Our study on analyzing students' strategies and difficulties in problem solving is considered indispensable to improve teaching and learning in mathematics classroom activities. It seems that these strategies and difficulties are influenced greatly by some social and cultural factors, such as languages, symbols and representations. This study is planned in order to make exact the effects of teaching and learning of teacher and students who engage in problem solving by means of the "Open-Approach" method, particularly with reference to sharing mathematical ideas of problem and the use of mathematical patterns involved in problem solving. We have to become more aware of the information processes, which consist of the communications and interactions between the teacher's explanations and student's approach to problem solving.

 

 

1. Mathematics Classroom Activities

Several difficulties concerning problem solving are, in our opinion, due to the narrow and isolated conceptions of the basic didactical category on "problem solving". We, therefore, attempt to reveal the global and relational character of "problem solving", which is to call attention to the necessity of dealing with a broad spectrum of activities related to Japanese culture and society.

These new demands can be found in Christiansen and Walter (1986), which necessitate changes in the teacher's roles and moves:

1. Changes in the distribution of emphasis on the different types of activity,

2. Changes in the types of teacher's moves and in the sequencing of these in the teaching process,

3. Changes in the ways in which the teacher serves as a mediator of mathematical meaning.

The process of problem solving becomes evident when teaching is seen as a process of interaction between the teacher and learner and among the learners in which the teacher attempts to provide learners with access to mathematical thinking in accordance with given problems. This teaching/learning process (like all processes between learners) is influenced by a number of social and developmental aspects and factors, which can be included in problem solving. The communication between teacher and learner is thus not only conditioned by formal decisions about goals, content and teaching methods, but it is also strongly dependent on even more informal aspects in initiative stages of problem solving, such as the teacher's words and explanations to the problem solver, and the students' motivation to solve the problem and to be concerned with it. Accordingly, communication through "problem solving" as an organizing principle in Japanese mathematics learning calls for meta-learning under the teacher's support. This communication in mathematics classroom teaching is considered as controlling the organization and dynamics of the classroom activities for the purposes of sharing and developing mathematical thinking.

2. What is the open-approach?

The aim of open-approach teaching is to foster both the creative activities of the students and their mathematical thinking in problem solving simultaneously. In other words, both the activities of the students and their mathematical thinking must be carried out to the fullest extent. Then, it is necessary for each student to have the individual freedom to progress in problem solving according to his or her own abilities and interests. Finally, it allows them to cultivate mathematical intelligence. Class activities with mathematical ideas are assumed, and at the same time students with higher abilities take part in a variety of mathematical activities, and also students with lower abilities can still enjoy mathematical activities according to their own abilities.

In doing so, it enables the students to perform the mathematical problem solving. It also offers them the opportunity to investigate with strategies in the manner they feel confident, and allows the possibility of greater elaboration within mathematical problem solving. As a result, it is possible to have a richer development in their mathematical thinking, and at the same time, foster the creative activities of each student. This is the idea of the "Open-Approach", which is defined as a method of teaching in which the activities of interaction between mathematics and students are open to varied problem solving approaches.

Next, it is necessary to make clear that the meaning of the activities of interaction between mathematical ideas and students’ behaviors is open in problem solving. This has been explained from three aspects:

(1) Students' activities are developed by the open-approach.

(2) A problem that is used in the open-approach involves mathematical ideas.

(3) Open-approach should be in harmony with interaction activities between (1) and (2).

3. Characterizations of the "Open-Approach" problem and method

We should become more aware of the information processes that consist in the "Open-Approach", that are the relationships between the problem and method. We use here "Open-Approach" problem as like non-routine problems: problem situations, process problems and open search problems (Christiansen & Walter, 1986). In actual practice, each teacher will have to take his or her own classroom conditions and teaching objectives into consideration. Therefore, the method we use in "Open-Approach" depends on the problems, which consist of problem situations, process problems and open-ended problem, and the procedures of these problems including classroom conditions and teaching objectives (Nohda, 1983, 1986).

We define a problem as follows: A problem occurs when students are confronted with a task, which is usually given by the teacher, and there is no prescribed way of solving the problem. It is generally not a problem when the students can immediately solve it. Here we use the problems as open-approach problems mentioned above. Treatments of these problems will depend on the teacher's intentions for his/her objectives:

A. What kind of problem does the teacher want the students to formulate from the given problem situations?

B.&emdash;How many ways of thinking does the teacher want the students to come up with concerning the problem given?

C. What kinds of advanced problems does the teacher want the students to make from the original problem?

 

 

4. Actual problem solving activity in a sixth-grade classroom

In everyday life, students are confronted with many problem situations where they can take a variety of solutions. The methods of solving the problem in daily life seem to include some regular rules or procedures.

To foster their mathematical thinking, mathematics teacher should emphasize problem solving, in which students would discover better way of thinking through discussions of various solutions of the problem.

Here, an example of actual problem solving activities in mathematics classroom is shown. In this study, a sixth-grade class (Male; 18, Female; 22, Total; 40) was taught by the "Open-Approach" method in a rural elementary school near Tsukuba City. The teacher was Ms. K. Mashiko. She is an excellent teacher, who came to me for studying mathematical problem solving for about three months. The lesson was held on January 26, 1987. A process problem was used in the lesson.

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