Theoretical Investigation And Empirical Research On Mathematical Understanding

Mathematical understanding has been identified as the central problem in mathematics education. From different perspectives, researchers have investigated mathematical understanding and given rise to several theories for interpreting the processes and mechanism of mathematical understanding. Previous researchs on mathematical understanding are mostly metaphysical academic study based on experience, absences from empirical research. All these research can interpret some phenomena of mathematical understanding, but the reseach can't be well connected with the problems of practical teaching, and can't solve the prevalent problems among the present practical teaching. therefore, The level of remembering, the level of interpretative understanding and the level of groping understanding, denoted the students' way and level of understanding mathematics, well connected with the problems among the present practical teaching. According to the frame of the level of the level of remembering and interpretative understanding and groping understanding, This study investigated the students' way and level of mathematical understanding. It is of great theoretic value and practical significance to study this problem.In order to invest the practical state of the students' achieving to the level of remembering, the level of interpretative understanding and the level of groping understanding, the researcher designed the testing problem and the questionnaire. By adopting the method of stochastic sample, and by the qualitative research methods, or by SPSS, the researcher invested the practical state and difference of the students' achieving to the level of remembering, the level of interpretative understanding and the level of groping understanding.The result showed that the level of remembering and interpreting has been achieved, but the level of inquiring has not been achieved. The result showed that the students didn't basically agree with sheer remembering, and the students depended on understanding by interpreting, and understanding by inquiring need to be strengthened. The students rarely understood mathematics by inquiring. Based on the findings, the researcher analyzed why understanding by inquiring was lost, and put forward the strategy of promoting students' groping understanding. that is, Pay attention to learning by one's own experience, pay attention to the process of mathematical discovery, pay attention to reflecting the process of learning mathematics. 2. Summarizing is an effective strategy of promoting the mathematical understanding. 3. Generalization is an important form of mathematical inquiring. Because promoting students' groping understanding is well connected with the teacher's instructional-design, the researcher designed the questionnaire to invest the patterns of instructional-design factors prompting students' understanding. The result showed that two instructional-design factors prompte middle school students' understanding, that is, facilitative scaffolding methods and constructivist learning environment. Based on the findings, the researcher carried out case analysis and didactical design based on exploratory understanding. Finally, the researcher illuminated that it was of vital importance for the teachers to understand mathematics deeply. that is, Only when a teacher can understand mathematics deeply, and get out the connection between the different knowledge, he can really teach mathematics for understanding.