A Relational Perspective on Issues of Cultural Diversity and Equity as They Play Out in the Mathematics Classroom

In this article, we present a relational perspective in which cultural diversity is viewed as a relation between people's participation in the practices of different communities. In the case at hand, the relevant practices were those of students' local, home communities, and the broader communities to which they belonged in wider society on the one hand and the specifically mathematical practices established by the classroom community on the other hand. In the 1st part of the article, we discuss how we might characterize the practices of these various communities by drawing on Wenger's (1998) notion of a community of practice and on Gee's (1997) notion of a Discourse. In doing so, we question the manner in which students are frequently classified exclusively in terms of the standard categories of race and ethnicity in investigations of equity in mathematics education. Later in the article, we clarify that in addition to focusing on the continuities and contrasts between the practices of different communities, the relational perspective also encompasses issues of both power and identity. As we illustrate, the gatekeeping role that mathematics plays in students' access to educational and economic opportunities is not limited to differences in the ways of knowing associated with participation in the practices of different communities. Instead, it also includes difficulties that students experience in reconciling their views of themselves and who they want to become with the identities that they are invited to construct in the mathematics classroom.     

An Investigation of Relationships between Seventh-Grade Students' Beliefs and Their Participation during Mathematics Discussions in Two Classrooms

As mathematics teachers attempt to promote classroom discourse that emphasizes reasoning about mathematical concepts and supports students' development of mathematical autonomy, not all students will participate similarly. For the purposes of this research report, I examined how 15 seventh-grade students participated during whole-class discussions in two mathematics classrooms. Additionally, I interpreted the nature of students' participation in relation to their beliefs about participating in whole-class discussions, extending results reported previously (Jansen, 2006) about a wider range of students' beliefs and goals in discussion-oriented mathematics classrooms. Students who believed mathematics discussions were threatening avoided talking about mathematics conceptually across both classrooms, yet these students participated by talking about mathematics procedurally. In addition, students' beliefs about appropriate behavior during mathematics class appeared to constrain whether they critiqued solutions of their classmates in both classrooms. Results suggest that coordinating analyses of students' beliefs and participation, particularly focusing on students who participate outside of typical interaction patterns in a classroom, can provide insights for engaging more students in mathematics classroom discussions.     

Which mathematics should we teach engineering students? An empirically grounded case for a broad notion of mathematical thinking

While many engineering educators have proposed changes to theway that mathematics is taught to engineers, the focus has oftenbeen on mathematical content knowledge. Work from the mathematicseducation community suggests that it may be beneficial to considera broader notion of mathematics: mathematical thinking. Schoenfeldidentifies five aspects of mathematical thinking: the mathematicscontent knowledge we want engineering students to learn as wellas problem-solving strategies, use of resources, attitudes andpractices. If we further consider the social and material resourcesavailable to students and the mathematical practices studentsengage in, we have a more complete understanding of the breadthof mathematics and mathematical thinking necessary for engineeringpractice. This article further discusses each of these aspectsof mathematical thinking and offers examples of mathematicalthinking practices based in the authors' previous empiricalstudies of engineering students' and practitioners' uses ofmathematics. The article also offers insights to inform theteaching of mathematics to engineering students.     

Individual and Collective Mathematical Development: The Case of Statistical Data Analysis

In the first part of this article, I clarify how we analyze students' mathematical reasoning as acts of participation in the mathematical practices established by the classroom community. In doing so, I present episodes from a recently completed classroom teaching experiment that focused on statistics. Against the background of this analysis, I then broaden my focus in the final part of the article by developing the themes of change, diversity, and equity.     

Home to School: Numeracy Practices and Mathematical Identities

Drawing on a perspective of mathematics as situated social practice, we focus on 4 children in an urban preschool classroom and follow those children between home and school sites to shed light on urban children's persistent underachievement in mathematics. In this article, we describe the ways in which numeracy practices travel with children between home and school and, within those contexts, shape complex and sometimes limited social identities for children. We found that school imperatives, such as assessments and socialization curricula, often obscure teachers' views of children's mathematical practices. Deficit assumptions about family and community support for children, and limited interaction between caregivers and teachers, further contribute to the tendency of school personnel to overlook the mathematical practices that children bring with them to school. We further suggest that vignettes drawn from ethnographic-type research such as this have potential for professional development for classroom teachers.     

Research, practice, uncertainty and responsibility

Three issues concerning the relationship between research and practice are addressed. (1) A certain ‘prototype mathematics classroom’ seems to dominate the research field, which in many cases seems selective with respect to what practices to address. I suggest challenging the dominance of the discourse created around the prototype mathematics classroom. (2) I find it important to broaden the school-centred discourse on mathematics education and to address the very different out-of-school practices that include mathematics. Many of these practices are relevant for interpreting what is taking place in a school context. That brings us to (3) socio-political issues of mathematics education. When the different school-sites for learning mathematics as well as the many different practices that include mathematics are related, we enter the socio-political dimension of mathematics education.On the one hand we must consider questions like: Could socio-political discrimination be acted out through mathematics education? Could mathematics education exercise a regimentation and disciplining of students? Could it include discrimination in terms of language? Could it include sexism and racism? On the other hand: Could mathematics education bring about competencies which can be described as empowering, and as supporting the development of mathematical literary or a ‘mathemacy’, important for the development of critical citizenship?However, there is no hope for identifying a one-way route to mathemacy. More generally: There is no simple way of identifying the socio-political functions of mathematics education. Mathematics education has to face uncertainty, and this challenge brings us to the notion of responsibility.     

Coordinating Theories of Learning to Account for Second-Grade Children's Arithmetical Understandings

This article coordinates social constructivism and socioculturalism orientations to explain 2nd-grade children's reasoning with 2-digit quantities. From a social constructivist position, we illustrate how the classroom teacher and the students constituted what counted as an acceptable mathematical explanation. As children offered informal and conventional ways of interpreting problem situations, they were expected to reason with quantities in sensible ways. From a sociocultural position, we explain how the teacher's and students' contributions were situated within the mathematical ways of knowing constituted by the community at large. Particular children's contributions were clarified in terms of the ways in which they participated in socially organized activities. By coordinating these lenses, we argue the local classroom mathematical practices constrained and enabled the mathematical practices of the wider society.     

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments.     

From the margin to the center: A framework for rehumanizing mathematics education for students with dis/abilities

Mathematics education for students with dis/abilities has a legacy of exclusion; its scholarship has been dominated by ableist perspective: a focus on rote mathematics facts, procedural instruction, and a narrow range of mathematics content. Grounded in critical pedagogy and critical disabilities studies, we argue for a shift in the focus away from the student as the site of defect and intervention to recognize the complex embodiment of dis/ability as a process produced within historical, political, social, and material contexts. The theoretical assumptions and guiding principles for a framework of culturally responsive and relational understanding of dis/ability will be shared together with a case of such an approach in classroom practice. The paper advances efforts to support all learners in robust mathematics learning communities and to situate research from perspectives that acknowledge the situated, interactional nature of ability, identity, and achievement.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

Researcher–teacher relationships and models for teaching development in mathematics education

This article offers theoretical and analytical approaches to investigating how researchers and teachers can work together to create knowledge in mathematics education. It argues that researchers and teachers are members of separate, but related, communities of practice, which create and value different types of knowledge. However, connections between communities can be established through discrete boundary encounters, longer term boundary practices, or peripheral participation by members of one community in the practices of another community. A framework for analyzing researcher–teacher relationships is presented and then used to compare ways in which I, as a university-based researcher, worked with teachers in three different types of research projects. The analysis indicates that successful research collaborations are characterized by mutuality of researcher and teacher motivations, roles, and purposes, and complementarity of their expertise and knowledge. Such collaborations build two-way connections between communities through practices that support mutual engagement across the boundaries that define them.     

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.     

Emergent pedagogies and the changing role of the teacher in the TI-Nspire Navigator-networked mathematics classroom

It is generally accepted that the introduction of networked technologies to the mathematics classroom can stimulate an irreversible change within the classroom concerning: the role of the teacher; the nature of the classroom tasks; and the way in which students engage in the process of learning mathematics. This article will use the context of a classroom-based study into teachers’ developing practices with the TI-Nspire Navigator-networked system of handhelds to explore the nature of these practices and the implications for the mathematics classroom. The emergence of a range of formative assessment practices is described and the implication of these practices on desirable learning opportunities (as described by the teachers themselves) is discussed.     

Identifying Cyclic Patterns of Interaction to Study Individual and Collective Learning

Our analysis of a college level mathematics course for prospective secondary mathematics teachers revealed that each student developed, at least to some degree, a conceptual orientation for teaching mathematics (A. G. Thompson, Philipp, Thompson, & Boyd, 1994). This initial finding led to a more in-depth question: If we assume an emergent perspective (Cobb & Bauersfeld, 1995; Cobb & Yackel, 1996) in which the values, practices, and social motivations of the classroom are believed to play critical roles in students' conceptual development, what social aspects emerged that supported these individual constructions? To address this question, we documented the emergence of a collective conceptual orientation and then used this construct to explore the reflexivity between its emergence and individual students' development of conceptual orientations.     

Exemplary mathematics instruction and its development in selected education systems in East Asia

What may teachers do in developing and carrying out exemplary or high-quality mathematics classroom instruction? What can we learn from teachers’ instructional practices that are often culturally valued in different education systems? In this article, we aim to highlight relevant issues that have long been interests of mathematics educators worldwide in identifying and examining teachers’ practices in high-quality mathematics classroom instruction, and outline what articles published herein can help further our understanding of such issues with cases of exemplary mathematics instruction valued in the Chinese Mainland, Hong Kong, Japan, Singapore, South Korea, and Taiwan.     

Characterizing exemplary mathematics instruction in Japanese classrooms from the learner’s perspective

This paper aims to examine key characteristics of exemplary mathematics instruction in Japanese classrooms. The selected findings of large-scale international studies of classroom practices in mathematics are reviewed for discussing the uniqueness of how Japanese teachers structure and deliver their lessons and what Japanese teachers value in their instruction from a teacher’s perspective. Then an analysis of post-lesson video-stimulated interviews with 60 students in three “well-taught” eighth-grade mathematics classrooms in Tokyo is reported to explore the learners’ views on what constitutes a “good” mathematics lesson. The co-constructed nature of quality mathematics instruction that focus on the role of students’ thinking in the classroom is discussed by recasting the characteristics of how lessons are structured and delivered and what experienced teachers tend to value in their instruction from the learner’s perspective. Valuing students’ thinking as necessary elements to be incorporated into the development of a lesson is the key to the approach taken by Japanese teachers to develop and maintain quality mathematics instruction.     

Students' gendered perceptions of mathematics in middle grades single‐sex and coeducational classrooms

With the introduction of single‐sex classroom settings in coeducational public schools, there is an ongoing debate as to whether single‐sex education may reduce or reinforce traditional stereotypes and gender roles. In this article we present findings from a study that investigated the extent to which mathematics is perceived as a gendered domain among adolescent students enrolled in single‐sex classes and coeducational classes. Further we analyzed the relationships between student characteristics, class‐type, and teacher variables on students' perceptions of gender in mathematics. Findings from this study challenge the traditional view of mathematics as a male domain. Female participants more frequently considered mathematics to be a female domain than the male participants. Male participants, on the other hand, typically did not stereotype the mathematics as a gendered domain. Results from this study do not indicate, for girls at least, that participation in single‐sex classes results in a greater propensity to stereotype mathematics as a gendered domain than would be the case in coeducational classes. This study contributes to the evolving discourse and understanding of adolescents' gendered attitudes and beliefs towards mathematics—especially in light of stereotyped assertions that have a bearing on efforts to promote the learning of mathematics and science.     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

“Mathematicians Would Say It This Way”: An Investigation of Teachers' Framings of Mathematicians

Although popular media often provides negative images of mathematicians, we contend that mathematics classroom practices can also contribute to students' images of mathematicians. In this study, we examined eight mathematics teachers' framings of mathematicians in their classrooms. Here, we analyze classroom observations to explore some of the characteristics of the teachers' framings of mathematicians in their classrooms. The findings suggest that there may be a relationship between a teachers' mathematics background and his/her references to mathematicians. We also argue that teachers need to be reflective about how they represent mathematicians to their students, and that preservice teachers should explore their beliefs about what mathematicians actually do.