The Research On Problem Solving Based Mathematics Instruction

Why should we study Problem Solving? What were the results of study in the past 20 years? How do we consider this topic today? In other words, what is the value of problem solving based mathematics instruction today? These are the problems to be discussed in this dissertation.We have considered the study value of problem solving from the history of mathematics and educational study. Then we focus on the theoretical study of mathematics education and practice in the school to find the problems need to be solved. They are the theoretical and practical background of this dissertation. We also review the theoretical study of problem solving in USA from 1980s, in order to find the clue from their study. From that point, we extend the opinion of problem solving from the learning theory. We also explain and analyze our opinions through two cases and show our image of this topic in the future. Such is the content of this dissertation:The first part of this dissertation is to review the theoretical background of problem solving. By reviewing the history of mathematics, especially the three times of crisis in the history, we find its logic of development is a process of problem posing and inquiring and solving. It has a long history since the problematic phenomena were considered in the instruction. The social change has new request to education. In mathematics education, "learn to value mathematics; become confident in their ability to do mathematics; become mathematical problem solvers; learn to communicate mathematically, and learn to reason mathematically" are the new goals. The study of mathematics philosophy promotes the change of beliefs and methods of mathematics instruction. The development of constructivism and situated learning and cognition call for "do mathematics ". The problems which lie in the theoretical study and practice imply the necessary, importance and urgence of problem solving study.In Chapter 2, we focus our attention on the theoretical study of problem solving in the past 20 years in USA, and try to find important clues from their opinions and methods. We are glad to find that most studies are based on the information -processing theory. The factors of problems (task variables, syntax variables, types of problems), strategies and methods of problem solving are all considered as static and objective knowledge, and the object of instruction is recognized as the transmission of such knowledge. The character of this kind of instruction is that it is the transmission of the skills (making a table, writing an equation) from teachers to students. The nature of this kind of instruction is "teaching problem solving ". In the end of 1980sand early of 1990s, with the deficiency of previous theoretical study and the development of study on the nature of learning, there has a request to change the old view of problem solving.From this point, we look for more understanding of problem solving from the metaphor of learning and teaching. Our opinion is that we should create open learning environment with cognition tools and rich resources in mathematics instruction. We should change our opinion of mathematics instruction from "teaching problem solving " to "learning through problem solving ", promote the construction of knowledge through the process of problem solving, and so promote the development of thinking of problem posing and problem solving of students. We also advocate building community of learners to break the isolate classrooms to promote interaction on problem solving. The learners are directed by teachers who as master learners, and will be changed from novice learners to master learners. We also explain and analyze two kinds of instructional models through two cases. They represent two kinds of learning environments. We express our opinion through this process of analyses.The last part of dissertation is the prospect of future study. We review the content of this dissertation and point out the new way in the future. In a word, the knowledge of "how people learn " has changed o...