The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

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.     

Implementation of IBL in Europe from an Australasian perspective

In this paper the situation in Australasia with respect to research, curricula and practice in inquiry-based learning in mathematics education is examined. As an organising frame the work of researchers in the Mathematics Education Research Group of Australasia (MERGA) was examined for points of intersection with several of the papers that have arisen from the international PRIMAS project, one of several projects funded by the European Union which follow the recommendations of the Rocard Report (Rocard et al. 2007). Although MERGA services researchers in countries such as Singapore in addition to Australia and New Zealand, a survey of outputs of members showed that the latter two countries had more of a tradition of inquiry-based learning practices and research in mathematics education than did Asian countries under the MERGA umbrella such as Singapore. For this reason the focus here is on mathematics education in the schools in these two countries and the classroom research in them that extends or complements the research or issues raised in other papers in this issue of ZDM.     

Views of cognition: different lenses for ‘looking-in’ on classrooms

The evolution of different theories of cognition over the years has given mathematics education researchers new tools for highlighting particular characteristics of classroom issues to enable more detailed investigation of how students learn mathematics. In the past there has been a strong dominance of the dualistic view separating body and mind. In recent times, however, the body has been given a more central role in shaping the mind. This has led to the situation where some long-standing conundrums of mathematics education have become more tractable to researchers. Similarly, using older ideas in new ways, light is being shone on what is possible at different stages of development or different levels of schooling. In this issue different views of cognition have been mined by the authors of different articles to frame studies and analyses or invent and apply new tools. They bring new lenses for looking-in on classrooms, a fresh view with old lenses and new methodological tools to the fore. Using a small selection of the articles of the issue the empowerment that differing views of cognition have enabled for mathematics education research is demonstrated.     

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.     

Students' Critical Awareness of Voice and Agency in Mathematics Classroom Discourse

This account of my extended conversation with a high school mathematics class focuses on voice and agency. As an investigation of possibilities opened up by introducing mathematics students to what Fairclough (1992) called “critical language awareness” (p. 2), I prompted the students daily to become ever more aware of their language practices in class. The tensions in this conversation proved parallel to the tension in mathematics between individual initiative and convention, a tension that Pickering (1995) called the “dance of agency” (p. 21). Participant students in this classroom-based research resisted the idea of linguistic reference to human agency, although their actual language practice revealed some recognition of human agency.     

基于“动态生成”的大学数学课堂教学

"动态生成"教学观的建立旨在摆脱课堂教学中以教师为中心、强调知识传授的传统教学模式的缺陷,从根本上正确理解课堂教学的复杂性和动态性,构建充满生命活力的大学数学课堂教学生态环境.在用动态生成的视角审视当前大学数学课堂教学中存在的问题的基础上,文章对在课堂教学中如何有效地进行动态生成提出了一些具体的策略:更新教学观念,精心预设弹性化的数学课堂教学方案;根据学生课堂反馈情况及时调整预设,并及时捕捉可利用的动态资源,为学生的生成创造可能的机会;加强教学研究,不断提高课堂教学智慧.     

On the process of gaining language as a resource in mathematics education

In this paper, we expand our prior work on mathematics education in contexts of language diversity by elaborating on the three perspectives on language described by Ruiz (NABE J 8(2):15–34, 1984): language-as-right, language-as-resource, and language-as-problem. We illustrate our arguments with data taken from research contexts in Catalonia-Spain and South Africa. In these two parts of the world, the language policy in education has long been an issue, with a monolingual orientation that values one language (i.e., Catalan in Catalonia and English in South Africa) over others. Throughout the introduction of specific examples of policy documents, classroom practices, and participants’ reports, our main point is that the right of using the students’ languages makes sense because it is itself more than an intrinsic human right; it is an option that potentially benefits the creation of mathematics learning opportunities. Especially for the instances of classroom practices, our examples can be considered as representative in that they point to a common situation in our data: despite the fact of the language of learning and teaching being fixed, there is room for the learners and the teacher to take or react to a decision on what language to use, with whom, and how in concrete moments of the interaction. However, on the basis of our studies and drawing on the literature in mathematics education and language diversity, we argue that language rights are not sufficiently connected to language as a pedagogical resource. The enactment of these rights is still contributing in many ways to the social and political construction of problems concerning the role of certain languages in classroom interaction. We conclude the paper by discussing some possibilities for framing language as a resource that provide effective support to all students’ learning of mathematics.     

Sociocultural perspectives in research on and with mathematics teachers: a zone theory approach

Sociocultural theories view teacher learning as changing participation in social practices that develop their professional identities rather than as acquisition of new knowledge or beliefs that are internal to the individual. Although sociocultural research on mathematics teacher education has tended to focus on understanding teachers’ learning, this article argues that sociocultural perspectives can also guide more interventionist research involving changing classroom practice. The approach illustrated here uses an adaptation of Valsiner’s zone theory to analyse teacher learning and development in two separate research studies. In one study the aim was to understand how teachers incorporated digital technologies into their practice, while the other study helped teachers implement an investigative approach to working mathematically consistent with a new syllabus. In both studies, productive tensions between teachers’ beliefs, contexts, and goals were a trigger for learning and development.     

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.     

An Evaluation of a Master's Degree in K‐8 Mathematics and Science: Classroom Practice

“Evaluation as a particular kind of investigated discipline is distinguished from, for example, traditional empirical research in the social sciences or from literary criticism, criminalistics, or investigative reporting, partly by its extraordinary multidisciplinarity” ( Scrivens, 1991 , p. 141). It is this unique multidisciplinary feature of evaluation that adds usefulness when determining the effectiveness of programs seeking to integrate mathematics and science teaching and learning across elementary and middle grade levels. In 2005, a K‐8 mathematics and science program celebrated its 15th year of service. The program was the result of education, business, and community partnership efforts focused on improving mathematics and science teaching and learning in schools throughout a metropolitan region in the southeastern United States. To date, over 350 K‐8 teachers have completed a master's degree through this mathematics and science education program. The director realized that an evaluation of the program would likely provide insights that would benefit not only the efforts of the program but the broader mathematics and science teaching and learning community. Hence, the National Science Foundation (award No. 9815931), which had provided start‐up funds for the program responded to this need and provided funding for a longitudinal evaluation of the program. The evaluation was conducted from 1999 to 2004. This article focuses on the evaluation results for years 1 and 2 and addresses the question related to changes in teachers' classroom practice.     

A teacher education experiment to challenge conceptions and practices

This study describes a teacher education experience with grade 5–6 teachers, based on a calculator module within a national program for mathematics in-service teacher education. The aim was to challenge the teachers’ conceptions about the role of the calculator in mathematics teaching and to promote their reflection about professional practices. The research methodology was qualitative and interpretive, with data collection through interviews and observation of teacher education and classroom supervision sessions, as well as analysis of teachers’ portfolios. The results indicate that some teachers are clearly against the use of the calculator in the mathematics classroom, while others allow students to use it in a passive way and some others are very affirmative about its use. The teachers who argue against the use of the calculator seem to predominate, suggesting a great distance between the curriculum orientations and classroom practice. The methodology of the course, combining collective sessions and individual classroom supervision, proved to be fruitful, providing new information, practice and discussion that allowed teachers to analyze different kinds of tasks in which the calculator might be useful, experiment using them in the classroom and reflect about the students’ work. The no imposing and questioning approach used in collective discussions encouraged teachers to assume their own positions; sharing and discussing in the collective reflections during the course stimulated a deeper reflection of their practice. Therefore, in this course, in-service teacher education focused on practice contributed to teachers to reflect on their conceptions and practices.     

Research in mathematics education— Who benefits?

For a long time mathematics education tries hard to win recognition as an academic discipline. In related classroom research and curriculum development one can find not rarely theories and methods in use that are adapted from other (and well established) disciplines. However, in many cases these adaptations do not serve the researcher's goals, yet more, their effects can contradict the stated purposes. The article discusses a few fundamental problems related to empirical research in mathematics education (e.g. the role of the teachers in experimental/control groups), using as concretization a printed research report that is chosen deliberately (and made nameless, therefore).     

Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms

Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed.     

Dynamics of change of mathematics education in Brazil and a scenario of current research

Mathematics education in Brazil, if we consider what one may call the scientific phase, is about 30 years old. The papers for this special issue focus mainly on this period. During these years, many trends have emerged in mathematics education to address the complex problems facing Brazilian society. However, most Brazilian mathematics educators feel that the separation of research into trends is a theoretical idealization that does not respond to the dynamics of the problems we face. We raise the conjecture that the complexity of Brazilian society, where pockets of wealth coexist with the most shocking poverty, has contributed to the adoption and generation of different strands in mathematics education, crossing the boundaries between trends. At a more micro level, we also raise the conjecture that Brazilian trends in research are interwoven because of the way that Brazilian mathematics educators have experienced the process of globalization over these 30 years. This tapestry of trends is a predominant characteristic of mathematics education in Brazil.     

Mimicry of proofs with computers: the case of Linear Algebra

In common teaching practice the habit of connecting mathematics classroom activities with reality is still substantially delegated to wor(l)d problems. During recent decades, a growing body of empirical research has documented that the practice of word problem solving in school mathematics does not match this idea of mathematical modelling and mathematization. If we wish to construct ‘real problems arising from real experiences of the child’ following the spirit of these new suggestions, we have to make changes. On the one hand we have to replace the type of activity in which we delegate the process of creating an interplay between reality and mathematics by substituting the word problems with an activity of realistic mathematical modelling, i.e. of both real-world based and quantitatively constrained sense-making; and, on the other hand, to ask for a change in teacher beliefs; furthermore, a directed effort to change the classroom socio-math norms will be needed. This paper discusses some classroom activities that takes these factors into account.     

Research into teacher knowledge: a stimulus for development in mathematics teacher education practice

In this paper, we document some developments in teacher education practice at one university, brought about by reflection on research into mathematics teacher knowledge. The authors are three members of the Cambridge-based research team who developed the Knowledge Quartet (KQ), a theory of mathematics teacher knowledge, with a focus on classroom situations in which this knowledge is applied. At the same time as being researchers, the authors were elementary mathematics teacher education instructors. They found that the KQ research brought about new awareness of the importance of some components of mathematics didactics, as well as providing new tools for undertaking some aspects of their teacher educator role. The paper explores some of these awarenesses and tools in detail.