Rajendran Govender

Pythagoras

A reflection on the use of artificial intelligence tools for the meaningful learning of mathematics

人工智能工具在数学有意义学习中的应用反思

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.

Editorial Office

Acknowledgement to reviewers

致谢审稿人

The narrative permeating higher education institutions globally is the intention to advance the quality of preservice mathematics teachers to gain better performance than mathematics school learners. In this article we report on a qualitative case study which explored the pedagogical content knowledge (PCK) of preservice teachers (PST) (n = 20) at a South African university of technology. These preservice teachers were provided with activity sheets with past examination questions in analytical geometry. Their written responses were analysed, and verification interviews were conducted. This was followed by an analysis of the preservice teachers’ lesson plans for teaching the topic to Grade 11 learners. The domains and subdomains of the Shulman’s PCK derived Mathematics Teacher’s Specialised Knowledge (MTSK) model were used as the theoretical framework. The study established that the PSTs had limited content knowledge, and knowledge for teaching analytical geometry at Grade 11 as their responses to the activities confirmed that they mostly had procedural understanding. The study was conducted in a context with multilingual PSTs who are training in multilingual classrooms.Contribution: The study will highlight the misconceptions and existing gaps in the PSTs’ knowledge for teaching analytical geometry in multilingual classrooms.

Sibongile Zulu

Deonarain Brijlall

Exploring preservice teachers’ pedagogical content knowledge for teaching analytical geometry in multilingual classrooms

探究多语课堂中职前教师教授解析几何的学科教学知识

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

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

八年级和九年级学生关于线性方程思维的共同认知视角

Various efforts are underway to improve achievement in high-stakes examinations in school mathematics. This article reports on one such initiative which focuses on the development of quality teaching of school mathematics by embedding it within an examination-driven emphasis. A quantitative approach was used to analyse the performance of Grade 10 learners in three consecutive end-of-year school-based examinations set by the initiative. Results indicate a trend in a positive direction over the three-year period. Nevertheless, there was a discernible decrease between the first and second administration of the examinations. It is concluded that examination-driven teaching holds a promise for enhancing achievement in high-stakes school mathematics examinations if sensibly and sensitively implemented.

Onyumbe Okitowamba

Cyril Julie

Monde Mbekwa

The effects of examination-driven teaching on mathematics achievement in Grade 10 school-based high-stakes examinations

考试驱动教学对高中10年级校内高风险数学考试成绩的影响

In this article we propose a framework for reporting mathematics results from national assessment surveys (NAS) such that effective use of the resulting reports can enhance teaching and learning. We explored literature on factors that may contribute to non-utilisation of assessment data as a basis for decision-making. In the context of South Africa, we identified the form and formats in which results of NAS are reported as a possible limiting factor to the effective use of summative assessment results for formative purposes. As an alternative, we propose a standards-based reporting framework that will ensure accurate measurement of, and meaningful feedback on, what learners know and can do. We illustrate how, within a properly designed reporting framework, the results of a NAS in mathematics can be used for formative purposes to enhance teaching and learning and, possibly, improve learner performance.

Meshack Moloi

Anil Kanjee

Beyond test scores: A framework for reporting mathematics assessment results to enhance teaching and learning

超越考试分数：提升教学与学习的数学评估结果报告框架

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

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

元认知意识与想象中的视觉化：不可见圆圈的案例

Textbook content has the ability to influence mathematical learning. This study compares how linear functions are presented in two textbooks, one of South African and the other of German origin. These two textbooks are used in different language-based streams in a school in Gauteng, South Africa. A qualitative content analysis on how the topic of linear functions is presented in these two textbooks was done. The interplay between procedural and conceptual knowledge, the integration of the multiple representations of functions, and the links created to other mathematical content areas and the real world were considered. It was found that the German textbook included a higher percentage of content that promoted the development of conceptual knowledge. This was especially due to the level of cognitive demand of tasks included in the analysed textbook chapters. Also, while the South African textbook presented a wider range of opportunities to interact with the different representations of functions, the German textbook, on the other hand, included more links to the real world. Both textbooks linked ‘functions’ to other mathematical content areas, although the German textbook included a wider range of linked topics. It was concluded that learners from the two streams are thus exposed to different affordances to learn mathematics by their textbooks.

Kathryn Mellor

Robyn Clark

Anthony A. Essien

Affordances for learning linear functions: A comparative study of two textbooks from South Africa and Germany

线性函数学习的功能性：南非和德国两本教材的比较研究

After a long six-year lapse, the Curriculum and Assessment Policy Statement introduced in 2012 included geometry as part of the South African Grade 12 Mathematics Paper 2. The first cohort of matriculation students wrote Paper 2 in 2014. This article reports on the understanding of geometry terminology with which a group of 154 first-year mathematics education students entered a rural South African university in 2015; 126 volunteered to be part of the study. Responses to a 60-item multiple-choice questionnaire (30 verbally presented and 30 visually presented items) in geometry terminology provided the data for the study. A concept’s verbal description should be associated with its correct visual image. Van Hiele theory provided the lens for the study. An overall percentage mean score of 64% obtained in the test indicated that the majority of the students had a fairly good knowledge of basic geometry terminology. The students obtained a percentage mean score of 68% on visually presented items against that of 59% on verbally presented items implying a lower level thinking as per Van Hiele theory. The findings of this study imply a combination approach using visual and verbal representations to enhance conceptual understanding in geometry. This has to be complemented and supplemented through scaffolding to fill student teachers’ content gap.

Jogymol Alex

Kuttickattu J. Mammen

Students’ understanding of geometry terminology through the lens of Van Hiele theory

学生通过范希尔理论理解几何术语

This article explores a conceptual relationship between learner choice and mathematical sense-making. It argues that when learners can exercise choice in their mathematical activities, mathematical sense-making can be enhanced. The literature around mathematical modelling suggests a link between sense-making and learner choice. A three-tiered conceptual analysis allowed ‘purposiveness to thinking’ from the author through engagement with selected literature. Research questions related to a three-tiered analysis: generic, context-specific, and conditional accounts of sense-making in mathematics classrooms were formulated. The analysis resulted in a framework showing how sense-making may be constrained or enhanced in mathematics classrooms through learner choice. This article may add to our holistic understanding of sense-making in mathematics classrooms. It may contribute to mathematics teacher education by proposing that teachers are resourced to facilitate learners’ conceptual and procedural choice in primary or secondary mathematics classrooms.

Piera Biccard

Mathematical sense-making through learner choice

通过学习者选择进行数学意义建构

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。

Journals	Pythagoras
ISSN	1012-2346, 2223-7895
h-index	12
Journal Impact	0.44
Quartiles(Global)	EDUCATION, SCIENTIFIC DISCIPLINES(Q4)