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毕业论文网 > 外文翻译 > 理工学类 > 数学与应用数学 > 正文

课堂数学传统的特征 ——互动分析外文翻译资料

 2022-12-29 11:39:32  

本科毕业设计(论文)

外文翻译

课堂数学传统的特征

——互动分析

作者:Paul Cobb,Terry Wood,Erna Yackel和Betsy McNeal

国籍:美国

出处:美国教育研究期刊

摘要:在本文中,我们试图阐明教授数学理解和学习数学的意义。为此,我们提供了对不同小学课堂中两个课程的转录录像的互动分析。选择了代表更大数据语料库的课程,因为它们都关注场所价值计算并涉及使用类似的操纵材料。该分析借鉴了Much和Shewder(1978)对五种质量上不同类型的课堂规范的识别,并特别关注课程中发生的数学解释和理由。在一个教室中,教师和学生似乎始终将数学视为在他们的时刻互动过程中遵循程序指令的活动。对另一个课堂的分析表明,教师和学生构成了数学真理,因为他们共同构建了一个由经验真实,可操纵但抽象的数学对象组成的数学现实。两个教室中数学活动之间的这些和其他差异是两种不同的课堂数学传统的特征,其中一种是通过所谓的理解来学习数学,而另一种则不是。

关键词:数学理解;数学学习;教学实践

最近美国数学教育改革的呼吁集中在促进教学实践的需要上,这些教学实践促进了通常所谓的有意义学习(例如,国家数学教师委员会,1989年国家研究委员会,1989)。在正在进行的改革努力中,阐明教导理解和学习理解的含义显然是至关重要的。在这方面,我们发现在鼓励有意义学习的教学情境与不鼓励有意义学习的教学情境之间的几种对比中具有相当大的价值。例如,Skemp(1976)讨论了关系学和工具性学习之间的区别; Brown,Collins和Duguid(1989)对比了真实和非真实的数学活动;和Richards(1991)发展了探究数学和学校数学的概念。我们试图进一步阐明促进学习与理解的教学情境与不通过对不同教室中建立的数学传统进行经验性分析的教学情境之间的对比。

我们提出的方法基于这样的假设,即通过分析教师和学生在课堂讨论中的数学解释和理由,可以突出这些课堂数学传统中的质的差异。我们通过首先发展我们的总体方向并澄清我们的理论结构来说明这种方法。然后,我们检查来自两个教室的课程成绩单,其中只有一个可以说是促进有意义的学习,这两个样本分析随后被用作重新考虑术语意义和理解的含义的背景。在讨论过程中,我们发现区分两种语境中术语的使用方式,教育改革的明确政治背景和数学教育研究的学术背景是有用的。最后,我们从信息处理的角度和强烈的社会学角度考虑了关于数学发展本质的主张。我们对质量上不同的课堂数学传统的分析使我们得出结论,这两种主张都反映了对数学活动的过度限制的概念。

课堂数学传统一些学者通过分析可能被称为课堂数学文化的东西,从社会学的角度研究了数学学习和教学的过程。在这样做的过程中,他们引入了各种密切相关的理论概念,包括课堂话语实践(Walker dine,1988)。课堂实践的传统(Solomon,1989),课堂亚文化或微观文化(Bauersfeld,Krummheuer,&Voigt,1988; Voigt,1985)。与注重定性差异的认知分析形成对比在个体教师或学生的信念和数学概念中,社会学分析集中于使得传播成为可能的共享或规范假设,假设和解释的演变(Gergen,1985)。这些通常是隐含的共识社会和数学理解构成我们称之为课堂数学传统(Cobb ,1991)。

正如该术语所暗示的,课堂数学传统在很多方面类似于科学研究传统(Cobb,Wood,&Yackel,1991)。为了我们的目的,只需要注意两者都是由一个社区创造的,并且通过限制可以算作问题,解决方案,解释和理由的因素来影响个人对科学或数学知识的建构我们将此视为我们的起点不好,分析数学活动的这四个方面 - 问题,解决方案,解释和理由 - 提供了一种经验基础的方式来表征在任何特定课堂中建立的数学传统。 Much和Shweder(1978)的工作在这方面具有相关性,因为他们通过分析教师和学生如何解释他们的行为,确定了五种质量上不同的规范活动。我们首先概述他们的分析方法,并试图通过考虑与数学活动有关的解释和理由来扩展它。接下来,我们认为解释和证明是联合的以及个别的活动,并说明了可能出现的沟通困难,除非对解释和证明的意义有共同的理解。完成这一初步定位后,我们将两个教室中师生建立的规范数学活动的质量进行比较和对比。

课堂规范

Much和Shweder(1978)确定了五种类型的课堂规范:规则,惯例,道德,真理和指示。 他们使用各种标准来区分这些规范,包括它们的历史性,它们的来源以及违反这些规范的后果。 在这个计划中,法规是历史性的,由可以改变它们的可指定权威机构建立。 违反规定的后果通常是某种惩罚。 例如,教师可能会在小组工作中告诉学生,每组中只有一名成员可以获取该组决定使用的操纵材料。 该指令是一项规定,因为它由可以改变它的教师建立。

公约也是历史性的,但它们的来源是不可预测的,并且违法的顺序是社会不赞成的。法规和惯例之间的区别类似于法律和习俗之间的区别。例如,在传统教学的背景下,学生习惯于回应教师的已知答案问题和教师评估他们的回答(参见Mehan,1979)。这些规范的潜在可变性是习惯性的。兰伯特(1990)对一个课堂的讨论证明了这一点,在这个课堂中,相互作用并没有遵循这种传统的启发 - 反应 - 评价模式。

与法规和惯例相反,被归类为道德,真理或指示的规范都被社区成员视为非历史性的。 Much和Shweder认为违背道德的后果是道德可耻性。一个例子是传统的课堂规范,学生不应该复制另一个人的答案,并将其作为自己工作的结果传递出去。教师可能会试图让违反这一规范的学生对自己所做的事感到内疚。正是这种有罪的感觉将道德与真理和指示区分开来。违背真理的后果本身就是错误,而违反指令的后果则是无效。

先验地,后两种类型的规范似乎与我们的分析具有最大的相关性,因为我们的重点是小学数学的一个更概念性的方面:地方价值计算。我们接受历史学家和准经验主义数学哲学家的观点,认为数学是一种不断发展的人类建构。尽管如此,与数学的其他方面相比,例如使用特定符号,当教师和学生在课堂上进行算术时,似乎不太可能将游戏价值视为可能被教师和学生修改。此外,虽然人们可以想象违反地方价值计算规范的各种后果,但如果道德责任只是其中之一,那似乎有些奇怪。因此,尽管我们不能提前忽略这种可能性,但我们利用Much和Shweder的工作的主要动机是调查他们在真理和指示之间所做的区分是否有助于描述有意义的数学学习发生的教室和教学之间的差异。它不是。

解释并证明是联合活动

我们可以通过对比我们对Brousseau(1984)的教学情境理论中使用的术语情况进行对比来进一步发展解释和辩解的概念,正如Balacheff(1990)和Laborde(1989)所阐述的那样。 Brousseau的理论受到教学考虑的推动,他所区分的各种情境可以通过教师希望学生接受数学活动的义务来表征。例如,行动的情况要求学生寻找特定任务情境的解决方案,以便制定要求学生明确表达他们的解释和概念,验证的情况要求学生证明他们明确的是什么,制度化的情况要求学生接受教师在课堂活动过程中发展起来的数学结构的合法性。这种分析是有价值的,因为它提供了一个理论框架,可以指导教学实验的发展,既考虑发布给学生的具体数学任务,也考虑学生应该在理想情况下尝试完成这些任务的社会环境。

我们对理由情境和解释情境概念的发展是出于一种不同的关注,即分析课堂数学传统的本质而不是制定教学情境。因此,两组结构之间没有直接的对应关系。例如,当学生解决给定任务,阐明他们的解释,验证他们明确表达的内容或讨论特定数学结构的合法性时,可能会出现解释或理由的情况。此外,虽然各种类型的教学情境可以通过研究人员或教师希望学生接受的义务来表征,但是理由和解释的情境是由教师和学生或一群学生在他们尝试时互动地构成的。协调他们的数学活动(Bauersfeld,1980; Bruner,1986; Mehan,1979; Voigt,1989)。从这个观点来看,解释和辩解被认为是集体或联合活动(cf.Blumer,1969; Mead,1934) )。

我们通过一个样本集来说明情况的这一方面,其中有两个女孩试图解决一项涉及找到三个九个总和的任务:

凯伦:我知道怎么做。你们不会听我的。

玛丽:一分钟!

凯伦:这是10分,20分,30分(她然后算上她的手指)31,32,33 ...... 55,56,57。

玛丽:不是那么远。

在这里,凯伦在没有要求的情况下解释了她的解决方案,大概是因为她推断她的方法对玛丽来说并不明显。玛丽为自己的角色做出了贡献,通过在挑战其合法性之前试图理解凯伦的解决方案,为解释情境的互动构成做出了贡献。为了使这种情况发展成为理由,凯伦必须提供一个理由来回应玛丽的挑战。她这样说,“我把所有数字加起来。”然而,玛丽忽视了凯伦的理由,并建议他们通过指望数百个董事会来解决冲突:

玛丽:我们来吧......

凯伦:W帽子18加30?

玛丽:(支持)我知道什么...我们可以得到数百个董事会......

凯伦:什么是18加30?

玛丽:数百板......

凯伦:玛丽!

玛丽:我要买几百块板......然后我们可以找......

凯伦:玛丽,我知道答案。

凯伦可能重新定义了她的活动,从30Dy计算三个九,30加18而不是30加27。她反复提出这个问题。为了说服玛丽在她最初的理由被忽略之后,她的解决方案的合法性已经被忽略了。“但是,玛丽并没有试图回答凯伦的问题,而是继续建议他们使用数百个董事会。结果,他们简短地确定了解释的理由,而没有得出关于凯伦解决方案合法性的结论。

正如这一短篇插图所说明的那样,具有挑战性或正当性的个人行为本身并不构成理由。 数学活动受到质疑的人必须将此解释为挑战并作出相应的回应。 同样,给予理由的人必须接受它或说出为什么不充分,从而进一步提出挑战(参见Mehan&Wood,1975)。 在解释的情况下可以说大致相同。 除非有人试图解释所提供的解释,否则我们不会说交互式构成解释的情况。 简而言之,只有当解释或解释解释或解释的理解被视为共享时,才能以交互方式形成解释或辩解的情况。

CHARACTERISTICS OF CLASSROOM MATHEMATICS TRADITIONS

——An interactional analysis

Author: Paul Cobb, Terry Wood, Erna Yackel and Betsy McNeal

Country: American

Source: American Educational Research Journal, 1992, 29, 3, 573-604.

In this paper, we attempt to clarify what it means to teach mathematics for understanding and to learn mathematics with understanding. To this end, we present an interactional analysis of transcribed video recordings of two lessons that occurred in different elementary school classrooms. The lessons, which are representative of a much larger data corpus, were selected because both focus on place value numeration and involve the use of similar manipulative materials. The analysis draws on Much and Shewders (1978) identification of five qualitatively distinct types of classroom norms and pays particular attention to the mathematical explanations and justifications that occurred during the lessons. In one classroom, the teacher and students appeared consistently to constitute mathematics as the activity of following procedural instructions in the course of their moment by moment interactions. The analysis of th

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CHARACTERISTICS OF CLASSROOM MATHEMATICS TRADITIONS

——An interactional analysis

Paul Cobb, Terry Wood, Erna Yackel and Betsy McNeal

Source: American Educational Research Journal, 1992, 29, 3, 573-604.

In this paper, we attempt to clarify what it means to teach mathematics for understanding and to learn mathematics with understanding. To this end, we present an interactional analysis of transcribed video recordings of two lessons that occurred in different elementary school classrooms. The lessons, which are representative of a much larger data corpus, were selected because both focus on place value numeration and involve the use of similar manipulative materials. The analysis draws on Much and Shewders (1978) identification of five qualitatively distinct types of classroom norms and pays particular attention to the mathematical explanations and justifications that occurred during the lessons. In one classroom, the teacher and students appeared consistently to constitute mathematics as the activity of following procedural instructions in the course of their moment by moment interactions. The analysis of the other classroom indicated that the teacher and students constituted mathematical truths as they coconstructed a mathematical reality populated by experientially real, manipulable yet abstract mathematical objects. These and other differences between mathematical activity in the two classrooms characterize two distinct classroom mathematics traditions, one in which mathematics was learned with what is typically called understanding and the other in which it was not.

Recent calls for reform in mathematics education in the United States have focused on the need to promote instructional practices that facilitate what is generally called meaningful learning (e.g., National Council of Teachers of Mathematics, 1989National Research Council, 1989). The explication of what it might mean to teach for understanding and to learn with understanding is clearly central in this ongoing reform effort. In this regard, we find considerable value in several of the contrasts that have been made between instructional situations that encourage meaningful learning and those that do not. For example, Skemp (1976) discussed the distinctions between relational and instrumental learning; Brown, Collins, and Duguid (1989) contrasted authentic and unauthentic mathematical activity; and Richards (1991) developed the notions of inquiry mathematics and school mathematics. We attempt to explicate further the contrasts between instructional situations that promote learning with understanding and those that do not by developing an empirically grounded analysis of the mathematics traditions established in different classrooms.

The approach we propose rests on the assumption that qualitative differences in these classroom mathematics traditions can be brought to the fore by analyzing teachers and students mathematical explanations and justifications during classroom discourse. We illustrate this approach by first developing our general orientation and clarifying our theoretical constructs. We then examine transcripts of lessons from two classrooms, only one of which can be said to foster meaningful learning these two sample analyses are subsequently used as a backdrop to reconsider what is meant by the terms meaning and understanding. In the course of this discussion, we find it useful to distinguish between the way the terms are used in two contexts, the explicitly political context of educational reform and the academic context of mathematics education research. Finally, we consider claims about the nature of mathematical development made from both the information-processing perspective and a strong sociological perpective. Our analysis of qualitatively distinct classroom mathematics traditions leads us to conclude that both claims reflect unduly restricted conceptions of mathematical activity.

Classroom mathematics traditions Several scholars have investigated the process of learning and teaching mathematics from a sociological perspective by analyzing what might be called the mathematical culture of the classroom .In doing so, they have introduced a variety of closely related theoretical notions including the classroom discursive practice (Walker dine, 1988).the tradition of classroom practice(Solomon,1989),and the classroom subculture or micro culture (Bauersfeld, Krummheuer, amp; Voigt, 1988; Voigt,1985).In contrast to cognitive analyses that focus on qualitative differences in the beliefs and mathematical conceptions of individual teachers or students ,sociological analyses concentrate on the evolution of taken-as-shared or normative suppositions, assumptions, and interpretations that make communication possible(Gergen, 1985).It is these often implicit consensual social and mathematical understandings that constitute what we will call the classroom mathematic tradition(Cobb,1991).

As the term implies, a classroom mathematics tradition is in many ways analogous to a scientific research tradition (Cobb, Wood, amp; Yackel, 1991). For our purpose, It suffices to note that both are created by a community and that both influence individuals construction of scientific or mathematical knowledge by constraining what can count as a problem, a solution, an explanation, and a justification We take this as our starting point ill that an analysis of these four aspects of mathematical activity-problems, solutions, explanations, and justifications-offers an empirically grounded way to characterize the mathematics tradition established in any particular classroom. The work of Much and Shweder(1978) is relevant in this regard in that they identified five qualitatively distinct types of normative activity by analyzing how teachers and students account for their actions. We first outline their analytical approach and attempt to extend it by considering both ex

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