【作者】 张玲；

【导师】 刘静；

【作者基本信息】 西南大学 ， 数学教育， 2016， 硕士

【摘要】 问题解决是国际数学教育的一个重要研究方向,而解应用题是小学生问题解决中重点且难点内容。作为新一代的测量理论——认知诊断理论,突破了传统测量仅对宏观水平的评估方式,为此本研究将其作为诊断小学生解应用题的内部心理加工过程的理论工具,研究六年级学生解应用题的知识结构的掌握水平及缺失,为教师补救性教学提供依据。本研究首先运用Mayer命题分析法,结合Shalin&Bee的符号网络分析法对小学应用题(代数应用题)在情境结构和数量结构上进行划分,建立小学应用题的题库资源;运用认知分析法,析出解应用题的2大成分所对应的7个认知属性,即操作成分中:A1(基本算术运算)、A2(多步运算)、A5(识别隐含条件)、A6(算式表征)、A7(正规代数策略);图式成分中:A3(基本数量关系)、A4(复杂知识图式);并建构7个属性间的层级关系(认知模型);基于认知模型和AHM思想编制测验卷,对5所城市11个班级的531名小六学生实施诊断;最后运用multilog7.0软件和属性不等权重多级评分AHM的matlab程序实现对被试的能力以及属性掌握模式的归类的诊断评估。根据诊断结果得出如下结论:(1)总体属性掌握模式归类上:531名被试共21种属性掌握模式;分布最多为全1模式和部分全1模式,较少的为仅掌握A1或全0模式;属性平均掌握水平上,A1(基本运算)和A3(基本数量关系)属性掌握得较为理想,而A4(复杂知识图式)和A7(正规代数策略)为整体学生的薄弱属性;(2)知识状态掌握上的城乡性别差异:城乡间原始总分差异较为明显,城市高分段的人数相对城市较为集中,超过一半的人集中于64-71分之间;在属性掌握模式占比上,大多数模式的男生占比高于女生;城乡差异在A1-A7属性上均较为显著,而男女性别差异仅在A4属性上显著;(3)知识状态掌握上的优差生差异:高分段被试对于A1-A7个属性的掌握概率较为稳定,均在0.962-0.994之间,低分段被试中,A2、A4、A7属性的平均掌握概率小于0.4,其中A4(复杂知识图式)和A7(正规代数策略)均低于0.3,各属性平均掌握概率上优生明显优于差生;鉴于此,提出补救性教学建议:(1)加强A4和A7属性的补救性教学;(2)加强基础性属性的突破教学演练以缩小城乡差异;(3)分析后进生知识状态,及时调整教学设计;更多还原

【Abstract】 Problem solving is an important research direction of international mathematical education,and it is the important and difficult contents for the primary school students.As a new generation of measurement theory——the cognitive diagnosis theory,has broken through the traditional measurement only on the macro level,therefore this study just recardsthe cognitive diagnosis theory as the theoritical instrument in the research of diagnosing the internal psychological process for the primary students’ problem solving,and mainly concentrates on the sixth grade students’ level and lack of knowledge structure on the problem solving and provide the basis for the remedial teaching.First, this study applied Mayer proposition analysis, combining Shalin&Bee symbol network analysis to classify the elementary algebra word problems in structure and quantity structure situation,and establishes an elementary algebra word problems bank resources.Then separated out two components corresponding to seven cognitive properties on solving the algebra word problems,The operating components: A1(basic arithmetic operations), A2(multi-step operation), A5(identify the implied conditions), A6(Mathematical representation), A7(the formalalgebraic strategy);and the schema components:A3(basic quantitative relationship),A4(complex knowledge schemata).Establish the hierachical relationship between the construction of seven attributes(cognitive model);Based on cognitive model and AHM prepare test paper and implement diagnosis for 531 students of grade six in about 11 classes.Finally using the software of multilog 7.0 and the matlab programming about the range attribute weight multistage score of AHM to realize the diagnostic evaluation of the ability of subjects and attributes master pattern classification. Draw the following conclusions according to the results of the diagnosis:(1)On the overall properties master pattern classification:the 531 subjects were divided into 21 kinds of attributes modes,just as only mastering the A1 attribute or failed all.On the average master level,A1(basic operation)and A3(basic quantitative relation)are belong to the ideal properties which had been acquired,and A4(complex knowledge schemata)and A7(formal algebra strategies)are of the weak properities for the whole students.(2)On the differences between genders and the urban and rural areas on the knowledge state:original scores differences between urban and rural is much obvious,the students number of rural areas on the high scores is relatively more concentrated,and more than half of the people focused on between 64-71 points.On the attribute mastering models, the number of boys is much higher than girls accounting for most of the models;Differences between urban and rural areas were more significant in the A1-A7 attributes,and the gender differences is only significant on A4 attribute.(3)On the differences between the outstanding students and the underachievers of the knowledge state: the A1-A7 properties are controled relatively stable by the high ability subjects for the probability between 0.962 to 0.994,for the lower scores subjects the average attributes probability of A2、A4 and A7 are less than 0.4,particularly the A4(complex knowledge schema)and A7(formal algebraic strategy)are less than 0.3.On the average probability of mastering attributes the outstanding students are obviously superior than the underachievers.In view of the conclusion above,come up with some remedial teaching suggestions:(1) strengthening the A4 and A7 properties of remedial teaching;(2) strengthening the teaching of basic properties to breakthrough the difference between urban and rural areas;(3)analysing the underachievers’ knowledge state,timely adjust to the teaching design;更多还原