Case Study About Grade One Senior High School Mathematically Excellent Students’ Problems-solving Process

With the rapid development of science technology and socio-economic, there is agrowing demand for talents in mathematics. Excellent students in mathematics arevaluable resources for national development. Therefore, schools must focus on theresearch and training of excellent students in mathematics. This paper, through casestudies for the senior1students in a certain middle school in Qidong city, investigatedthe features which the old and new students have during the Polya problem-solvingsteps and summarized their similarities and differences.Using the thinking aloud method, this research case studied four excellent studentsin mathematics in middle school. Two of them were new students and the other twowere old students. They were told to solve the problem using thinking aloud method.And by analyzing the record of their thinking, this study investigated the problemsolving features of the students and summarizes the similarities and differences betweenold and new students. The features during their subject understanding, programdevelopment, program implementation and review are summarized as follows:1. Subject understanding: both the old and new students had strong perceptionability towards the mathematical materials. They were able to understand the structureof the subject once they got a mathematic problem. The information they need wasquickly summarized from the problem. And they were able to enter the problem-solvingstate very quickly. Moreover, it was easy for them to extract the information needed tosolve the problem even for complex problems. They would ignore the irrelevantinformation and focus their attention on useful information.2. Program development: the students were able to use a variety of commonproblem-solving strategy, such as classified discussion, substitution, the combination ofnumber and shape, etc. They can easily determine the type of the problem even when it comes to unfamiliar problems. Furthermore, they were able to tell the similarities duringthe preliminary analysis stage, grasp the general characteristics of the problem structurebefore answering the questions, and summarize the general problem on this basis.Therefore, the problem-solving methods can be easily applied to other issues, and thegeneral type of questions can be determined.3. Program implementation: the students were able to solve the problems correctlyand apply the mathematic formulas skillfully. The old students tended to shorten thereasoning and mathematic operation quickly and thoroughly. They would try to writethe process clearly, and simplify the process of writing and calculation. On the otherhand, the new students tended to follow the step-by-step problem-solving process, andwrite clearly and specifically. For problems that are not difficult or moderately difficult,the students tended to develop the problem-solving program while implementing it, andthey were able to solve the problem quickly with high accuracy. For problems that aremore difficult, they would try to develop the problem-solving program after a certainamount of analysis. While trying, they would often realize the reason why the methodwas adopted and what results they would like to see. At the mean time, combined withthe use of a variety of problem-solving strategies, the problem was transformed into afamiliar type, and the problem was solved in the end.4. Review: the students would not solve the problem again by a new way, they justreview the problem-solving process once again. The new students did better when itcomes to the details of the subject.