Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors
Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors
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线性推理的不当运用：深入研究中学生错误的本质和不可抗拒性
DOI：
10.1023/a:1021205413749
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发表时间：
2002
影响因子：
3.2
通讯作者：
L. Verschaffel
中科院分区：
文献类型：
--
作者：
Dirk De Bock;W. Van Dooren;D. Janssens;L. Verschaffel
最近的几项调查研究表明，在12-16岁的学生中，有一种根深蒂固的、几乎无法抗拒的倾向，即在涉及相似平面图形和立体的长度、面积和体积的应用题中，不恰当地应用线性或比例模型。虽然这些先前的研究表明，在何种程度上学生的线性推理的不当使用是由不同的任务特点的影响，它仍然在很大程度上不清楚他们的知识基础的哪些方面是负责这种现象的发生和强度，以及这些方面如何与其他更普遍的误解和错误的规则在文献中确定。本文报道了一项通过个人半标准化访谈进行的深入调查，旨在分析学生不当线性推理背后的思维过程，以及这一过程如何受到他们的数学概念、信仰和习惯的影响。在这些采访中，学生的解决方案的过程中，揭示了通过一些明确的问题，由面试官就一个单一的非线性应用问题，以及通过他们的反应，随后的各种认知冲突。访谈提供了很多关于学生陷入“线性陷阱”的实际解题过程及其背后机制的信息。虽然有些学生似乎真的“相信"量总是成比例联系的，但他们对线性的不当使用往往是由于肤浅和直觉的推理，受到特定数学概念、习惯和信念的影响，导致建模过程不完善。
Several recent ascertaining studies revealed a deep-rooted and almost irresistible tendency among 12–16-year old students to improperly apply the linear or proportional model in word problems involving lengths, areas and volumes of similar plane figures and solids. While these previous studies showed to what extent students' improper use of linear reasoning is affected by different characteristics of the task, it remained largely unclear what aspects of their knowledge base are responsible for the occurrence and strength of this phenomenon and how these aspects relate to other more general misconceptions and buggy rules identified in the literature. This paper reports an in-depth investigation by means of individual semi-standardised interviews aimed at analysing the thinking process underlying students' improper linear reasoning and how this process is affected by their mathematical conceptions, beliefs and habits. During these interviews,students' solution processes were revealed through a number of well-specified questions by the interviewer with respect to one single non-linear application problem, as well as through their reactions to subsequent kinds of cognitive conflict. The interviews provided a lot of information about the actual process of problem solving from students falling into the ‘linearity trap’ and the mechanism behind it. Although some students seem to really ‘believe’ that quantities are always linked proportionally, their improper use of linearity often results from superficial and intuitive reasoning, influenced by specific mathematical conceptions, habits and beliefs leading to a deficient modelling process.