摘要
解决数学问题可以分为四个过程：理解问题、选择算子、应用算子、结果评价。与此对应,其认知过程分别为：问题表征、模式识别、解题迁移、解题监控。这里的“模式”是指数学模式,即“形式化的采用数学语言,概括的或近似的表述某种事物系统的特征或数量关系的一种数学结构”。各种基本概念、理论体系、定理、法则、公式、算法、命题、方法都是数学模式；在问题解决中,具有共同结构或相同解法的一类问题也称为一种模式。所谓模式识别,指当主体接触到数学问题之后,能将该问题归类,使得与自身认知结构中的某种数学模式相匹配的过程。在此系统中,模式识别作为问题解决过程中的第二环,以问题表征为基础,又是实现解题迁移的前提条件,可见模式识别在问题解决过程中的地位。在现实的数学问题解决中,学生对已经习得的模式是怎样的一种识别过程、其影响因素有哪些,以此进行模式识别的教学的探索。在这样的意义上,问题解决中模式识别的研究成为现实的需要。
本研究采用文献分析法、量化研究方法、质性研究方法、结构方程模型方法等多种方法相结合,着眼于研究模式识别的影响因素,来试图研究以上问题。
首先,在综述了国内外关于模式识别在知觉领域和数学问题解决领域的相关研究的基础上,提出“数学问题解决中的模式识别”概念的界定,即,当主体接触到数学问题后,与自身认知结构中的某问题图式最佳匹配的思维与认知过程。
其次,第3、4、6章分别通过质性方法(问卷调查、访谈)、结构方程模型、实验研究三种方法探寻了数学问题解决中模式识别的影响因素,为有较好的理论说服力,论文基于不同的角度,采用不同的方法,以达到较为稳固的三角互证。第3章研究一,通过问卷调查的研究方法,分析解题者模式识别的策略,质性寻求模式.识别的某些影响因素。研究二、三、四,通过三则访谈,分别考察个体在数学问题解决中模式识别的具体认知过程,可以分别用不同的模型来解释。第4章研究五,通过结构方程模型探讨个体模式识别能力、自我监控能力、思维品质、问题解决成绩之间的关系。在相关研究的基础上,建立假设模型,通过对被测者各变量的测查,验证模型。第6章研究六、七、八,通过实验的方法,探索了模式识别的影响因素。研究六研究模式习得方式(结构学习方式、一般学习方式)对个体不同类型问题(同型问题、变式问题、叉联问题)解决中模式识别的影响。研究七研究不同自我解释水平(自发自我解释、诱发回忆自我解释、诱发概念映射自我解释、诱发数字映射自我解释)对不同类型问题(同型问题、变式问题、叉联问题)解决中模式识别的影响。研究八在研究七的基础上,研究对不同特征间叉联性的意识及加工水平(高、中、低)对叉联问题模式识别的影响。
再次,第7章通过考察与分析一节优秀数学课堂实例,提出反思性实践是数学问题解决中的模式识别的教学实践路径。
研究的主要结论：(1)模式质量是模式识别的基础与先决因素,自我监控能力和数学思维品质是模式识别的条件因素。(2)不同问题类型的模式识别的具体认知过程,可以分别用不同的模型来解释；数学问题解决中模式识别过程具有自下而上与自上而下的双向加工特点。(3)数学思维品质、自我监控对模式识别产生直接影响；自我监控能力对个体数学问题解决成绩的影响部分是直接效应,部分通过模式识别间接影响；数学思维品质对个体数学问题解决成绩的影响部分是直接效应,部分通过模式识别间接影响。(4)习得方式显著影响模式质量,结构学习条件下模式质量显著高于一般学习条件下模板质量。习得方式显著影响学生问题解决中模式识别,结构学习条件下模式识别显著优于一般学习条件下模式识别。问题类型显著影响学生问题解决中模式识别。习得方式与问题类型的交互作用对模式识别影响显著。模式质量对模式识别影响显著。(5)自我解释水平显著影响学生问题解决中模式识别,诱发概念映射自我解释和诱发概念数字映射自我解释条件下模式识别平均成绩明显高于自发自我解释与诱发回忆自我解释两种条件下的成绩。自我解释水平与问题类型的交互作用对模式识别影响显著。对于同型问题和变式问题,模式识别的成绩依自发自我解释、诱发回忆自我解释、诱发概念映射自我解释、诱发数字映射自我解释的顺序而提高；而对于叉联问题而言,没有这种趋势。(6)对于叉联问题的模式识别,高叉联性意识及加工水平组与中叉联性意识及加工水平组之间不存在显著差异,高叉联性意识及加工水平组与低叉联性意识及加工水平组之间存在显著差异,中叉联性意识及加工水平组与低叉联性意识及加工水平组之间存在显著差异。
Cognitive process of solving mathematical problem can be divided into four stages: problem presentation, pattern recognition, transfer, and self-control. Here, the "pattern" refers to mathematical patterns, such as kinds of concepts, systems, theory, rules, and methods. In mathematical problem solving, a kind of problems which have the same structure or solving method can be called a pattern. Pattern recognition refers that when confronting a mathematical problem, indivduals can sort the problem so that it matches a kind of mathematical pattern in his/her cognitive struction. In this systerm, pattern recognition, as the second stage of mathematical problem solving process, is based on problem presentation, and is the precondition of transfer, through which we can see the important role of pattern recognition in mathematical problem solving. In mathematical problem solving, how individuals recognize the patterns learned, and what factors influnence the pattern recognition, and then instruction strategy of pattern recognition is explored. In these sense, the study on pattern recognition in mathematical problem solving has realistic need.
This study employs documentary analysis, quantitative analysis, qualitative analysis, and structural equation model to study the factors influencing pattern recognition.
First, on the basis of summarizing researches on pattern recognition in the field of sensory perception and mathematical problem solving at home and abroad, the concept of "pattern recognition in mathematical problem solving" is proposed, that is, the thinking and cognitive process of matching the problem and a kind of mathematical pattern in his/her cognitive struction when individuals confronting a mathematical problem.
Second, chaper3,4,6respectively explored the factors influencing pattern recognition in mathematical problem solving by qualitative analysis, structural equation model and quantitative analysis, which achieve the stable methodological triangulation. Study1in chaper3, analysizes problem solvers'strategy of pattern recognition by questionnaire survey, and study2,3,4interview problem solvers'cognitive process of pattern recognition. Study5in chaper4explored the relationship between individuals' pattern recognition, self-control, thinking quality and mathematical problem solving achivevment using structural equation model. Study6,7,8in chaper6explored the factors influencing pattern recognition by experimentation.
Thirdly, chaper7examines an outstanding classroom and propose that reflective practice is the practice pathway of pattern recognition in mathematical problem solving.
Results:(1) pattern quality is the basis and predetermination element of pattern recognition, and self-control and thinking quality is the conditional factor.(2) the cognitive process of pattern recognition of different kinds of problems can be explained with different models; pattern recognition in mathematical problem solving has the characteristic of bottom-top and top-bottom.(3) self-control and thinking quality have direct influence on pattern recognition. Their influence on mathematical problem solving achievement is partly direct and partly indirect through pattern recognition.(4) acquisition way and problem kind significantly influence pattern quality, and their interaction influence pattern quality significantly.(5) self-explanation level significantly influence pattern quality in mathematical problem solving, and interaction of self-explanation level and problem kind influence pattern quality significantly.(6)for cross-over problems, there is no significant difference in pattern recognition between high-sense group and mid-sense group, while significant difference exist between high-sense group and low-sense group, and between mid-sense group and low-sense group.
引文
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