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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. [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]

This paper presents a curriculum, textbook and test result analysis for the new (to California) elementary school “Key Standard” mathematics curriculum, transplanted in 1998 from it's foreign roots in Asia and Europe, locations with far different cultural and economic backgrounds. Based on topic analysis methods developed by Michigan State University, this curriculum is a “quality” curriculum, since it is closely aligned with the curriculum of the six leading TIMSS math countries. Five-year test results are presented for two cohorts totaling over 13,000 students, all from four “early adoption” urban districts where 68% of the students were economically disadvantaged. Included are two cohorts of English learning immigrants totaling over 4,400 students. Performance was found to be statistically superior to similar (control) districts which continued with the old 1991 curriculum and textbooks (0.003 < p < 0.015). The focus of this paper is on the transition from far-below to above average learning performance of these students over the 1998–2002 period. [ABSTRACT FROM AUTHOR]

This paper is part of a broader study that draws on Wenger's (Wenger, E.: 1998, Communities of Practice: Learning, Meaning, and Identity, Cambridge University Press, New Work) social practice perspective to investigate teacher learning. The study extends Wenger's complex model of interrelated components of leaning (as meaning, practice, identity and community) to describe and explain teacher learning that occurs within a mathematics senior-phase in-service program that was stimulated by curriculum change. The study uses qualitative ethnography in which the researcher performs the dual role of both coordinator and researcher of the in-service practice. In a longitudinal study the phenomenon of confidence emerged in teachers' descriptions and explanations of their learning. In this paper I explore this phenomenon both empirically and theoretically. The extension of Wenger's (1998) theory to include the overarching and interacting component of confidence is embedded in and derived from data analysis of 10 teachers' learning, over a 2-year period, during a time of radical curriculum change. Since it would be incoherent within this framework to draw on psychological explanations of confidence I set out to explore confidence from within a social practice frame in a way that is grounded in data of the teachers in this study. The paper offers a concept of confidence in relation to teacher learning as 'learning as mastery', and confidence as both a product and a process of learning. Teachers can at once state their confidence as mathematics teachers, and their confidence to admit to what they do not know and still need to learn. It is argued that this is a primary condition for ongoing learning in a profession like mathematics teaching. In addition, the paper provides a critique of the applicability of Wenger's work to the context of teacher education and in particular highlights the absence of the notion of confidence within his work. [ABSTRACT FROM AUTHOR]

The supposed apolitical nature of mathematics is an institutional frame that functions to sustain specific power structures within schools. This paper disrupts the common assumption that mathematics (as a body of knowledge constructed in situated historical moments) is free from entrenched ideological motives. Using narrative inquiry, the paper examines the ways in which novice mathematics teachers negotiate the intersection of curriculum and institutional politics. After outlining a theoretical framework, the paper performs a story entitled `political text' which reveals the plaintive, dissenting voice of a novice teacher as she negotiates the friction between her life history and the ossified canonicity of mathematics curriculum. Through the critical and reflexive voice of the novice teacher, the naturalized concepts of counting and measurement are playfully revisioned. The frequent silencing of such oppositional readings underscores the deeply embedded conflict between voice and text in the contemporary mathematics classroom. [ABSTRACT FROM AUTHOR]

In the recent evolution of Portuguese basic education, the concept of competence and the process of curriculum development became major aspects of an innovative movement at a national level. Concerning school mathematics, this evolution provided an opportunity to explore the way in which mathematical competence for all could be defined and interpreted with the purpose of abandoning a technique-oriented view of the curriculum and the process of developing it. After describing the context and exploring the concept of competence, this paper presents a formulation for the characteristics of mathematical competence for all and discusses some of its implications. Finally, the paper tries to identify the obstacles that tend to emerge when such an approach is proposed in the context of a national curriculum. [ABSTRACT FROM AUTHOR]

This paper presents findings from a comparative analysis of three similar secondary geometry texts, one critical unit, one standards-based reform unit, and one specialist chapter. I developed the critical unit as I took the tenets of critical mathematics (CM) and substantiated them in printed curricular materials in which to teach as part of a larger study. The reform and specialist texts were both sections from commercially available textbooks. The primary goal of the textual analysis was to examine how the incorporation of critical or political themes into the required secondary mathematics curriculum transformed this curriculum. Key findings were that the CM text exchanged political for mathematical content and sent class-based messages about students' academic potential. I conclude by arguing that the CM agenda proposed by advocates in the USA is problematic in terms of promoting equity in mathematics education. [ABSTRACT FROM AUTHOR]

Mathematics teaching in Burkina Faso is faced with major challenges (high illiteracy rates, students’ difficulties, and high failure rates in mathematics, which is a central topic in the curriculum). As evidenced in many of these studies, mathematics is reputed to be tough, inaccessible, and far from what students live daily. Students here look as though they are living in two seemingly distant worlds, school and everyday life. In order to better understand these difficulties and to contribute in the long run to a more adapted teaching of mathematics, we tried to document and elicit the “mathematical resources” mobilized in various daily life social practices. In this paper, we focus on one of them, the counting and selling of mangoes by unschooled peasants. An ethnographic approach draws on the observation of the situated activity of counting and selling mangoes (during harvesting) and on “eliciting interviews” of the involved actors. The analysis of results highlights a richness of structuring resources mobilized and distributed through this practice, related to what Lave () call “the experienced lived-in-world” and “constitutive order.” The mathematical resources take the form of “knowledge in action” and “theorems in action” (Vergnaud, Rech Didact Math 10(23):133–170, ), embedded in the social, economic, and even cultural structures of actors. [ABSTRACT FROM AUTHOR]

This paper investigates the nature of the interaction between the teacher and students as they worked on different mathematics activities in a single classroom over a 10-month period. Sociocultural theories and the Vygotskian zone of proximal development provide the main framework for examining the teaching and learning processes and explaining the incorporation of a four-phase lesson plan as increasing participation of the teacher and students in the teaching and learning process. Drawing on the analyses of discourse from videotaped lessons and the interviews with the teacher and students, five different types of interactions that emphasized mathematical sense-making and justification of ideas and arguments were identified. Excerpts from transcriptions of such interactions are provided to illustrate the learning practices, either academic or non-academic, that students developed in response to these interactions. [ABSTRACT FROM AUTHOR]

In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through a symbiosis of induction and abduction from experience and deduction from the relationships generated in exemplification. In other words, experience can be organised in such a way that shifts of understanding take place as a result of learners’ own actions. Actions, in this context, include mental acts of organisational reflection. [ABSTRACT FROM AUTHOR]

In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles. [ABSTRACT FROM AUTHOR]

The geometry textbooks of the new system of secondary education in Iran differed dramatically from the old ones considering the aims, the visions, the content, the approach, and the educational purposes. Four hundred eighty mathematics teachers participated in a nationwide professional development program conducted by the author to facilitate the implementation of the new changes. Based on this program, a research study was designed to investigate the teachers’ beliefs about a new reform in high school geometry as a result of the professional development program. The data of the study were collected through teachers’ reflective writings, project works, group discussions, reflective questionnaires, and oral communications. The purpose of this paper is to draw upon this study and to document mathematics teachers’ beliefs about new changes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity, we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes. Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning. Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically gathered: some empirical evidence is presented in this paper. [ABSTRACT FROM AUTHOR]

Many mathematics departments usually teach a variety of courses for students from different science departments and even from different faculties. These “service” courses are usually taught in the same way as the courses for mathematics major students. However, in science, because of the need to better analyse and interpret experimental data and the increased use of mathematical tools in chemistry and geology textbooks, it is becoming necessary to teach these science students quantitative skills beyond the scope of first year mathematics courses. This paper describes the design of a one-semester second year mathematics course, mainly for chemistry and geology students, with three specific objectives: to develop students’ ability to quantitatively analyse problems arising in their own field, to illustrate the great utility of mathematical models to provide answers to key chemistry and geology problems, to develop students’ appreciation of the diversity of mathematical approaches potentially useful in the chemical and geological sciences. [ABSTRACT FROM AUTHOR]