Lexical bundle analysis in mathematics classroom discourse: the significance of stance

In this article, we introduce the lexical bundle, defined by corpus linguists as a group of three or more words that frequently recur together, in a single group, in a particular register (Biber, Johansson, Leech, Conrad, & Finegan, 2006; Cortes, English for Specific Purposes 23:397–423, 2004). Attention to lexical bundles helps to explore hegemonic practices in mathematics classrooms because lexical bundles play an important role in structuring discourse and are often treated as “common sense” ways of interacting. We narrow our findings and discussion to a particular type of lexical bundle (called a “stance bundle” or bundles that relate to feelings, attitudes, value judgments, or assessments) because it was the most significant type found. Through comparing our corpus from secondary mathematics classrooms with two other corpora (one from university classrooms (not including mathematics classrooms) and one from conversations), we show that most of the stance bundles were particular to secondary mathematics classrooms. The stance bundles are interpreted through the lens of interpersonal positioning, drawing on ideas from systemic functional linguistics. We conclude by suggesting additional research that might be done, discussing limitations of this work, and pointing out that the findings warrant further attention to interpersonal positioning in mathematics classrooms.     

Challenges of critical colleagueship: examining and reflecting on mathematics teacher study group interactions

This article examines mathematics teacher collegiality by focusing on both the ways in which teachers interacted as critical colleagues in a long-term professional development project and the evolving role of the teacher–educator–researcher as the facilitator of this project. The professional development collaboration comprised two phases: one focused on reading classroom discourse literature and one focused on supporting each other through cycles of action research related to mathematics classroom discourse. Lord’s (1994) critical colleagueship framework is used to examine how a study group of middle-grades (ages 11–16) mathematics teacher–researchers took (or did not take) a more critical stance toward their own teaching practice and that of their colleagues. We found that challenging interactions were related to instances in which the teachers interacted as critical colleagues and were marked by particular features including the use of particular words and the use of personal experience as a form of evidence. We present the ways in which we came to understand what it might look like to scrutinize one’s practice and findings related to the development of this type of collegiality across the two different phases of this project. We end with a section in which the teacher–educator–researcher who facilitated the professional development project reflects on the ways in which the analysis caused her to reconsider both the nature of argumentation in mathematics study group settings and what implications this has with respect to her own practice as a facilitator.     

UNDERSTANDING STUDENTS�� OUT-OF-SCHOOL MATHEMATICS AND SCIENCE PRACTICE

This study examined standard 6 and 8 (Standards 6 and 8 are the sixth and eighth years, respectively, of primary level schooling in Kenya.) students’ perceptions of how they use mathematics and science outside the classroom in an attempt to learn more about students’ everyday mathematics and science practice. The knowledge of students’ everyday mathematics and science practice may assist teachers in helping students be more powerful mathematically and scientifically both in doing mathematics and science in school and out of school. Thirty-six students at an urban school and a rural school in Kenya were interviewed before and after keeping a log for a week where they recorded their everyday mathematics and science usage. Through the interviews and log sheets, we found that the mathematics that these students perceived they used outside the classroom could be classified as 1 of the 6 activities that Bishop (Educ Stud Math 19:179–191, 1988) has called the 6 fundamental mathematical activities and was also connected to their perception of whether they learned mathematics outside school. Five categories of students’ perceptions of their out-of-school science usage emerged from the data, and we found that 4 of our codes coincided with 2 activities identified by Lederman & Lederman (Sci Child 43(2):53, 2005) as part of the nature of science and 2 of Bishop’s categories. We found that the science these students perceived that they used was connected to their views of what science is.     

The role of persuasive research in educational progress

This article describes how scaffolded instruction during whole-class mathematics lessons can provide the knowledge, skills, and supportive context for developing students' self-regulatory processes. In examining classroom interactions through discourse analysis, these qualitative methods reflect a theoretical change from viewing self-regulation as an individual process to that of a social process. This article illustrates how studying instructional scaffolding through the analyses of instructional discourse helps further the understanding of how self-regulated learning develops and is realized in mathematics classrooms. Qualitative methods, such as discourse analyses, and their underlying theoretical frameworks have great potential to help "unlock" theories of learning, motivation, and self-regulation through exploring the reciprocity of teaching and learning in classrooms.     

Mathematics teachers’ representations of authority

Issues of authority abound in education and schooling but have not been interrogated sufficiently. We describe a tool that we have developed to initiate dialogue with teachers about authority in their classrooms—using a diagram to represent authority in their classrooms. Our analysis of the diagrams mathematics teachers created and discussed in our work with them illustrates the importance of understanding teachers’ perspectives about authority. To understand better how mathematics teachers think about the authority in their classrooms, we investigated what sources of authority they represented in their diagrams, and how the teachers related these sources to each other. The diversity in the teachers’ representations exceeded our anticipations, indicating that research on authority in classrooms has merely scratched the surface of understanding the ways mathematics teachers think about authority in their classrooms.     

教师变量对小学生数学学习成绩影响的多水平分析

32名小学数学教师与这些教师所教班级的1691名学生参与了本研究。两个测量工具评价了教师的数学学科知识与学科教学知识,对教师的55节数学课进行了录像,并按照学习任务的认知水平与课堂对话的特点进行了编码,同时测查了学生的期末数学学习成绩。多水平分析结果表明:教师的学科教学知识、课堂学习任务的认知水平、课堂对话中教师提问问题的类型与对话的权威来源对学生的数学成绩具有显著的预测作用;而教师的学科知识对学生数学成绩的影响未达到显著性水平。     

On negotiating White science: a call for cultural relevance and critical reflexivity

This article is a response to Randy Yerrick and Joseph Johnson’s article “Negotiating White Science in Rural Black America: A Case for Navigating the Landscape of Teacher Knowledge Domains”. They write about research conducted by Yerrick in which videos of his teaching practice as a White educator in a predominately Black rural classroom were examined. Their analysis is framed through Shulman’s (1986) work on “domains of teacher knowledge” and Ladson-Billings’ (1999) critical race theory (CRT). Although we appreciate a framework that attends to issues of power, such as CRT, we see a heavier emphasis on Shulman’s work in their analysis. We argue that a culturally relevant pedagogy (CRP) framework has the potential to provide a more nuanced analysis of what occurred in Yerrick’s classroom from a critical lens. Thus we examine Yerrick and Johnson’s work through the five main CRP components (as defined by Brown-Jeffy and Cooper 2011) and ultimately argue that science educators who want to promote equity in their classrooms should engage in continuous critical reflexivity, aid students in claiming voice, and encourage students to become not only producers of scientific knowledge but also users and critics of such knowledge.     

Constructivism in Mathematics Classrooms: Listening to Ghanaian Teachers’ and Students’ Views

One of the challenges of implementing a new curriculum is how to bridge the gap between the underlining principles of the curriculum and the cultural and social orientations of the society which includes teachers and students. This article reports on a study that explored how the cultural and social orientations of teachers and students can influence the implementation of a constructivist curriculum in mathematics classrooms. The data for the study came from 250 students and 41 mathematics teachers, using questionnaires, observations, and interviews. The results showed that inasmuch as mathematics teachers and their students acknowledge the importance of student’s active participation and teamwork, these practices have not been fully conceptualised into the Ghanaian mathematics classroom due to some cultural factors. Two main cultural factors were discovered from the analyses of the results. Firstly, the culture of acknowledging only correct answers in class has a negative impact on individual students’ confidence and participation during mathematics lessons. Also, the culture of teamwork is not fully accepted within Ghanaian classrooms as most students find it difficult working in groups and accepting and appreciating each other’s view. It was evident in all lessons that students were ridiculed by their peers when they provide a wrong answer to a question and this affected individual students’ participation in the classroom. Therefore, we suggested that teachers should be pro-active in promoting a classroom environment which is free from fear and intimidation to motivate students to be actively involved in the classroom discourse.     

Peer relations and access to capital in the mathematics classroom: a Bourdieusian social network analysis

The aim of this article is to explore the structure of social capital in peer networks and its relation to the unequal access of educational resources within mathematics classrooms. We hypothesise that learners can gain access to mathematics through friendship networks which provide more or less help from peers that might sustain (or curtail) their mathematics learning based on a Bourdieusian framework. We report a social network analysis of mutually-recognised helping within friendship groups. This is complemented by observation and interviews that illustrate how different classrooms have different network structures, positioning learners of different ethnic minorities and genders in significantly different ways regarding access to learning. We argue that friendship networks mediate social capital and access to further cultural capital, and that this may help explain structural differences in attainment. Because ethnicity and gender, inter alia, mediate friendship networks, they also mediate access to capital in the classroom.     

Using History to Teach Mathematics: The Case of Logarithms

Many authors have discussed the question why we should use the history of mathematics to mathematics education. For example, Fauvel (For Learn Math, 11(2): 3–6, 1991) mentions at least fifteen arguments for applying the history of mathematics in teaching and learning mathematics. Knowing how to introduce history into mathematics lessons is a more difficult step. We found, however, that only a limited number of articles contain instructions on how to use the material, as opposed to numerous general articles suggesting the use of the history of mathematics as a didactical tool. The present article focuses on converting the history of logarithms into material appropriate for teaching students of 11th grade, without any knowledge of calculus. History uncovers that logarithms were invented prior of the exponential function and shows that the logarithms are not an arbitrary product, as is the case when we leap straight in the definition given in all modern textbooks, but they are a response to a problem. We describe step by step the historical evolution of the concept, in a way appropriate for use in class, until the definition of the logarithm as area under the hyperbola. Next, we present the formal development of the theory and define the exponential function. The teaching sequence has been successfully undertaken in two high school classrooms.     

Constructing competence: an analysis of student participation in the activity systems of mathematics classrooms

This paper investigates the construction of systems of competence in two middle school mathematics classrooms. Drawing on analyses of discourse from videotaped classroom sessions, this paper documents the ways that agency and accountability were distributed in the classrooms through interactions between the teachers and students as they worked on mathematical content. In doing so, we problematize the assumption that competencies are simply attributes of individuals that can be externally defined. Instead, we propose a concept of individual competence as an attribute of a person's participation in an activity system such as a classroom. In this perspective, what counts as “competent” gets constructed in particular classrooms, and can therefore look very different from setting to setting. The implications of the ways that competence can be defined are discussed in terms of future research and equitable learning outcomes.

Notes on contributors

Authority, a fundamental part of the teaching–studying–learning process, is a problematic and poorly understood component of classroom life. It can be said, in practical terms, that pedagogical authority is constructed in classrooms, in teacher–student interaction and in the spirit of their physical presence, confidence, appreciation, responsibility and respect, and in the way they both relate to the content and norms. Based on the German Didaktik tradition and classroom interaction, an approach for the analysis of pedagogical authority is proposed providing an analytical tool for examining and understanding its constitutive elements and explaining its construction. It posits the existence of three types of interaction or relation from which pedagogical authority emanates: pedagogical interaction, deontic interaction and didactic interaction. Depending on how these relations are enacted in the classroom, they, in turn, may evolve into pedagogical authority. Data collected from four teachers’ interviews in Finnish comprehensive schools are analysed and some empirical evidence is provided to show the characteristics of these relations and how they construct this positive view of authority. More research on pedagogical authority resulting from these relations in classroom settings is needed to account for this critical educational phenomenon.     

“Noticing” as an Alternative Transfer of Learning Process

As transfer researchers have begun to investigate a broader range of phenomena, they have correspondingly put forward new processes to provide explanatory accounts for the occurrence of transfer. This move coincides with a call to acknowledge the contribution of social interactions, language, cultural artifacts, and normed practices to the generalization of learning. In this article, we posit “noticing” as a plausible transfer process and investigate both individual noticing and the social organization of noticing via the focusing framework. Specifically, we relate the nature of students' reasoning on transfer tasks with what students notice mathematically in classrooms when many sources of information compete for their attention, and then we account for noticing as socially situated in classroom discourse practices, features of mathematical tasks, and the nature of mathematical activity.     

Practice and conceptions: communicating mathematics in the workplace

The study examined the experience of communication in the workplace for mathematics graduates with a view to enriching university curriculum. I broaden the work of Burton and Morgan (2000), who investigated the discourse practices of academic mathematicians to examine the discourse used by new mathematics graduates in industry and their perceptions of how they acquired these skills. I describe the different levels of perception of discourse needs and of how they gained the necessary skills. At the lowest level, they learnt through trying out different approaches. At the next level, they were assisted by colleagues or outside situations. At the highest level, a small group viewed communication and interpersonal skills as a scientific process and stood back and used their “mathematical” observation skills to model their behaviour. These graduates did not appear to have systematically studied communication as part of their degree and they were unaware of the power of language choices in the workplace. Those who were working as mathematicians had to come to grips with explaining mathematical concepts to a wide range of people with varying mathematical skills but who generally were considerably less skilled in mathematics. The study revealed that these graduates were seriously underprepared in many aspects for joining the workforce. Many found it difficult to adapt to dealing with colleagues and managers, and developing communication skills was often a matter of trial and error.     

From Getting “Fired” to Becoming a Collaborator: A Case of the Coconstruction of Identity and Engagement in a Project-Based Mathematics Classroom

This article investigates the coconstruction of student identity and engagement in the case of a 9th grader in a project-based algebra classroom that afforded students a great deal of autonomy. The focal student, Terrance, utilized classroom resources to serve both project-related and social functions as he interacted with his peers during multiweek projects. As a result, his positioning within his group and patterns of engagement in the mathematics projects shifted dramatically across the academic year. The article ends with a discussion of student autonomy as a potentially powerful feature of hybrid classrooms.     

The teacher–learner relationship in the management of discipline in public high schools

Abstract

This article discusses data from a case study involving Grade 8–12 teachers in 14 classrooms. In all the schools that were identified, one teacher from each of the schools was identified for interviewing. After the interview the teacher was used as an ‘informant’ to identify other teachers who could provide additional information on the issue of classroom discipline in public high schools. Teachers are uncertain about how to relate to the learners and still maintain discipline in the classrooms. Reasons for the persistence of poor teacher–learner relationship include lack of knowledge regarding the effective use of alternatives to corporal punishment and the use of power to establish teacher authority. The results of the study showed that teachers, who are successful in managing misbehaviour in the classrooms, maintain good relations with the learners, encourage self-discipline and dignity, and involve the parents, learner peers as well as other teachers in the learning process. Involving all people who are close to the learner is essential in encouraging the learner to accept the teacher's authority and establish the required interpersonal classroom relationships.     

Social representations as mediators of mathematics learning in multiethnic classrooms

Drawing on socio-cultural theory, we understand the norms regulating the practices within the mathematics classroom as resulting from the social representations of the socially dominant groups and of the school culture related to what constitutes learning mathematics. Immigrant studients, having their own personal histories as members of particular social groups, and having been in school traditions other than the one predominant in the host society, have their own images of what mathematics in school is about. Individuals interacting in the classroom are all re-interpreting the different episodes from the perspective of the social representations of the larger groups with which they identify themselves. In multiethnic classrooms different re-interpretations of the same norms clash. The lack of negotiation gives rise to obstacles to immigrant students’ participation in the mathematical conversations and, therefore, interferes with the students’ learning process.     

The TI-Nspire CAS: A happy-medium mobile device for grades 8–16 mathematics classrooms

This article justifies classifying the TI-Nspire CAS Touchpad as a mobile device for grades 8–16 mathematics classrooms equipped with a Navigator system. The 2010 Horizon Report: K-12 indicates virtually every secondary school aged child has some sort of mobile device. Yet, many school policies ban the use of mobile devices, preventing students from using these devices in an educational setting. Further, some teachers have been slow to adopt and use mobile devices to advance the educational setting of their classroom to a 21^st century classroom. A classroom equipped with wireless Navigator technology and students equipped with Nspire handhelds presents an engaging 21^st century mathematics classroom for all learners. These classrooms present a happy-medium wireless communication setup that prevents students from communicating with each other or browsing the web. Students can instantly communicate with the teacher and vice-verse. Students stay on task while using a mobile device.     

What Does Distributed Cognition Tell Us about Student Learning of Science?

This paper reports multi-layered analyses of student learning in a science classroom using the theoretical lens of Distributed Cognition (Hollan et al. 1999; Hutchins 1995). Building on the insights generated from previous research employing Distributed Cognition, the particular focus of this study has been placed on the “public space of interaction” (Alac and Hutchins 2004, p. 639) that includes both participants’ interaction with each other and their interaction with artefacts in their environment. In this paper, a lesson from an Australian science classroom was examined in detail, in which a class of grade-seven students were investigating the scientific theme of gravity by designing pendulums. The video-stimulated post-lesson interviews with both the teacher and the student groups offered complementary accounts (Clarke 2001a) that assisted the interpretation of the classroom data. The findings of this study provide supporting evidence to demonstrate the capacity of Distributed Cognition for advancing our understanding of the nature of learning in science classrooms.