Case Studies Of High School Mathematical Gifted Students On Their Mathematical Conjecture Ability

This article makes case studies of five high school mathematical gifted students focusing on their mathematical conjecture abilities, mainly pays attention to the following questions:1. For high school mathematical gifted students, what attitude do they hold to mathematical conjecture ability, how about their current mathematical conjecture abilities and where the knowledge comes from.2.What’s their understanding of the classical mathematical conjecture? Where does the knowledge comes from and what inspiration and insights can they get from the classical mathematical conjecture? What suggestions can be provided to high school mathematical teachers in order to cultivate the ability of mathematical conjecture better for high school students?In these case studies, the author combines a variety of research methods: literature review, questionnaire and interview, through these methods the author can make full understanding of the five mathematical gifted students’ mathematical conjecture abilities. In the questionnaire, there are five questions. Three of them are heuristic questions and another two are subjective questions. During the questionnaire, through "think aloud" method, the author lets the five students show their thinking process of mathematical conjecture from all aspects. During the analysis of the questionnaire, the ability of mathematical conjecture is divided into five levels according to the SOLO theory. After the survey, the author also interviewed the five mathematical gifted students and their teachers. Through the above research, the author mainly gets the following results:1. The attitudes of high school mathematical gifted students toward mathematical conjecture ability:All the five students agree that the mathematical conjecture ability is very important, in their opinions, with the help of the mathematical conjecture ability, they can be provided the research direction in the process of problem solving. In the daily process of problem solving, they can make mathematical conjecture independently and verify the conjecture consciously.2. The mathematical conjecture ability of high school gifted students at present:all the five mathematical gifted students have a good number sense and they have good abilities of space imagination and image observation. They can find the rules quickly and set up the connection between the unknown questions and known experience. They can make use of the common methods of mathematical conjecture skillfully. Most of the five students can make mathematical conjecture correctly and prove the conjecture strictly. But for some of them, the consciousness of verifying the conjecture remains to be strengthened.3. The cognitional situation of high school mathematical gifted students for famous mathematical conjectures:They can roughly talk about the relative questions about "Goldbach’s Conjecture". But they can’t accurately write down the complete content. They also know some classic mathematical conjectures, for example, "Four-color Conjecture", "Mason Prime Number Conjecture", "Fermat Conjecture" and "Twin Prime Number Conjecture". The main learning approaches are taking part in extracurricular interest-oriented classes, reading extracurricular books and searching relative websites. These contents are seldom involved in daily teaching.4. The sources of mathematical conjecture ability for high school mathematical gifted students:The mathematical conjecture ability relies on the complete mathematical knowledge system and abundant experience in problem solving. During the process of mathematical conjecture, the most important step is building up proper relation between unknown questions and known experience, at the same time, building bridges through all kind of methods and auxiliary means while conjecturing. All the knowledge accumulation, experience in problem solving and the structuring of connection compose the affluent origins of mathematical conjecture ability.5. Teaching suggestions:1). Heuristic education can be adopted through designing teaching situation.2). Teachers should encourage students to ask questions and make discussions.3). Make some introduction about some classic mathematical conjecture knowledge at suitable opportunities in classes.4). Give students enough time to think independently, meanwhile, provide guidance and reminding at right moments.