Search

This paper examines a Special Issue of Educational Studies in Mathematics comprising research reports centred on Peircian semiotics in mathematics education, written by some of the major authors in the area. The paper is targeted at inspecting how subjectivity is understood, or implied, in those reports. It seeks to delineate how the conceptions of subjectivity suggested are defined as a result of their being a function of the domain within which the authors reflexively situate themselves. The paper first considers how such understandings shape concepts of mathematics, students and teachers. It then explores how the research domain is understood by the authors as suggested through their implied positioning in relation to teachers, teacher educators, researchers and other potential readers. [ABSTRACT FROM AUTHOR]

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 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]

This Special Issue of Educational Studies in Mathematics honoring the life and work of Leen Streefland is based on Research Forum 5, Realistic Mathematics Education Research: Leen Streefland's Work Continues, at the 25th Annual Meeting of the International Group for the Psychology of Mathematics Education (PME), held in Utrecht, The Netherlands, July, 2001 (see PME-25 Proceedings, Vol. 1, pp. 221­253). The papers in this issue address the work of Leen Streefland at the Freudenthal Institute, and its continuation by his former colleagues and by international researchers who were influenced by his ideas. The essential problem addressed is how to develop a school mathematics curriculum that is grounded in the ex- periential reality of the learners (hence Realistic Mathematics Education ­ RME). In honoring Leen Streefland in this way, we are also honoring Hans Freudenthal, who was the first director of the Institute that after his death was named after him. Freundenthal was the originator of many of the ideas on which the continuing research is based, although (as elaborated in sev- eral of the papers) these ideas evolved and were modified and expanded in the subsequent and ongoing research. Some of these ideas include `math- ematizing', `mathematics as human activity', `anti-didactical inversion', `phenomenological didactical analyses', `guided reinvention', `long-term learning processes', `levels in learning processes', and `developmental re- search'. Freudenthal's work was an important source of inspiration for Stree- fland, as it also was for many others. In `The Legacy of Hans Freudenthal', a Special Issue of Educational Studies in Mathematics (Vol. 25, 1­2) that was published ten years ago, Streefland emphasized how Freudenthal cre- ated the context for further development. Paradoxically, he sometimes did this by putting people back on the track of their previously held ideas. For Streefland this happened for example with the issue of interweaving learning strands, which was an early part of his work. By higlighting the importance of this issue, Freudenthal revived it. Incidentally, this aspect of interweaving was by no means isolated within Streefland's work. Like no other at that time, he was focused on making connections, such as those between the different phases in long-running learning processes, between the current didactics of mathematics and the history of mathematics, between the different roles ­ student, teacher and researcher ­ which all who are involved in the learning-teaching process Educational Studies in Mathematics 54: 1­4, 2003. [ABSTRACT FROM AUTHOR]

This paper initially discusses recent changes in mathematics education and assessment in England and Wales against the background of research on mathematics performance and assessment. It then reports findings from qualitative research with 10–11 year olds undertaken with the object of exploring the validity of the pilot pencil and paper tests in mathematics developed as part of the assessment programme for the English and Welsh National Curriculum in 1993 and 1994. Fifteen children from across the ’ability‘ range were asked to work through a selection of items in the situation of an individual clinical interview. This enabled in-depth data to be collected about their interpretation of and performance on the items. This paper focuses in particular on items where the ambiguity of the boundary between everyday knowledge and mathematics might be expected to lead to problems for children in interpreting the tasks required of them by the test items. The results show that the nature of the items might well have prevented some of these children, in the context of a paper and pencil testing situation, from demonstrating mathematical capacities and understandings they seem to have possessed. Their initial performance does not always seem to have reflected their underlying competence as demonstrated in the extended interview. [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]

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431-459, ) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies. [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]

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can be used to create new constructions. Newly formed constructions are fragile entities. In the course of consolidating a construction this student creates connections between the new construction and already established mathematical knowledge and develops a language to describe and guide mathematical actions related to the construction. The resulting abstraction is a more resilient form of the construction and the student is able to justify his assertions. The paper also examines issues in designing tasks to consolidate a construction. [ABSTRACT FROM AUTHOR]

The sequential organization of actions necessary to construct a figure in a dynamic geometry environment (DGE) introduces an explicit order of construction. Such sequential organization produces what is, in effect, a hierarchy of dependences, as elements of the construction depend on something created earlier. This hierarchy of dependences is one of the main factors that determine the dynamic behavior (DB) within a DGE, and the order is often explicitly stated by terms such as parent and child. This article is a part of a larger study that examines various instruments developed by users when they use dragging. It addresses one aspect of dragging: the connection between dynamic behavior and the sequential order of construction. Junior high students and graduate students in mathematics education were asked to predict the DB of points that were part of a geometric construction they had executed using a DGE according to a given procedure, and to explain their predictions. The study reveals that while hierarchy in geometric constructions in a DGE is mirrored by the DB, user actions and perceptions of DB indicate that users often grasp a reverse hierarchy in which dragging a child affects its parent. The study reveals four categories of reverse-order predictions, and suggests that these may be caused by terms and knowledge built into paper-and-pencil geometry, which are part of the resources users bring to dragging. Examining various DGEs1 in detail reveals that in some cases the DB is in fact compatible with the reverse-order predictions. The paper concludes with a brief discussion of some implications for learning activities and software design. [ABSTRACT FROM AUTHOR]

Provides a commentary on the papers presented in the issue of 'Educational Studies in Mathematics.' Range of general theories underpinning the emerging social paradigm; Development in mathematical learning; Analyses of tool mediation.

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 results of a longitudinal study in the transition to independent graduate studies in mathematics. The analysis of the data collected from 24 students doing a PhD in mathematics revealed the existence of three transitional stages within the PhD degree, namely Adjustment, Expertise and Articulation. The focus is on the first two transitional stages, since the data collection focused mainly on these. Based on the first two transitional stages and the students’ ways of dealing with them, which were called ‘survival strategies’, three types of students were identified. The importance of motivation for each transitional stage and the successful transition overall are considered as well. [ABSTRACT FROM AUTHOR]

The construct of identity has been used widely in mathematics education in order to understand how students (and teachers) relate to and engage with the subject (Kaasila, ; Sfard & Prusak, ; Boaler, ). Drawing on cultural historical activity theory (CHAT), this paper adopts Leont’ev’s notion of leading activity in order to explore the key ‘significant’ activities that are implicated in the development of students’ reflexive understanding of self and how this may offer differing relations with mathematics. According to Leont’ev (), leading activities are those which are significant to the development of the individual’s psyche through the emergence of new motives for engagement. We suggest that alongside new motives for engagement comes a new understanding of self—a leading identity—which reflects a hierarchy of our motives. Narrative analysis of interviews with two students (aged 16–17 years old) in post-compulsory education, Mary and Lee, are presented. Mary holds a stable ‘vocational’ leading identity throughout her narrative and, thus, her motive for studying mathematics is defined by its ‘use value’ in terms of pursuing this vocation. In contrast, Lee develops a leading identity which is focused on the activity of studying and becoming a university student. As such, his motive for study is framed in terms of the exchange value of the qualifications he hopes to obtain. We argue that this empirical grounding of leading activity and leading identity offers new insights into students’ identity development. [ABSTRACT FROM AUTHOR]

This paper comments on the expanded repertoire of techniques, conceptual frameworks, and perspectives developed to study the phenomena of gesture, bodily action and other modalities as related to thinking, learning, acting, and speaking. Certain broad issues are considered, including (1) the distinction between “contextual” generalization of instances across context (of virtually any kind—numeric, situational, etc.) and the generalization of structured actions on symbols, (2) fundamental distinctions between the use of semiotic means to describe specific situations versus semiosis serving the process of generalization, and (3) the challenges of building generalizable research findings at such an early stage in infrastructure building. [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]

This empirical paper considers the different purposes for which teachers use examples in elementary mathematics teaching, and how well the actual examples used fit these intended purposes. For this study, 24 mathematics lessons taught by prospective elementary school teachers were videotaped. In the spirit of grounded theory, the purpose of the analysis of these lessons was to discover, and to construct theories around, the ways that these novice teachers could be seen to draw upon their mathematics teaching knowledge-base in their lesson preparation and in their observed classroom instruction. A highly-pervasive dimension of the findings was these teachers’ choice and use of examples. Four categories of uses of examples are identified and exemplified: these are related to different kinds of teacher awareness. [ABSTRACT FROM AUTHOR]

Students mainly perceive the transition to secondary school as an opportunity to start their school career anew. Reality often proves them wrong, especially in the case of mathematics. In our paper, we briefly discuss children’s transition to secondary school as both an opportunity and a problem, with reference to the Greek context. In discussions about the transition to secondary school and its effect on the teaching and learning of mathematics, primary and secondary school teachers in Greece often depict school mathematics as a “chain” of concepts and procedures. With this metaphor as our reference point, we discuss how ideologies about the teaching of mathematics are enacted in both school levels. We will base our discussion on the analysis of extracts taken from dialogues in primary and secondary school mathematics classrooms in Greece. In our analysis, we employ a Peircean view of semiotics in an attempt to conceptualize students’ rushed introduction to rigor in justifying mathematical statements in secondary school. These extracts are part of a longitudinal study that aimed, on the one hand, to pinpoint discontinuities and continuities in the teaching of geometry between primary and secondary schools and, on the other, to investigate whether a set of curriculum material that we designed could serve as a link in the teaching of geometry between the two school levels. [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]

How do teachers conceptualize and deal with context of mathematics word problems in their teaching? This question is discussed based on a study of 14 experienced teachers at the elementary, junior high and senior high school levels. Bruner’s notions of paradigmatic and narrative modes of knowing formed the basis of analysis of data from sources that include interviews and classroom observations. The findings highlight the teachers’ conceptions of problem context and teaching approaches for each of these modes of knowing. All of the teachers used the paradigmatic mode in their teaching but with different depth and most engaged in some form of the narrative mode to create a classroom environment that was motivational for students to learn word problems. The paper also highlights characteristics of these two modes as they relate directly to word problems and discusses implications for instruction, learning and teacher development. [ABSTRACT FROM AUTHOR]

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity. [ABSTRACT FROM AUTHOR]

This paper examines part of a set of students who were followed during their first-term, first-year studies in formal definition-based real analysis at a British university. It explores the approaches to problems about convergence of sequences and series made by students who have a tendency to include visual imagery in their reasoning. We explore links between the students' mathematical behavior in solving these problems and their perception of their roles as learners. We develop a theory in which the tendency to visualize, coupled with the students' view of their role, can be used to account for their mathematical behavior. [ABSTRACT FROM AUTHOR]

Niels Bohr's term `complementarity' has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972, stated ``the real problem which confronts mathematics teaching is not that of rigor, but the problem of the development of `meaning', of the `existence' of mathematical objects''. Student's insistence on absolute `meaning questions', however, becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all, an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice, mathematical practice requires a complementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux' profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. [ABSTRACT FROM AUTHOR]

This paper presents a preliminary analysis from a study designed to investigate the participation of students learning English as an additional language (EAL) in primary school mathematics. A key question for the study concerns how such students make sense of mathematics classroom interaction. Primary data consists of two transcripts in which a 9-year-old EAL student works on a word problem task with a monolingual peer. The discursive analysis explores the two students' use of patterns of attention in their interaction. The analysis describes three patterns of attention: attention to narrative experience, to the genre of word problems and to the mathematical structure of their problems. These patterns of attention are used in the process of the two students' sense-making together in ways which support the participation of the EAL student. [ABSTRACT FROM AUTHOR]

Implication is at the very heart of mathematical reasoning. As many authors have shown, pupils and students experience serious difficulties in using it in a suitable manner. In this paper, we support the thesis that these difficulties are closely related with the complexity of this notion. In order to study this complexity, we refer to Tarski's semantic truth theory, which contributes to clarifying the different aspects of implication: propositional connective, logically valid conditional, generalized conditional, inference rules. We will show that for this purpose, it is necessary to extend the classical definition of implication as a relation between propositions to a relation between open sentences with at least one free variable. This permits to become aware of the fact that, in some cases, the truth-value of a given mathematical statement is not constrained by the situation, contrary to the common standpoint that, in mathematics, a statement is either true or false. In the present paper, the didactic relevance of this theoretical stance will be illustrated by an analysis of two problematic situations and the presentation of some experimental results from our research on first-year university students' understanding of implication. [ABSTRACT FROM AUTHOR]

This paper presents a semiotic analysis of the (often claimed) potential of computer algebra systems (CAS) for enabling mathematical activity on a conceptual level higher than usual. Theoretical points are illustrated by examples from a development project in the context of a first year university course on calculus. We also discuss how they may be used in `a posteriori' didactical analysis and in didactical engineering. [ABSTRACT FROM AUTHOR]

This paper is a critical review of the ethnomathematics literature and classifies ethnomathematics according to where it might stand in relation to the teaching of formal, academic mathematics. This paper investigates what it sees as four possibilities: ethnomathematics should replace academic mathematics, ethnomathematics should be a supplement to the mathematics curriculum, ethnomathematics should be used as a springboard for academic mathematics and ethnomathematics should be taken into consideration when preparing learning situations. We argue that it is only through the lens of formal, academic mathematics sensitive to cultural differences that the real value of the mathematics inherent in certain cultures and societies be understood and appreciated. [ABSTRACT FROM AUTHOR]

Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult questions, I propose to base research on a metaphor of thinking-as-communicating . This conceptualization entails viewing learning mathematics as an initiation to a certain well defined discourse . Mathematical discourse is made special by two main factors: first, by its exceptional reliance on symbolic artifacts as its communication-mediating tools, and second, by the particular meta-rules that regulate this type of communication. The meta-rules are the observer's construct and they usually remain tacit for the participants of the discourse. In this paper I argue that by eliciting these special elements of mathematical communication, one has a better chance of accounting for at least some of the still puzzling phenomena. To show how it works, I revisit the episodes presented at the beginning of the paper, reformulate the ensuing questions in the language of thinking-as-communication, and re-address the old quandaries with the help of special analytic tools that help in combining analysis of mathematical content of classroom interaction with attention to meta-level concerns of the participants. [ABSTRACT FROM AUTHOR]

The topic of this paper is mathematical identity or self-concept. This elusive construct is explored through two classroom episodes, in the public domain of whole classwork, in a senior high-school class. The structures in the dialogue in both episodes could constitute enhancement of self-concept for some students and mediate against the confidence in mathematics of others, but intersecting effects of students’ one-to-one conversations are shown to be important to final outcomes. [ABSTRACT FROM AUTHOR]

This paper examines perspectives on values, purpose and methodology in mathematics education in schools in the light of the Third International Mathematics and Science Survey (TIMSS) and current debates on standards. It argues that standards of attainment in school mathematics are closely connected to belief systems regarding value and purpose; that these systems do not always collectively offer a credible and coherent vision for mathematics education which can be effectively implemented in school classrooms; and that this coherence of vision is what to a large extent characterises the higher performing TIMSS countries. The paper forms part of a wider investigation into the processes of change in education, with a particular focus on mathematics. [ABSTRACT FROM AUTHOR]

This paper is an attempt to add to the foundation of our understanding of meaning making in mathematics education. This attempt seems to be necessary as a growing body of research, primarily in developmental psychology, begins to change our view of early human development. Empathy, reciprocity, and implicit understanding seem to be more suitable concepts to describe human development and learning than the ones previously employed, based, e.g., on aggression as natural instinct or competitiveness as genetically wired basic drive. [ABSTRACT FROM AUTHOR]

The semiotic approach to mathematics education introduces the notion of 'semiotic system' as a tool to describe mathematical activity. The semiotic system is formed by the set of signs, the production rules of signs and the underlying meaning structures. In this paper, we present the notions of system of practices and configuration of objects and processes that complement the notion of semiotic system and help to understand the complex nature of mathematical objects. We also show in what sense these notions facilitate the description and comprehension of building and communicating mathematical knowledge, by applying them to analyze semiotic systems involved in the teaching and learning of some elementary arithmetic concepts. [ABSTRACT FROM AUTHOR]

This paper offers a typology of forms and uses of abduction that can be exploited to better analyze abduction in proving processes. Based on the work of Peirce and Eco, we describe different kinds of abductions that occur in students' mathematical activity and extend Toulmin's model of an argument as a methodological tool to describe students' reasoning and to classify the different kinds of abduction. We then use this tool to analyze case studies of students' abductions and to identify cognitive difficulties students encounter. We conclude that some types of abduction may present obstacles, both in the argumentation when the abduction occurs and later when the proof is constructed. [ABSTRACT FROM AUTHOR]

Developing deep conceptual understanding of what Ma () calls fundamental mathematics is a well-accepted goal of teacher education. This paper presents a microanalysis of an intriguing episode within a course designed to encourage such understanding. An adaptation of Krummheuer's () elaboration of Toulmin's (1958/) diagrams is used to examine video recordings and transcripts of a group of graduate students in secondary mathematics education grappling with the idea of a three-dimensional line having negative slope. The graduate students' understandings of slope are examined using an expansion of Stump's (, ) categories of conceptions of slope. The episode ends in an interesting impasse, in which the graduate students agree to pursue the idea no further, purposely ignoring the question of negative slope, despite the clear intention of the task. The analysis explores the argumentation, factors of the learning environment, and conceptions of slope that may have contributed to this impasse. [ABSTRACT FROM AUTHOR]

This paper focuses on fifth-grade students’ use and preference for mathematically (MB) and practically based (PB) explanations within two mathematical contexts: parity and equivalent fractions. Preference was evaluated based on three parameters: the explanation (1) was convincing, (2) would be used by the student in class, and (3) was one that the student wanted the teacher to use. Results showed that students generated more MB explanations than PB explanations for both contexts. However, when given a choice between various explanations, PB explanations were preferred in the context of parity, and no preference was shown for either type of explanation in the context of equivalent fractions. Possible bases for students’ preferences are discussed. [ABSTRACT FROM AUTHOR]

The introduction of the concept of the variable represents a critical point in the arithmetic–algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of “unknown” interferes with the interpretation of the variable “in functional relation” and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16–19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16–17 years. [ABSTRACT FROM AUTHOR]

The study reported here is the third in a series of research articles (Harkness, S. S., D’Ambrosio, B., & Morrone, A. S.,in Educational Studies in Mathematics 65:235–254, ; Morrone, A. S., Harkness, S. S., D’Ambrosio, B., & Caulfield, R. in Educational Studies in Mathematics 56:19–38, ) about the teaching practices of the same university professor and the mathematics course, Problem Solving, she taught for preservice elementary teachers. The preservice teachers in Problem Solving reported that they were motivated and that Sheila made learning goals salient. For the present study, additional data were collected and analyzed within a qualitative methodology and emergent conceptual framework, not within a motivation goal theory framework as in the two previous studies. This paper explores how Sheila’s “trying to believe,” rather than a focus on “doubting” (Elbow, P., Embracing contraries, Oxford University Press, New York, ), played out in her practice and the implications it had for both classroom conversations about mathematics and her own mathematical thinking. [ABSTRACT FROM AUTHOR]

This paper reports the outcomes of an empirical study undertaken to investigate the effect of students’ cognitive styles on achievement in measurement tasks in a dynamic geometry learning environment, and to explore the ability of dynamic geometry learning in accommodating different cognitive styles and enhancing students’ learning. A total of 49 6th grade students were tested using the VICS and the extended CSA-WA tests (Peterson, Verbal imagery cognitive styles and extended cognitive style analysis-wholistic analytic test—Administration guide. New Zealand: Peterson, ) for cognitive styles. The same students were also administered a pre-test and a post-test involving 20 measurement tasks. All students were taught a unit in measurement (area of triangles and parallelograms) with the use of dynamic geometry, after a pre-test. As expected, the dynamic geometry software seems to accommodate different cognitive styles and enhances students’ learning. However, contrary to expectations, verbalisers and wholist/verbalisers gained more in their measurement achievement in the environment of dynamic geometry than students who had a tendency towards other cognitive styles. The results are discussed in terms of the nature of the measurement tasks administered to the students. [ABSTRACT FROM AUTHOR]

The article presents instructions for authors or contributors to the journal.

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]

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention. [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]

History of mathematics occupies itself describing processes of growth and development, whereas philosophy of mathematics is concerned with questions of justification. Both play an essential role within the educational context. But there is a problem because genuine historical studies necessitate ever greater particularity whereas mathematics and philosophy require generality and abstraction. The paper offers some methodological reflections about these matters together with two case studies from nineteenth century history of arithmetic and integration theory, respectively, which try to strike a balance between the directly opposed requirements. [ABSTRACT FROM AUTHOR]

The CAME1 project was inaugurated in 1993 as an intervention delivered in the context of mathematics with the intention of accelerating the cognitive development of students in the first two years of secondary education. This paper reports substantial post-test and long-term National examination effects of the intervention, yet, by discussing the methodology used, questions the assumptions implicit in the original intention. It is now suggested that a better view is to regard CAME as a constructive criticism of normal instructional teaching, with implications for the role of mathematics teachers and university staff in future professional development. [ABSTRACT FROM AUTHOR]

As a first step of a project on learning difficulties in elementary set theory, we carried out a study aimed at identifying some of the difficulties encountered by a group of 21 mathematics majors taking a course on logic and set theory. In this paper, we shall discuss one of the difficulties that we observed, namely a difficulty in conceiving a set as an object distinct from its elements, i.e. failing to fully grant sets the status of objects. This difficulty may be related to the following three types of errors that we observed in the students' first assignment: (1) confusing belonging and inclusion, (2) confusing the union of sets A, B, C, ... with the set whose elements are A, B, C, ... and (3) adding or deleting curly brackets. We shall present excerpts from the students' work illustrating these three types of errors and explain how they may be related to a difficulty in conceiving a set as an object distinct from its elements. [ABSTRACT FROM AUTHOR]

ABSTRACT. This study investigates how mathematics secondary schools' teachers in \hbox{Jordan} perceive critical thinking and compares teachers' perceptions before and after educational reform. Data were collected from 12 schools twice: in 1988 and in 2004 by interviewing 47 Mathematics teachers. The interview included questions related to teachers' understanding of critical thinking, its role and importance in learning Mathematics, and instructional strategies that could help improve students' critical thinking skills. Results found no improvement in secondary Mathematics teachers' perceptions of critical thinking despite 15 years of educational reform. The majority of Mathematics teachers seemed not to have a clear and adequate understanding of critical thinking. Though most of the teachers claimed they have to teach critical thinking, more than half of them could not suggest any learning situation that could help in fostering critical thinking in Mathematics classes. Also most of the teachers claimed that critical thinking would help students in learning Mathematics; yet less than half were able to give a convincing justification for that. A small percentage of the secondary Mathematics teachers believed they can help all students to foster their critical thinking. It is thus recommended that the concept of critical thinking be transferred from the realm of rhetoric to the field of practice. The paper ends with some recommendations related to the Mathematics teachers training programs in view of the results of this study. [ABSTRACT FROM AUTHOR]

This paper proposes a genetic development of the concept of limit of a sequence leading to a definition, through a succession of proofs rather than through a succession of sequences or a succession of εs. The major ideas on which it is based are historical and depend on Euclid, Archimedes, Fermat, Wallis and Newton. Proofs of equality by means of inequalities precede the notion of limit. For example, the determination of the volume of a pyramid precedes the definition of limit. The algebraic details given here are anachronistically modern, and the notion of a vice between two inequalities which provides the distinctive perspective of this paper would not have been recognized in this form by any of the named mathematicians since it presumes the existence of negative numbers and their comparability in a modern sense with positive numbers. [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]