| Questions to be solved function as the internal component of the mathematical science itself, in which the diversified ways for resolving the mathematical problems are of great educational value for its contribution to the thought development of students. Under the requirements of the present compulsory education in China, the mathematics curriculum standards aim to realize the mathematical problem solving in diversified ways. In practice, the application of the diversified ways for solving mathematical problems is the general ways in the mathematics teaching praxis and the textbooks in usage. Nowadays, it has been more than ten years since the implementation of the mathematics curriculum standards. Some scholars, however, point out that it remains be a vulnerable area for students’development of solving mathematical problems in diversified ways. Besides, there are lack of the systematic theory guidance for diversified ways for the mathematical problems solution in the present teaching praxis. The mathematics teachers therefore rely on individual understanding and temporary performance. Thus the realistic contradiction sticks out between the general teaching practices and the research deficiency on the diversified ways for resolving mathematical problems.This study try to explore the theory on diversified ways for solving mathematical problems in primary school and think about the assessment on the effectiveness of the teaching practice and the empirical analysis in particular. Hence, it’s necessary to assess the students’development of solving mathematical problems in diversified ways. Aiming for the improvements to the universal application of the diversified ways for solving mathematical problems in mathematical teaching, it is the urgent need to carry out the concerning studies on diversified ways for resolving the mathematical problems.This dissertation, including the theory exploration and the empirical analysis, studies the diversified ways for resolving the mathematical problems in the primary school by adopting the qualitative research methods and quantitative research methods. The studies contents are stated as following:1) Through the exploration on the theoretical foundation on the mathematical problem solving in diversified ways, the dissertation attempts to reach the elementary rational cognition. After the collation process of the relevant reference, the dissertation gives the exact definition for the mathematical problem solution, the diversified ways for solving mathematical problems as well as the students’development of solving mathematical problems in diversified ways, which would help to clarify the chaos caused by the confusion among the definition, relative concepts and mutual relations. The dissertation also achieves in the new interpretation on the education value of the creative thinking ability that students acquired by settling the mathematical problem in diversified ways. Besides, the classification, which reveals the mechanism and influencing factors behind the diversified ways for solving mathematical problems, contributes to probe the precautions and attention points concerning its teaching praxis.2) In order to assess the effectiveness of the present teaching praxis on the diversified mathematical solutions which would in turn helps the exploration of the theoretical foundation for the subject, the dissertation attempts to invent the assessment tools for estimating the students’ cognitive development after the acquisition of the mathematical problem solving in diversified ways. At the same time, the Empirical Analysis(the test survey) are applied to detect the current situation of the students’abilities on the diversified mathematical solutions, to research the rules and characteristics for training students’competence on the mathematical problem solving in diversified ways, which would further confirms the theory exploration that was stated in the previous part the dissertation.3)and to obtain the implications on the compilation of the mathematics courses and teaching arrangements.The main findings and conclusions of this study:"The ways of mathematical problem solving" refers to the specific way for solving mathematical problems (the way closely related to problem situations and specific knowledge) and it is a complex used to solve a mathematical problem which involves production system and internal stipulation of problem situations. It consists of two parts:(1) the production system to solve mathematical problems (the combinations of basic quantitative relation), which can be written on dissertation explicitly.(2)"Algorithm" of the solutions to mathematical problem solving, a part lurking in the background, which means problem situation’s inherent regulation of production systems. Among them, the direct result of production system is the mathematical algorithms used to get questions answered. The concept of "the solutions to mathematical problem solving" commonly includes "solution"("mathematical problem-solving approach") and "algorithm" lurking behind, which is an expansion. According to the structure analysis on the ways of mathematical problem solving, this research clearly identified the relationship about "diversification of the ways of mathematical problem solving" and "one problem with multiple solutions","the ways of mathematical problem solving" and "algorithm", and "diversification of solutions to problems" and "algorithm diversification" to clarify chaos phenomenon about the teaching of "one problem with multiple solutions", reinterpret the function to cultivate student’s creative ability in mathematics by using a variety of ways to solve mathematical problems, also have a discussion about mechanism and influencing factors to develop multiple solution ways, and even to help people identify whether the students really have understood or really have solved the problem, which is often so-called "true understanding" or "acting understanding".The nature of "diversification of the solutions to mathematical problem solving" is to construct a variety of production systems used to solve mathematical problems."Diversification of the solutions to mathematical problem solving" is also on behalf of the teaching mathod that using different ways to solve mathematical problems. The basis to judge whether a solution is different from another solution is that the regularity of problem situation the two solutions embodied is different, and ultimately the combinations of basic quantitative relation the two solutions embodied is different, or the mathematical structure of the two solutions is different. The origin of "diversification of the solutions to mathematical problem solving" is the variability of the combinations of basic quantitative relation of problem situation, while the development of various solutions is based on the inherent regularity in problem situation. Due to the change(flexibility) in a variety of ways to solve problems, therefore, to solve math problems in a variety of ways is not a skill can be automated, which highlights the value of "diversification of the solutions to mathematical problem solving" on cultivating students’creative ability in mathematics."Diversification of the solutions to mathematical problem solving" not only can be used as a teaching form to develop thinking and promote creativity, but also can be used to drill down a specific topic, or as a new technique to inspire new topic and expand the field of knowledge. The teaching of "diversification of the solutions to mathematical problem solving" is necessary and reasonable."The students’development in diversification of the solutions to mathematical problem solving" refers to student’s understanding, mastering and application (cognitive outcomes) of various solutions they obtained after the teaching of "diversification of the solutions to mathematical problem solving" and the application of it to solve mathematics problems. It includes the composition of the solutions’quantity (diversification) and quality (perception or understanding of the entire solutions to the problem, which means the complete collection of solutions) when students dominate them to solve problems. In teaching process of "diversification of the solutions to mathematical problem solving", whether "group diversity" form or "individual diversity" form is adopted, the ultimate goal is to make students understand and experience the nature, difference and characteristic of each way so as to improve and enhance students’cognitive level.This study tries to divide students’development in "diversification of the solutions to mathematical problem solving" into five levels:level1, students cannot correctly solve the problem given; level2, students can correctly solve the problem given; level3, students can correctly solve the problem given in2ways; level4, students can summarize the2ways and express their connection between based on the2ways they find; level5, students can correctly solve the problem given in3ways.The result of empirical research illustrates that students can obtain certain cognitive development in "diversification of the solutions to mathematical problem solving" after the learning of mathematics courses, and although the effect is not very obvious, and the current teaching of "diversification of t the solutions to mathematical problem solving" is not entirely valid. There is no significant gender difference in students’development in "diversification of the solutions to mathematical problem solving"; while there is significant difference between grade, and significant difference exists in each dimension for different grade.To sum up the study conclusions, this dissertation makes such recommendations on mathematics curriculum and instruction to elementary school:(1) To pay attention to students’overall construction in the teaching of "diversification of the solutions to mathematical problem solving";(2) to arrange for mathematics curriculum and content layout reasonably and plan students’development processes in the teaching of mathematics ability. Students of grade four and before should focus on the numeracy skills development; while grade five or six on algebra and geometry for priority development; meanwhile, the teaching for grade five or six should focus more on internalization and overall construction of knowledge and self-reflection on learning so as to promote the internal construction of knowledge;(3) to develop different solutions based on the problem situations;(4) to guide students to develop a variety of solutions independently;(5) to focus on the process of developing new ways and understanding of multiple resolutions;(6) to pay a attention to the "degree" in the teaching of "diversification of the solutions to mathematical problem solving";(7) to implement the teaching of "diversification of the solutions to mathematical problem solving" in three ways.The legislation Figure innovation in this dissertation is:To explore the theory on diversified ways for solving mathematical problems in primary school for the first time,and Supplement the test survey on students’cognitive development in the problem-solving method diversifying.So, this dissertation give an rudimentary definition on the levels of the students’development in "diversification of the solutions to mathematical problem solving", and preparation of the corresponding rudimentary test volume These are original researches with certain exploratory and pioneering. Hope that the conclusions and recommendations can provide some theoretical and practical reference for the further development of the elementary school mathematics curriculum and it’s teaching。The areas for improvement:1) The exploration of the theory on diversified ways for solving mathematical problems in primary school is only at the early-stage. There are many problems to be studied further to get a systerm theory on diversified ways for solving mathematical problems, including the teaching principles for apply diversified ways for solving mathematical problems, and the corresponding teaching mode and so on.2) Since the example came from a specific region only, The result of empirical research might not represent the full conclusion. |
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