摘要
有关问题解决的心理学研究是近年来认知心理学研究的热点。然而，有关问题解决策略的研究却一直是一个研究得相对薄弱和不充分的领域。因此，本论文将小学生数学应用题解决过程中的认知策略和元认知策略及其教学训练作为主要课题，进行系统的实验研究。实验的具体目标是：第一，探索小学生解决应用题的认知和元认知策略的一般模式以及表征阶段的策略的特点；第二，探讨几种不同方式的认知策略训练和元认知策略训练的效果；第三， 自编一套用于诊断小学生数学应用题解决的元认知水平的有效的测量工具。以小学中高年级的学生为被试，以小学数学应用题为材料，进行了六个系列的研究。本研究的结果可以概括为以下三个方面：
1、关于小学生数学应用题解决的认知与元认知策略
本研究首先揭示了与小学生数学应用题解题成绩密切相关的一系列认知与元认知因素：问题情境的理解、问题表征、问题归类、结果估计、解题计划、对列式的自我评价。通过进一步的路径分析发现：首先是问题情境理解会影响问题表征、问题归类、解题计划和对列式的自我评价，而问题表征、问题归类、解题计划和对列式的自我评价进而又会直接影响解题成绩。问题情境理解、问题表征、问题归类、解题计划和对列式的自我评价是小学生在解决数学应用题过程中的不同阶段相继采取的一系列认知与元认知策略，它们共同构成了小学生解决数学应用题的整体策略。另外还发现，我国小学生对和差应用题的表征存在两种策略，即直译策略和问题模型策略。
2、关于小学生数学应用题解决策略的训练
首先，本研究发现，三种方式的解题策略训练，即问题情境理解训练、画图表征策略训练以及整体模型策略训练均能够显著提高小学生解决数学应用题的策略水平，进而显著提高小学生数学应用题解决的成绩。其次，三种方式的解题策略训练分别与元认知训练相结合，即问题情境理解训练与元认知训练相结合、画图表征策略训练与元认
知训练相结合以及整体模型策略训练与元认知训练相结合均能够显著
提高小学生数学应用题解决的成绩，但与单纯的问题情境理解训练组、
画图表征策略训练组、整体模型策略训练组相比，与元认知训练相结
合的策略训练组在数学应用题解决的成绩、策略水平以及元认知的总
体水平的提高上没有更突出的效果。第三，策略训练的效果与学生原
有的成绩地位有一定的关系，即策略训练与学生的原有成绩地位之间
出现了若干显著的交互作用效应。其中，画图表征策略训练、整体模
型策略训练以及它们分别与元认知训练的结合对于低分组的学生比对
高分组的学生在某些方面有更好的训练效果。第四，策略训练的效果
与应用题的类型有一定的关系，即策略训练与应用题的类型之间出现
了若干显著的交互作用效应。其中，在行程应用题上，画图表征策略
训练显著地提高了学生画图表征策略的水平，而在分数和工程应用题
上的提高不显著;整体模型策略训练能够显著地提高小学生解决倍数
应用题的整体策略水平以及解决工程应用题的整体策略水平和理解策
略水平:整体模型策略训练与元认知训练的结合能够显著提高小学生
解决工程应用题的整体策略水平和理解策略水平。从中可以发现在某
些类型的应用题之间出现了一些训练的迁移效果。
3、关于小学生数学应用题解决的元认知问卷
本研究自编了小学生数学应用题解决元认知问卷。通过多种方法的
检验表明，该问卷具有较好的内部一致性信度、分半信度、稳定性信
度和区分度，具有良好的结构效度和效标关联效度。因此，小学生数
学应用题解决元认知问卷是可靠的，并且是有效的，可以用来测量小
学中高年级学生的数学应用题解决的元认知水平。
Nowadays problem-solving has been a heated topic of cognitive psychology. However, it is also a category with so many problems unsolved or unclear. Hence, the main purpose of this study was to investigate cognitive and metacognitive strategies, as well as correspondent trainings, in mathematical problem solving, confronting primary students when solving arithmetic word problems. The specific aims are, firstly, to observe cognitive and metacognitive strategies in general and representation strategies, secondly, to observe different effects resulting from applying different types of cognitive and metacognitive training, and thirdly, self-designed method was employed to evaluate metacognitive behaviors of primary students, which was hoped to provide insight into metacognitive aspects of mathematical problem solving. The subjects are middle and high-level graders in primary schools. Six experiments were conducted by taking arithmetic word problems as its material. The results of this study were as follows:
( 1 ) Cognitive and metacognitive strategies employed by students coping with mathematic problems. At the very beginning, it revealed that cognitive and metacognitive aspects were closely related to their achievements in arithmetic word problems. They were related to text comprehension, problem representation, problem categorization, result estimate, solution plan, procedure self-evaluation. If we went further, we could know that text comprehension would, in a way, influence
problem representation, problem categorization, solution plan and procedure self-evaluation. Those strategies would, in turn, have a direct effect on the achievements of students. Self-evaluation of all those strategies mentioned above, which constituted metacognitive strategies for solving word problem as a whole, were a series of ones employed by students at different stages. In addition, we found that, the choice of using direct translation strategy or problem model strategy had every reason to distinguish successful and unsuccessful students. (2) Strategic training during problem solving. First of all, three different types of strategic training were found, namely text comprehension training, representation strategy training and whole model strategy training, which had been proved to enhance students' abilities of problem solving, and as a result to help improve their achievements. Secondly, these three different types of strategic training went, respectively, with metacognitive training, that is, text comprehension training combined with metacognitive training, representation strategy training combined with metacognitive training, and finally whole model strategy problem training combined with metacognitive training. By so doing, students made great progress in their achievements concerning problem solving. However, we should draw our attention to the fact that, those trainings combined with metacognitive training did not show greater progress comparing with the above three types which were conducted independently. Thirdly, the effects of strategic trainings were consistent with students' original achievements. That is to say, strategic trainings and their original achievements were correlated by having an interaction with each other. To put it more explicit, those who had had lower scores made more progress than those with
higher scores in representation strategy training, whole model training and metacognitive training mixed with the above two. Fourthly, we also found that the effects of strategic training were in line with different types of word problem, i. e, either side had an obvious interaction with the other. Representation strategy training enhanced students' abilities of representation strategy when solving distance problem, but made no greater progress when solving fraction and project problem. On the other hand, whole model strategy training could greatly enhance students' abilities of whole model strategy when solving multiplication and project problem. At the same time, whole model strategy training, if combined with met
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