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$LLLLLLChinese Teachers' Constructions of Their Roles in Developing Curriculum
Li Jun
East China Normal University
Abstract
This paper is mainly concerned how Chinese mathematics schoolteachers design their teaching based on textbooks. Take the teaching of distance formula from a point to a straight line as an example, teachers active role in developing the new curriculum is illustrated. To seek concise and new solutions, to prepare a sequence of questions to guide the students investigations, to pose a realistic related problem to create an atmosphere of active learning are identified as three common ways to show teachers creativity in teaching. Some inappropriate changes made by schoolteachers in using textbooks are also indicated and discussed.
Introduction
Curriculum has several dimensions (Goodlad et al., 1979; Robitaille et al., 1993; Valverde et al. 2002). It could be expected as a set of documents, such as syllabi or standards, highlighting official intentions, aims and goals. It could be considered as the textbooks and other organized resource materials used by schoolteachers and students. It could be teaching strategies, practice, and activities that actually conduct by teachers in the classroom. And, it could be considered as student achievement, assessed after learning. This paper is mainly concerned with the practical curriculum to show how Chinese mathematics teachers play their essential role in the current curriculum reform in China.
Overall, the Chinese education system is a national system governed by the Ministry of Education. From 1952 to 1986, only one series of mathematics textbooks, compiled and published by the Peoples Education Press (PEP), which is directly affiliated with the Ministry of Education (MOE) in China, were used at the same time in all schools in mainland China. It could be imagined that the unified textbooks could not meet various demands arose from the whole country, who has the largest population in the world. So we always take textbooks as our teaching base not teaching Scripture. Plan ones own lessons to meet his/her students particular needs is one of essential teaching skills that each qualified teacher should have. In China, schoolteachers usually teach only one subject, the major he/she studied during the preservice teacher training. In 1952, the MOE required every school to set up teaching research group in all subjects. Since then, mathematics teachers who teaching in the same grade level in the same school are organized to form a gradebased teaching research group. All grades groups in the school will form the schoolbased teaching research group. Interchange and discuss the next week teaching plans is the main activity for gradebased teaching research group. Such a teaching research system is also available in town/city, province, and nationwide levels and plays a very important role in teacher professional development in China. Meanwhile, more than fifty mathematics teaching and learning journals publish teaching research papers year by year. Most of the papers are written by schoolteachers and related school teaching closely. In summary, Chinese teachers are always encouraged and supported to develop their own teaching plans on the basis of textbooks.
New Standardsbased Textbooks in China
The current curriculum reform was under the guide of two national curriculum standards issued by the MOE. One is The Standards of Mathematics Curriculum for Compulsory Education (MOE, 2001) and the other is The Standards of Mathematics Curriculum for Fulltime Senior High Schools (MOE, 2003). For mathematics teaching, we now have 6 series of national new Standardsbased textbooks being used in primary schools. Another 9 series and 6 series of national new Standardsbased textbooks are being used in junior high schools and senior high schools, respectively. Usually, all schools within a city select the same textbooks in the same year. All textbooks, except the new senior high school textbooks, have two volumes for two semesters in a school year.
The new Standardsbased textbooks are roughly 120~210 pages (7871092) each volume, a bit thicker than the previous ones. As spiral curriculum is advocated and strongly recommended for textbooks writing by the new standards (MOE, 2001), the new textbooks always include topics from algebra, geometry, sometimes as well as data and chance together in one volume, a very different structure from that of the previous textbooks. The Standards also indicate that the new textbooks should tie with practical life intimately, and provide opportunities for students to appreciate the value of mathematics in everyday living. These requirements lead the textbooks appeared recently have a great improvement on context (Bao, 2004). When editing textbooks, the writers also set up a variety of columns (e.g. Observation, Manipulation, Inquisition, Conjecture, Think it Over, etc.) to prompt students active learning, or in H. Freudenthals word, to show the process of progressive mathematization( Freudenthal, 1991).
Compared to the previous sets of textbooks, the new ones focus on the learner and learning, rather than on mathematical knowledge. This significant change reflects the current curriculum reform goal of improving students understanding and helping them to engage actively in mathematics learning. The new teaching materials deploy a variety of formats (e.g. colour printing, pictures, cartoon, and games) to increase students reading interests. Considering the individual differences of students cognitive development and imbalance in regional development, textbook writers are conscious of exhibiting some degree of flexibility. For example, they prepare optional contents for the students who want to gain indepth insight. They classify chapter review exercises into A (fundamental level), B (enhanced level), and C (advanced level) groups for selections. In the new textbooks, some valuable materials but not required for all students at present are included in Reading Material. For instance, many teachers and students are eager to learn how to solve mathematics problems efficiently by using computer and calculator but only some of them have access to modern technology. Some textbooks introduce the knowledge in Reading Material. Since knowledge introduced in Reading Material is beyond of examinations, such an arrangement is welcome for the current situation that many schools in rural areas are still not equipped with computers and calculators, but making use of new technology has become a world trend. When design of Project Learning, they choose themes carefully to allow all students to participate and suggest various tasks at different levels to different students.
The new Standardsbased textbooks usually have the following format: Words from the writers, Tables of Contents, Preface of Chapter, Text (problems with contexts, definitions, columns such as Observation, Thinking, Manipulation, Inquisition, Conjecture, Think it Over), Examples, Drill in class, Reading Material, Exercise, Chapter Review, Chapter Review Exercises (Groups A, B, and C), and Project Learning. The new textbooks give prominence to the construction of realistic contexts, exploration of knowledge, and attraction for learners. These changes are all in accordance with the overall goals of the current reform.
Teachers Constructions of Their Roles in Developing Curriculum
As indicated earlier, Chinese teachers were encouraged and supported to design their own teaching based on textbooks in the years when only one series of textbooks were used in China. Today, the tradition is still alive, although several series of textbooks with different features are available to meet various teaching and learning needs. In the part of Teaching Recommendations, the new Standards indicated: During teaching, teachers should deploy teaching materials creatively, exploit actively, and use all sorts of teaching materials, so as to provide students rich and multifarious learning materials. They should pay attention to students individual differences and practice individualized teaching effectively, to help every student receive sufficient development assistance (MOE, 2001).
Compared to many countries in the West, China has a different culture of teaching where mathematics teaching is taken as a professional activity that is open to public scrutiny and evaluation. Teachers often sit in others classrooms and discuss teaching with fellow teachers. It is a great honor for both the teacher and the school if they are selected to give a public lesson to teachers outside. In particular, teaching contests organized by government also serve as a platform to value and promote mathematics classroom instruction excellence. The last, but not the least, publish papers in Journals, as well as in web in recent years, is another platform for teachers to contribute and share their creativities.
In China, teachers, especially those who work in highperforming schools, are willing to and specialized in teaching design based on textbooks. To seek concise and new solutions, to prepare a sequence of questions to guide the students investigations, to pose a realistic related problem to create an atmosphere of active learning are identified as three common ways to show teachers creativity in teaching. I would like to take the teaching of distance formula from a point to a straight line as an example to illustrate teachers active role in developing the new curriculum.
The distance formula from a point to a straight line is a learning topic should be learned by all senior high school students. The students are required to explore and master the formula. The textbooks are usually start from the task directly: To find the distance from a point EMBED Equation.DSMT4 to the line EMBED Equation.DSMT4 .
In the earlier versions, the following method was used in the textbooks:
Method 1: To construct a perpendicular of l through P and label the foot of perpendicular as point Q(a,b). So line PQ will have slope EMBED Equation.DSMT4 and the equation of line PQ can be obtained. Solve it with EMBED Equation.DSMT4 simultaneously to get the coordinates of Q. Using the distance formula of two points we get the distance of EMBED Equation.DSMT4 , which is the answer of the problem. Last, to check when A=0 or B=0, the formula is still true.
Figure 1
After teaching, many schoolteachers found that this method involves numerous calculations, although it is easy to obtain. They try to find out new ways to solve the problem. In the past several decades, more than 10 solutions were reported by schoolteachers and it makes the textbooks change. Now, in all textbooks, method 1 is usually described briefly and concluded with the comments: this method involves numerous calculations so we will introduce you another method.
To such a comment, a schoolteacher argued in his paper (Shao, 1995) that the numerous calculations are not necessary at all and easily to be avoided. Once we solve EMBED Equation.DSMT4 instead of a, EMBED Equation.DSMT4 instead of b, the solution would become concise. To set an unknown only for a bridge not for solve it is a very useful skill in solving problems and familiar to students. The teacher also recommended that the new method will make the discussion of the values of A and B unnecessary. There are some other methods have been suggested, such as extreme method, vector method, parameter method, Cauchy inequality method, complex number module method, coordinates transformation method, and the trigonometric method. Anyway, the following deduction is adopted by many textbooks.
Method 2: To calculate the area of EMBED Equation.DSMT4 twice, that is, EMBED Equation.DSMT4 , where EMBED Equation.DSMT4 , and EMBED Equation.DSMT4 are easy to gain (see Figure 1). Solve the equation to get the answer of d.
EMBED Equation.DSMT4
Obviously, Method 2 is more direct, very limited preknowledge is required, and could be understood by all students. Comparatively, the other methods mentioned above are seemed not so popular and more creativity is needed. Now, vector geometry is introduced into the new curriculum. It will definitely influence the teaching of plane analytic geometry. In 2004, Vectors and Equations of Straight Line, Vectors and Positional Relationships of two Straight Lines were included in the Reading Material in a textbook published by PEP. Follow it, several papers arguing about how to make use of vector tool to deduce the distance formula from a point to a straight line appeared. They extend the knowledge further, beyond the textbook explanation, but still quite related to the common teaching contents.
To seek concise and new solutions is greatly valued. Many schoolteachers are happy to send a better solution or correct an error in textbooks to the Journals and once they are published more feedback will follow in most of time. Not only textbooks editors but also teachers benefit a lot from this kind of comments and suggestions. In highperforming schools, teachers usually discuss various methods with students in their teachingeven though the methods are not included in textbooks.
Knowing all of the above methods is not enough for teaching. Besides of being prepared in mathematics, teachers also should be armed with Pedagogical Content Knowledge (PCK) (Shulman, 1986). In the following teaching plan prepared by a teacher from Yunnan Province (Li, 2006), you will see the teacher how to use his PCK to assist students move forward.
At the beginning of the lesson, the teacher asked the students to solve a set of exercises first, which was not ready in the textbook. The set of the exercises is:
Find out the distance d from the Point P(1,2) to the straight line l: x=3;
Find out the distance d from the Point to the straight line l: By+C=0 (B`"0);
Find out the distance d from the Point to the straight line l: Ax+C=0 (A`"0);
Find out the distance d from the Point to the straight line l: 3x4y+5=0;
Find out the distance d from the Point to the straight line l: Ax+By+C=0 (AB`"0).
Clearly, Item (1) is the easiest and it could be answered mentally. Items (2) and (3) are easy as well, because the positions of the straight lines are special, either horizontal or vertical. The straight line in item (4) is inclined and caused a lot of calculation work. Anyway, it is still solvable. The students will definitely be frustrated in numerous calculations in solving the item (5). The teacher asked 5 students to give their answers in front of the class and wrote down their solutions on the blackboard. All of the other students were required to solve the problems independently in their notebooks at the same time. Then the teacher made comments to every solution on the blackboard and indicated that the main task for todays class was to find concise solutions to item (5). When we read the five tasks carefully, we can feel the Chinese way of promoting effective mathematics learning: teaching with variation (Gu, Huang, and Marton, 2004).
The teacher asked the students whether they could get any hints from the experience they gained just now to solve item (5). The expected answer was similar saying that the problem was much easier when the positions of the given lines were parallel to xaxis or yaxis but a lot of calculations were needed if the lines were in other positions. Then the teacher asked:Is there a better solution for (5) available, which does not involve so much calculations? Can we solve it using any horizontal distance, vertical distance and plane geometry knowledge we have learned in junior high school? After the students discover the Method 2, the teacher continued to encourage the students: There are many methods could be used to get the distance formula from a point to a straight line. We have learned Function, Trigonometric Function, Vector, Inequality etc. Could you use what we have learned to deduce the formula in different approaches? Please think about it yourself for a few minutes, and then exchange your ideas in groups. The leader of each group please makes the notes. After 10 minutes, each group sends a student to present the best method discovered in your group.
The new textbooks describe the Method 1 briefly and show the Method 2 step by step, then followed by Examples and Drills, totally edited in a traditional, formal style. So teachers have a broad space to design their own teaching. The teaching plan illustrated above is in accordance with the principles advocated in the new curriculum. Especially, the teacher recognized his role in teaching is an organizer, a facilitator, and a collaborator. He planned the lesson according to students general process of cognitive development. His teaching was emphasis on students understanding and cooperative exchange and so on.
The new Standards encourage textbook writers and schoolteachers to present the teaching contents in a realistic, meaningful, and challenge way (MOE, 2001). Almost all teachers arranged to give public lessons will agree that they spend a lot of time in designing an attractive realistic problem as the start of the teaching, especially when the textbooks do not provide such a problem, or, the problem provided is not appropriate for their students. A schoolteacher from Shanghai shares the problem that she developed for her public lesson of teaching the distance formula (Zhang, 2006). Actually, the paper reported the detailed process of the teaching plan formed, what was the original plan, why it was changed, and what it was after several times modifications. Here is the context problem that she designed:
Look at the map of our campus. We labeled the Gym, Student Flat, and the Water Lily Pond as three points. Now we are plan to build a straight road, to let the distances from all the three points to the road are the same. Is it possible to build such a road in theory? Give us your idea.
The answer of course is Possible and three roads can be built in theory. Once the students have learned the distance formula, they could solve the problem completely. Connecting mathematics with the real world is becoming an attractive research topic in these years in China. Mathematics is always a subject that all Chinese students spend plenty of time to learn. But for most students, this is not because they are interested in mathematics or they are conscious of its special contributions to the modern civilization. Examinations, high expectations from parents, or wish to be a good student might be real motivations. Some educators have appeal strongly that the teaching of mathematics should spread mathematical culture as well. A good teacher should raise his/her students interests for learning mathematics, to enjoy the beauty of mathematics, at least should let the students value the contributions of mathematics. To initiate students active, longterm motivation of learning is becoming an urgent big problem that every Chinese mathematics teacher should answer seriously. A survey on teaching contexts in middle school textbooks shown that the students hope their textbooks use more interesting context questions which are close to social and daily life (Hang, 2007).
Discussion
In China, mathematics is a required course for all students from Grade 1 to Grade 10, and its instructional contents are officially mandated by National Curriculum Standards issued by the MOE. However, the contents required by National Curriculum Syllabus/Standards or textbooks are basal and they are insufficient for students to solve problems appeared in exams. In the past University entrance examinations, about 30% scores are allotted to the items that with equivalent difficulty to the exercises in textbooks. Therefore, school teachers always add some more complicated materials in their teaching, especially in highperforming schools. For the traditional teaching contents, teachers usually have some ideas of what makes concepts difficult or easy to learn after several years of teaching. So changes and complementarities are encouraged in China as long as the basic requirements prescribed by National Curriculum Standards are reached. The changes occurred not only in mathematical contents, but also in teaching approaches. We always believe that a teaching plan conducted successfully in Class A may not be successful in Class B. It is natural and necessary to form ones own teaching plan. Since teachers have freedom of making their own decisions, sometimes the changes they made are not appropriate.
The new curriculum has reduced the requirements for some topics in both algebra and geometry, but not a few educators question to this decision. Some teachers argue that although some math contents may not stay in foreign textbooks, as long as students can learn and without bringing too much learning burdens, should still be in China math curriculum. They believe the learning will enable students to gain a comprehensive understanding of mathematics. In a survey on the use of standardsbased textbooks, Yang found that over 90% of the junior high school teachers that he investigated admitted that they did pick some contents back when they teach the new curriculum. About 75% of the students investigated answer that the time they spend doing exercises outside textbooks was as long as or longer than the time they spent on doing exercises in textbooks (Yang, 2005). Yangs study was conducted in 2004 when the teachers and students he surveyed only had 1~3 years of using the new standardsbased textbooks. Once the teachers taught the whole set of the textbooks and learned what examinations would look like, they might keep in line with the new curriculum more willingly.
In the past 50 years, textbooks published in China were mainly knowledgeoriented and straight line processing structured. Sometimes, algebra and geometry were even taught separately by two different teachers to the same class in a semester. However, the new curriculum Standards advocates the unification of mathematics and requires new textbooks displaying the upward spiral structure which is very different from the traditional structure. Some teachers feel uncomfortable to the new structure and often neglect such columns as Observation, Doit, and Inquisition, which arranged to enlighten the students mathematical thought and to improve their learning styles. The real situation is that students are happy to do activities in class but teachers are not willing to and they believe such kind of teaching effects is impossible to be reflected in a timelimited paper examination. Solving various problems prepared by teachers is more efficient to get higher marks in examinations. In my opinion, it is unfair to criticize these teachers for their reluctant behavior in the reform. Different philosophies should be respected. George Polya is well known in China and his books have been translated into Chinese and spread widely among educators. Many schoolteachers in China believe what Polya has said in the Preface of the book Mathematical Discovery: Solving problems is a practical art, like swimming, or skiing, or playing the piano; you can learn it only by imitation and practice (Polya, 1962). We need time to see the results of the reform. We need time to find an appropriate way to keep the balance between reform and tradition.
References
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Gu, L., Huang, R., & Marton, F. (2004). Teaching with Variation: A Chinese Way of Promoting Effective Mathematics Learning. In Fan, L., Wong, N.Y., Cai, J.,& Li. S. (Eds.), How Chinese Learn Mathematics (pp.309347). Singapore, World Scientific.
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Zhang, L. (2006). A teaching case of the distance formula from a point to a straight line. (In Chinese). [online] Available: HYPERLINK "#$)*12=HIPmvwx = > E F H u üyqyhy_W_NhmnHo(tHhmnHtHh@EnHo(tHhR#nHo(tHhhnHtHhhnHo(tH"hLhL5OJQJnHo(tH"hLhH5OJQJnHo(tH*hj LhH5;B*KHnHo(phtHhj LhoVhj LhoVCJhj LhoV5;CJo(hj LhP5;CJo(hj LhZ5;CJhj LhZ5;CJo(IPmx
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