Case Study About Junior Middle School Mathematically Gifted Students' Feature In Solving Mathematical Problems

This research is mainly concerned the characteristics of the gifted students in mathematics of junior high school when they solve mathematics problems. According to the experimental subject system classification method in P.A.Крутецкий's book Psychology of Primary and Middle School Students'Mathematical Ability, this paper provides four types of questions, including information collection, information processing, information preservation and observation or imagination types, which form a 11-question test paper.In this study, we chose six eighth-grade students from two famous junior high schools in Shanghai as the case study objects. We asked them to do the test paper mentioned before by way of thinking aloud, and then we interviewed them. This paper made a qualitative analysis of their problem-solving process and a quantitative analysis of their scores in the mathematics test according to the rating criteria in this paper.Based on the Carlson's VAMS questionnaire about students' mathematical learning and mathematics knowledge, this paper established its questionnaire and determined the "expert view" and "folk view", and analyzed the students' mathematical belief with 26 gifted students in mathematics as its study objects. The main conclusions are as follows:1. About information collection:The gifted students in mathematics of junior high school have strong math materials perceptive ability, who can quickly summarize the necessary information in solving mathematics problems. Even if encountering difficulties, they would accurately summarize the key information after some analysis process, and classify the topics subconsciously.2. About information processing:The gifted students in mathematics of junior high school can transform the problems into familiar topic type by using all sorts of problem-solving strategies, such as changing element method, chemical method and so on. They would try to simplify the writing and calculation process. When solving mathematics, they don't give up easily. On the contrary, they have a plan to try all kinds of methods, from simple to complex, and continuously think about it. If they still have no ideas, they choose to consult teachers or discuss with their classmates.3. About information preservation:The gifted students in mathematics of junior high school can easily remember the complex information in the math problems, or they take some tools such as forms to assist them to memorize it. For mathematical formulas, they generally don't forget them, and they always consolidate them by solving problems. It is not a problem for them to memorize by rote. For previously done topics, they don't deliberately memorize them. It will leave some impression in their mind once the topics are done again.4. About observation imagination:The gifted students in mathematics of junior high school can quickly observe the rule or key characteristics in topics and graphics. They can imagine basic three-dimensional graphs, and some of complex three-dimensional graphs by analysis.5. About mathematics belief:The gifted students in mathematics of junior high school have a correct understanding about mathematics, who think that it is more important to comprehend the thought and methods of mathematics than to memorize topics in problem-solving. They think that the law in mathematics is an interconnected system, and that mathematics formulas are meaningful, not just problem-solving tools, and that they can find some conclusions by themselves. They prefer to solve difficult mathematics problems, and they will concentrate on the math problem when they solve it. When encountering difficulties in the process of learning mathematics, they tend to rely on their own efforts to solve them, rather than looking for help or giving up immediately. They think that learning mathematics needs natural talent and hard works. They like mathematics, and their self-confidence is improved through mathematics learning. They think that mathematics is very important in daily life, and it can train one's clear thinking in daily life. Being good at mathematics can bring them great achievements.