Search

Emotions and affect have become an area of interest to mathematics research. However, both tend to be intellectualized and approached as external to and separate from intellect. As far back as in the 1930s, Vygotsky considered the split between affect and intellect as psychology’s greatest defect. To address that defect, over the last 18–24 months of his life, he sketched the barebones of a different psychology but died before he could develop his ideas into a theory of emotions. His ideas provided us with a roadmap for developing an approach in which intellect and affect are united to constitute a central part in the life of the whole person. Such a position, conceived here in dramatic terms, remains virtually unknown, in part because of the unavailability of specific texts and personal notes until a few years ago. In this study, we develop Vygotsky’s seeds and go beyond what he actually stated. We focus our conceptualization of affect on mathematics classroom practices and exemplify our theoretical perspective with empirical data drawn from a number of studies.

Inspired by the entering of computer algebra systems (CAS) in the Danish upper secondary school mathematics program, this article addresses, from a theoretical stance, what may happen when traditional procedures are outsourced to CAS. Looking at the commands “solve” and “desolve,” it is asked what happens when such CAS procedures are objectified in the students’ minds, and what the nature might be of the resulting “objects.” The theoretical analyses draw on a selection of classical mathematics education frameworks on conceptualization and are related to the research literature on technology in mathematics education. The article suggests the following characteristics as elements of negative effects: loss of distinctive features of concept formation, a consequential reclassification of mathematical objects, instability of CAS solutions as objects, and prevailing a posteriori reasoning on students’ behalf when relying solely on CAS in their mathematical work.

This paper presents the results of a systematic review of the empirical literature, reported in research journals, on Mathematics Learner Identity (MLI). In the mathematics education research field, arguments have been made that the conceptualization of Mathematics Identity (MI) is inconsistent and this makes the literature as a whole incoherent. This study aims to summarize how the concept has been employed and to develop an integrative model for analyzing concept-definitions and concept-operationalizations of MLI. An analysis of 69 papers revealed 3 main dimensions that characterized how identity is conceptually defined in any research literature (social/subjective, enacted/representational and change/stability) and 5 main categories that describe how the literature has implemented these dimensions operationally (identity as individual attributes; identity as narratives; identity as a relationship with specific practices; identity as ways of acting; and identities as afforded and constrained by local practices). An emphasis on representational aspects of identities, particularly during higher education, and enacted and practice-related identities during primary and secondary education research was found. The discussion shows how the field would be clarified if studies made their choices of conceptualization clear in terms of these dimensions and categories, and if research as a whole becomes more aware of restrictions/limitations in identity research, particularly at different points of the educational trajectory of students.

Learning experiences regarding the line of best fit are typically students’ first encounters with the fundamental topic of statistical association. Students bring with them into these learning experiences prior knowledge and experiences about mathematical lines and their properties, namely slope. This study investigated the role students’ conceptions of slope play in their conceptualization of the line of best fit and approaches to placing it informally (i.e., by eye). Task-based interviews concerning the meaning and placement of the line of best fit conducted with seven grade 8 students were analyzed for this study. The results showed that students’ conceptualizations of slope can play a significant role in their reasoning about the line of best fit, in both productive and unproductive ways. Analysis of associations between slope conceptualizations and students’ criteria for placing the line, accuracy of the placed line, and meaning of the line of best fit are presented. The discussion highlights implications of the study for the teaching of lines in both mathematical and statistical settings.

This article is intended for inclusion in the Special Issue on Affect in Mathematical Problem Posing: Conceptualization, Advances, and Future Directions of Research.

This paper aims to explore the educative power of an experienced mathematics teacher educator-researcher (MTE-R) who displayed his insights and strategies in teacher professional development (TPD) programs. To this end, we propose a framework by first conceptualizing educative power based on three constructs—communication, reasoning, and connection—and then we extend the conceptualization with another two dimensions: the reciprocal facilitator-learner relationships involving educators, teachers, and students, as well as a bridge between research and practice. Based on both self-study and case-study approaches, we further elaborate features specific to the MTE-R’s educative power which includes communication using an approach of creating educative phenomenology, reasoning by mapping teachers’ ideas onto emergent models to solve problems in educative challenges, and connection between research and practice by coordination. In particular, the core of the educative power that supported the MTE-R to initiate at-the-moment actions was his insights into the essence of mathematics, and the learning of students and teachers. We believe that the conceptual framework in this study offers a powerful tool that could guide the analyses of educative power, especially for those studies related to the initiation of at-the-moment actions and the implementation of TPD programs.

The articles in this Special Issue arose originally from the work of the Research Forum on Sociological Frameworks in Mathematics Education Research at the conference of the International Group for the Psychology of Mathematics Education (PME) in 2009. The acceptance of this Research Forum at PME was indicative of the shift within mathematics education research during the last two decades involving what has been called the “social turn” (Lerman, 2000). Researchers concerned with a wide variety of issues within mathematics education have come increasingly to see the inseparability of culture, context and cognition. Even within research that focuses primarily on cognitive aspects of learning and knowledge, notions of situated learning and distributed knowledge (Lave & Wenger, 1991) are widely used, as well as other theoretical perspectives that emphasise social aspects of learning, drawing in particular on Vygotskian psychology (Vygotsky, 1978). Moving beyond seeing mathematics learning solely as the endeavour of individual students and teachers has been reflected in a broader conceptualization of the subject matter of the field of mathematics education research. Valero (2010) has drawn our attention to the complexity of the networks of communities, interest groups and practices relevant to mathematics education and to the need for research to address this multiplicity of social practices and the connections between them. We are thus aware of the importance of studying the various communities and practices in which students and teachers participate, both within the classroom and beyond. We recognise the influence of policy and institutional structures and constraints at local, national and international levels. We appreciate the impact of the various discourses available inside and outside the school—discourses in the sense written with a capital D by Gee (1996 )a nd defined as incorporating “theories” about what is normal and right and structuring the kinds of identities available to participants. This increasing attention to social aspects of learning has been accompanied by a growth in research foregrounding issues of social justice. Differing levels of achievement in mathematics in particular as well as in education as a whole have been associated with membership of various social groups and the effects of such factors as gender, ethnicity, class and linguistic background on the achievement of students in school mathematics have long been a focus of study. However, our ways of understanding the phenomenon of school failure have developed. In particular, there has been a move from locating the reasons for failure in the characteristics of the individuals concerned or of their communities towards seeking to understand how the

The data reported in this article come from a large project whose goal was to explore how Latin American students in Catalonia, Spain use their two languages—Spanish and Catalan—to support their learning of mathematics in small groups with other students who are Spanish- or Catalan-dominant. For 5 years, lessons from bilingual mathematics classrooms in three public secondary schools were video-recorded and transcribed. In the presentation of findings, I discuss three language practices that emerged from the analyses of several classroom instances as follows: (1) caution with mathematical vocabulary, (2) invention of terms, and (3) word-for-word translation. One example is chosen to represent each practice and some of its situated effects. The first two examples support the view that the experience of language difficulties, either real or presumed, contributes to generating opportunities that may be beneficial to mathematics learning. The third example, where the focus on the mathematics is hindered, points to a contrasting finding. Unlike other studies in the field, which have reported the difficulties and obstacles that arise in learning and teaching due to bilingualism in the classroom, I propose a change of focus through the conceptualization of language as a potential for thinking and doing, and particularly for learning and teaching mathematics.

This article presents a theoretical conceptualization of mathematics homework as a social practice. Rather than considering homework as a task or an artifact, this approach frames homework in terms of the social contexts in which students participate and how students participate in those contexts. This perspective has long been suggested by homework researchers but has not been developed as a framework for understanding homework. Drawing from Wenger’s (1998) social theory of learning and research grounded in sociocultural theory, this conceptualization makes central meaning making and identity development, and puts forth meaning and identity as lenses for understanding students’ participation in the practice of mathematics homework. To further develop this conceptualization of homework, I draw on data from an ethnographic study of the role and meaning of mathematics homework in the lives of middle school students. Case studies of two students are presented to demonstrate the relationships among the meaning of homework, students’ identities, and their participation in the practice of homework. One student’s experiences support him in identifying as a capable mathematics student who is bound for continued academic success through high school to college. Thus, he comes to take on mathematics homework as a means to learn and succeed. The other student’s experiences support him in building an identity that leads him to reject homework. This conceptualization of homework and the case study data have implications for the practice of homework and for theories of students’ motivational dispositions in the context of mathematics homework and learning in general.

In this article, I present and build on the ideas of John Threlfall [(Educational Studies in Mathematics 50:29–47, 2002)] about strategy development in mental mathematics contexts. Focusing on the emergence of strategies rather than on issues of choice or flexibility of choice, I ground these ideas in the enactivist theory of cognition, particularly in issues of problem posing, for discussing the nature of the solving processes at play when solving mental mathematics problems. I complement this analysis and conceptualization by offering two examples about issues of emergence of strategies and of problem posing, in order to offer illustrations thereof, as well as to highlight the fruitfulness of this orientation for better understanding the processes at play in mental mathematics contexts.

This study describes the types of explanations one student, Sharon, gives and prefers at different ages. Sharon is interviewed in the second grade regarding multiplication of one-digit numbers, in the fifth grade regarding even and odd numbers, and in the sixth grade regarding equivalent fractions. In the tenth grade, she revisits each of these concepts again. The study investigates the different forms Sharon’s explanations take at different ages as well as how she perceives the nature of mathematical explanations at different ages. Sharon’s explanations are also used to investigate her conceptualization of the number zero, a concept which runs across the curriculum at different ages. Finally, the study explores a method for investigating the long-term mathematical development of one student. Implications for future research are discussed.

In this paper, we examine sixth grade students’ degree of conceptualization of fractions. A specially developed test aimed to measure students’ understanding of fractions along the three stages proposed by Sfard (1991) was administered to 321 sixth grade students. The Rasch model was applied to specify the reliability of the test across the sample and cluster analysis to locate groups by facility level. The analysis revealed six such levels. The characteristics of each level were specified according to Sfard’s framework and the results of the fraction test. Based on our findings, we draw implications for the learning and teaching of fractions and provide suggestions for future research.

With our conceptualization of Harre and van Langenhove’s (1999) positioning theory, we draw attention to immanent experience and read transcendent discursive practices through the moment of interaction. We use a series of spatial images as metaphors to analyze the way positioning is conceptualized in current mathematics education literature and the way it may be alternatively conceptualized. This leads us to claim that changing the way mathematics is talked about and changing the stories (or myths) told about mathematics is necessary for efforts to change the way mathematics is done and the way it is taught.

I investigate the contributions of three theoretical frameworks to a research process and the complementary role played by each. First, I describe the essence of each theory and then follow the analysis of their specific influence on the research process. The research process is on the conceptualization of the notion of limit by means of the discrete continuous interplay. I investigate the influence of the different perspectives on the research process and realize that the different theoretical approaches intertwine. Moreover, I realize that the research study demanded the contribution of more than one theoretical approach to the research process and that the differences between the frameworks could serve as a basis for complementarities.

Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.

This paper addresses issues linking research into the classroom teaching and learning of mathematics with the growth of knowledge in mathematics teaching, developments in the practice of teaching and the enhanced learning of mathematics by students in classrooms. A basic premise is that research promotes development. The paper considers both insider andoutsider research and co-learning between teachers and educators in promoting classroom inquiry. Through a consideration of elements of theory such as knowledge and inquiry in teaching and of learning as knowledge growth through research/inquiry leading to enhancement of students' learning of mathematics, a framework is suggested. Its purposes include analysis of a research project's contribution to teaching development and conceptualization of research which has teaching development as one of its aims. Use of the framework is exemplified through its application to reports of three mathematics education research projects in the public domain. A brief afterword links the framework to concepts in activity theory.

I will analyse the individual and collective activities of fourth grade students while trying to explain a schoolfellow's mistake in the use of a geometrical schema (shared by the class) of the sunshadows phenomenon. The study aims at detecting productive links between context-related argumentation, mathematical modelling (here concerning sunshadows) and conceptualization (inclination of a straight line to a plane), as well as discussing their relevance for mathematics education.

This article reports on a study1 that investigated the teaching of mathematical problem solving from a teacher's perspective. The study focused on three teachers and their way of making sense of teaching problem solving. Data collected through interviews and classroom observations were analyzed in the context of an interpretive qualitative study to understand the meanings of the participants' classroom processes. The findings indicated that the participants unconsciously constructed personal metaphors that became the basis of their conceptualization of problems and making sense of their teaching. “Community”, “adventure” and “game” were determined to be the key metaphors of the three participants, respectively. These metaphors embodied their personal experiences and personal practical beliefs that provided the unique meanings associated with their classroom processes. The outcome suggested that the study of such metaphors could be a promising avenue in enhancing mathematics teacher education and in problem solving research in the quest to make the teaching of problem solving more meaningful and effective in a classroom context.

The different stages in the conceptual evolution of actual infinity are analyzed. We show how this concept occurs in high-school students at a prior, or at best, at the same level, as in Bernard Bolzano's conceptualization (as defined in his work The Paradoxes of Infinity). Furthermore, in terms of the historical development of the concept, the students' conceptualization has intra-objectal stage characteristics. Within the same framework, we show how Cantor's work fits into the inter-objectal stage, as well as presenting the difficulties encountered by students in order to reach this stage, given the current curricular structure. We conclude that such evidence should bring about a reconsideration of Piagetian ideas in education.