Study On Advanced Mathematical Thinking For Freshman Of Science And Engineering

During the transition from elementary mathematics to higher mathematics, students may encounter with all kinds of difficulties. The changes of thinking modes play a fundamental role in this transition. Therefore investigating the development of mathematical thinking has been an important issue in tertiary education. The teaching practice indicates that students often use imitative reasoning and few constructive reasoning when trying to solve mathematical tasks at the beginning of university. So it is necessary to study mathematical thinking created by reasoning sequences, which is helpful for developing students' advanced mathematical thinking.Firstly, the research reviews the literature abroad on advanced mathematical thinking. A new concept of advanced mathematical thinking and framework are formulated based on previous definitions and theories. It is the thinking process that needs exact definition and logical deduction proof based on it. (Tall,1992; Edwards et al,2005)According to this definition, exact definition and logic deduction proof are two important components. The article puts forward theory framework from the aspects of concept construction, the concept construction is divided into four developmental processes according to APOS theory. In the modes of concept usage, the ability to using definitions is sorted out four developmental stages. On the other hand, with the models of students'proof difficulties, the article gives four levels of difficulty in view of definitions generating proof.Secondly, research method such as questionnaire,classroom investigation and interviews are adopted for studying students'advanced mathematical thinking. The questionnaire at the beginning of semester is to find out their perception of mathematics and reasoning modes, however, the questionnaire in the process of teaching is to analyze the difficulties of advanced mathematical thinking, which is carried out from perspectives of concept definition usage. With classroom recording, we investigate the cognition action of students on the assumption of teachers' higher cognition teaching, it is hopeful for students to develop advanced mathematical thinking to some extent. Also what caused the deficiency of advanced mathematical thinking is explored thorough case study of interviews.It is showed that students'way of mathematical thinking seems to be difficulty to develop, such as:(1) Students'understanding of concepts is lacking in relational thinking, however relational structure is the key to develop the level of reflexive abstraction.(2) Students'understanding of concepts is short of technical thinking modes, which is important for constructing proof.In the end, the article tries to advancing mathematical thinking at university. Teachers should develop different means of didactical action toward the specific Calculus topics, some ideas of teaching designs are put forward. Also several cases of classroom teaching are listed.In a word, teachers must provide opportunity for students in order to foster their advanced mathematical thinking, so that they might truly understand mathematics and own higher lever skills. During the transition from elementary mathematical thinking to advanced mathematical thinking, the advanced topics in mathematics may result in the students'difficulties, which is not avoidable. However, the pedagogical conflict is caused by the teaching or teachers, so the mathematics action is crucial to develop advanced mathematical thinking.Also, some constructive suggestions are derived in this research.