Dialogue on mathematics education: two points of view on the state of the art

On many fronts, the field of mathematics education does not speak with a single voice. There appears to be no firm consensus regarding the scientific character of mathematics education, the research methodologies it deems legitimate, the kinds of questions it addresses, the appropriate preparation for its practitioners, and its relationship with other disciplines, including, ironically, mathematics itself. Our field seems to be going through a new phase of self-definition, a crisis from which we shall have to decide who we are and what direction we are going. The authors of the present paper themselves tend towards different positions on these questions. The paper, then, takes the form of a letter in which one of us raises issues about the current state of mathematics education and the other responds. We see this as an attempt to initiate a dialogue on our field, which we consider urgently needed.     

Creativity and school mathematics: some modest observations

In spite of its importance, research into mathematical creativity has not received the same attention as research into student difficulties in learning mathematics. In this paper, I review some of the history, policies and research in this area from the last 40 years, and then critique the papers in this issue, noting progress that has been made in the field.     

Mathematical art and artistic mathematics

ABSTRACT

The purpose of this note is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of geometry and algebra.     

Visualization and art in the mathematics classroom

In this paper we summarize our concepts and practice on computer-aided mathematical experimentation, and illustrate them byMathematica projects that we have developed for our research and the courses “Computer-aided mathematical modelling” and “Computer Algebra I–II” held for students of life sciences at University of Szeged and computational engineering at TFH Berlin, University of Applied Sciences.     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

Economy: the absent centre of mathematics education

Social and political turns in mathematics education research have brought into the field postmodern theorisations that researchers have been using to dismantle traditional philosophies of mathematics, to posit mathematics in the sociocultural terrain, and to spell out the role mathematics has in school exclusion. Sociopolitical perspectives constitute a privileged field of research to address the influence of economy on mathematical achievement. However, instead of investigating the role of economy in students’ achievement, sociopolitical studies have been contributing to a disavowal of the economic dimension of school mathematics. This paper synthesises a set of investigations carried out by the author in the last 5 years endeavouring to posit mathematics education in the political and economic spectrum of our time. It takes advantage of the contemporary combination of Hegel’s dialectics, Lacanian psychoanalysis and Marx’s critique of political economy, carried out by Slavoj ?i?ek, to develop a critique of the way research within the so-called ‘sociopolitical turn’ deals with the issue of equity; and marks out the contours of mathematics education’s ideological belonging.     

Metacognition and mathematics education

The role of metacognition in mathematics education is analyzed based on theoretical and empirical work from the last four decades. Starting with an overview on different definitions, conceptualizations and models of metacognition in general, the role of metacognition in education, particularly in mathematics education, is discussed. The article emphasizes the importance of metacognition in mathematics education, summarizing empirical evidence on the relationships between various aspects of metacognition and mathematics performance. As a main result of correlational studies, it can be shown that the impact of declarative metacognition on mathematics performance is substantial (sharing about 15–20% of common variance). Moreover, numerous intervention studies have demonstrated that “normal” learners as well as those with especially low mathematics performance do benefit substantially from metacognitive instruction procedures.     

Teachers learning the registers of mathematics and mathematics education in another language: an exploratory study

Many mathematics teachers around the world teach in a language different from the one in which they studied or completed their teacher education. Often these teachers must learn both the registers of mathematics and of mathematics education to teach in the additional language. This paper examines the factors that help teachers to learn these registers in Māori, the Indigenous language of New Zealand. Many of these teachers are second-language learners of the Māori language and attended English-medium schools and teacher-education programmes. After a brief discussion about the key role of language in teaching mathematics, this paper examines data from teachers at two Māori-immersion schools and a professional development facilitator. The analysis provides initial understanding of the factors that support or hinder their learning of the mathematics registers. Finally, a research agenda is suggested for further investigation of this issue.     

Interval linear systems: the state of the art

Summary  Linear systems represent the computational kernel of many models that describe problems arising in the field of social, economic as well as technical and scientific disciplines. Therefore, much effort has been devoted to the development of methods, algorithms and software for the solution of linear systems. Finite precision computer arithmetics makes rounding error analysis and perturbation theory a fundamental issue in this framework (Higham 1996). Indeed, Interval Arithmetics was firstly introduced to deal with the solution of problems with computers (Moore 1979, Rump 1983), since a floating point number actually corresponds to an interval of real numbers. On the other hand, in many applications data are affected by uncertainty (Jerrell 1995, Marino & Palumbo 2002), that is, they are only known to lie within certain intervals. Thus, bounding the solution set of interval linear systems plays a crucial role in many problems. In this work, we focus on the state of the art of theory and methods for bounding the solution set of interval linear systems. We start from basic properties and main results obtained in the last years, then we give an overview on existing methods.     

Arnold Kirsch and mathematics education

This article pays tribute to the German mathematics educator Arnold Kirsch (1922–2013), especially for his contributions to calculus education. The main aim of his work was to make mathematics accessible to learners so that they are able to genuinely understand the subject.     

Conceptual mathematics: An application to education

This paper presents an alternative proposal concerning the teaching of mathematics. The present paper can be placed within the broader framework of the teaching of mathematics, but also within the more specific framework of category theory (CT). In other words, new ways will be investigated in which CT can be best developed within the broader framework of the teaching of mathematics. Following the research at the end of this paper, the outcome of this investigation is that CT can successfully be used as a background for the foundation and teaching of mathematics.     

Interviewing in mathematics education research: Choosing the questions

With the development of qualitative methodologies, interviewing has become one of the main tools in mathematics education research. As the first step in analyzing interviewing in mathematics education we focus here on the stage of planning, specifically, on designing the interview questions. We attempt to outline several features of interview questions and understand what guides researchers in choosing the interview questions. Our observations and conclusions are based on examining research in mathematics education that uses interviews as a data-collection tool and on interviews with practicing researchers reflecting on their practice.     

Research on mathematics education

This study addresses aspects that should be considered in every investigation concerning the reality of the subject being investigated, which in turn provide the basis for the procedures adopted to carry out the research. It speaks about the analysis of the procedures chosen to carry out the research. It is assumed that this care should be taken by the researcher at the moment the research procedures are being defined and made explicit. It is argued that the consonance between the ontological and epistemological dimensions of “what” and “how” to investigate the subject of investigation confers a degree of confidence to the research findings. The search for that confidence transcends analyses based only on calculations and explanations of methodological procedures, regardless of how well founded they are. This study addresses mathematics education specifically, adopting a phenomenological perspective. It is focused on the constitution of mathematical idealities and of mathematics as a science under the perspective of the Husserlian phenomenological conception of reality and knowledge. Characteristics of a phenomenological pedagogy are presented, which is carried out through work that is always intentional, with the educator taking account of what occurs with himself/herself, with the life world of the school, and with the student. The student is seen as a person and as being with others, his/her classmates, and the theme is addressed in the context of the field of inquiry under focus, with the teacher and with his/her “surroundings”.