Truth and the renewal of knowledge: the case of mathematics education

Mathematics education research must enable adjustment to new conditions. Yet such research is often conducted within familiar conceptualisations of teaching, of learning and of mathematics. It may be necessary to express ourselves in new ways if we are to change our practices successfully, and potential changes can be understood in many alternative, sometimes conflicting, ways. The paper argues that our entrapment in specific pedagogic forms of mathematical knowledge and the styles of teaching that go with them can constrain students’ engagement with processes of cultural renewal and changes in the ways in which mathematics may be framed for new purposes, but there are some mathematical truths that survive the changing circumstances that require us to update our understandings of teaching and learning the subject. In meeting this challenge, Radford encountered a difficulty in framing notions of mathematical objectivity and truth commensurate with a cultural–historical perspective. Following Badiou, this paper distinguishes between objectivity, which is seen necessarily as a product of culturally generated knowledge, and truth, as glimpsed beyond the on-going attempt to fit a new language that never finally settles. Through this route, it is shown how Badiou’s differentiation of knowledge and truth enables us to conjure more futuristic conceptions of mathematics education.     

Preservice Elementary Teachers’ Voices Describe How Their Teacher Motivated Them to do Mathematics

In this study, data in the form of (preservice teacher) student voices taken from mathematical autobiographies, written at the beginning of the semester, and end-of-semester reflections, were analyzed in order to examine why preservice elementary school teachers were highly motivated in a social constructivist mathematics course in which the teacher emphasized mastery goals. The findings suggest that students entered the course with a wide variety of feelings about mathematics and their own mathematical ability. At the end of the semester, students wrote about aspects of the course that “led to their growth as a mathematical thinker and as a mathematics teacher…” Student responses were coded within themes that emerged from the data: Struggle; Construction of meaning [mathematical language; mathematical understanding]; Grouping [working in groups]; Change [self-efficacy; math self-concept]; and the Teacher’s Role. These themes are described using student voices and within a motivation goal theory framework. The role of struggle, in relation to motivation, is discussed.     

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.     

Applying Lakatos' theory to the theory of mathematical problem solving

In this paper, the relation between Lakatos' theory and issues about mathematics education — especially issues about mathematical problem solving — is reinvestigated by paying attention to Lakatos' methodology of a scientific research programme. By comparing the same findings about mathematical problem solving with the discussion in Lakatos' theory — e.g. research programmes' hard cores, their negative and positive heuristics, and their goals — we establish the correspondence between research programmes and solver's structures of a problem situation, i.e. structures given by a solver to a problem situation. After establishing this, the implications of Lakatos' theory, i.e. the nature of selection from competing programmes and the social origins of the cores of programmes, are applied to the discussion about mathematical problem-solving, with indications of the related evidence in the theory of mathematical problem solving which seems to support the application of those implications. Such an application leads to one view of mathematical problem solving, which reflects the irrational nature and social aspects of problem-solving activities, both in solving problems and in selecting better solutions.     

Investigating Forms of Children's Writing in Grade 7 Mathematics Classrooms

Recent changes in mathematics curricula, both in South Africa and elsewhere, have begun to change the overwhelmingly symbolic nature of mathematics in schools (in the sense of use of mathematical symbolism), promoting more use of the oral and written language. Engaging students in `Writing-to-Learn' activities in mathematics classrooms has been identified and claimed by various mathematics education researchers as having a positive impact on the learning of mathematics. In this paper, I report on a piece of research, which is part of a broader study, on forms of mathematical writing and written texts produced by learners in grade 7 (12–13-year-olds) classes in six junior high schools in KwaZulu-Natal, in South Africa.     

The Dialectic Relationship between Research and Practice in Mathematics Teacher Education

This paper addresses issues linking research in student teacher learning with reflection on practice in mathematics teacher education. From a situated perspective on learning and practice, we explore our own practice as teacher educators while researching student teacher learning in our classrooms. We describe a study on student teacher learning, considering student teacher learning as a “process of becoming”, and how the results of this research have affected our development as mathematics teacher educators and members of a community of inquiry. Our work shows how in the mathematics teacher education context the relationship between theory and practice becomes an element of both teacher educator and researcher development.     

Subjectivity and cultural adjustment in mathematics education: a response to Wolff-Michael Roth

In this volume, Wolff-Michael Roth provides a critical but partial reading of Tony Brown’s book Mathematics Education and Subjectivity. The reading contrasts Brown’s approach with Roth’s own conception of subjectivity as derived from the work of Vygotsky, in which Roth aims to “reunite” psychology and sociology. Brown’s book, however, focuses on how discourses in mathematics education shape subjective action within a Lacanian model that circumnavigates both “psychology” and “sociology”. From that platform, this paper responds to Roth through problematising the idea of the individual as a subjective entity in relation to the two perspectives, with some consideration of corporeality and of how the Symbolic encounters the Real. The paper argues for a Lacanian conception of subjectivity for mathematics education comprising a response to a social demand borne of an ever-changing symbolic order that defines our constitution and our space for action. The paper concludes by considering an attitude to the production of research objects in mathematics education research that resists the normalisation of assumptions as to how humans encounter mathematics.     

The Art of Mathematics: Bedding down for a new era

Comparisons made between art and mathematics so often centre on the beauty of mathematics and how its forms might be seen as aesthetically pleasing. Yet the prominence of beauty as an attribute is less prevalent in contemporary art. Rather, art has a much broader scope of concern, perhaps with a greater emphasis on providing apparatus through which we might better understand who we are. This paper considers some performative aspects of contemporary art and draws parallels with some examples of mathematical activity within educative contexts. It argues that the performative dimension of mathematics is underemphasised in school activity and that the social and linguistic conditioning of mathematics within performance is a crucial aspect of the discipline being addressed in school and vocational courses. In particular, proficiency with concretisations of mathematics and the social dynamics that attend these is integral to the broader proficiency of moving between concrete and abstract domains.     

What does Social Semiotics have to Offer Mathematics Education Research?

Social Semiotics, based on the work of the linguist Michael Halliday, emphasises the ways in which language functions in our construction and representation of our experience and of our social identities and relationships. In this paper, I provide an introduction to the theory and its analytic tools, considering how they can be applied in the field of mathematics education. Some research questions that may be raised and addressed from this perspective are identified. An illustrative example is offered, demonstrating a social semiotic approach to addressing questions related to construction of the nature of school mathematical activity in writing produced by secondary school students.     

An Ode to Imre Lakatos: Quasi-Thought Experiments to Bridge the Ideal and Actual Mathematics Classrooms

This paper explores the wide range of mathematics content and processes that arise in the secondary classroom via the use of unusual counting problems. A universal pedagogical goal of mathematics teachers is to convey a sense of unity among seemingly diverse topics within mathematics. Such a goal can be accomplished if we could conduct classroom discourse that conveys the Lakatosian (thought-experimental) view of mathematics as that of continual conjecture-proof-refutation which involves rich mathematizing experiences. I present a pathway towards this pedagogical goal by presenting student insights into an unusual counting problem and by using these outcomes to construct ideal mathematical possibilities (content and process) for discourse. In particular, I re-construct the quasi-empirical approaches of six!4-year old students’ attempts to solve this unusual counting problem and present the possibilities for mathematizing during classroom discourse in the imaginative spirit of Imre Lakatos. The pedagogical implications for the teaching and learning of mathematics in the secondary classroom and in mathematics teacher education are discussed.     

Participating in Classroom Mathematical Practices

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students' diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.     

The computer as a cultural influence in mathematical learning

My starting point in this paper is that there is a cultural gap between the mathematics that children do as part of their everyday experience and the mathematics that they learn at school; my thesis is that the computer has (perhaps uniquely) the potential to bridge this divide. The paper will examine the cultural impact-both actual and potential-of the computer on children's mathematical education; at the ways in which the introduction of the computer does and will changes the ambient space in which children learn mathematics.I begin with a brief discussion of the cultural context of mathematics learning and the relationship between informal, everyday mathematical activity, and formal, school mathematies. This perspective leads to a closer examination of what it means to do mathematics, and on the relationship of a technology to the mathematics embedded within a given culture. I discuss the issue of injecting meaning into mathematical activity, and then examine some ways in which the computer might offer a solution to this central problem. Next, I give some examples of the influence of the computer on the culture of the mathematics classroom. Finally, I suggest some of the outstanding issues of research and curriculum development which remain.This paper is based on substantially the same data as is discussed in an article inCultural Dynamics.     

Beyond Motivation: Exploring Mathematical Modeling as A Context for Deepening Students' Understandings of Curricular Mathematics

Views of mathematical modeling in empirical, expository, and curricular references typically capture a relationship between real-world phenomena and mathematical ideas from the perspective that competence in mathematical modeling is a clear goal of the mathematics curriculum. However, we work within a curricular context in which mathematical modeling is treated more as a venue for learning other mathematics than as an instructional goal in its own right. From this perspective, we are compelled to ask how learning of mathematics beyond modeling may occur as students generate and validate mathematical models. We consider a diagrammatic model of mathematical modeling as a process that allows us to identify how mathematical understandings may develop or surface while learners engage in modeling tasks. Through examples from prospective teachers' mathematical modeling work, we illustrate how our diagrammatic model serves as a tool to unpack the intricacies of students’ mathematical experience while engaging in modeling tasks.     

An awareness of epistemological assumptions: the case of gender studies

Perceptions of differences in the participation and achievement of girls and boys in school science and mathematics have given rise to considerable curriculum research and professional development. This research, and the practices arising from it, has aimed at increasing the participation of girls in ‘non‐traditional’ school subjects, with the ultimate goal of enhancing girls’ post‐school options. Mathematics and science have been seen as critical in this respect, particularly the higher levels of mathematics and the physical sciences.

This research and professional development have contributed to our understanding of the issues and have no doubt affected the post‐school options of some girls. However, there is also dissatisfaction at the lack of significant change that has flowed from this work. A more critical analysis is now emerging, leading to a deeper questioning of the assumptions that underlie research and development in the field. Ironically, at the same time there are reports that funding for research into aspects of gender issues is now taking an even lower profile than in recent years (Tisdall 1992).

This paper has emerged from debates relating to gender issues from the perspective of two teachers, one of science and the other of mathematics. The debates were precipitated by a move from work as school teachers to work in higher education, where there is an explicitly stated responsibility for research as well as teaching. The change in our labour served to highlight personal questions about the legitimation of different accounts of education in research, science and mathematics education and gender issues. It was in the process of grappling with the apparent plethora of research methods that we began to look at the epistemological assumptions of research into gender issues in science and mathematics education.

For us then, there are three major interrelated and overlapping areas of concern. One is related to the epistemological assumptions that pervade science and mathematics in schools. The second also concerns epistemological assumptions, those relating to educational research, particularly into gender issues. The third raises the ontological question of the conceptualization of gender within the research. We needed a conceptual framework to enable us to critique the normalization of the debates and uncover the problematics within each of these three concerns. At the same time, the framework had to allow an examination of the interrelatedness and continuities between them. It was the normalization and apparent consensus surrounding these issues that was disturbing for us. This we recognized as a depoliticization of social institutions which are essentially political. It was against this background, and in acknowledgement of some feminist critiques (Fraser 1989), that we use a framework developed by Habermas (1972), which recognizes the political dynamic of epistemological positions, and allows us to make explicit the politics of educational research.

Our starting point, in this paper, is Habermas's work. We outline the three ‘knowledge‐constitutive’ interests of this schema, and describe their implications for education. We then explore the dominant perspective within science and mathematics education. Having described this context, we examine the gender research with particular attention to the conceptualization of gender. With reference to Habermas's framework the epistemological and ontological assumptions of this research are explored, and the possibilities and limitations are discussed. This is used to examine the epistemological basis of different research methodologies.

We contend that there is a relationship between the scientific paradigm, the organization of education and the framing of research. Further this relationship has constituted the field of research about gender and science and mathematics and limited its potential for explanation.     

Shedding light on and with example spaces

Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct of example spaces and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our way of working, we delay definition until after appropriate reader experience has been brought to the surface and several ‘examples’ have been discussed. Of special interest is the notion of accessibility of examples: an individual’s access to example spaces depends on conditions and is a valuable window on a deep, personal, situated structure. Through the notions of dimensions of possible variation and range of permissible change, we consider ways in which examples exemplify and how attention needs to be directed so as to emphasise examplehood (generality) rather than particularity of mathematical objects. The paper ends with some remarks about example spaces in mathematics education itself.     

数学全息方法及其在数学科研和教学中的应用

宇宙全息律作为客观世界的一种普遍规律,被反映到数学这种人们认识世界的基本工具上,形成数学科学的一种普遍规律,即数学全息律.以数学全息元为起点,利用数学全息律进行数学科研与教学的方法就是数学全息方法.利用周期研究时间函数和利用“生成元”研究数学结构等均是应用数学全息方法的生动事例;同时,数学全息方法还是促进数学创造的有力工具.数学全息方法在数学教学中也有着广泛的应用.     

Mathematics-for-Teaching: an Ongoing Investigation of the Mathematics that Teachers (Need to) Know

In this article we offer a theoretical discussion of teachers' mathematics-for-teaching, using complexity science as a framework for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching. We label these aspects “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”. Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for teachers.     

Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students’ problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.