The Origin of Bianshi Problems: A Cultural Background Perspective on
the Chinese Mathematics Teaching Practice
Department of Curriculum and Instruction
The Chinese University of Hong Kong
Department of Curriculum and Instruction
The Chinese University of Hong Kong
Integrated systems that work well together are the essence of civilization.
------Irving Wladawsky Berger, general manager, Internet division, IBM
Recently Bianshi problems, math isomorphic problems by changing the conditions, conclusions or deduction process of the three components of the example problem, are consistently identified as an important element in Chinese math education, a characteristics in Chinese math education culture and form a hot point in China. Five main factors related to reasons is regarded as CULTURE characteristics in Chinese math education are discussed: (a) Exam goal; (b) the curriculum objectives; (c) teaching tradition; (d) Chinese situation; (e) Chinese math tradition .
The brilliant performance of the Chinese learner having attributions of such a success is the balance of ―content ― and ―process‖ in the Chinese style of teaching and learning (Wong , 2004). In this regards, Bianshi teaching, or teaching of variation (Gu , Huang & Marton , 2004), which has been very popular in China in these decades, is seen to serve as a bridge between basic skills and high order thinking ,in Wong`s (2004) term, from ―entering the way‖ to ―transcending the way‖. In this paper ,the authors would like to give comprehensive review of the
following issue: why is Bianshi problem regarded as a cultural characteristic in Chinese math education！
Bianshi teaching evolves from the experience and gets its name from the experience of mathematics teaching reform, with salient practical effect from the passing rate 2.8% in 1979 to 85% in 1986, headed by Professor Gu in Qingpu county in Shanghai, China (Qingpu educational reform group, 1991). Today Bianshi teaching is adopted broadly by classrooms around the whole country.
We will illustrate what is a typical Bianshi problem by the following example. If a student does not know how to solve this problem: How many is 3/4 times 2/3 then his teacher will perhaps give him a group of problems like this:
How many is 1 times 2? How many is 2 times 2? How many is 2 times 1/2? How many is 1/2 times 2? How many is 1 times 2/3? How many is 3 times 2/3? How many is 1/2 times 2/3? How many is 3/4 times 2/3?
Students may understand this problem step by step despite of changing the number only in the problem. Mathematical understanding may be deepened step by step at the same time.
Besides this, changing a variation in the problem below was used broadly: 2Factorization of polynomial ，x+5x+6= (x+m)*(x+n) m+n=5, m*n=6
What are the possible values of a and b, such that the polynomial can be factored? 2 2 2 3 nx+ax+6? x+5x+b ? x+ax+b ? x+ax+b? x+ax+b?
Students may extend their understanding broader and broader despite of adding a variation in the problem. Having solved an example problem, we can get a group of subproblems (isomorphic problems) under students control by changing the conditions, conclusions or deduction process of the three components of the example problem in order to make students extract the essential elements of the mathematical problem-solving method of the example and adapt them to other problems easily. The subproblems are called Bianshi problems (Qingpu educational reform group, 1991). Bianshi problems is the media of Bianshi teaching. The word of Bianshi could be translated
in to Chinese ―变式‖ and English ―Variation‖.
2. The Origin from the Culture: Exam
It is well-known that the culture of examination is one of the characteristics of Chinese community. China, a country that is rich in population but relatively poor in resources, makes examination crucial in employment, graduation and ascendance in the society hierarchy. The subject of mathematics is the best differentiator among all the subjects in terms of sifting or filtering students through the education ladder in China. Therefore, for the sake of reaching the goal, math teaching under the pressure of society developed into the teaching and exercise of Bianshi problems, linking textbook problems to exam problems by the special position of exams in Chinese society and the distinctive role that math’s exams play in all the exam system.
We will illustrate the situation by the following example: the group of Bianshi problems below (designed by Shen , 2001), made up from (Math editing group of primary school, 2002) a textbook problem and 3 exam problems, show the root of Bianshi problems tied with exams.
The textbook problem: There are 25 questions in a test. Each of them has 4 answers. Only one is right among them. A pupil will score 4 if he chooses the right one. A pupil will score -1 if he chooses the wrong one or does not choose. How many questions did a student choose right if he scores 90? How many questions did a student choose right if he scores 60? (Algebra 1 new edition Vol. 1 the fifth question in B group) Bianshi problem 1: During a football competition in China in 1996, Dalian Wanda Team scored 3 points and had no defeat record. How many times did it win? According to the rule of competition: The team will score 3 points if it wins 1 time otherwise it will score 1 point if it ends with a draw ？upper secondary entrance exam
problem in Ningxia province,1996？.
Bianshi problem 2: During a football competition (each team should have 11 games (?)), Beijing Hongan Team scored 14 points. The number of wins is twice that of defeats. How many times did it end with a draw? According to the rule of competition: The team will score 2 points if it wins 1 time, otherwise it will score 1 point if it ends with a draw and it will score 0 point if it defeats？upper secondary entrance exam problem in
Wuhan city, 1997？.
Bianshi problem 3: During a middle school football competition (each team should have 4 games), the Team scored 17 points and had no defeat record. How many times did it win? According to the rule of competition: The team will score 3 points if it wins 1 time otherwise it will score 1 point if it ends with a draw and it will score 0 point if it defeats？upper secondary entrance exam problem in Zhuhai city , 2001？.
The example shows that Bianshi problems serve as the bridge between the original problems in the textbooks and real problems in examinations. Besides, exercise is highly regarded as an ideal learning way by Chinese society, which makes Bianshi problems with simultaneously daily school teaching an even more crucial part in mathematical curriculum and a major activity in mathematics learning after school.
For example, ―Bianshi problems zhen, Huanggang bingfa‖ (a success way of examination out of a package of Bianshi problems ) - the book was granted National Excellent Educational Books Award and National Bestseller Book Award - has been one of the bestsellers in China in educational series (Zhu, et, 2003). The preface on the book cover reads ―If you can solve examples in the textbook, you have learned 50% of the solutions .If you can
solve Bianshi problems, you have learned 100% of the solutions ---grasp the laws‖. It also shows Chinese math
beliefs beyond surface.
It is well known that examinations have been criticized for the reason that they narrow down curriculum and limit students` interest by most scholars from west to east for a long time. We say nothing of the exercise of Bianshi problems for the exam.
But the top-performing countries in the TIMSS and PISA were more likely to have high-stakes examination systems than were others. For example, the list of top-performing countries: Singapore, Korea, Japan, Hong Kong, The Netherlands, all of these countries have high-stakes examination systems. (Mullis, 1997). Here we need to rethink the role of exam.
In fact, exams` advantages were taken for granted and the exams` role as well: the quality control of the education systems was neglected. Some studies identified that examinations are mechanics to enforce the curriculum and instruction system (Phelps, 2001; Bishop, 1997).Examinations are superstructures of the education systems that support curriculum and instruction and hold the implemented and the attained system together. The relationship between the degree of quality control and student achievement appears to be positive and exponential conclusion: The more quality control measures are employed in an educational system, the greater is students` academic
achievement (Phelps, 2001; Bishop, 1997). Phelps (2001) mentioned the discoveries are all remarkable even in states and provinces.
Bishop (1997) further called the alleged effect of the high stakes upper secondary exit exams on the behavior of students and teachers at the lower secondary level a ―backwash‖ effect. From this angle, Bianshi problems may be
regarded as the means to achieve the ―backwash‖ effect of the quality control of the education systems.
Besides, examinations also set ―national achievement standards‖ for students to pursuit. China always maintains
curriculum quality horizontal coherence primarily through frequent administrations of nationally standard high-stakes examinations each year. At the same time exams also keep curriculum quality vertical coherence primarily by adjusting curriculum between the intended and attained curriculum. So the exam with high reasoning load and high order thinking added plays an important ―ladder‖ role to make students ―climb‖.
By and large, we can say that it is the special role of exams in China predetermines the cultural characteristics of Bianshi problems. Bianshi problems are turned into the practical curriculum designed to the ―bridge problems‖
with depth so as that students to realize the goal of exam by solving it .The special practical curriculum just fill in the gaps between the intended and attained curriculum and between the national and local curriculum .
3 The Origin from the Curriculum: Two Bases
―Two Bases‖ refers to basic knowledge and basic skill stipulated by the Ministry of education. It also means to recognize the invariables in the changing situation and to apply each basic skill in answering problems in different conditions. For example, the General Outline of National Mathematical Teaching (Ministry of education, 1999) stipulated: the concepts, formulae, theorems are the foundation for continued learning. Students should understand these completely. As it is the official objective of Chinese mathematical teaching and core curriculum stipulated in the General Outline of National Mathematical Teaching (now known as curriculum standard) in China. Each element involved in the education process (e.g. teachers , parents , school and so on ) form a integrated system
that work well together to carry out this education goal. Therefore Bianshi problem solving acts like an effective curriculum implementing model characterized by ―dispersed and progressive difficulties‖ based on curriculum standard. ―Bianshi teaching and exercise‖ has become the pathway to the ―double foundation‖.(Zheng , 2003).
We drew a picture to clarify the character of Bianshi problems in the curriculum development above.
Goal of exam
—invariant “axis ”
Two bases Example problems
Figure 1:Bianshi problems in the
Based on the analysis mentioned, Bianshi problems should be designed to the ―bridge problems‖ with depth under
the vertical direction so that students can realize the goal of exam by solving it .On the other hand, Bianshi problems should also be designed to the ―bridge problems‖ with width under the horizontal direction so that
students can realize the goal of double base by solving it. All the Bianshi problems form a Bianshi problem space with the spiral uptrend by circling math understanding.
4. The Origin from the Circumstances: the Chinese Situation
There are so many students in each classroom. Therefore, teaching in China is very different from that in foreign countries. Bianshi teaching, however, helps address individual difference by Bianshi problems including from low level to upper level.
2For example, Bianshi problem above: Factorization of polynomial x+6x+8? (Level 1)What is the possible value 2 2of a and b, the polynomial can be factored. x+ax+6? (Level 2) x+5x+b? (Level 3)
The group of Bianshi problems includes tree level of ―factorization‖ concept development. Each pupil should
learn ―factorization‖ concept and extend their understanding from surface content to structure content through the three stages. At last grasp the abstract essence .On the other hand, there are tree levels of difficulty problems. The students in different levels of cognition will find the starting points from the group of Bianshi problems. So the group of Bianhsi problems may be regarded double scaffolding in the individual level and in the classroom level.
5. The Origin from the Tradition
5.1 Tradition of Chinese Teaching
The old tradition of Chinese teaching emphasizes teachers to be learning examples and problems in the textbook as a workable example to follow. This tradition that epitomized as ―initiating principles, imparting knowledge, and solving problems‖ (―传道授业解惑‖) affirms the dominant role examples play in teaching. The new tradition
of Chinese teaching mainly was influenced by Russian Kailuofu mode after the liberation of China, a pattern expressed as a process of preparation----introduction—explanation----exercise----conclusion. This mode made it
an easy common model that workable examples enter and Bianshi problems consolidate after that. The mode confirms the dominant role example and exercise tradition in the Chinese math’s teaching.
5.2 Chinese math tradition
The Chinese ancient math developing system has its differentiation from the western traditional teaching system. It was more algorithm-oriented and application-oriented. There is a certain fundamental pattern for the Oriental mathematics. ―That is to produce new methods from practical problems, promote them up to the level of general
method, generalize them into “shu” (“术”similar to ―arithmetic‖ or general methods ) and deploy this shu to
solve various similar problems which are more complicated, more abstruse‖ ( Wu, 1996). For example , the fifth
problem in the eight volume of ―Jiuzhang Suanshu‖ (九章算术)which was the earliest math textbook in print in
the world (Wu, 1996), follows a route like this ―the example problem –answer—solution--the Bianshi
problems(―a group of isomorphic math problem with variation‖). Bianshi teaching displays the traditional
continuity of structure and function of Chinese ancient math tradition to some extent in modern classrooms. In other words, it is the ―replay‖ of a specific form of education, a Chinese math tradition that works under today’s mathematical teaching environment.
In short, Bianshi problems are not just isolated teaching actions and skills, but a collective system driven by the curriculum object and exam goal. It is realized through individual actions driven by goals. It in turn is realized by means of routinized operations, dependent on the conditions such as Exam Goal, Curriculum Objectives, Chinese Situation, and Tradition. In order to portray the character of Bianshi problems in the implementation of curriculum in Mainland China, we integrate the various features of historical and cultural context of Bianshi problems behind into a network by a holistic view (see Figure 2). However, the purpose of this paper is not to justify the practice of Chinese mathematics education but to show that there exist distinctive features of Bianshi problems as the byproduct of chinese culture, and that those features have distinctive significance for underlying cultural and historical context.
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Figure 2: The model of Bianshi problems
Two Teachers Exam Goal Bases
The model of Bianshi
Chinese Situation Chinese math tradition Tradition