Open-ended problem is a problem that has several or many correct answers, and several ways to the correct answer(s). The Open-Ended Problem Solving is based on the research conducted by Shimada S., which is called "The Open-Ended Approach". The Open-Ended Approach provides students with "experience in finding something new in the process"(Shimada 1997). The Open-Ended Approach started in 1970s. Since then, Japanese teachers have developed many open-ended problems and lesson plans using open-ended problems. These problems are being used in mathematics lessons elementary through high school grades, and the lessons are called the Open-Ended Problem Solving now. Open-Ended problems are also used as assessment tasks because "In responding to such (open-ended) items, students are often asked not only to show their work, but also to explain how they got their answers or why they chose the method they did" (Schoenfeld, A. et al., 1997). The Open-Ended Problem Solving also has been widely regarded as an advanced style of teaching mathematics in the U.S. recent years.
There are 5 advantages that can be summarized, based on what Sawada mentioned in 1977 (Sawada, 1997),
1). Students participate more actively in lessons and express their Ideas more frequently.
The Open-Ended Problem Solving provides free, responsive, and supportive learning environment because there are many different correct solutions, so that each student has opportunities to get own unique answer(s). Therefore, students are curious about other solutions, and they can compare with and discuss about their solutions each other. As students are very active, it brings a lot of interesting conversation to the classroom.
2). Students have more opportunities to make comprehensive use of their mathematical knowledge and skills.
Since there are many different solutions, students can choose their favorite ways toward the answer(s) and create their unique solution(s). Activities can be the opportunities to make comprehensive use of their mathematical knowledge and skills.
3). Every student can respond to the problem in some significant ways of his / her own.
There are various kinds of students in a mathematics classroom, since there are no tracking in Japanese classroom. Therefore, it is very important for every student to be involved into the classroom activities, and the lessons should be understandable for every student. The open-ended problems provide every student with the opportunities to find his / her own answer(s).
4). The lesson can provide students with a reasoning experience.
Through the comparing and discussing in the classroom, students are intrinsically motivated to give reasons of their solutions to other students. It is a great opportunity for students to develop their mathematical thinking.
5). There are rich experiences for students to have pleasure of discovery and to receive the approval from fellow students.
Since every student has each solution based on each unique thinking, every student is interested in fellow students solution.
There are also some disadvantages that Sawada mentioned in 1977 (1997), such as difficulty of posing problems successfully, difficulty of developing meaningful problem situations, and difficulty of summarizing the lesson.