Senior Middle School Students' Unserstanding About The Mathematical Difinitions

The fostering of basic knowledge including mathematical concepts and basic skill should be emphasized in high school mathematics teaching and learning, which is ordained in curriculum standards all over our country. Understanding mathematical concepts correctly is the premis of mastering mathematics basic knowledge. A mathematical concept appears mostly in the form of definition. Mathematical definitions provide the foundation for the edifice of mathematical knowledge. Therefore, exploring students' understanding of a mathematical definition is of great significance.The understanding of students of grade 12 about a mathematical definition was investigated through questionnaires, group discussions and interviews in this study. The former two activities were carried at Guangde high school of Anhui Province and Jiading No.l high school in Shanghai as well, while only twelve teachers from Jiading No.l high school took part in the interviews. In the research activities participants were asked to consider several possible definitions of seven mathematical concepts, two of which were geometric, two of which were analytic and the others were analytic geometric. They made judgments, gave the explanations and were asked to answer six open questions.According to written response of questionnaires, tape-record of group discussion and interviews with teachers, some results are acquired:1. Students' conception of a mathematical defition depends on their own understanding at the very start, which is concerned with concept images and personal concept defitions of their own and much related to their knowledge, their ideation etc.As far as conciseness and procedural quality of mathematical definitions, students' view are as follows: a mathematical definition should be concise, but should not be extracted of conciseness; a definition can be procedural; a concept can be defined by another basic concept, which may not be the original one; a definition with the form of "genus + difference generically" is supposed to be both sufficient and necessary, while a proposition with other forms should be analysed individually; a geometric concept can be defined by its connotative properties; all propositions equivalent to a mathematical definition can be treated as definitions of the corresponding concept; definitions are helpful for students to classify concepts and understand concept deeply.2. Students mainly show three different definition conceptions about the understanding of a mathematical definition: mathematical, communicative, and figurative; two kinds of reasoning are adopted when students make judgements of whether a proposition can become a definition of certain concept: one of which is based on definitions and the other is based on examples.3. Different students have different conceptions for the same mathematical definition.4. Changes take place in students' understandings of mathematical definitions and mathematical concepts through group discussion.Finally, some teaching suggestions are proposed based on the findings of this study.