The 2001 annual meeting of the Executive Committee of ICMI was held last April in Shanghai at the invitation of Jianpan Wang, member of the ICMI EC. On that occasion, a research group of East China Normal University collaborated with a Grade four local schoolteacher on a concrete mathematics project for the pupils, namely the detailed planning of a half-a-day visit to a famous tourist venue in downtown Shanghai by a group of foreign visitors (the members of the ICMI EC!). A report on this project was presented at an international symposium on mathematics education held at ECNU during the ICMI EC visit. The editor is pleased that the researchers involved in this project have kindly accepted his invitation to prepare a brief account of this activity for the ICMI Bulletin.
The members of the ICMI Executive Committee wish to express their gratitude to Linmin Ma, director of Wan Yu Street Primary School (Shanghai), as well as to Peiqing Feng, teacher, and to all the pupils involved in this project, for their warm hospitality over a memorable lunch meal in the classroom.
IntroductionMathematics correlates closely with human life. It is suggested that mathematics can and should be learned in connection with the authentic life of the learners or their future life. Mathematics instruction should be based on the introduction of situations containing problems originating from learners' life or real tasks they are likely to face in the future. Learners can thus have opportunities to learn to identify and solve mathematical problems so to adapt to the complex world and experience social responsibility. But the traditional mathematics instruction in China is more or less isolated from learners' life and overemphasise learning mathematical concepts and theorems and exercising repeatedly. So Chinese students are well known for their excellence in standardised test, but also their weakness in analysing and solving authentic problem.
The present case study aims at challenging and innovating traditional mathematics instruction and trying to improve the situation through concerted efforts of researchers and school teachers. Situated instruction is selected as the theoretical support in our case study, because the related instruction theory underlines the importance of learners exploring in problematic situation and thus corresponds to our intention.
The present case study aims at challenging and innovating traditional mathematics instruction and trying to improve the situation through concerted efforts of researchers and school teachers. Situated cognition and learning (cf. McLellan, Ed., 1996; Lave & Wegner, 1991) is selected as the theoretical support for the instructional design in our case study, because the cognitive theory underlines that "knowledge is situated, being in part a product of the activity, context, and culture in which it is developed and used." (Brown, Collins & Duguid, 1989, p. 32)
Theoretical background
1. The principle of instructional designAccording to the theory of situated instruction, students should have the desire to study in an integrated and authentic problem situation; through interaction, to communicate among the members of the community; through initiative learning and productive study, to experience the process of identifying objectives and independent learning. The students learn to adapt to daily life, learn to identify and try to solve authentic problems.
As many research works show us, different authentic learning situations support all kinds of learning activities. The design of a 'situation' aims at promoting the kind of study activity which is emphasised by constructivism. These activities are different from those supported by other kinds of textbook. For example, in mathematics instruction, the regular word problems always have clear objectives and data required for solving the problem. In this kind of problem, the student has nothing to do except simple direct computations. In other words, there are two important principles of design: first, learning activity and instruction should be designed around one authentic 'situation'; and second, learners should have the chance to probe independently.
Local culture provides learners with familiar and exciting situations to explore actively and independently. Learners are inclined to find various problems and apply what they have learned to solve them or identify what knowledge is lacking. This kind of internal thinking activity acts as an essential element for creative learning. And the assigned task encourages the sense of responsibility.
2. The method of instruction
Based on situated cognition theory and the principles mentioned above, we underlined several points as follows:
What is most challenging in this kind of instruction is the necessary conversion in the role of the teacher, which should change from information provider to "coach" and "learning partner". The teacher must be good at encouraging and supporting students to learn productively. The teacher should not follow strictly prepared instructional plan, and need not be an expert of each problem. The teacher also needs to learn, together with the students.
A Case StudyThis case study is part of the research program "Research on situated mathematics instruction: theory and practice" supported by the Ministry of Education of China. This program aims at doing research and development on mathematics instruction models that challenge and innovate the so-called cramming method of teaching.
The researchers first put forward a protocol related to the project. Then school teachers and researchers exchanged opinions on the plan. We discussed repeatedly the possible difficulties and finally worked out an implementation schedule in details. During this process, the school teachers became more familiar with the theory and method of situated instruction.
In the project pupils face a situation in which some foreign experts need suggestions for a half-day tour in Cheng-Huang-Miao, a well-known venue in Shanghai famous for its great number of antique and China-characteristic old houses. They then made the necessary preparation for a guided tour. They suggested plans for all kinds of possible activities in the tour and estimated the time and the money needed. The pupils involved in this project come from Wan Yu Street Primary School (Shanghai, China), near Cheng-Huang-Miao. They are from one regular class in Grade four and many of them are familiar with the scene.
The mathematics teacher of the class guided the pupils' activities. Researchers went to observe and took videos of the whole process. They were also participants in the activity.
Every phase of the project was followed by reflection from the teacher, researchers and some students.
2. Project objectives
The objectives of the project were as follows: to identify and solve mathematical problems embedded in a tour to Cheng-Huang-Miao in Shanghai; to improve the habit and ability to estimate; to promote information literacy.
3. Project implementation
The whole process consisted of four phases: (i) Entering the problematic situation, identifying and decomposing the problem; (ii) Collecting, selecting, organising and applying information to solve the problem; (iii) Solving the whole problem; (iv) Solving new problems put forward by the students themselves.
A detailed description follows.
Phase One: Entering the problematic situation, identifying and decomposing the problem
An authentic problem is selected as an anchor into the project. We have given the pupils photos about our guests' visit to the Cheng-Huang-Miao, and explained our problems and wishes. The pupils are listening to our story.
Some guests from America, Argentina, Australia, Canada, France, Japan and Russia will come to East China Normal University for an academic meeting, and they want to have a half-a-day tour in Cheng-Huang-Miao. Please make a tour plan for the foreign guests. It can be anticipated that the guests want to see some attracting places, buy some souvenirs and enjoy a light meal. Of course, they also want to know how much they will spend (in foreign currency).
The pupils became interested in this "authentic problem", but felt perplexed. Some warm-up questions were put forward to remind the pupils of their own correlated experience, as followed:
Are you familiar with Cheng-Huang-Miao, one of the most famous venues in Shanghai? With whom have you ever been there? How many times? How long did you stay there each time? What did you buy? What did you do also?
The pupils then talked in groups about their experience. The teacher suggested that the pupils could present their experience with the help of mathematical tools they have learned. The pupils selected tables and statistical bar charts.
During the discussion about their experience in the group, they made various statistical graphs, for example, one group counted what places and how often they have visited in the Cheng-Huang-Miao, and then they gave the resulting statistics. Another group reported what they had eaten and how much time it took them to eat. And so on. After recollecting their experience, the pupils began to make a detailed schedule from the viewpoints of foreign experts. They summarised their activities in Cheng-Huang-Miao and classified them into several kinds (nosh, place, etc.)
Several students have written down their ideas and identified three settings: places, shopping, and refection.
The pupils experienced the concept of classification. During the discussion, some pupils realised that several elements might be listed in both refection and place. The students made disputes on the problem. Then the teacher felt it was a good chance to make an introductory explanation about the new concept of "intersection", and did so. The pupils showed desire and interest to learn the concept. In such situation, they experienced the situation nature and application nature of mathematics.
After interaction between the pupils and the teacher, the pupils realised the complexity of the problem and the necessity to divide the work among groups of pupils and collaborate. Information is also precondition to solve the problem, so they decided to go to Cheng-Huang-Miao and collect information there.
We have seen that the pupils know they must prepare some instruments for counting or measuring.
We have constituted four groups, the first dealing with the project of visiting Yu Yuan Garden; the second, with the project of visiting Jiu-Qu-Qiao and some other scenes; the third, with project of having a light meal; and the fourth with the project concerning shopping.
Phase Two: Collecting, selecting, organising and applying information to solve the problem
Collecting information
With sub-problems in mind, the class went to Cheng-Huang-Miao to collect information in groups. Let us note here that the ability to collect information for problem solving is required in the new curriculum standards of our country.
Selecting and applying information
After going back into the classroom, the pupils clarified their problems under guided discussion and then set about to make a plan for the selected sub-problems by groups.
The pupils found the irrelevance of some information and realised they must spend some time in selecting those relevant information and data. They then began to solve the problem.
In this phase, the pupils applied their knowledge and ability to deal with questions such as the unit conversion (from renminbi to foreign currency, and vice versa), the algebraic relations between unit, number and total price, and the ability to estimate.
The pupils then presented their designs in vivid ways. Two groups made use of the computer.
Phase Three: Solving the whole problemThe next step was to solve extended authentic problems applying all the information. Every group was asked to design a three-hour-tour in Cheng-Huang-Miao on the basis of the information gathered by all the groups. The guests should be able to:
The pupils were asked to arrange the sequence and estimate the money spent and the time needed.
The pupils tried to make many designs but many set their own sub-problems at the centre. They found that was inappropriate after discussion and revised their designs according to opinions from other groups.
In this phase, the pupils experienced the necessity to verify the reasonableness of approaches to mathematical problems. What is very important is to verify and reflect.
Phase Four: Solving new problems put forward by the students themselves
Extending problems to other subjects
The experience can be extended to other subjects such as science, society studies, history or language. The pupils became interested in many other aspects including:
During the project, students found some problems they were interested in and hoped to solve. The representative of each group represented the problems on the blackboard and the rest of the group were having heated discussions about the problems.
Teachers and parents took an active part in the project organised by the students themselves. The pupils interacted with each other and with teachers and parents who tried their best to provide information.
We noticed that the pupils were very active in the problems put forward by themselves. They made full use of all resources (internet, books, adults around them, peers, etc.) to solve those problems collectively.
4. Reflections
From the teacher
It was an absolutely new instructional activity to the teacher and she was not assured of the result. But she was very glad to find that the pupils were very active and interested in the whole process. They could put forward problems and try to solve them. She found that many pupils themselves could apply what they had learned the previous term.
From the pupils
Many of the pupils said with deep feelings that they had been regarding mathematics learning as doing exercises, but now they realised that mathematics is relevant to and very useful in authentic life. "We are very interested in this kind of activity. We keep the problem in mind even after class and try every possible way to solve it." A pupil said: "We become more independent and brave."
From the parents
"They (the pupils) look excited and interested in the work. They keep asking me all kinds of questions about Cheng-Huang-Miao and also try to collect information from books and the web." The children like putting forward problems themselves and often discuss them after they finish their lessons. "They even hope to arrange our family's summer holiday travel."
From the researchers
"We find the pupils experienced radical change during the process. They were first perplex at the situation and then they became interested and confident in the problem. Finally they initiated problems related to mathematics and other subjects." "But the teacher finds it difficult to escape from conventional method despite her understanding in instruction theory. We must enter schools and classrooms to convert theoretical research achievements into teachers' practical behaviour. It seems that mathematics teachers' training is necessary."
Conclusion
Contrasted to the cramming method of mathematics teaching, this kind of activity integrates life in mathematics learning and make the pupils experience the application value in authentic world. They realised the importance of internal connection of mathematical knowledge. Learners learned to find, identify, analyse and solve mathematical problems actively. Learners improved their social responsibility, their ability of collaboration with peers and adults and critical thinking.
This kind of mathematics instruction gives an impulse to reflection on instruction models from both school teachers and researchers. School teachers recognise the effect of this innovative instructional model and want to make some experiments, but hesitate to do it owning to pressure from uniform examination.
Supported by school director, school teachers and parents, we achieved many significant revelations in this case study. We plan to enrich our theoretical framework and instructional design and develop corresponding learning situations based on Chinese culture to reform mathematics instruction in our country.
Acknowledgements
During the whole process, the director, teachers, pupils and their parents in Wan Yu Street Primary School (Shanghai, China) provided us with all necessary support. All the members in the Curriculum and Instruction Development Lab of East China Normal University have been devoting themselves to the project. Particular thanks are due to the members of the Executive Committee of ICMI, who showed great interest in the project and went to the primary school to exchange their viewpoints with the pupils and teachers and had a lunch meal with them, during their EC meeting held in Shanghai in April 2001.
References
Brown J. S., Collins A. & Duguid P., (1989) "Situated cognition and the culture of learning", Educational Researcher 18(1)(1989), 32-42. [See also in McLellan H., Ed. (1996): Situated Learning Perspectives. Englewood Cliffs, New Jersey: Educational Technology Publications, pp. 19-44.]
Hilary McLellan. Ed. (1996): Situated Learning Perspectives. Englewood Cliffs, New Jersey: Educational Technology Publications.
J.K.Lave & E. Wenger, (1991): Situated Learning: Legitimate Peripheral Participation. New York: Cambridge University Press.
E. Wenger (1998): Communities of Practice: Learning, Meaning and Identity. New York: Cambridge University Press.
XU Binyan, ZHENG Tainian, QIAO Lianquan
Curriculum and Instruction Development Lab
Institute of Curriculum and Instruction
East China Normal University
ZhongShan Road (North) 3663
200062 Shanghai, China
xubinyb@online.sh.cn