Gender justice,human rights and measurement in the mathematics classroom

In line with international trends, the new South African mathematics curriculum implores mathematics educators to realize a pedagogy in their classrooms that is more practical, activity-oriented, and connected to their learners' lives. Drawing on data from a larger study that explores theory–practice relations in mathematics education, this paper shows how such progressive practices, when interpreted with respect to the teaching of measurement which required learners to use different measuring instruments for measuring the school grounds in learning about length and perimeter, were found to be deeply gendered. In two different contexts of an ‘African' township school and a predominantly ‘Indian' suburban school, girls in a grade 6 mathematics classroom faced direct sexism as they struggled to take the opportunity to participate in the activity and learn how to measure – an important mathematical competence and everyday knowledge and skill. The article analyses the data with reference to the human rights imperatives of the new national curricula and approaches to addressing disadvantage and discrimination for girls in mathematics classrooms.     

Classroom mathematical practices in differential equations

This article describes the results of a design experiment conducted in one differential equations classroom. The purpose of the article is to present an analysis of the classroom mathematical practices that were established over the first half of the semester including instruction on first order differential equations. We discuss and illustrate our use of Toulmin’s model of argumentation to develop an analytical technique for documenting the emergence and stability of classroom mathematical practices. This analysis is significant in that it contributes to an emerging body of research on students’ learning in social context, in particular at the undergraduate level where such analyses are lacking. Our analysis also serves as a case to examine the construct of classroom mathematical practices in new light and to extend prior research by documenting two theoretical ideas; that practices can emerge in a non-sequential fashion with regard to both time and structure.     

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.     

Problem solving in France: didactic and curricular perspectives

In this paper, we address the issue of problem solving in classrooms in France through two different and complementary approaches: didactic research and curricular choices. These two approaches correspond to two different, but not independent perspectives on problem solving and we investigate the existing links between them. We show that in France, the solving of problems is given a central role both in didactic research and curricular choices and that problem solving, as generally understood, is an object taking controversial positions, and we try to elucidate the rationale behind such positions. This paper is structured into two main parts: the first part devoted to didactic research, the second part to curricula, and one last part, discussion, wherein we question didactic research and curricula as regards their potential for influencing the reality of problem solving in classroom practices.     

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.     

From Classroom to Online Teaching: Experiences in Improving Statistics Education

This study used reflective practitioner methodology to investigate how to improve the quality of statistical education. During the study, this methodology, curricula, pedagogical practices, assessment and a framework for learning to learn statistics were all developed as means of improving the quality of statistical education. Also documented was the move from being a classroom teacher of statistics to a teacher who is developing learning resources for online delivery to students. For a classroom teacher, flexible delivery has meant drawing on the sights, sounds, movement, quiet and live shows. By contrast, the online teacher feels the constraints of translating activity based programs to technologically based programs. As more students have chosen to rely on online materials rather than classroom activities, the focus of improving quality has been extended to the enrichment of online resources, so that the learning experience is not second to that of the classroom.     

TRACING FOUCAULT'S ‘POWER’ IN THE MATHEMATICS CLASSROOM

In this paper I discuss some reflections on power and how its workings may be traced in the mathematics classroom. I explore some work of Michael Foucault on the nature of power and attempt to use his analytics of power to consider the formation and organisation of interactions in mathematics classrooms. I offer this as a strategy to start to (re)examine the practices of teachers and children in this arena. I move on to explore a number of concerns including the ‘regulation’ of teachers’ and children's practices: how power, in this sense, might legitimate certain classroom actions and limit others.     

Connections and confusion: teaching perimeter and area with a problem-solving oriented unit

Problem-solving-oriented mathematics curricula are viewed as important vehicles to help achieve K-12 mathematics education reform goals. Although mathematics curriculum projects are currently underway to develop such materials, little is known about how teachers actually use problem-solving-oriented curricula in their classrooms. This article profiles a middle-school mathematics teacher and examines her use of two problems from a pilot version of a sixth-grade unit developed by a mathematics curriculum project. The teacher's use of the two problems reveals that although problem-solving-oriented curricula can be used to yield rich opportunities for problem solving and making mathematical connections, such materials can also provide sites for student confusion and uncertainty. Examination of this variance suggests that further attention should be devoted to learning about teachers' use of problem-solving-oriented mathematics curricula. Such inquiry could inform the increasing development and use of problem-solving-oriented curricula.     

Practicing Reform‐Based Science Curriculum in an Urban Classroom: A Hispanic Elementary School Teacher's Thinking and Decisions

This study explores the thinking and decisions of Vera (pseudonym), a Hispanic elementary teacher, while she enacted a reform‐based science curriculum in an urban school in the southern United States. Vera's thinking, decisions, experiences, and practices were documented over a 2‐year period. Using the data collected from semistructured interviews, participant observations and classroom documents, a rich and complex case study of Vera is developed in this paper. This case study describes how Vera makes curricular choices from reform‐based science curricula such as the LiFE curriculum; how she enacts those choices to empower poor urban minority students; how Vera believes that preparing students for the high‐stakes test is empowering because it ensures continued schooling for students; how, for Vera, teaching connected science using students' lived experiences is a risky act; and how she uses negotiation in her science teaching.     

Inequalities,assessment and computer algebra

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students’ own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students’ answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.     

Practicing Reform‐Based Science Curriculum in an Urban Classroom: A Hispanic Elementary School Teacher's Thinking and Decisions

This study explores the thinking and decisions of Vera (pseudonym), a Hispanic elementary teacher, while she enacted a reform‐based science curriculum in an urban school in the southern United States. Vera's thinking, decisions, experiences, and practices were documented over a 2‐year period. Using the data collected from semistructured interviews, participant observations and classroom documents, a rich and complex case study of Vera is developed in this paper. This case study describes how Vera makes curricular choices from reform‐based science curricula such as the LiFE curriculum; how she enacts those choices to empower poor urban minority students; how Vera believes that preparing students for the high‐stakes test is empowering because it ensures continued schooling for students; how, for Vera, teaching connected science using students' lived experiences is a risky act; and how she uses negotiation in her science teaching.     

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.     

How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving?

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers’ self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the ‘what I know, what I look for’ method. These results led us to hypothesize that the observed gap may have an impact on students’ comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students’ literal and inferential understanding of word problems. A total of 278 fourth graders (9–10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students’ understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.     

Visage—Visualization of algorithms in discrete mathematics

Which route should the garbage collectors' truck take? Just a simple question, but also the starting point of an exciting mathematics class. The only “hardware” you need is a city map, given on a sheet of paper or on the computer screen. Then lively discussions will take place in the classroom on how to find an optimal routing for the truck. The aim of this activity is to develop an algorithm that constructs Eulerian tours in graphs and to learn about graphs and their properties. This teaching sequence, and those stemming from discrete mathematics, in particular combinatorial optimization, are ideal for training problem solving skills and modeling—general competencies that, influenced by the German National Standards, are finding their way into curricula. In this article, we investigate how computers can help in providing individual teaching tools for students. Within the Visage project we focus on electronic activities that can enhance explorations with graphs and guide studients even if the teacher is not available—without taking away freedom and creativity. The software package is embedded into a standard DGS, and it offers many pre-built and teacher-customizable tools in the area of graph algorithms. Until now, there are no complete didactical concepts for teaching graph algorithms, in particular using new media. We see a huge potential in our methods, and the topic is highly requested on part of the teachers, as it introduces a modern and highly relevant part of mathematics into the curriculum.     

Mathematics textbooks and their use in English, French and German classrooms:

After a through review of the relevant literature in terms of textbook analysis and mathematics teachers' user of textbooks in school contexts, this paper reports on selected and early findings from a study of mathematics textbooks and their use in English, French and German mathematics classrooms at lower secondary level. The research reviewed in the literature section raises important questions about textbooks as representations of the curriculum and about their role as a link between curriculum and pedagogy. Teachers, in tunr, appear to exercise control over the curriculum as it is enacted by using texts in the service of their own perceptions of teaching and learning. The second and main part of the paper analyses the ways in which textbooks vary and are used by teachers in classroom contexts and how this influences the culture of the mathematics classroom. The findings of the research demonstrate that classroom cultures are shaped by at least two factors: teachers' pedagogic principles in their immediate school and classroom context; and a system's educational and cultural traditions as they develop over time. It is argued that mathematics classroom cultures need to be understood in terms of a wider cultural and systemic context, in order for shared understandings, principles and meanings to be established, whether for promotion of classroom reform or simply for developing a better understanding of this vital component of the mathematics education process.     

Classroom mathematical practices and gesturing

The purpose of this paper is to illustrate a methodological approach for empirically investigating the function of gesturing in the collective development of knowledge. We extend the earlier work of Stephan and Rasmussen [Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior 21, 459-490] who analyzed classroom discourse and symbolizing to document the emergence of six classroom mathematical practices over the course of 22 days of instruction on first-order differential equations. We complement and extend this previous analysis by re-examining the same data for gesturing and coordinate this analysis with the evolution of the classroom mathematical practices as they developed in this particular community of learners. Our illustration of the methodology we developed suggests that (1) gestures and argumentation can function as a unit that supports the establishment of one or more taken-as-shared ideas, and (2) that a gesture/argumentation pair that develops while establishing one practice can change function to support the establishment of ideas embedded in other classroom mathematical practices.     

A Vision Educators Can Put Into Practice: Portraying the Constructivist Classroom as a Cultural System

Constructivist perspectives on learning have helped math and science educators better understand how students make sense of their experiences. Unfortunately, the intuitively appealing explanations of how learners construct knowledge have not been translated into a systematic body of pedagogical methods or a coherent curricular approach. Constructivist teaching is often portrayed in the literature as an alternative to traditional instructional approaches or as a toolbox of pedagogical techniques. These incomplete images do little to help practitioners understand constructivism or how it should be integrated into the life of the classroom. There may, however, be help for teachers in conceptualizing constructivism as a foundation for classroom practice. Recent anthropological investigations of learning have directed attention to the culture of classroom environments and the characteristic norms, beliefs, and practices that participants share in their dealings with one another. This article contends that envisioning the classroom as an articulated system of beliefs and practices not only serves an explanatory function for learning theorists but, more importantly, serves as a heuristic for teachers in conceptualizing constructivism and offers a starting point for teachers in implementing constructivist practices.     

Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.