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.

Divan Jagals

Martha Van der Walt

Pythagoras

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.

Pythagoras is a scholarly research journal that serves as a platform for presenting and critically discussing current research and developments in mathematics education at both national and international levels. Pythagoras publishes articles that significantly enhance our understanding of mathematics teaching, learning, and curriculum studies. This includes research reports (experiments, case studies, surveys, philosophical and historical studies, etc.), critical analyses of school mathematics curricula and teacher development initiatives, literature reviews, theoretical analyses, expositions of mathematical thinking (mathematical practices), and commentaries on issues related to the teaching and learning of mathematics at all levels of education.

《Pythagoras》是一本学术研究期刊，旨在提供一个平台，以展示和深入讨论当前在数学教育领域的研究和发展，涵盖国内外的相关内容。《Pythagoras》发表的文章显著推动了我们对数学教学、学习和课程研究的理解，包括研究报告（实验、案例研究、调查、哲学和历史研究等）、对学校数学课程和教师发展计划的批判性分析、文献综述、理论分析、数学思维的阐述（数学实践）以及关于各级教育中数学教学与学习相关问题的评论。

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

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Journal Info

Category	Quartile
EDUCATION, SCIENTIFIC DISCIPLINES	4