paid to the learning process for the students. Many kinds of mathematics learning have been accepted in mathematics class. For example, selflearning, cooperatelearning, practice activities have occurred in the class. Teachers pay more attention to encouraging students to discover patterns in the mathematical phenomena.
4. Challenges Students are Facing
Middle school mathematics is simpler than before, however high school mathematics has become more complex.
The new content and learning requirement are not easily adapted.
More concepts and theorems are to be learned, yet students do not have enough time to develop an understanding of the content. Although the content of high school mathematics has been expanded, the teaching hours of mathematics class has decreased from 6 hours per week to 4 hours per week. The speed of teaching is very quick, more students are unable to keep up with the speed. Because content is covered so quickly students cannot understand the content they are being taught. Students must then spend time outside the class to finish the learning tasks. Based on a survey in Guangzhou, high schools students always spend half an hour on average on the exercises of mathematics per day.
The degree of abstraction of High School Mathematics is increasing.
Students are not prepared to cope with the increasing level of formalism in upper secondary mathematics. For instance, Required module 1 is the most abstract for high school students, but it was arranged at the beginning of first year, so that a large number of students could not satisfy the course. Students tend to rely on methods used in primary middle school to handle new problems they encounter in high schools.
The situation of applying mathematics is new and challenging.
The environment of the practical problems is complex for students and even for teachers. High school students should recognize and experience the close relationship between mathematics and daily life, between mathematics and other subjects, between algebra and geometry, but they do not have enough time to experience connections between mathematics and other subjects. Lack of such experience is the main cause of their difficulties in solving practical problems. There is limited articulation and coherence across different stages. For example, when students study monotonicity of function, they should recognize some knowledge of inequality, but they did not encounter the concept of inequality in middle grade of mathematics, so students struggle .
The mental pressure of unified examination is a great challenge.
National uniform entrance examination of High Education Institutes is very important for high school students. The examination is a serious challenge for them. If students are successful in the exam, they can be accepted as university students. The examination is a key stage of their future career. This kind of examination is not easy to pass. Therefore the exam puts a lot of heavy pressure on students. Students also spend a lot of time to prepare for the examination. Because of the high stakes nature of the examination, teachers and students spend a lot of time reviewing previously covered topics but they have only limited time to study the new topics.
5. Challenges Mathematics Teachers are Facing
Mathematics teachers have to handle a variety of contradictions which are not only from the new curriculum, but also from other aspects.
5.1 The content of the course was increased, but the teaching time was decrease.
The Teaching tasks are not easy to complete, and a series of new principles of teaching is not easy to implement. Therefore, schools have arranged more teaching hours for mathematics class. Some maths teachers have to teach even on the weekend.
5.2 The time is limited, but the expectation is being enhanced.
Some of the new topics have been added in the high school mathematics, with which they are not familiar. Teachers also have struggled to adapt to the requirements of new curriculum. Teachers need the opportunity for further learning, but they are burdened with the teaching task. It is not easy for them to find time for their professional development.
They play the role not only as mathematics teachers, but also as educators.
Most of mathematics teachers are also in charge in a class. As mathematics teachers, they need to prepare their lesson, to help students understand concepts and principles of mathematics; as head teachers they need to administrate their classes, to organize meaningful activities, to cultivate scientific spirit on the class. They always feel that they do not have enough time for their selflearning.
The common view of teaching evaluation is unilateral.
The results of national uniform examination are emphasized as the most important standard of teaching and learning mathematics. If any topics are not listed on the testing syllabus, then attention will not be paid to those topics. In fact, the freely optional series 3 and 6 topics and freely optional series 4 were not taught in most of high schools.
Conclusion
A largescale reformation of high school mathematics is being implemented in China. We have made progress in some aspects, but we continue to encounter some problems that need to be solved. We would like this study of our experiences to be useful to those from overseas.
Acknowledgements
I would like to express my hearty gratitude to Dr. Gwen ZimmermannUSA , Professor Denis Tanguay (Canada) and Professor Neil Eddy (South Africa) for their warm help of my paper. ]
References
[1] Chinese Ministry of Education: Graft National Curriculum Standards of Compulsory School Mathematics,Draft for experiment Press of Beijing Normal University, 2001, Beijing
[2] Chinese Ministry of Education: National Curriculum Standards of High School Mathematics,Draft for experiment Peoples Education Press, 2003, Beijing
[3] Chinese Ministry of Education: Graft National Curriculum Standards of Compulsory School Mathematics,Recention Draft for experiment Press of Beijing Normal University, 2007, Beijing
[4] Wang Linquan: constitution and cultivation of Spatial Sense of High School Students, Bulletin of mathematics, Beijing Normal university, October 2007, Beijing
[5] Wang Linquan: To deal with formulization in High School Mathematics Bulletin of mathematics, Beijing Normal University, October 2005, Beijing
[6] Wang Linquan and Wu Yuezhong: Comparison of Teaching Geometry Between China and USAFrom an Oriental Perspective, Research in Mathematics Education, Vol. 6 Num 12 September 2002
Team chairs:
Denis Tanguay (Canada)tanguay.denis@uqam.ca
Neil Eddy (South Africa)neddy@bishops.org.za
Team members:
Thomas Jahnke (Germany)jahnke@math.unipotsdam.de
Gwen Zimmermann (USA)gzimmerm@district125.k12.il.us
Luis Ricardo Garza (Mexico)
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