Estimating Iraqi deaths: a case study with implications for mathematics education

In this paper, I present an account of attempts to quantify deaths of Iraqis during the occupation by US and other forces since the invasion of March 2003, and of the reactions to these attempts. This story illuminates many aspects of current socio-political reality, particularly, but by no means exclusively, in the United States. Here, these aspects are selectively discussed in relation to the overarching themes of what the story illuminates about the uses of statistical information in society and about shortcomings in mathematics education.     

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.

     

A global survey of international perspectives on modelling in mathematics education

In this article we survey the current debate on modelling and, describe different perspectives on this debate. We relate these perspectives with earlier perspectives and show similarities and differences between these different approaches.     

How some research mathematicians and statisticians use proof in undergraduate mathematics

The paper examines the roles and purposes of proof mentioned by university research faculty when reflecting on their own teaching and teaching at their institutions. Interview responses from 14 research mathematicians and statisticians who also teach are reported. The results suggest there is a great deal of variation in the role and purpose of proof in and among mathematics courses and that factors such as the course title, audience, and instructor influence this variation. The results also suggest that, for this diverse group, learning how to prove theorems is the most prominent role of proof in upper division undergraduate mathematics courses and that this training is considered preparation for graduate mathematics studies. Absent were responses discussing proof's role in preparing K-12 mathematics teachers. Implications for a proof and proving landscape for school mathematics are discussed.     

Theories of Mathematics Education: A global survey of theoretical frameworks/trends in mathematics education research

In this article we survey the history of research on theories in mathematics education. We also briefly examine the origins of this line of inquiry, the contribution of Hans-Georg Steiner, the activities of various international topics groups and current discussions of theories in mathematics education research. We conclude by outlining current positions and questions addressed by mathematics education researchers in the research forum on theories at the 2005 PME meeting in Melbourne, Australia.     

The mutual influence of theory and practice in mathematics education: implications for research and teaching

Questions related on how to connect theory and practice in school mathematics have been under debate for several years. Also, different forms of co-operation between academic researchers and school teachers are widely discussed. In the search for boundary conditions to mediate knowledge between the two poles, there is evidence that any conception which assigns to “theory” the place of instructin, “practice” is doomed to fail, and the necessity of developing the notion of cooperation comes as a consequence. Following this assumption, existing literature provides interesting contributions supporting the idea of blending mathematical content with pedagogical knowledge. This contribution focuses on the role that theoretical models, as emerged from the observation of students at work, can play on instructing practice. In particular, we will approach algebraic thinking and refer to a theoretical model based on the distinction between sense and denotation of algebraic expressions. We will then discuss how this theoretical model can shed light on students’ difficulties when solving equations and inequalities. Finally, we will point out how findings coming from research can suitably orient teachers and promote further development.     

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.     

History of Greek mathematics: A survey of recent research

This survey reviews research in four areas of the history of Greek mathematics: (1) methods in Greek mathematics (the axiomatic method, the method of analysis, and geometric algebra); (2) proportion and the theory of irrationals (controversies over the origins of the theory of incommensurables); (3) Archimedes (aspects of controversies over his life and works); and (4) Greek mathematical methods (including discussion of Ptolemy's work, connections between Greek and Indian mathematics, the significance of Greek mathematical papyri, Arabic texts, and even archaeological investigations of scientific instruments).     

A simple proof of some ergodic theorems

Some ideas of T. Kamae’s proof using nonstandard analysis are employed to give a simple proof of Birkhoff’s theorem in a classical setting as well as Kingman’s subadditive ergodic theorem.     

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).     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

Dog and tricks: some mathematics for students of graphics

This paper describes how parametric cubic splines and cubicBezier curves may be used in designing a two dimensional shape.A simple aerofoil shape is designed using both methods. Themathematics is described and the shape drawn using Excel. Theeffect of varying parameters is shown in both methods.     

Linking mathematics education and democracy: Citizenship, mathematical archaeology, mathemacy and deliberative interaction

The relationship between mathematics education and democracy is discussed in terms of citizenship, mathematical archaeology, mathemacy and deliberative interaction. The first issue concentrates on the learner as a member of society, the second on the social functions of mathematics and on how to get to grips with mathematics in use, the third refers to an integrated kind of competence including different forms of reflection (mathematics-oriented, model-oriented, context-oriented and lifeworld-oriented reflections), the fourth issue considers the classroom as a micro-society and deals with the nature of the teaching-learning process. These four issues are discussed with reference to an example of educational practice. “Our Community”, carried out among sixteen-year-old students as an interdisciplinary project including a one-week trainee service. Finally, it is indicated that a discussion of mathematics education and democracy is essential to a further development of social theory, as the notions of citizenship, mathematical archaeology, mathemacy and deliberative interaction become part of the discussion about modemity, reflexive modemity and other constructs from recent social theory.     

Handheld technology for mathematics education: flashback into the future

In the 1990s, handheld technology allowed overcoming infrastructural limitations that had hindered until then the integration of ICT in mathematics education. In this paper, we reflect on this integration of handheld technology from a personal perspective, as well as on the lessons to be learnt from it. The main lesson in our opinion concerns the growing awareness that students’ mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on instrumentation and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion, extrapolation of these lessons to a future with mobile multi-functional handheld technology leads to the issues of connectivity and in- and out-of-school collaborative work as major issues for future research.     

A simple proof of some generalized principal ideal theorems

Using symmetric algebras we simplify and slightly strengthen the Bruns-Eisenbud-Evans ``generalized principal ideal theorem' on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwiecinski, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

     

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.     

Orbit equivalence of Cantor minimal systems: A survey and a new proof

We give a new proof of the classification, up to topological orbit equivalence, of minimal AF-equivalence relations and minimal actions of the group of integers on the Cantor set. This proof relies heavily on the structure of AF-equivalence relations and the theory of dimension groups; we give a short survey of these topics.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.