A Multi-case Study Of High School Mathematically Gifted Students On Solving Mathematical Problems With Constructive Method

Constructive method is a method for defining conceptions or verifying propositions,which can be realized in a confirmed way through a finite number of steps.In high school mathematical problem solving,constructive method is common although not universally used.It is applicable for the construction of examples or counterexamples when necessary,or for the construction of auxiliary objects in order to transform the original problem.It is generally accepted that the constructive method in mathematical problem solving is characterized by non-routine and creativity.And as recognized based on experiences,mathematically gifted students in high schools turn out to have great creativity when solving such problems.It is noticed that domestic studies on constructive method in problem solving are mainly out of the interest in the method itself yet lacking effective empirical researches,while foreign studies on mathematical problem solving are not focused on the subject of constructive method.In view of the facts above,this study is focused on the process of solving difficult mathematical problems with constructive method by high school mathematically gifted students,aiming to shed light on the nature and the characteristics of their thinking process,thus expand the understanding of mathematically gifted students as well as their higher-order mathematical thinking process,so as to provide a basis for the future practice in mathematics gifted education as well.Four mathematically gifted students of grade 12,distinguishing in their growing-up experiences and individual characters,were selected as subjects.This study was mainly concerned with the nature and characteristics of their problem solving processes.Particular attention was paid to the following two aspects:(1)How were the heuristic strategies applied?(2)How did the metacognitive monitoring perform?With multiple procedures from several perspectives,the author has established,in the context of problem solving with constructive method,a list of heuristic strategies,which include special cases,association and connection,propositional transformation,indirect construction,with a few of relevant sub-strategies,and meanwhile selected a set of materials to test the use of these strategies.Developing the list of heuristic strategies is an attempt to introduce a framework for constructive method in mathematical problem solving.On the other hand,metacognitive monitoring was mainly investigated from the perspectives of orientation,path control,process monitoring,and verification in this paper.Think-aloud method was applied in the tests of each subject,supplemented with observation and interviews.The whole process was audio-taped.All the pieces of verbal data in problem solving were completely transcribed and further analyzed with regard to problem solving episodes,heuristic strategies,and metacognitive monitoring,followed by a qualitative analysis of the problem solving processes.The characteristics of the problem solving processes by each subject were then summarized.In general,all the four students were able to use various heuristic strategies creatively in a variety of ways in problem solving,although one of them did not manage to utilize two of the sub-strategies.Whether they were able to utilize a certain strategy in a flexible way,not only depended on their previous experiences of the strategy,but was also affected by the knowledge basis and the breadth of thinking.Rich metacognitive monitoring behaviors were observed,which mainly include alternating thinking from different perspectives,self-reflection on paths,and verification on results,and distinct characters were reflected.The positive effects and negative influences of metacognitive monitoring behaviors turned out to be complicated and varied with each individual.In addition,their knowledge basis,affects and beliefs were also reflected in the process of problem solving.The concluding of analysis of each subject was specifically carried out in the text.The limitations of this study were fully discussed.Finally,the author presented several proposals for future studies,and elaborated the potential value and possibility of utilizing constructive method for problem solving in mathematics gifted education.This study can be taken as a reference for the future practices of identification,evaluation,and instructional intervention of mathematically gifted students.