From solving equations to the meaning of life: mathematics, rationality and values

An attempt is made to suggest some ultimate goals for education beyond the narrow framework of the mathematics curriculum. In addition, there is an attempt to link mathematics education to these goals by pointing at some principles which direct mathematical behavior as well as educated behavior. The main claim is that both mathematical behavior and educated behavior are supposed to be directed by rational thinking.     

Word problems in mathematics education: a survey

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

     

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.     

Animating an equation: a guide to using FLASH in mathematics education

Macromedia's FLASH development system can be a great tool for mathematics education. This article presents detailed Flash tutorials that were developed and taught by the author to a group of mathematics professors in a summer course in 2005. The objective was to educate the teachers in the techniques of animating equations and mathematical concepts in Flash. The course was followed by a 2-year study to assess the acceptance of the technology by the teachers and to gauge its effectiveness in improving the quality of mathematics education. The results of that 2-year study are also reported here.     

From lessons to lectures: NCEA mathematics results and first-year mathematics performance

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.     

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).     

The origins of mathematics education research in the UK: a tribute to Brian Griffiths

The meeting of the British Society for Research into Learning Mathematics held at the University of Southampton on 21st June 2008 was dedicated to the memory of Brian Griffiths, who had died earlier that month. At the meeting it was suggested that I write a paper tracing the development of research in mathematics education in the UK up to the writing of Mathematics: Society and Curricula, a book that Brian and I co-authored, published in 1974. This paper is dedicated to Brian's memory: I hope that it will be considered a fitting tribute to a great friend and colleague, and to an outstanding mathematician and mathematics educator.     

The onto-semiotic approach to research in mathematics education

In this paper we synthesize the theoretical model about mathematical cognition and instruction that we have been developing in the past years, which provides conceptual and methodological tools to pose and deal with research problems in mathematics education. Following Steiner’s Theory of Mathematics Education Programme, this theoretical framework is based on elements taken from diverse disciplines such as anthropology, semiotics and ecology. We also assume complementary elements from different theoretical models used in mathematics education to develop a unified approach to didactic phenomena that takes into account their epistemological, cognitive, socio cultural and instructional dimensions.     

Theories of mathematics education: Is plurality a problem?

In this developed contribution to the Research Forum, held at the recent meeting of the International Group for the Psychology of Mathematics Education, the theme being “Theories of Mathematics Education”, I focus on the call by the organisers: ‘the time seems ripe for our community to take stock of the multiple and widely diverging mathematical theories’. I examine empirically the diversity of theories and I draw on the sociological theories of Basil Bernstein to relate the developments to the nature of intellectual communities and their productions. In particular, I suggest that the multiplicity and divergence are not surprising nor are they necessarily damaging to the field. I end by discussing concerns about accountability in relation to research in education.     

A brief history of mathematics education in Turkey: K-12 mathematics curricula

In this study, we survey the history of mathematics education in Turkey starting with its historical roots in the foundation of the republic. The changes in mathematics education in Turkey over the last century are investigated through an analysis of changes in curricular documents for K-12 schools. We consider the factors and reasons affecting curriculum developments, changes in philosophy and structure in terms of standards, objective and instructions. This article utilizes archival research techniques by examining original sources and illustrates the nature of the changes benefiting from a historical perspective. As a result of such analysis of the aforesaid sources, we have seen that the main reasons for changing mathematics curricula are: to build up a modern civilization in Turkey; the reports of John Dewey and the recommendations of Kate Wofford, William C. Varaceus and Watson Dickerman; the desire to become a member of the European Union; international factors and political situations.     

From the margin to the center: A framework for rehumanizing mathematics education for students with dis/abilities

Mathematics education for students with dis/abilities has a legacy of exclusion; its scholarship has been dominated by ableist perspective: a focus on rote mathematics facts, procedural instruction, and a narrow range of mathematics content. Grounded in critical pedagogy and critical disabilities studies, we argue for a shift in the focus away from the student as the site of defect and intervention to recognize the complex embodiment of dis/ability as a process produced within historical, political, social, and material contexts. The theoretical assumptions and guiding principles for a framework of culturally responsive and relational understanding of dis/ability will be shared together with a case of such an approach in classroom practice. The paper advances efforts to support all learners in robust mathematics learning communities and to situate research from perspectives that acknowledge the situated, interactional nature of ability, identity, and achievement.     

CAS in general mathematics education

A rational discussion of the use of Computer algebra systems (CAS) in mathematics teaching in general education needs an explicit image of (general) mathematics education, an explication of global perspectives and goals on mathematics teaching focusing on general education (chapter 1). The conception of general education according to the «ability of communication with experts» described in chapter 2 can be such an orientation for analysing, considering, classifying and assessing the didactical possibilities of using CAS. CAS are materialised mathematics allowing for more or less exhaustive outsourcing of operative (also symbolically) knowledge and skills to the machine. This frees up space of time as well as mental space for the development of those competences being in our view relevant for general mathematics education. In chapter 3 the idea of outsourcing and the role of CAS for it is discussed more detailed as well as consequences being possible for the CAS-supported teaching of mathematics. Beyond, CAS can be didactically used and reflected as a model of communication between (mathematical) experts and lay-persons (chapter 4). Chapter 5 outlines some research perspectives.     

Conceptualizing inquiry-based education in mathematics

The terms inquiry-based learning and inquiry-based education have appeared with increasing frequency in educational policy and curriculum documents related to mathematics and science education over the past decade, indicating a major educational trend. We go back to the origin of inquiry as a pedagogical concept in the work of Dewey (e.g. 1916, 1938) to analyse and discuss its migration to science and mathematics education. For conceptualizing inquiry-based mathematics education (IBME) it is important to analyse how this concept resonates with already well-established theoretical frameworks in mathematics education. Six such frameworks are analysed from the perspective of inquiry: the problem-solving tradition, the theory of didactical situations, the realistic mathematics education programme, the mathematical modelling perspective, the anthropological theory of didactics, and the dialogical and critical approach to mathematics education. In an appendix these frameworks are illustrated with paradigmatic examples of teaching activities with inquiry elements. The paper is rounded off with a list of ten concerns for the development and implementation of IBME.     

Teaching values and valued teaching in the mathematics classroom: toward a research agenda

In this paper, we first discuss the teaching of values by focusing on the kinds of values that have been discussed and studied in the other papers in this special journal issue and elsewhere. Then we raise a number of issues about the product-based values in mathematics education, which we identify as teaching values and which can be realized through classroom instruction. In the second section, we discuss the process-based valued teaching methods used to maximize the realization of the teaching values in the classroom. As valued teaching may be perceived differently by different people, in the discussion we analyze how it is seen from both students?? and teachers?? perspectives. We end this paper by discussing a number of methodological issues in studying teaching values and valued teaching as well as offering suggestions for future research.