【作者】 刘颖；

【导师】 黄兴丰；

【作者基本信息】 上海师范大学 ， 小学教育（专业学位）， 2021， 硕士

【摘要】 在数学理解这个领域,概念性理解与程序性理解一直是国内外热议话题。我国课标指出:“不仅要使学生掌握技能操作的程序和步骤,还要使学生理解程序和步骤的道理。”(1)竖式除法不仅涉及程序性知识,也涉及乘法、减法以及除法的相关概念性知识,是学生学习的重难点,也是进一步学习与理解运算的关键。学生在学习该算法时存在诸多困难,教师一般会通过大量的练习帮助学生记住运算的程序,并不强调算理,导致学生只会计算,却不知为何可以这样计算,也不知道计算过程的实际意义,归根到底源于对算理的不理解。从竖式除法内容来看,整数竖式除法是学生学习小数除法的基础。整数竖式除法的内容包括“被一位数除”和“被两位数除”。学习“被一位数除”是学习“被两位数除”以及“小数除法”的基础。而“三位数被一位数除”相较“两位数被一位数除”而言,计算的程序不变,难度有所增加,因此本文以“三位数被一位数除”为例考察学生对竖式除法的理解现状,发现学生理解存在的困难之处以及背后的原因,从而提出一些建议与措施。本文采用测验法、访谈法、内容分析法进行研究。首先广泛阅读相关文献资料,了解数学理解以及竖式除法的研究现状,将数学理解分为“程序性理解”与“概念性理解”两个维度,然后分别构建研究框架,考察学生在这两个维度上对于竖式除法的理解情况。“程序性理解”维度框架的构建借鉴沃斯的分类方法,主要研究三个问题:学生程序性理解现状如何;存在哪些计算错误;计算错误背后的原因。“概念性理解”维度框架的构建借鉴斯莱斯尼克的问卷,调查学生对5部分内容的理解现状:位值概念;商、余数、部分积的实际意义;横式与竖式的转化;竖式与实际情景的转化以及为何从高位算起。本文参考特蕾莎的理解水平构建“三位数被一位数除”的评价框架,用以评价学生的理解水平。最后选取上海市某所小学四年级全体学生作为研究对象,发放测验卷进行测验并对学生、任课教师进行访谈。主要的研究结论如下:总体来说,学生对于竖式除法的程序性理解情况较好,概念性理解情况较差,即计算正确率较高,但对算理理解情况较差。在程序性理解层次上,学生在计算时主要出现了6种错误:减法错误;乘法错误;余数错误;书写格式错误;抄写错误;关于“0”的错误。其中最后一种错误率较高。究其原因,学生犯程序性错误除了粗心之外,多是概念性理解不佳造成的。在概念性理解层次上,学生对于竖式除法中“位值”概念以及“竖式与实际情景的转化”理解情况较好,对“商、部分积、余数实际意义”以及“横式与竖式的转化”理解情况较差。并且学生对“为何从高位算起”缺乏思考。在理解水平上,绝大部分学生都处于2水平,即能够较好的理解竖式除法中的“位值”概念,但不能理解数的拆分与组合,对于除法模型理解不深刻,无法体会竖式除法的便捷性。影响学生理解困难的原因主要有4点:学生原有知识经验不足;算法规则本身难度大;教师教学不足;教科书呈现方式不足。综上所述,笔者提出5点帮助小学生更好的理解竖式除法的建议:让学生在动手操作中体会算理;在教学中渗透算法多样化的思想;在教学中使用多种表征方式;教科书编排高质量的现实情景例题;教科书中增加多样表征帮助学生理解算理。更多还原

【Abstract】 In the field of mathematical understanding,conceptual understanding and procedural understanding have been hot topics at home and abroad.The curriculum standard in China points out that students should not only master the procedures and steps of skill operation,but also make them understand the principles of procedures and steps.Vertical division not only involves procedural knowledge,but also involves the related conceptual knowledge of multiplication,subtraction and division.It is a key point for students to learn and understand the operation.Students have many difficulties in learning this algorithm.Teachers usually help students remember the program of calculation through a lot of practice,and do not emphasize the calculation,which leads to students only can calculate,but they don’t know why they can calculate this way,and do not know the practical significance of the calculation process.In the end,it comes from the incomprehension of the calculation.From the vertical division content,integer vertical division is the basis for students to learn decimal division.The contents of integer vertical division include "divided by one digit" and "divided by two digits".Learning "divide by one digit" is the basis of learning "divided by two digits" and "decimal division".Compared with "the double digit is divided by one digit",the calculation procedure is unchanged and the difficulty is increased.Therefore,this paper takes "three digits are divided by one digit" as an example to investigate the understanding status of students for vertical division,and finds out the difficulties and reasons behind the understanding of students,and puts forward some suggestions and measures.This paper uses the test method and the interview method to carry on the research.First,read the relevant literature,understand the current situation of mathematical understanding and vertical division,divide mathematical understanding into two dimensions: procedural understanding and conceptual understanding,then construct the research framework respectively,and investigate the students’ understanding of vertical division in these two dimensions.The construction of the framework of "procedural understanding" is based on the vertical division classification method of worth,which mainly studies three problems: how to understand the program of students;what are the calculation errors;the reasons behind the calculation errors.The construction of the "conceptual understanding" dimension framework is based on the questionnaire of slasnick to investigate the students’ understanding of the five parts: the concept of position value;the practical significance of quotient,remainder and partial product;the transformation of horizontal and vertical;the transformation of vertical and actual situation;and why from the high level.This paper constructs an evaluation framework of "three digits are divided by one digit" by referring to Theresa’s understanding level,so as to evaluate students’ conceptual understanding level.Finally,the fourth grade students of a primary school in Shanghai were selected as the research object,and the test papers were distributed for testing and interviews were conducted.The main conclusions are as follows:Generally speaking,students have better understanding of vertical division program,and poor conceptual understanding,that is,the accuracy of calculation is higher,but the understanding of calculation is poor.At the level of procedural understanding,students have six errors in calculation: subtraction error;multiplication error;remainder error;writing format error;copying error;error about "0".The last one is high error rate.The reason is that the students make procedural mistakes,besides carelessness,are caused by poor conceptual understanding.At the conceptual level,students have better understanding of the concept of "position value" and "transformation of vertical and actual situations",and have a poor understanding of "quotient,partial product,and remainder practical significance" and "transformation of horizontal and vertical".And students lack of thinking about "why from high level".In terms of understanding level,most students are at 2 levels,that is,they can better understand the "bit value" in vertical division,but they can not understand the split and combination of numbers.They have no deep understanding of division model and cannot understand the convenience of vertical division.The main reasons that affect students’ understanding are: the lack of original knowledge and experience of students;the difficulty of algorithm rules themselves;the lack of teaching of teachers;the lack of textbook arrangement system and presentation.In conclusion,the author puts forward five suggestions to help pupils better understand vertical division: let students understand the calculation in the hands-on operation;infiltrate the idea of diversity of algorithms in teaching;use multiple representation methods in teaching;arrange high-quality practical situation examples in teaching materials;add multiple representations in teaching materials to help students understand calculation.更多还原