A Study On Cognitive Mechanism And Teaching Strategies Of Mathematical Modeling

The study on cognitive mechanism and teaching strategies of mathematical modeling to students major in mathematics and applied mathematics (normal) is of great theoretical, methodological and practical significance.With 518 students in mathematics and applied mathematics (normal) major from five universities/colleges as the research subjects, adopting the research methods of verbal protocol analysis, the depth interview, questionnaire test and theoretical analysis etc., this study investigates the general cognitive process in mathematical modeling of subjects, reveals the differences on the cognitive process of mathematical modeling between subjects with different levels, explores the influential factors and its path and degree on mathematical modeling achievement of the subjects, and puts forward the teaching strategies of mathematical modeling in mathematics and the applied mathematics (normal) majors.This study achieved basic conclusions as follows:(1) This study made an attempt to set up the pattern of general cognitive process in mathematical modeling of subjects. The pattern reflects each concrete stage and its dynamic relations of the process of subject's mathematical modeling, and expounds cognitive operation and its way of each concrete stage of the process of subjects' mathematical modeling.(2) There are significant differences between expert subjects and novice subjects in problem representation, strategy applying, train of thought, result and efficiency in mathematical modeling; there are no significant differences between expert subjects and novice subjects in symbol representation and method representation, but there are significant differences in mechanism representation; expert subjects tend to make use of multiple representation, novice subjects tend to make use of simplicity representation; expert subjects tend to make use of circular representation strategy, novice subjects tend to make use of one-way representation strategy; expert subjects tend to make use of equilibrium assumption strategy, novice subjects tend to make use of accurate assumption strategy; expert subjects tend to make use of example analogy strategy, novice subjects tend to make use of immediate generation strategy; expert subjects tend to make use of immediate monitoring strategy, novice subjects tend to make use of looking-back monitoring strategy; expert subjects tend toward to make use of adjustment strategy in assumption and modeling method, novice subjects tend toward to make use of adjustment strategy in solving; compared expert subjects with novice subjects, expert subjects have more conversion in thinking and get more right (reasonable) answers to the questions related to mathematical modeling; the student subjects have more set thinking and make more mistakes (unreasonable answers) significantly; the verbal protocol of experts subjects are brief with logical expressions, their analysis to mathematical modeling problems is deep and thorough, the thinking of mathematical modeling is faster and flexible, the usage of mathematical modeling methods are revealed in heuristic searching. The verbal protocol of novice subjects are cumbersome with illogical expressions, the analysis to mathematical modeling problems is shallow and general, the way of thinking of mathematical modeling is one-way and stereotyped, the usage of mathematical modeling methods are revealed in blind searching; expert subjects are faster than novice subject in mathematical modeling, spending much less time on each question.(3) Achievement motive, creative inclination, cognitive type, the mathematics cognitive structure and self-monitor ability to mathematical modeling have significant positive correlation with mathematical modeling achievement of subjects; the mathematics cognitive structure, self-monitor ability to mathematical modeling present significant regression effect on mathematical modeling achievement of subjects, directly influencing mathematical modeling achievement of subjects, and two factors can explain 55. 8% variations of mathematical modeling achievement of subjects; achievement motive, creative inclination, cognitive style, the mathematics cognitive structure have significant regression effect on self-monitor ability to mathematical modeling, directly influencing self-monitor ability to mathematical modeling, and four factors can explain 70. 1% variations of self-monitor ability to mathematical modeling; achievement motive, creative inclination, cognitive type have significant regression effect on mathematical cognitive structure, directly influencing mathematical cognitive structure, and three factors can explain 40. 9% variations of mathematical cognitive structure, which influence the achievement in mathematical modeling through influencing mathematics cognitive structure and elf-monitor ability to mathematical modeling.(4) The teaching strategies of mathematical modeling for mathematics and applied mathematics (normal) majors include: the integration of example teaching, variation practicing, open training; the integration of general thinking strategy, mathematical modeling strategy and mathematical modeling method; the integration of independent exploration, mutual exchange and guided introspection; the integration of multidimensional objective, varied methods, and diversified subjects of evaluation.The innovation of this research lies in the following aspects: trying to set up the general pattern of cognitive process of mathematical modeling; revealing the differences in mathematical modeling cognitive process between subjects with deferent levels; building up the path analysis model of influential factors on mathematical modeling; and putting forward the teaching strategies of mathematical modeling for mathematics and applied mathematics (normal) major.