The novice mathematician’s encounter with mathematical abstraction: tensions in concept-image construction and formalisation
The novice mathematician’s encounter with mathematical abstraction: tensions in concept-image construction and formalisation
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数学新手遇到数学抽象：概念图像构建和形式化中的张力
DOI：
10.1080/1380361960020103
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发表时间：
1996
影响因子：
1.4
通讯作者：
Elena Nardi
中科院分区：
文献类型：
--
作者：
Elena Nardi
数学被定义为一种抽象的思维方式。抽象是最难理解的心理活动之一。在教育背景下，数学抽象的遭遇是从非正式学校数学过渡到大学数学形式主义的关键一步。这种转变的特点是认知紧张。本研究的目的是识别和探索新手数学家遇到的数学抽象的紧张局势。为了这个目的20个一年级的数学本科生观察到他们的每周教程在牛津大学四个学院在米迦勒节和希拉里学期的第一年。在观察期间，磁带记录并保存现场记录。在每个观察期结束时，学生们也接受了采访。观察到的教程和访谈的录音被转录并提交给一个分析过程，过滤出阐明新手认知的片段。一个分析框架，包括认知和社会文化理论的学习被应用在集的情节内的数学领域的基础分析，微积分，线性代数和群论。在主题分析之后，我们对被试的认知进行了跨主题的综合。初学者与数学抽象的相遇被描述为一个个人意义建构过程和一个文化适应过程：新的文化是由专家（导师）介绍的高等数学。新手与新概念定义的互动受到他们不稳定的先前知识的阻碍。概念意象建构被描述为一种有意义的隐喻的建构，以及对新概念和新推理的“存在理由”的探索，其特征在于非正式/直观/口头与正式/抽象/象征之间的紧张关系-这是从语义和推理方面进行讨论的。初学者在正式的数学推理机制以及应用这些机制在一个contextualised方式的困难。这种脱离语境的行为与他们的知识在形式数学的严谨性方面的脆弱性有关。
Mathematics is defined as an abstract way of thinking. Abstraction ranks among the least accessible mental activities. In an educational context the encounter with mathematical abstraction is the crucial step of the transition from informal school mathematics to the formalism of university mathematics. This transition is characterised by cognitive tensions. This study aimed at the identification and exploration of the tensions in the novice mathematician's encounter with mathematical abstraction. For this purpose twenty first-year mathematics undergraduates were observed in their weekly tutorials in four Oxford Colleges during Michaelmas and Hilary Term of Year 1. Tutorials were tape-recorded and fieldnotes kept during observation. The students were also interviewed at the end of each term of observation. The recordings of the observed tutorials and the interviews were transcribed and submitted to an analytical process of filtering out episodes that illuminate the novices' cognition. An analytical framework consisting of cognitive and sociocultural theories on learning was applied on sets of episodes within the mathematical areas of Foundational Analysis, Calculus, Linear Algebra and Group Theory. This topical analysis was followed by a cross-topical synthesis of themes that were found to characterise the novices' cognition. The novices' encounter with mathematical abstraction was described as a personal meaning-construction process and as an enculturation process: the new culture is Advanced Mathematics introduced by an expert, the tutor. The novices' interaction with the new concept definitions was obstructed by their unstable previous knowledge. Concept image construction was described as a construction of meaningful metaphors and an exploration of the 'raison-d'-etre' of the new concepts and the new reasoning and was characterised by the tension between the Informal/Intuitive/Verbal and the Formal/Abstract/Symbolic — which was discussed in terms of semantics and reasoning. The novices were in difficulty with the mechanics of formal mathematical reasoning as well as with applying these mechanics in a contextualised manner. This decontextualised behaviour was linked to the fragility of their knowledge with regard to the nature of rigour in formal mathematics.