Metacognitive awareness and visualisation in the imagination: The case of the invisible circles

IF 0.3 Q4 EDUCATION, SCIENTIFIC DISCIPLINES

Pythagoras Pub Date : 2018-08-13 DOI:10.4102/PYTHAGORAS.V39I1.396

D. Jagals, Martha Van der Walt

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引用次数: 5

Abstract

Awareness of one’s own strengths and weaknesses during visualisation is often initiated by the imagination – the faculty for intuitively visualising and modelling an object. Towards exploring the role of metacognitive awareness and imagination in facilitating visualisation in solving a mathematics task, four secondary schools in the North West province of South Africa were selected for instrumental case studies. Understanding how mathematical objects are modelled in the mind may explain the transfer of the mathematical ideas between metacognitive awareness and the rigour of the imaginer’s mental images. From each school, a top achiever in mathematics was invited to an individual interview (n = 4) and was video-recorded while solving a mathematics word problem. Participants also had to identify metacognitive statements from a sample of statement cards (n = 15) which provided them the necessary vocabulary to express their thinking during the interview. During their attempts, participants were asked questions about what they were thinking, what they did and why they did what they had done. Analysis with a priori coding suggests the three types of imagination consistent with the metacognitive awareness and visualisation include initiating, conceiving and transformative imaginations. These results indicate the tenets by which metacognitive awareness and visualisation are conceptually related with the imagination as a faculty of self-directedness. Based on these findings, a renewed understanding of the role of metacognition and imagination in mathematics tasks is revealed and discussed in terms of the tenets of metacognitive awareness and imagination. These tenets advance the rational debate about mathematics to promote a more imaginative mathematics.

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想象中的元认知意识和可视化:看不见的圆圈的情况

在可视化过程中，对自己的长处和弱点的认识通常是由想象力发起的-直观地将物体可视化和建模的能力。为了探索元认知意识和想象力在促进可视化解决数学任务中的作用，南非西北省的四所中学被选中进行工具性案例研究。理解数学对象是如何在大脑中建模的，可以解释数学思想在元认知意识和想象者心理图像的严谨性之间的转移。每所学校邀请一名数学成绩最好的学生进行单独面试(n = 4)，并在解决数学单词问题时进行录像。参与者还必须从陈述卡样本(n = 15)中识别元认知陈述，这为他们提供了必要的词汇来表达他们在面试过程中的想法。在他们的尝试过程中，参与者被问及他们在想什么，他们做了什么以及他们为什么这么做。先验编码分析表明，与元认知意识和视觉化相一致的三种想象类型包括启动想象、构思想象和转化想象。这些结果表明，元认知意识和可视化在概念上与想象作为一种自我指导的能力相关的原则。基于这些发现，本文从元认知意识和元想象的原则出发，揭示和讨论了元认知和想象在数学任务中的作用。这些原则推动了关于数学的理性辩论，促进了更具想象力的数学。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Pythagoras EDUCATION, SCIENTIFIC DISCIPLINES-

CiteScore

1.50

自引率

16.70%

发文量

审稿时长

20 weeks

期刊介绍： Pythagoras is a scholarly research journal that provides a forum for the presentation and critical discussion of current research and developments in mathematics education at both national and international level. Pythagoras publishes articles that significantly contribute to our understanding of mathematics teaching, learning and curriculum studies, including reports of research (experiments, case studies, surveys, philosophical and historical studies, etc.), critical analyses of school mathematics curricular and teacher development initiatives, literature reviews, theoretical analyses, exposition of mathematical thinking (mathematical practices) and commentaries on issues relating to the teaching and learning of mathematics at all levels of education.