【作者】 张勇；

【导师】 熊斌；

【作者基本信息】 华东师范大学 ， 课程与教学论， 2021， 博士

【摘要】 错误是一笔宝贵的学习资源.从错误中学习日益引起研究者的兴趣和关注.高中生数学学习的很大一部分时间用于解各种数学题和阅读数学解答,两者都蕴含着从错误中学习的丰富机会.学生阅读的数学解答可能隐含错误,求解的数学题可能本身是道错题.实践中不乏学生发现解答错误甚至题目有错的情形,但是鲜有实证研究对学生发现和纠正上述错误的思考过程进行探究.本研究的主要目的是探究高中生数学侦错的认知机制,揭示不同数学能力高中生解数学侦错题的认知特点和差异,了解影响被试侦错的数学观念和学习习惯以及学生对数学侦错题价值的看法和使用建议.研究使用的数学侦错题包括两种类型:解答或证明过程隐藏错误的解答侦错题;题目条件自相矛盾的条件侦错题.具体研究问题如下:问题1高中生是如何解数学侦错题的?即解侦错题的一般认知过程是什么?问题2普通生和优秀生解数学侦错题的表现如何?有何差异?具体包括下面两个子问题:问题2.1普通生和优秀生在解答侦错题上的表现如何?有何差异?问题2.2普通生和优秀生在条件侦错题上的表现如何?有何差异?问题3影响高中生侦错表现的数学观念和学习习惯是什么?高中生如何看待数学侦错题的价值?对其使用有何建议?本研究采用出声思考口语报告法和个案研究法,借鉴专家—新手比较的研究范式,招募某重点中学高二年级的8名数学资优生和8名数学普通生作为被试,每名被试以出声思考的方式完成侦错任务,并接受后续的访谈.对于研究问题1,以被试的口语报告和访谈数据为基础,提炼出5类侦错情节:阅读、分析、检验、修正、判断.被试侦错的认知过程是始于阅读题目的不同情节的组合.由于任务难度、任务特点和被试特征的不同,5类情节的具体组合方式有所不同.两组被试整体上呈现出两种不同的侦错风格,优秀组被试的侦错更加主动,普通组被试的侦错相对被动,更加依赖侦错文本.对于研究问题2.1,普通生更关注解答过程的表面特征,主要进行的是计算检查,对概念的意义和推理的有效性关注不够,在概念性错误和逻辑性错误的发现与解释上存在困难;优秀生更关注解答过程的逻辑结构,主要进行的是推理检查,且具备更多有关解题错误的知识,可以快速地发现概念性错误和逻辑性错误.优秀生均可以写出正确解答,给出的解法更多样且对正确解法的使用更灵活,既可以在发现错误之后的修正阶段写出正确解法,也可以在阅读侦错文本的检验阶段写出正确解法,进而通过对比文本解法发现错误.普通生写出正确解法的意愿不强、能力不足,在独立写出解法的过程中容易犯错且难以自我发现和纠正.对于研究问题2.2,优秀生更关注问题的深层结构和对知识的关系性理解,对侦错任务已知条件之间的联系与制约关系有更深刻的感知,对条件的分析和加工更深刻,可以更快速地揭示矛盾并修改条件,给出的修改方案更丰富.普通生更关注对文本解答过程的分析和检验,对题目条件的分析比较表面化,难以察觉条件之间隐含的制约关系,在识别多余条件和揭示矛盾上存在困难,多依据直觉给出修改方案,修改方案比较单一或者因违反条件之间的制约关系而出错.对于研究问题3,访谈结果表明,两组被试侦错表现的差异与平常的学习习惯和侦错经历有关.主要结果包括下面几点.被试对条件侦错题目印象最深的人数多于解答侦错题.数学侦错题的形式比较新颖.平常的题目都是直接去做,解出答案即可.条件侦错任务与解答侦错任务相比更有问题解决的色彩,需要根据已有条件推断出一些新的结论,导出矛盾的归谬过程具有一定的探索性,条件的修改具有一定的开放性.优秀生与普通生在发现解答错误的人数上无显著差异,在发现题目错误的人数上具有显著差异;优秀生与普通生在具有一题多解和检验反思习惯的人数上具有显著差异;在独立作答的情况下,优秀生比普通生更有可能发现题组2的题目条件矛盾.被试普遍持有如下观念:题目一般不会有错,拿到题目后往往直接解;如果题目错了,对错都能拿到分数.被试处理错误的方式主要有三种:错题本整理、笔记本整理、原题订正;优秀生均无错题本,均采取原题订正或直接记住的方式,普通生三种方式都有;平常数学教学进度很快,教师很少专门讲评学生错误,通常都是直接提供正确答案;两组被试的教师在教学中都会使用错解,但是对错题利用很少.被试均认可数学侦错题的价值,主要价值包括锻炼思维、加深印象、增进理解、培养阅读侦错能力、对数学教师教学和命题有用.同学认为可以在课堂教学、练习、复习和习题讲评中使用数学侦错题,对于在考试中是否可以使用,同学的意见并不一致.最后,对本研究的启示、局限性、以及未来的研究建议进行了探讨.更多还原

【Abstract】 Errors are valuable learning resources.Learning from errors has attracted increasing interest and attention from researchers.A significant portion of high school students’ math learning time is spent solving various math problems and reading math solutions,both of which contain rich opportunities to learn from errors.A student may read a solution that implicitly contains an error,or solve a mathematical problem whose conditions are contradictory.In practice,there are cases where students find answers wrong or even questions unrealistic.However,few empirical studies have explored the thinking process of students in detecting and correcting these errors.The main purpose of this study is to explore the cognitive mechanism of mathematical error detection,to reveal the cognitive characteristics and differences of senior high school students with different mathematical abilities in solving mathematical error-detecting problems,and to understand students’ mathematical beliefs and learning habits that affect the performance of error detection,as well as students’ views on the value of mathematical error-detecting problems and their suggestions on their use.Two types of mathematical error-detecting problems are used in this study: type 1 error-detecting problems which require the subjects to judge the correctness of the solution and then correct the error,and conditional error-detecting problems(type 2)which require the subjects to reveal the contradiction of conditions and modify conditions to make the problem correct.Research questions are as follows:Question 1.How do high school students solve mathematical error-detecting problem? Namely,what are the general cognitive processes for solving error-detecting problems?Question 2.How do average ability students and high ability students perform in solving mathematical error-detecting problems? What’s the difference? Specifically,it includes the following two sub-questions:Question 2.1 How do average ability students and high ability students perform in solving type 1 mathematical error-detecting problems? What’s the difference?Question 2.2 How do average ability students and high ability students perform in solving type 2 mathematical error-detecting problems? What’s the difference?Question 3 What are the mathematical beliefs and learning habits that affect the performance of high school students in solving error-detecting problems? What are the views of high school students on the value of mathematical error-detecting problem?Any suggestions on its use?In this study,using think-aloud method and case study method,and referring to the research paradigm of expert-novice comparison,8 students with high mathematical ability and 8 students with average mathematical ability in the second grade of a key middle school were recruited as the subjects.Each subject solved error-detecting task through thinking aloud and accepted the follow-up interview.For research question 1,based on the think aloud protocols and interview data,five types of error-detecting episodes are extracted: read,analyze,inspect,correct,and judge.The cognitive processes of error detection are combination of different kinds of error-detecting episodes which begins with reading question.Due to the differences in task difficulty,task characteristics and subjects’ personal characteristics,error-detecting episodes have different ways of combination.On the whole,the two groups present two different debugging styles.High ability students are more active in debugging,while average ability students are more passive and more dependent on the debugging text.For research question 2.1,the results show that average ability students pay more attention to the surface features of the solution process,and mainly check the calculation.They pay less attention to the meaning of concepts and the validity of reasoning and have difficulty in finding and explaining conceptual errors and logical errors.High ability students pay more attention to the logical structure of the solution process,and mainly check reasoning process.They have more knowledge related to errors,and can quickly find out the conceptual and logical error.High ability students are all able to write correct solutions,with more varied solutions and more flexible use of correct solutions.They can not only write correct solutions after detecting errors,but also independently write correct solutions and compare them with the solutions in the text after reading the questions or solutions.Average ability students are not willing to write solutions by themselves,and their ability is insufficient.When writing solution independently,they are easy to make mistakes and difficult to self-correct.For research question 2.2,the results show that high ability students focus more on the deep structure of the problem and the relational understanding of knowledge,have a more profound perception of the conditions of the questions and the connections and restraints between the conditions,are more active in the analysis of the conditions,can reveal the contradictions and modify the conditions more quickly,and propose more abundant modification approaches.Average students’ analysis of the conditions are more superficial,and spend more time in the analysis and verification of the solution.Because they do not carefully analyze the implicit restrictive relationship between the conditions,average ability students are slow to reveal the contradictions,and they give a relatively simple modification approach or make mistakes because of neglecting the constraint relationship between conditions.For research question 3,the interview results show that the performance and its differences between the two groups of subjects are related to their daily error-detecting experience and learning habits.The main results include the following.More subjects are most impressed by type 2 task.Mathematical error-detecting problem is relatively novel to them.Daily problems are all done directly to get answers.Compared with type 1 task,type 2 task is more challenging.Students need to infer some new conclusions according to conditions.The process of derivation of contradiction is exploratory,and the modification of conditions is open to some extent.There is no significant difference in the number of subjects who find wrong answers.There is a significant difference in the number of subjects who find unrealistic problems.There are significant differences in the number of students who have the habit of solving one problem with multiple solutions and verification and reflection.Without being told beforehand,high ability students are more likely to find that conditions of type 2 task are contradictory.The subjects generally held the following beliefs: questions are generally solvable,their goal is to solve them directly and get right answer,and that if questions are wrong in itself,they can get points anyway.There are three ways for students to deal with mistakes: sorting out using wrong problem book,sorting out using notebook and correcting directly.High ability students all correct error directly and remember them,ordinary students may use three ways.Usually,the progress of mathematics teaching is very fast,and teachers seldom comment on students’ mistakes and usually provide correct answers directly.Their teachers would make use of type 1 error-detecting problems in teaching,but rarely make use of type 2 error-detecting problems.All the subjects agree on the value of mathematical error-detecting problems,and the main values include developing mathematical thinking,deepening impression,enhancing understanding,cultivating reading error detection ability,and being useful for mathematics teachers’ teaching and propositions.Students suggest that mathematical error-detecting tasks can be used in classroom teaching,mathematical practice,review and exercise comment,but there is no agreement on whether mathematical error-detecting tasks should be used in the examination.Finally,the implications and limitations of this study and suggestions for future research are discussed.更多还原