Research On Learning And Teaching For Mathematical Understanding

Learning and teaching for understanding has become a more remarkable researching problem in international education-researching domain. The research on learning and teaching for mathematical understanding show the weakness in the integrity of system and depth of theory because of the difference of inquiring orientation, the lacking of deep understanding of international advanced theory on learning and teaching, the deficiency in assimilating correlative researching fruit. In this paper, the author tries to construct a more integrate frame of theoretical analysis and guidance strategy of teaching practice, based on the absorbing of international researching fruit, synthesizing of a number of viewpoints in several other subjects, and reflecting on the Chinese native circumstance of theory research and practice in mathematical education.The paper applies both qualitative research and quantitative research. Base on the expounding of key word, and analysis from several angles including knowledge, learning and teaching, the author made a systematic and comprehensive discuss on the several correlated thesis, such as, meaning of understanding, rethinking of the viewpoint on mathematical knowledge, rationale of learning for mathematical understanding, instructional design frame and strategy, and case experimental research.In general, primary productions and characteristic in the paper are as follows:(1) Inquiry of the meaning of understanding based on several angles, including: hermeneutic; cognitive and constructivist point; brain-based learning; theory on the society, context and culture.(2) Rethinking the mathematical knowledge. From the viewpoint of mathematical thought development, the author explores the significance of understanding; analyze the meaning and essence of mathematics based on the mathematical culture.(3) Construct the framework of the development and dimensionality of understanding in leaning for mathematical understanding. The development framework includes four main levels: experiential understanding; formalized understanding; structure-based understanding; culture-based understanding. The framewok of understanding dimensions include six components: explain; interpret; application; reflection; empathizing; insight.(4) Construct the grounded instructional design framework that point to learner. This framework comprises three interdependent key elements: the design for aim and knowledge; the design for problem situation and activity situation; the design for assessment and feedback.(5) As far as the case experiment concerned, the author analyzes several important parts, such as, hypothesis, basic status, researching problems, primary conclusions and limiting factors. In addition, the author made elementary theoretical analysis and expectation to the future researching direction.