Analyzing Mathematical Teaching-Learning Situations — the Interplay of Communicational and Epistemological Constraints

This is a commentary paper in the volume on “Teachings situations as object of research: empirical studies within theoretical perspectives”. An essential object of mathematics education research is the analysis of interactive teaching and learning processes in which mathematical knowledge is mediated and communicated. Such a research perspective on processes of mathematical interaction has to take care of the difficult relationship between mathematics education theory and everyday mathematics teaching practice. In this regard, the paper tries to relate the development in mathematics education research within the theory of didactical situations to developments in interaction theory and in the epistemological analysis of mathematical communication.     

Rate your course! Student teachers’ perceptions of a primary pre-service mathematics education programme

Although research suggests that many pre-service mathematics education programmes are weak interventions having a negligible effect on student teachers’ knowledge, beliefs and attitudes, there is consensus that programmes that model and engage student teachers in reform teaching and learning approaches have the potential to effect positive change in attitudes and knowledge of mathematics. This paper focuses on the perceptions of 345 student teachers regarding their mathematics education programme. The authors consider that in order to further our understanding of the characteristics of an effective pre-service mathematics education programme, the opinions of participating student teachers are fundamental. Participants reported improved mathematical subject matter knowledge and pedagogical content knowledge in addition to greater confidence and decreased anxiety towards mathematics. Programme features such as small group tutorials and interactive engagement were identified as impacting this positive change.     

Knowledge for Teaching Mathematics: some student teachers' views

ABSTRACT

This investigation examines the views of pedagogical content knowledge of 12 student teachers of mathematics drawn from three different training groups. Attitudes towards and understandings of pedagogical content knowledge were elicited using six Likert‐item statements whilst information on students' learning was obtained through open‐ended questioning. Differences between groups were apparent only in tendencies to give extreme responses and to adopt outlying positions. These differences may be attributable to differences in values or prior knowledge and to training parallels with the social construction of mathematical meaning. Indications of the acquisition of pedagogical content knowledge appeared in all training groups and seemed to be linked to a hierarchical view of mathematics; possible limitations for student learning are implied. Students' views also highlight the importance of reflection upon subject matter knowledge and awareness of learning processes when transforming mathematical ideas for teaching purposes. It is suggested that the role of prior learning is worthy of further investigation.     

On mathematics teaching for future engineers—in the digital era

Current changes, especially the wide application of information technology, in all fields of our life, mean that mathematical knowledge becomes necessary in almost every domain. It implies new expectations for mathematical education. An urgent need of a new mathematical literacy for all—also a new mathematical literacy for engineers—is evident. It is necessary to consider a process of mathematics learning at tertiary level from the epistemological perspective and to investigate students’ ways of mathematical thinking. This epistemological knowledge is especially indispensable when students use information technology. In this article current requirements regarding mathematical education are discussed, especially those for future engineers. Analysis of examples of learning elementary statistics, using graphing calculators as supporting tools, leads to the formulation of essential aims for mathematics educators concerning mathematics teaching for future engineers.     

Teacher characteristics and contextual factors: links to early primary teachers’ mathematical beliefs and attitudes

This study examines how various teacher characteristics and contextual factors are related to early primary teachers’ beliefs about mathematical teaching and learning and teachers’ attitudes toward their own learning of mathematics. A total of 396 early primary teachers across Nebraska participated in the study. Teacher characteristics and contextual factors were grouped into four sets: teacher professional background, teacher mathematical knowledge for teaching, teaching contexts, and students’ experiences. Multiple regression analyses were conducted with each set of predictors separately, as well as with all four sets together. The results showed significant relationships between teachers’ mathematical knowledge for teaching and teacher-centered beliefs, motivation in learning mathematics, and anxiety toward learning mathematics. Teacher certification level, the number of college math courses taken, and perceived support from colleagues and administrators were also related to some aspects of teachers’ mathematical beliefs and attitudes. The findings suggest the potential role of teachers’ mathematical knowledge for teaching in improving teachers’ mathematical beliefs and attitudes.     

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.     

梳·理·化,三段式复习结构探微

复习课教学是根据学生的认知特点和规律,在学生学习数学的某一阶段,以巩固、梳理已学知识、技能,促进知识系统化,提高学生的学习能力和解决实际问题的能力,是小学数学教学中的重要课型之一,在数学教学中占有重要的地位。在复习教学中,要引导学生自主"梳"、教师点拨"理"、练习中"化",完善学生的认知结构与自我学习能力,发展学生的数学综合素养。     

Connecting with the horizon: developing teachers’ appreciation of mathematical structure

A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’ reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support their students’ present and future learning of mathematics. Further research is needed about professional learning approaches and tasks that may enable teachers to imbed and develop awareness of structure in their practice.     

Teachers' Interpretations of Effective Whole-class Interactive Teaching in Secondary Mathematics Classrooms

Prompted by ongoing concerns over standards of numeracy, eight mathematics teachers from four secondary schools in south Wales formed a teacher inquiry group to research the impact of introducing whole-class interactive teaching strategies into their own practice. Possible teaching approaches and activities were developed and agreed during group meetings. These were trialled by the teachers during their normal mathematics lessons. Lesson observations, interviews with teachers and discussions at the group meetings, however, indicated that the implementation of the approaches varied significantly across teachers. This paper analyses the variation in interpretations and teaching styles, and discusses their impact on the quality of the classroom discourse. The determining features discerned within the classroom dialogue included the extent to which pupils were encouraged to reflect on their mathematical knowledge, and the ways in which teachers were able to scaffold pupils' learning.     

学生所知道的 教师所知道的——理论、工具和结果

评估——无论采用何种形式——不得不建立在对不同水平和质量的理论思考基础之上。其中至关重要的是测验内容领域要依据教与学的过程来构建。其次,任务是评估相关内容结构的核心工具。因此,在设计对数学的教育评估时,按照合理的标识和特征对任务进行分类就成为一个息息相关的问题,也是一个巨大的挑战。本文重点研究PISA(国际学生评估项目的缩写,是以数学素养为目标的国际比较研究,包括在国家层面德国所做的许多调整)和COACTIV(德国教师数学知识和数学教学知识的一个代表性研究)。两者都旨在实现区分性评估(differentiated assessment),都考虑了评价内容和任务分析的不同方面。区分性评估结果使得我们能够获取有用的信息,进一步推动课堂教与学的发展。这些信息主要表现为两个方面:评估学生不同类型的知识,这些知识类型直接指向课堂教学与学习中存在的各种问题;探索教师知识对学生学习进展的影响,表明哪种教学行为是有效的。     

What Is a “Semiotic Perspective”, and What Could It Be? Some Comments on the Contributions to This Special Issue

This comment attempts to identify different “semiotic perspectives” proposed by the authors of this special issue according to the problems they discuss. These problems can be distinguished as problems concerning the representation of mathematical knowledge, the definition and objectivity of meaning, epistemological questions of learning and activity in mathematics, and the social dimension of sign processes. The contributions are discussed so as to make visible further research perspectives with regard to “semiotics in mathematics education”.     

数学应用题教学之我见

应用题是数学的重要组成部分,也是教学的重点和难点,学生应用题解题能力可以在一定程度上代表数学知识掌握水平。教师应把教授学生怎样解答应用题作为教学重点,巩固学生的数学基础知识,培养学生的数学思维和逻辑思维能力。文章对数学教学中教师怎样对学生进行应用题教学进行论述。     

重视数学阅读,打造深度学习课堂

许多小学数学教师已然认识到了开展小学数学阅读活动和制定科学的教学方法,对提高学生的计算能力和思维能力的重要作用。所以,教师在教学过程中要积极探索科学的教学方法培养学生的有效阅读能力,这样学生才能够更好地理解和学习数学知识,提高学生的解题思维能力和学习有效性。以下便对数学阅读活动的开展和深度学习课堂的构建进行分析和探究。     

有效融入多媒体手段 提升数学课堂效率

如今,多媒体已经成为日常教学的重要手段,能够有效促进教学水平的提高。随着新课程改革的深入推进,将多媒体手段与课堂教学相结合越来越受到教师的重视。在高中数学课堂上运用多媒体技术可以将抽象的数学知识形象化,将烦琐的数学定理简单化,改变学生被动学习的状态,使学生可以更好地投入数学学习中。本文将从五个方面进行论述,旨在构建高效的数学课堂。     

Epistemological investigation of classroom interaction in elementary mathematics teaching

In everyday teaching, the mathematical meaning of new knowledge is frequently devalued during the course of ritualized formats of communication, such as the funnel pattern, and is replaced by social conventions. Problems of understanding occurring during the interactively organized elaboration of the new knowledge require an analysis of the interplay between the social constraints of the communicative process and the epistemological structure of the mathematical knowledge. Specific aspects of the problem of meaning development are investigated in the course of two exemplary second-grade teaching episodes. These are then used to develop and discuss decisive requirements for the maintenance of an interactive constitution of meaning for mathematical knowledge.     

Engaging with issues of emotionality in mathematics teacher education for social justice

This article focuses on the relationship between social justice, emotionality and mathematics teaching in the context of the education of prospective teachers of mathematics. A relational approach to social justice calls for giving attention to enacting socially just relationships in mathematics classrooms. Emotionality and social justice in teaching mathematics variously intersect, interrelate or interweave. An intervention, using creative action methods, with a cohort of prospective teachers addressing these issues is described to illustrate the connection between emotionality and social justice in the context of mathematics teacher education. Creative action methods involve a variety of dramatic, interactive and experiential tools that can promote personal and group engagement and embodied reflection. The intervention aimed to engage the prospective teachers with some key issues for social justice in mathematics education through dialogue about the emotionality of teaching and learning mathematics. Some of the possibilities and limits of using such methods are considered.     

Approaches to teaching in thematic work: early childhood teachers’ integration of mathematics and art

Thematic work that integrates different knowledge areas is considered suitable for developing young children’s knowledge and skills in early childhood education. This paper reports evidence from a survey of early childhood teachers’ work with mathematics and art integrated in thematic work. In this study, we aim to explore how teachers perceive mathematics learning and teaching integrated with art. Data for analysis were collected from a questionnaire answered by 27 early childhood teachers from Finland and Sweden. Influenced by a sociocultural theoretical framework, the analysis concerns the teachers’ expressed goals for learning and their strategies for achieving these goals. The analysis reveals qualitatively different approaches to the learning goals addressed by the teachers – process-oriented, product-oriented, development-oriented and providing awareness – which, respectively, seem to generate different meanings of mathematical learning integrated with art. This study offers an overview of teachers’ approaches to teaching mathematics and art in thematic work, and of how different approaches may influence what mathematical meaning is offered to children to explore.     

Academic continuity through online collaboration: mathematics teachers support the learning of pupils with chronic illness during school absence

Recent applications of technology to mathematics education have been designed with cognitive and constructivist theoretical perspectives in mind, viewing mathematical learning as the acquisition of knowledge through the construction of meanings and connections between concepts. With the advent of increasingly flexible communication technologies, there is both the need and opportunity to consider how they might be utilised, particularly since emergent socio-cultural theories advocate learning in mathematics as an inherently social activity where understanding is developed and negotiated collaboratively. The need to examine effective technology-facilitated learning arose in the context of a research project, currently underway in a number of secondary schools in the state of Victoria and funded by the Australian Research Council. It is investigating the learning needs of pupils who are absent from school for prolonged or intermittent periods owing to chronic illness yet continue with their school studies. An emerging understanding of the significant difference between computer-mediated contact for mere information exchange and communication for teaching and learning has led to a consideration of socio-cultural perspectives on effective mathematical learning and a focussed investigation of technologies able to facilitate them. Early data have demonstrated the potential of videoconferencing, online whiteboarding and interactive whiteboard application sharing, but which require particular resources, aligned infrastructure and teacher support. This article explores issues surrounding the use of such technologies for collaborative mathematical learning in a context where online interaction is being considered for the learning support of pupils unable to attend school.     

美国《中学数学教与学》教材的特点及其对中国数学教师教育的启示

美国《中学数学教与学》是初中年级数学教师教育"数学教学法"课程的教材,其特点表现为:单元主题以数学教学理论为线索呈现,单元内容以具体数学知识为载体展开并且以参与式活动的形式组织教学活动.该教材对我国的数学教师教育有如下启示:数学教师教育课程应以数学知识为载体学习数学教育理论;数学教师教育教材应突出教学方式的示范性,体现做中学,在数学教学活动中学习数学教学方法;数学教师教育应注重教师学科专业素质的提升.     

外语情感教育中的沟通策略

外语学习既是一个认知过程，又是一个情感过程。它们互相联系、互相作用和互相制约，是一个和谐的统一体。本讨论了情感对外语学习的重要作用以及外语情感教育中有效的沟通策略。