Concerns have been expressed that although learners may solve linear equations correctly they cannot draw on mathematically valid resources to explain their solutions or use their strategies in unfamiliar situations. This article provides a detailed qualitative analysis of the thinking of 15 Grade 8 and Grade 9 learners as they talk about their solutions to linear equations in interviews. The article stems from a study that describes whether learners use mathematically endorsable narratives to explain and justify their solutions. Sfard's theory of commognition is used to develop a framework for analysis of their discourse. The findings show that all learners use ritualised rather than explorative discourse, characterised by applying strict rules to operations with disobjectified entities. The only mathematical objects they produce endorsed narratives about are positive integers. Thus they do not meet the relevant curriculum requirements. Nevertheless, the analytic tools – adapted from Sfard specifically for the study of linear equations – give a particularly nuanced account of differences in the learners' ritualised discourse. For example, some learners used endorsed narratives about negative integers, algebraic terms and the structure of an equation when prompted by the interviewer. There is not sufficient evidence to suggest that any learners are in transition to explorative discourse. However, the article shows that learner discourse is a rich resource for teachers to understand the extent to which learners are thinking exploratively, and offers suggestions for how their thinking can be shifted. This is an opportunity for teacher professional development and further research.

Anthea Roberts

Kate le Roux

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。

A commognitive perspective on Grade 8 and Grade 9 learner thinking about linear equations

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Journal Info

Category	Quartile
EDUCATION, SCIENTIFIC DISCIPLINES	4