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This paper draws on data from the development of annual national mathematics assessment materials for 7-year-old pupils in Wales for use during the period 2000–2002. The materials were developed in both English and Welsh and were designed to be matched. The paper reports on item analyses which sought items that exhibited differential performance in relation to whether the materials were English medium or Welsh medium. The items that exhibited consistent differential item functioning in relation to language during pre-testing are reviewed in order to discuss the linguistic factors that could affect such behaviour. [ABSTRACT FROM AUTHOR]

This paper develops an understanding of the issues, interests and concerns within the mathematics education community related to the use of computers and other digital technologies in the teaching and learning of mathematics. It begins by arguing for the importance of understanding this landscape of interests and concerns, and then turns to the theoretical and methodological choices made in this study, explaining how it has drawn on the approach developed by the STELLAR European Network of Excellence. By analysing the titles and abstracts of a conference chosen to represent the mathematics education community, it maps out the landscape framed by three 'Grand Challenges', finding that an understanding of orchestrating learning is at the heart of the interests of the community, and that the community is interested in exploring new and different contexts for the teaching and learning of mathematics. However, there is currently less interest in investigating and exploiting the increasing connectedness of learners within this community. Further, while the 'Grand Challenges' framing is useful in mapping the landscape, it fails to take into account both the personal concerns of teachers and students, such as attitude and confidence, and issues related to doing research and understanding research concerns. [ABSTRACT FROM AUTHOR]

The basic assumption of this paper is that mathematics and history of mathematics are both forms of knowledge and, therefore, represent different ways of knowing. This was also the basic assumption of Fried (2001) who maintained that these ways of knowing imply different conceptual and methodological commitments, which, in turn, lead to a conflict between the commitments of mathematics education and history of mathematics. But that conclusion was far too peremptory. The present paper, by contrast, takes the position, relying in part on Saussurean semiotics, that the historian's and working mathematician's ways of knowing are complementary. Recognizing this fact, it is argued, brings us to a deeper understanding of ourselves as creatures that do mathematics. This understanding, which is a kind of mathematical self-knowledge, is then proposed as an alternative commitment for mathematics education. In light of that commitment, history of mathematics assumes an essential role in mathematics education both as a subject and as a mediator between the aforementioned ways of knowing. [ABSTRACT FROM AUTHOR]

Textbooks have the potential to be powerful tools to help students develop an understanding of mathematics. However, many students are unable to use their textbooks effectively as learning tools. This paper presents a framework that can be used to analyze factors that impact the ways students read textbooks. It adapts ideas from reader-oriented theory and applies them to the domain of mathematics textbooks. In reader-oriented theory, the reader is viewed as actively constructing meaning from a text through the reading process; this endeavor is shaped and constrained by the intentions of the author, the beliefs of the reader, and the qualities the text requires the reader to possess. This paper also discusses how reading mathematics textbooks is further constrained by the authority and closed structure of these textbooks. After describing the framework, the paper discusses recommendations for future avenues of research and pedagogy, highlighting the importance of teachers' roles in mediating their students' use of textbooks. [ABSTRACT FROM AUTHOR]

This case study deals with a solitary learner’s process of mathematical justification during her investigation of bifurcation points in dynamic systems. Her motivation to justify the bifurcation points drove the learning process. Methodologically, our analysis used the nested epistemic actions model for abstraction in context. In previous work, we have shown that the learner’s attempts at justification gave rise to several processes of knowledge construction, which develop in parallel and interact. In this paper, we analyze the interaction pattern of combining constructions and show that combining constructions indicate an enlightenment of the learner. This adds an analytic dimension to the nested epistemic actions model of abstraction in context. [ABSTRACT FROM AUTHOR]

This paper presents the results of a study developed in the context of indigenous teachers education from Xingu Indian Park, Brazil. The indigenous bilingual (or multilingual in some cases) teachers that participated in this education program were from 14 ethnic groups. The research focused on a mathematics textbook production, written in indigenous language by indigenous teachers to be used at schools in the Park. The paper discusses the numeracy-literacy practices in this process and focuses on the meanings, values and ways of use that are related to numbers, writing and drawing. In particular, mathematics problems written by the indigenous teachers (in indigenous language and Portuguese) are analyzed. The analysis shows how aspects of orality influence the writing of these problems. [ABSTRACT FROM AUTHOR]

A common approach used for introducing algebra to young adolescents is an exploration of visual growth patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of these difficulties at an early age. A teaching experiment was conducted with two classes of students with an average age of eight years and six months. From the results it appears that young students are capable not only of thinking about the relationship between two data sets, but also of expressing this relationship in a very abstract form. [ABSTRACT FROM AUTHOR]

This paper reports a survey of research in mathematics teacher education from 1999 to 2003 done by an international team of five mathematics educators and researchers. The survey included published research in international mathematics education journals, international handbooks of mathematics education and international mathematics education conference proceedings. Some regional sources from various parts of the world were also included. We investigated who was writing, from and in what settings, with what theoretical frameworks, and with what sorts of study designs for what core questions. We also examined the range of findings and conclusions produced in these studies. Our analysis presented here focuses on four themes that stood out from our initial investigation of almost 300 published papers, and systematically elaborated through a focused study of a 160 papers across key journals and conference proceedings in the field. From this vantage point, the paper offers a reflection on the current state of the field of mathematics teacher education research. Our aim is to stimulate discussion that can support the development of the field, not make final pronouncements about its nature. [ABSTRACT FROM AUTHOR]

My purpose in this paper is to present a didactic tool – a set of specially designed problems and questions for discussion – that can help making students better aware of the various aspects of the formal notion of limit of a sequence. The didactic tool will be justified using results from my own and other authors' research on students' naïve or erroneous conceptions of limit of a sequence. [ABSTRACT FROM AUTHOR]

In this paper we describe an ontological and semiotic model for mathematical knowledge, using elementary combinatorics as an example. We then apply this model to analyze the solving process of some combinatorial problems by students with high mathematical training, and show its utility in providing a semiotic explanation for the difficulty of combinatorial reasoning. We finally analyze the implications of the theoretical model and type of analysis presented for mathematics education research and practice. [ABSTRACT FROM AUTHOR]

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof. [ABSTRACT FROM AUTHOR]

In the international community of mathematics and science educators the intuitive rules theory developed by the Israeli researchers Tirosh and Stavy receives much attention. According to this theory, students' responses to a variety of mathematical and scientific tasks can be explained in terms of their application of some common intuitive rules. Two major intuitive rules are manifested in comparison tasks: `More A–more B' and `Same A–same B'. In this paper, we address two important questions for which the existing literature on intuitive rules does not provide a convincing research-based answer: (1) are the reasoning processes of students who respond in line with a given intuitive rule actually affected by that rule or by essentially other misconceptions (leading to the same answer), and (2) are individual students consistent in their choice of one of the intuitive rules when confronted with different, conceptually unrelated tasks? A test consisting of five comparison problems from different mathematical subdomains was administered collectively to 172 Flemish students from Grades 10 to 12. An analysis of students' written calculations and justifications suggested that the students were considerably less affected by the intuitive rules than their multiple-choice answers actually suggested. Instead, essentially different misconceptions and errors were found. With respect to the issue of individual consistency, we found that students who made many errors did not answer systematically in line with one of the two intuitive rules. [ABSTRACT FROM AUTHOR]

Visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education. This paper is an attempt to define visualization and to analyze, exemplify and reflect upon the many different and rich roles it can and should play in the learning and the doing of mathematics. At the same time, the limitations and possible sources of difficulties visualization may pose for students and teachers are considered. [ABSTRACT FROM AUTHOR]

This article reports on young children's self-efficacy beliefs and their corresponding performance of mathematical and nonmathematical tasks typically encountered in kindergarten. Participants included 132 kindergarten children aged 5-6 years old. Among the participants, 69 children were identified by the social welfare department as being abused and/or neglected. Individual interviews were conducted where children were asked to assess their self-efficacy regarding sorting tasks, mathematics tasks, and reciting the alphabet. Children were then requested to perform each of the tasks. Results revealed that no significant differences were found between the abused and neglected children and their peers regarding their self-efficacy beliefs and performances for any of the tasks. For some of the tasks, children were able to correctly assess their performance, while for other tasks, children overestimated their performance. Possible reasons for these outcomes are discussed. [ABSTRACT FROM AUTHOR]

The paper is a reflective account of the design and implementation of mathematical tasks that evoke uncertainty for the learner. Three types of uncertainty associated with mathematical tasks are discussed and illustrated: competing claims, unknown path or questionable conclusion, and non-readily verifiable outcomes. One task is presented in depth, pointing to the dynamic nature of task design, and the added value stimulated by the uncertainty component entailed in the task in terms of mathematical and pedagogical musing. [ABSTRACT FROM AUTHOR]

From time to time, the International Commission on Mathematical Instruction (ICMI) mounts studies to investigate in depth and detail particular fields of interest in mathematics education. This paper is the Discussion Document of the forthcoming ICMI Study 16, Challenging Mathematics in and beyond the Classroom. [ABSTRACT FROM AUTHOR]

Examines the concept of ethnomathematics. Consideration of replacing mathematics with ethomathematics; Failure of formal mathematics to conform to the disciplinary protocols of mathematical proof; Advantages of ethnomathematics for students.

This paper offers for discussion and critique a conceptual framework that applies a situative perspective on learning to the study of learning to teach mathematics. From this perspective, such learning occurs in many different situations -- mathematics and teacher preparation courses, pre-service field experiences, and schools of employment. By participating over time in these varied contexts, mathematics teachers refine their conceptions about their craft -- the big ideas of mathematics, mathematics-specific pedagogy, and sense of self as a mathematics teacher. This framework guides a research project that traces the learning trajectories of teachers from two reform-based teacher preparation programs into their early teaching careers. We provide two examples from this research to illustrate how this framework has helped us understand the process of learning to teach. [ABSTRACT FROM AUTHOR]

Literacy educators have been actively theorising the demands of literacy in New Times yet mathematics educators have taken little of this debate up. If, as literacy educators suggest, literacy demands are different in these New Times, what are the implications for numeracy or mathematics educators? This paper explores perceptions of young and older people who are engaged in work practices. It was found that there were statistically significant differences in a number of areas. Numeracy was found to be an important variable in discerning differences between older and younger people, and that technology was also seen to differentiate the two cohorts. Within numeracy, older people were more like to see number as important, whereas younger people were more likely to identify statistics, applied areas of mathematics and the use of technology to support numeracy as being important. These findings have implications for theorising and practice in mathematics education. [ABSTRACT FROM AUTHOR]

Teachers in the UK and elsewhere are now expected to foster creativity in young children (NACCCE, ; Ofsted, ; DfES, ; DfES/DCMS, ). Creativity, however, is more often associated with the arts than with mathematics. The aim of the study was to explore and document pre-service (in the UK, pre-service teachers are referred to as ‘trainee’ teachers) primary teachers’ conceptions of creativity in mathematics teaching in the UK. A questionnaire probed their conceptions early in their course, and these were supplemented with data from semi-structured interviews. Analysis of the responses indicated that pre-service teachers’ conceptions were narrow, predominantly associated with the use of resources and technology and bound up with the idea of ‘teaching creatively’ rather than ‘teaching for creativity’. Conceptions became less narrow as pre-service teachers were preparing to enter schools as newly qualified, but they still had difficulty in identifying ways of encouraging and assessing creativity in the classroom. This difficulty suggests that conceptions of creativity need to be addressed and developed directly during pre-service education if teachers are to meet the expectations of government as set out in the above documents. [ABSTRACT FROM AUTHOR]

In this article, I make a case for the inputs that Martin Heidegger's theoretical perspective offers to current concerns about the nature of mathematics, its teaching and learning, and the problem of subjectivity. In particular, I consider Heidegger's notion of positive science and discuss both its applicability to mathematics and its importance to mathematics education. I argue that Heidegger's ontological position is consonant with some sociocultural approaches in mathematics education and that Heidegger's work can shed some light on the problem of knowing and being. Finally, I raise some questions concerning subjectivity and the link between language and mathematical objects. [ABSTRACT FROM AUTHOR]

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction. [ABSTRACT FROM AUTHOR]

Bilingual students have, at times, been thought to be at a disadvantage in learning mathematics because of an assumed interference between their two languages. Earlier research, confirmed again in this study, shows that this is a naive view to take. Although some bilingual students do have a harder time, others seem to be at an advantage. This study explores the use that bilingual students who are succeeding in mathematics make of their two languages. These students seem to have better metalinguistics skills that allow them to self-correct when solving problems, and are perhaps more confident in their approach to solving difficult problems. It also appears that students in this study switched between languages in early years of schooling, but only used English by the time they were completing elementary school. [ABSTRACT FROM AUTHOR]

This study examined changes in 26 fourth-grade students' early conceptions of rational number representations as a function of receiving one of two curricular interventions. The first group of 12 students received a curriculum that emphasized constructing knowledge through extended problem solving with a single perspective of the rational number domain based on part-whole relations. A second group of 14 students received a curriculum that emphasized a more conceptually diverse multiple perspective view of the domain through problem solving with operator and ratio relations. Analyses of the students' rational number knowledge before and after the interventions indicated that students in the single perspective group produced organizations of knowledge that more frequently diverged from a formal domain analysis than those produced by students in the multiple perspective group. Further, students in the single perspective group increased their focus on superficial surface features. Alternatively, students in the multiple perspective group demonstrated an increased focus on operations that more frequently reflected the underlying mathematical relation conveyed by the representation. The findings indicate that an early exposure to more diverse perspectives of rational numbers assists students in developing more interconnected and viable representation knowledge for rational numbers. [ABSTRACT FROM AUTHOR]

Forty-one students in two third grade classes, including special education students, participated in an action research project conducted jointly by two university supervisors, three teachers, and three student teachers. The "Minute Math" project involved students in predicting and graphing their test scores on a weekly conventional timed test of the 0-9 multiplication facts. Students also reflected each week on their progress and the success of their studying and problem-solving strategies. Student self-assessment was successful at turning the rote memorization task of learning the times tables into a deeper experience for students about monitoring their own mathematics learning. [ABSTRACT FROM AUTHOR]

A broad view of mathematics education takes it as the study of how people learn and do mathematics. Starting with this view, the actual and potential relationships of mathematics education as a research discipline to mathematics as a field of knowledge and activity and to the mathematicians carrying out that activity are analyzed. This leads to the picture of a gulf between the two scientific communities which are based in different cultures of thinking and research. A (meta-)study of mathematics and all its facets termed here mathematicology is proposed. It could serve as common ground for cooperative studies by mathematicians and mathematics educators. Thereby the gulf will not necessarily become narrower but a bridge over the differences and mutual misunderstandings could be built. [ABSTRACT FROM AUTHOR]

Reports on a study which examined the anxiety of mathematics in nine to eleven-year old children. Comparison of the anxiety of children who were taught in the traditional style to that of pupils whose teachers adopted an alternative teaching approach; Details on discussion and problem-solving of the children's own informal strategies; Findings of the study.

The use of writing as a learning activity in mathematics has been the subject of many publications. However, little evidence has been presented to support the claims that writing enhances learning in mathematics. One difficulty in such research has been the lack of a detailed method for analysing the writing products of students. In the present study, a scheme for coding the parts of written mathematical presentations was developed. At the grade 8 level at which the study was conducted, a limited style of expository writing was found to predominate. The writing was shown to closely resemble the style of the typical mathematics textbook used by the students. [ABSTRACT FROM AUTHOR]