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.     

Critical Conversations on Social Justice in Undergraduate Mathematics

Abstract

This article explores how critical conversations engage undergraduate mathematics faculty in a community of practice that enhances their knowledge about teaching and learning mathematics for social justice. More broadly, critical conversations are defined as a cooperative learning strategy that can be used to identify, explore, and respond to various interests and issues situated across differing values and beliefs. We present a case study of a critical conversation that took place at a 2016 Mathematics for Social Justice workshop organized by a group of junior faculty. Participant reflections situate perspectives that can help novice and experienced instructors design conversations about teaching mathematics for social justice. Specifically, individual and group reflections highlight the importance of: (i) framing and reflecting on the conversation; (ii) exploring implications and content connections; and (iii) identifying barriers. Implications for faculty members and mathematics departments are provided.     

中小学"数学情境与提出问题"教学的实验研究

This research tends to make the experimental study on the mathematics teaching model of “situated creation and problem-based instruction” (SCPBI), namely, the teaching process of “creating situations—posing problems—solving problems—applying mathematics”. It is aimed at changing the situation where students generally lack problem-based learning experience and problem awareness. Result shows that this teaching model plays a vital role in arousing students’ interest in mathematics, improving their ability to pose problems and upgrading their mathematics learning ability as well.      

The meaning of mathematics instruction in multilingual classrooms: analyzing the importance of responsibility for learning

In the multilingual mathematics classroom, the assignment for teachers to scaffold students by means of instruction and guidance in order to facilitate language progress and learning for all is often emphasized. In Sweden, where mathematics education is characterized by a low level of teacher responsibility for students’ performance, this responsibility is in part passed on to students. However, research investigating the complexity of relations between mathematics teaching and learning in multilingual classrooms, as well as effect studies of mathematics teaching, often take the existence of teachers’ responsibility for offering specific content activities for granted. This study investigates the relations between different aspects of responsibility in mathematics teaching and students’ performance in the multilingual mathematics classroom. The relationship between different group compositions and how the responsibility is expressed is also investigated. Multilevel structural equation models using TIMSS 2003 data identified a substantial positive influence on mathematics achievement of teachers taking responsibility for students’ learning processes by organizing and offering a learning environment where the teacher actively and openly supports the students in their mathematics learning, and where the students also are active and learn mathematics themselves. A correlation was also revealed between group composition, in terms of students’ social and linguistic background, and how mathematics teaching was performed. This relationship indicates pedagogical segregation in Swedish mathematics education by teachers taking less responsibility for students’ learning processes in classes with a high proportion of students born abroad or a high proportion of students with low socio-economic status.     

USING COMBINATORIAL APPROACH TO IMPROVE STUDENTS’ LEARNING OF THE DISTRIBUTIVE LAW AND MULTIPLICATIVE IDENTITIES

This article reports an alternative approach, called the combinatorial model, to learning multiplicative identities, and investigates the effects of implementing results for this alternative approach. Based on realistic mathematics education theory, the new instructional materials or modules of the new approach were developed by the authors. From the combinatorial activities based on the things around daily life, the teaching modules assisted students to establish their concept of the distributive law, and to generalize it via the process of progressive mathematizing. The subjects were two classes of 8th graders. The experimental group (n = 32) received a combinatorial approach to teaching by the first author using a problem-centered with double-cycles instructional model, while the control group (n = 30) received a geometric approach to teaching, from the textbook by another teacher who uses lecturing. Data analyses were both qualitative and quantitative. The findings indicated that the experimental group had a better performance than the control group in cognition, such as for the inner-school achievement test, mid-term examination, symbol manipulation, and unfamiliar problem-solving: also in affection, such as the tendency to engage in the mathematics activities and enjoy mathematical thinking.     

Motivating adults to learn mathematics in the workplace: a trade union approach

ABSTRACT

The aim of this qualitative study is to explore adults’ motivation to learn mathematics in the workplace and the role that the trade union education approach promoted in the United Kingdom plays in that motivation. The findings draw on data from 20 semi-structured in-depth interviews with adults learning mathematics, organised through their trade union representatives. Trade unions promote education which uses teaching and learning approaches based on collectivist and activist principles, which is different from mainstream education, so the research explores what might be learnt from this approach. The findings show that the trade union-led approach enables even long-held negative feelings towards mathematics to become positive. The research finds a strong link between supportive social networks, including Union Learning Representative, as well as positive social and emotional encounters in the classroom that develop adult learners’ confidence, increasing their motivation to both learn and use mathematics in their everyday lives. This change in feelings and motivation is termed an Affective Mathematical Journey. These findings, while taking place in a non-traditional context, nevertheless are relevant to practitioners working with adults in both traditional (school and college) and non-traditional (workplace and community) settings. There are also indications of the positive influence this learning has on the relationship between individual members and their trade union organisations.     

Assisting teachers and students to reform the mathematics classroom

This study examines the usefulness of selected aspects of Tharp and Gallimore's (1988) theory of assistance as a theoretical framework for describing and analyzing change efforts in a middle school mathematics reform project. Drawing upon Tharp and Gallimore's redefinition of teaching as assisting performance and learning as the result of assisted performance, the social organization of a school-based mathematics reform effort in which teacher educators, mathematics teachers, and students both assist and are assisted is analyzed. In addition, one particular classroom assistance activity is presented and analyzed in terms of characteristics of assistance that, according to the theory, should lead to significant learning.     

Explanations of attitudes to change: Colombian mathematics teachers’ conceptions of the crucial determinants of their teaching practices of beginning algebra

This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers’ conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers’ understandings of their teaching practices were explored with a view to unravelling their conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the powerful role that the teachers’ conceptions of social/institutional factors of teaching played in their conceptions of their practices. The degree to which the teachers attributed these (external) factors as crucial reasons for what they do in their teaching was the basis of a categorisation of their conceptions of the crucial determinants of their teaching practices into three types. The findings are particularly relevant to our understanding of the stability of mathematics teaching approaches in the Colombian context but have likely implications for a range of international education contexts. Specific implications for the development of the research into teachers’ conceptions of mathematics and its teaching, and for teacher education programmes are presented.

Mythmaking and mythbreaking in the mathematics classroom

Constructivism has become a major focus of recent pedagogical reform in mathematics education. However, epistemological reform that is based on the constructivist referent of learning as conceptual change has a very limited viability in traditional mathematics classrooms because of its cultural insensitivity. By contrast, the social epistemology of critical constructivism addresses the socio-cultural contexts of knowledge construction and serves as a powerful referent for cultural reform. From this perspective, the social reality of traditional mathematics classrooms is governed by powerful cultural myths that restrain the discursive practices of teachers and students. The power of the repressive myths of cold reason and hard control is evident in the ways in which they act in concert to create a highly coherent and seemingly natural social reality. Epistemological reform of traditional mathematics classroom learning environments is, therefore, synonomous with cultural reconstruction. Critical constructivism, which has a central concern with discourse ethics and the moral agency of the teacher, draws on the social philosophy of Jurgen Habermas and argues for an alternative culture of communicative action to be established in mathematics classrooms. Teachers are expected to work collaboratively as agents of cultural change in forums beyond their classrooms.Religions, philosophies, arts, the social forms of primitive and historic man, prime discoveries in science and technology, the very dreams that blister sleep, boil up from the basic, magic ring of myth.(Joseph Campbell, The Hero With a Thousand Faces, 1968, p. 8.)     

IDENTIFICATION AND ASSESSMENT OF TAIWANESE CHILDREN’S CONCEPTIONS OF LEARNING MATHEMATICS

The aim of the present study was to identify children’s conceptions of learning mathematics and to assess the identified conceptions. Children’s conceptions are identified by interviewing 73 grade 5 students in Taiwan. The interviews are analyzed using qualitative data analysis methods, which results in a structure of 5 major conceptions, each having 2 subconceptions: constructivist (interest and understanding), interpretivist (liberty and innovation), objectivist (academic goal and perseverance), nativist (confidence and anxiety (reverse)), and pragmatist (vocational goal and application). The conceptions are assessed with a self-developed questionnaire, titled “the Conception of Learning Mathematics Questionnaire” (CLMQ), which is administered to 513 grade 5 students in Taiwan and examined with a reliability measure, confirmatory factor analysis, and correlations with 2 criteria: mathematics achievement and approaches to learning mathematics. The results show that the CLMQ has desirable internal consistency reliability and construct validity. The conceptions are also sensibly in relation to the 2 criteria, suggesting that the CLMQ is a valid measure for evaluating the quality of children’s learning mathematics in relation to teaching contexts.     

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.     

‘It’s getting me thinking and I’m an old cynic’: exploring the relational dynamics of mathematics teacher change

Actor-network theory is a way of describing and understanding the complexity of social change. This article explores its relevance to understanding teacher change in mathematics education by considering a single teacher change narrative. This is centred on a veteran teacher of mathematics who participated in a teacher led, teacher-educator-supported professional development project. The project had two foci: investigating forms of school-based collaborative professional development in the context of developing a dynamic approach to teaching and learning geometry. Three conceptual tools appropriated or adapted from actor network theory are used to describe and analyse features of this teacher narrative. These are relationality, translation and fluidity. Some implications are considered for developing accounts of, and actions for, mathematics teacher change.     

The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish

Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí) in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and applications for mathematics teaching and learning are presented.

An Account of a Teacher's Perspective on Learning and Teaching Mathematics: Implications for Teacher Development

This report presents an account of one teacher's mathematics teaching and a perspective that underlies his teaching. Nevil was a fifth grade teacher participating incurrent mathematics education reforms in the United States. Through the account, we make distinctions about teachers' thinking and practice that can inform teacher education efforts. We constructed an account by analyzing four sets of classroom observations and interviews. We observed that Nevil decomposed his understandings of the mathematics into smaller components and connections among those components. He created situations that he believed made those components and connections transparent and attempted to elicit those connections from the students. This account illustrates a practice that is different both from traditional practice and the type of practice that we would envision as a goal for teacher development. We contribute two important aspects of mathematics teacher development from traditional to reform-oriented teaching. In particular, we describe teachers' perspectives – assimilatory structures that constrain and afford (a) the sense they make of professional development opportunities and (b) their potential learning in teacher education settings. This revised version was published online in August 2006 with corrections to the Cover Date.     

Students’ images and their understanding of definitions of the limit of a sequence

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students’ understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students’ images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the ɛ–strip activity. This paper illustrates how these students’ understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.     

A Framework for Analysis of Teachers’ Geometric Content Knowledge and Geometric Knowledge for Teaching

Current reform-driven mathematics documents stress the need for teachers to provide learning environments in which students will be challenged to engage with mathematics concepts and extend their understandings in meaningful ways (e.g., National Council of Teachers of Mathematics, 2000, Curriculum and evaluation standards for school mathematics. Reston, VA: The Council). The type of rich learning contexts that are envisaged by such reforms are predicated on a number of factors, not the least of which is the quality of teachers’ experience and knowledge in the domain of mathematics. Although the study of teacher knowledge has received considerable attention, there is less information about the teachers’ content knowledge that impacts on classroom practice. Ball (2000, Journal of Teacher Education, 51(3), 241–247) suggested that teachers’ need to ‘deconstruct’ their content knowledge into more visible forms that would help children make connections with their previous understandings and experiences. The documenting of teachers’ content knowledge for teaching has received little attention in debates about teacher knowledge. In particular, there is limited information about how we might go about systematically characterising the key dimensions of quality of teachers’ mathematics knowledge for teaching and connections among these dimensions. In this paper we describe a framework for describing and analysing the quality of teachers’ content knowledge for teaching in one area within the domain of geometry. An example of use of this framework is then developed for the case of two teachers’ knowledge of the concept ‘square’.     

Working for learning: teaching assistants developing mathematics for teaching

This article derives from a case study of 10 secondary school teaching assistants (TAs) who did not have conventional pre-qualifications in mathematics but who undertook an honours degree in mathematics education studies at a Higher Education Institution in England whilst continuing to work as TAs in school. Work-based learning was thus undertaken in parallel with advancement through the hierarchical undergraduate mathematics curriculum. Lave and Wenger’s work on communities of practice is used as a framework to explore the TAs’ learning of mathematics alongside their professional work in schools. This case illustrates how and where institution-based undergraduate teaching relates to work in school, and where it does not, thus signalling the importance of the TAs’ informal learning strategies in bringing together these experiences.