Research On Learning The Concept Of Monotonicity Among The Ninth Grade Students

Monotonicity is of great value, for example, in theory of advanced mathematics, and it is also one of the fundamental concepts in school mathematics too. For instance, knowing that it is a prerequisite for a deeper understanding about the concept of function, the ultimate idea of monotone function is that order is always either preserved or reversed in monotone mappings. In addition, Monotonicity contains the ideas of symbolic-graphic combination, classified discussion.Our research has selected 262 Grade 9 students in six high schools from Shanghai and Shenzhen as participants. We focused on their representatives of monotonicity, performance of monotonicity proof and performance of monotonicity application. Based on pencil-paper test, questionnaires and interviews, we investigated their motivation and belief on studying monotonicity. After the discussion of the most typical misconceptions and deficiencies when the participants learned monotonicity, we analyzed the possible reasons for their difficulties.The research results are as below:First, symbolic aspect turned out to be dominant in most students’concept representatives of monotonicity. In terms of individual, most students have a better grasp of the literal representative than the symbolic representative or graphic representative.Second, we divided the performance of proof into 3 levels:proof the monotonicity of the specific functions (level 1), proof the monotonicity of the abstract functions (level 2), proof the monotonicity of the abstract functions which is in the complex form (level 3). The research found that students from the key school at city level, the key school at district level, and school at average level could respectively reach level 3, level 2 and level 1. Problems in the proof of monotonicity are:(1) Some students proved in a wrong way; (2) Some students proved loosely.Third, according to the APOS theory, we divided the performance of application into 4 levels:Direct Application (level 1), Inverse Application (level 2), Integrated Application (level 3), Problem Solving (level 4). The research found that students from the key school at city level, the key school at district level, and school at average level could respectively reach level 4, level 2 and level 1. Problems in monotonicity application are:(1) The weak awareness of monotonicity; (2) The improper use of monotonicity.Fourth, we analyzed the reasons of students’difficulties in learning monotonicity. The reasons for the difficulties in concept image are:(1) Confusion mathematics concepts with concepts in daily life; (2) The negative impact of prototype. The reasons of the difficulties in conceptual links are:(1) Concepts are isolated; (2) Concept network is inflexible. Difficulties of monotonicity application come from the inappropriate generation.Fifth, from the investigation of the motivation and the belief, we found that (1) Students have a strong motivation to learn. However, motivation results from the exams rather than the concept itself; (2) Students’ learning interest is not strong; (3) Students’ self-efficiency of learning monotonicity is relatively high, while the self-efficiency of monotonicity proof and monotonicity application is low.