【作者】 李金富；

【导师】 宋乃庆； 常永才；

【作者基本信息】 西南大学 ， 课程与教学论， 2012， 博士

【摘要】 计算是小学生数学学习的重要内容,小学生数学学习成效主要决定于进行正确运算的效率。彝族农村小学生数学计算上的困难相比城市小学生较大。目前,对数学计算的研究大多停留在计算方法的简单呈现,计算技巧的直接展示,而利用数学表征系统深入到学习者心理层面的研究并不多见,深入到心理层面的有关彝族农村小学生数学计算思维研究更为少见。本研究结合信息加工理论、情境认知、学习理论和解题分析的信息论,并结合计算思维的表征过程观点,利用数学表征系统深入分析彝族农村小学高段学生数学计算过程中思维差异,并对差异背后的文化、观念、习惯等影响因素进行探析。具体说来,研究了下面两个问题：(1)彝族农村小学高段学生数学计算思维现状是怎样的?(2)彝族文化背景、数学问题解决观念、数学学习习惯如何影响数学计算表征?研究按照以下过程展开：(1)梳理文献,依据文献及数学计算的一般过程提出计算表征的分类、分层及编码；(2)根据研究内容设计问卷,选取测试学校和测试学生,对学生进行调查,根据调查结果,进行访谈；(3)用定量的方法和定性的方法对收集到的数据进行分析,对调查问卷进行编码,用SPSS软件进行统计分析,对访谈进行分析。本研究发现彝族农村小学高段学生的数学计算思维有以下特点：(1)彝族农村小学高段学生在计算时,学困生频繁使用实物表征、言语表征,而良好生、优秀生频繁使用命题表征,主要表现为关系表征、“口诀”表征,其中个别优秀生使用图式表征。(2)彝族农村小学高段学生计算表征一般常见：口语表征、概念表征、关系表征；同一题目(不含信息复杂性题目)近75%同学能够正确运用1种计算办法,能运用2种以上计算办法的同学不足15%,还有约10%同学不能给出一种正确计算办法,对问题的表征在读题阶段一般表现为口语表征、符号表征。具体体现在以下方面：①整数、小数、分数与”+,－,×,÷”直接相连的计算题,彝族小学生习惯使用法则表征,比如,一般先出现有关运算顺序的法则、后出现“口诀”,在“口诀”中间常有“竖式”等表征形式。在这类计算题上,学困生与良、优生在表征上的最大差异主要表现在这三种表征是否正确。②彝族农村小学高段学生用列式计算时频繁使用言语表征与关系表征之间的转译,如：“和、差、商、除、乘、积、倍、半、…分之…、比…多(少)…增加、减少、比…大(小)…、剩余、平分…份、包含”等言语表征与运算符号表征之间的转译。③彝族农村小学高段学生在解答几何计算题时频繁使用言语表征,相比较少使用图形表征。④彝族农村小学高段学生解答真实情境计算题时频繁使用言语表征和关系表征,近80%彝族学生对真实情境用了关系表征和符号表征。这种表征现状5年级比6年级表现得更好。⑤彝族农村小学高段学生用列算术计算求解信息关系复杂的应用题时频繁使用言语表征、视觉表征和关系表征,但较多表现为零乱、分散、无逻辑状态。这种表征现状在年级间无显著差异。(3)解答计算题的正确率与学生使用合理计算表征水平显著正相关。更多使用关系表征、图式表征的同学成功计算的比率大于更多使用视觉表征、语言表征计算的同学。(4)在数学学习优秀、良好、困难三类学生间和四种类型学校的学生间数学计算思维差异明显。(5)彝族农村小学高段学生计算表征受包括彝族语言、民族心理、风俗习惯、宗教信仰、家庭教育等的文化背景影响,特别是彝族语言的动宾结构明显影响学生正确表征问题。明确、合理的数学问题解决观念、数学学习习惯促进彝族小学生在解答计算题的过程中产生优势表征,进而提高计算正确率。反之,产生相反影响。综合研究结论,对提高彝族农村小学数学计算的教学提出了以下建议：(ⅰ)在数学课程标准理念指导下,开发民族文化背景的数学课程。(ⅱ)提倡母语、汉语与数学语言互译的教学。(ⅲ)将数学计算教学建立在学生数学学习心理基础上。(ⅳ)教师在“教”计算时,应在改善彝族学生数学学习的计算表征、语言环境、数学问题解决观念、数学学习习惯等方面付出积极努力。比如：①计算教学体现计算表征的过程,关注不同表征之间的相互转化；②要有一些具体办法改善汉语理解的相对困难,注意力不稳定、不持久,兴趣不持久等表现。(ⅴ)在“学”计算时,彝族学生也要为改善自己的语言表达、数学问题解决观念、数学学习习惯等方面的现状付出努力。(ⅵ)将民族文化、地区差异等相应的内容纳入数学教师的相关训练。更多还原

【Abstract】 Calculation is an important aspect of math learning for primary school students, and the accuracy of calculation even makes great differences in students’math learning. Yi rural primary school students face more difficulties than urban counterparts in calculation. Till now, most research about calculation lie in the stage of simp presentation of calculation methods or skills. However, whilst not many research go deep into students’psychological level of use of calculation representation system, there are even fewerresearch on Yi rural primary students.Combined with Information Processing Theory, Situated Cognition and Learning Theory, this study looked into the calculation thinking differences among upper grade students of Yi rural primary school in making use of calculation representation system, and analyzed relative culture, concept and habits that may cause these differences. Specifically speaking, the following two main problems were focused on:(1) What’s the calculation thinking mode of uppergrade students of Yi rural primary school?(2) How could Yi culture, mathematical problem resolve methods and learning habits affect calculation representation?This study went as following steps:(1) Sort out relative documents, and bring forward the categories, levels and codes of calculation representation.(2) Design questionnaires according to our research content, and choose suitable schools and students as subjects, then interview subjects based on the result of the survey.(3) Code the questionnaires. Analyze the collected data with quantitative methods and qualitative methods separately, processing these data with the help of SPSS. Analyze the information got in interview.As a result, this study found following features of calculation thinking mode of higher grade students of Yi rural primary school.(1) Basically, in higher grade students of Yi rural primary school, those who have humble performance tend to make use of physical and verbal calculation representations, while those who have good performance tend to make use of propositional calculation representations, such as relationship, formula, and even chart.(2)The frequently used calculation representations in higher grade students of Yi rural primary school are:verbal representation, concept representation and relationship representation. For the same question, nearly75%of students can use one calculation method, nearly15%of students can use two calculation methods, and approximately10%of students cannot find a way to work it out. They tend to make use of oral language and symbolic calculation representations only in reading mathe problem.Specifically shows in the following aspects:①For the arithmetic of integer, decimal and fraction, Yi primary students prefer to use rule. For example, there’re rules of calculation sequence, formulas, and vertical. The difference between differently performed students is whether they can understand these three calculation representations properly.②When formulating, higher grade Yi rural primary students frequently use representation translation between language and calculation relationship, for instance, they may translate such terms and relationships as plus, minus, quotient... into corresponding operators.③Higher grade Yi rural primary students usually use more language representation than graphic representation when working on geometric calculation problems.④Higher grade Yi rural primary students tend to use language and when working on real-life situation calculation problems. Nearly80%of students apply relationship and symbolic representation to real-life situation, and5-grade students do better than6-grade students in this way.⑤When formulating to resolve application problems with complex relationships, higher grade Yi rural primary students frequently use language representation, visual representation and relationship representation, but their representations appear to be messy and lack of logic. There is no big difference within different grades.(3)The accuracy of answering shows significant positive correlation with the rationalness of calculation representation. That is to say, the answer accuracy of those who frequently use relation and graphic representation is higher than that of those who tend to use visual and language representation.(4) There is remarkable difference in mathematical computation thinking among three kinds of students---mathematics learning is excellent, good or difficult, and also among those in four types school.(5)The calculation representation of higher grade Yi rural primary students is affected by the language, national psychology, customs, religion, and family education of Yi-people. An explicit, reasonable conception and habit of math problem-solving can help Yi rural primary students get better calculation representation, and then get higher accuracy. Reverse is also true.All stated above, we put forward the following suggestions to enhance the calculating education of Yi rural primary schools:(ⅰ)Develop math course with Yi culture background under the guidance of the mathematics curriculum standards.(ⅱ)Promote math teaching in both Chinese and Yi-language.(ⅲ)To carry out calculation education based on the students’ math study psychology.(ⅳ)Teachers should make efforts to improve Yi rural primary students’calculation representation, math problem-solving concept, learning habits when teaching calculation.①Show students the process of calculation representation, and concern about the mutual conversion between the different representations.②Come up with particular ways to resolve the problem of Chinese understanding and lacking of long-term attention and interesting of Yi rural primary students.(ⅴ)When learning calculation, Yi students should also try to enhance their own language ability, calculation representation, math problem-solving concept and learning habits.(ⅵ)Consider national culture, regional differences into math teacher training.更多还原