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.     

On forests, trees, elephants, and classrooms: a brief for the study of learning ecologies

The classroom intervention studies in this volume, ranging from the study of gestures to that of systemic implementation, are at very different grain sizes. A challenge is to see the forest for the trees – to see how these studies, focusing on different aspects of mathematical activity at different grain sizes, can be seen as aspects of a coherent whole. I propose an ecological metaphor for the study of mathematical activity. In ecological terms, the biosphere is comprised of interconnected and interrelated ecosystems. I argue that, analogously, there are nested and interrelated mathematical activity systems and structures in which “mathematical sense-making” plays the role of “health” in ecosystems. Moreover, what happens in the classroom environment shapes and is shaped by what happens in sub-ecologies of the classroom (e.g., sociomathematical norms, participation structures, communicational forms such as gesture, and representational tools and their use) and the larger social and organizational ecologies of which it is a part (building culture, support structures for teachers and teaching, external pressures such as testing and accountability systems etc).     

Mathematics teachers’ specialized knowledge: a secondary teacher's knowledge of rational numbers

Recognising teachers’ knowledge as one of the main factors influencing their practices and student learning, we aim to contribute to obtaining a better and deeper understanding of the specificities of teachers’ mathematical knowledge. A case study involving one 8th-grade Chilean mathematics teacher is presented in the context of rational numbers. Using video and audio recordings of classroom practices, questionnaires, and an interview, we sought to characterise, and better understand the content of the Knowledge of Topics from the perspective of the Mathematics Teachers’ Specialized Knowledge (MTSK) theoretical framework. The results reveal some critical aspects that teacher education should focus on, while also identifying lost opportunities and examples of “good” practices, thus contributing to the refinement of the MTSK conceptualisation. The conclusions can be considered in a broader perspective, with implications for teacher education in other contexts.     

Students' Alternate Conceptions on Acids and Bases

Knowing what students bring to the classroom can and should influence how we teach them. This study is a review of the literature associated with secondary and postsecondary students' ideas about acids and bases. It was found that there are six types of alternate ideas about acids and bases that students hold. These are: macroscopic properties of acids and bases, microscopic properties of acids and bases, neutralization, acid strength, pH, and titration.     

Structure of multi-criteria decision-making

Multi-criteria decision-making (MCDM) is presented as an eight-stage process of shaping information that satisfies the following criteria. The information should be accessible, differentiable, abstractable, understandable, verifiable, measurable, refinable and usable. For some stages, the decision-advisor should emphasize doing the stage convincingly by carrying out first its technical aspects, then relating to the context of the problem, and finally by taking into account the particular situation of the decision. For others, the decision-advisor should emphasize evincing information from the decision-maker first by relating to the situation of the decision, then seeing it in its context, and finally in its technical aspects. Methods for supporting the first four stages are shown to be personal construct theory for accessing the information, grounded theory for differentiating clusters of constructs, critical realism for abstracting their real meaning, and Nomology to understand how they fit into the criteria tree. An illustration is given.     

Handling errors as they arise in whole-class interactions

There has been a long history of research into errors and their role in the teaching and learning of mathematics. This research has led to a change to pedagogical recommendations from avoiding errors to explicitly using them in lessons. In this study, 22 mathematics lessons were video-recorded and transcribed. A conversation analytic (CA) approach was then taken to examine how mathematical errors are treated by teachers and students when they arise in interaction. Despite pedagogical recommendations, in these interactions, errors continue to be predominantly treated as something to avoid. There is a tension between the affective aspects of managing errors in interactions and the cognitive aspects. Close examination of classroom interactions enable us to see how these tensions are managed both by teachers and students.     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

“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.     

Epistemic management in mathematics classroom interactions: Student claims of not knowing or not understanding

An important role of classroom interaction is the processes involved in knowing or coming to know. Following a conversation analytical approach, this study investigates how students’ claims of not knowing, not remembering or not understanding are handled by mathematics teachers in whole class interactions. The study draws on video recordings of 42 mathematics lessons from 8 secondary schools in England. It is argued that claims of not knowing and claims of not remembering perform different social actions and are consequently treated differently by teachers. Claims of not knowing can challenge the assumption that knowledge can be taken-as-shared in a way that claims of not remembering do not. This contributes to the research field of mathematics classroom interaction as it nuances the epistemic management within these interactions and how this can contribute to the norms around the negotiation of meaning.     

Model‐Based Inquiry and School Science: Creating Connections

Much has been made in recent years of inquiry approaches to science education and the promise of such instruction to alleviate some of the ills of science education, yet in some ways this construct is still unclear to many in the field. In this paper we explore one view of inquiry in science that is based on the development, use, assessment, and revision of models and related explanations. Because modeling plays a central role in scientific inquiry it should be a prominent feature of students’ science education. We present a framework based on this view that can serve as a guide to curriculum development and instructional decision‐making with the goal of creating classroom environments that mirror important aspects of scientific practice. Specifically, the framework allows us to emphasize that scientists: engage in inquiry other than controlled experiments, use existing models in their inquiries, engage in inquiry that leads to revised models, use models to construct explanations, use models to unify their understanding, and engage in argumentation. Here, we discuss how these practices can be incorporated into science classrooms and illustrate that discussion with examples from our research classrooms.     

The theory and practice of boundary critique: developing housing services for older people

This paper begins by presenting the theory of boundary critique, which is a key aspect of current work in the area of critical systems thinking. The theory suggests that researchers should remain aware of the need to access a diverse variety of stakeholder views in defining problems, and to ‘sweep in’ relevant information. It also offers an understanding of how conflicts between stakeholders can become stabilised, leading to the marginalisation of some stakeholder groups and the issues that concern them. This indicates the importance of taking processes of marginalisation into account during interventions, promoting and revaluing the contributions that can be made by marginal groups. The theory of boundary critique is illustrated through a case study in which the researchers supported the multi-agency development of housing services for older people. Reflection upon this case study reveals that the principle means by which the theory of boundary critique informs intervention is through the design of methods. Methods can be developed specifically to explore the boundaries of problems. Also, the design of methods to address these problems can take account of the need to preserve the contributions of marginalised groups.     

Technological Tools in the Introductory Statistics Classroom: Effects on Student Understanding of Inferential Statistics

While technology has become an integral part of introductory statistics courses, the programs typically employed are professional packages designed primarily for data analysis rather than for learning. Findings from several studies suggest that use of such software in the introductory statistics classroom may not be very effective in helping students to build intuitions about the fundamental statistical ideas of sampling distribution and inferential statistics. The paper describes an instructional experiment which explored the capabilities of Fathom, one of several recently-developed packages explicitly designed to enhance learning. Findings from the study indicate that use of Fathom led students to the construction of a fairly coherent mental model of sampling distributions and other key concepts related to statistical inference. The insights gained point to a number of critical ingredients that statistics educators should consider when choosing statistical software. They also provide suggestions about how to approach the particularly challenging topic of statistical inference. This revised version was published online in July 2006 with corrections to the Cover Date.     

The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem

This paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction-diffusion problem to hold. It explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension. As a result, it can be computed how small the acute simplices should be for the discrete maximum principle to be valid. Numerical experiments suggest that the bound, which considerably improves a similar bound in [P.G. Ciarlet, P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973) 17-31.], is in fact sharp.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

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.     

Reply to Machina and Deutsch on vagueness, ignorance, and margins for error

In their paper “Vagueness, Ignorance, and Margins for Error” Kenton Machina and Harry Deutsch criticize the epistemic theory of vagueness. This paper answers their objections. The main issues discussed are: the relation between meaning and use; the principle of bivalence; the ontology of vaguely specified classes; the proper form of margin for error principles; iterations of epistemic operators and semantic compositionality; the relation or lack of it between quantum mechanics and theories of vagueness.     

OR Enactment: the Theatrical Metaphor as an Analytic Framework

Theatre provides a powerful metaphor to support thinking about human interactions in organizations. This paper indicates how a dramaturgical approach can illuminate what goes on in OR interventions (as well as, incidentally providing guidance in their prosecution). This perspective emphasizes aspects of the conduct of our work with clients that are too easily neglected, and help us to anticipate how things may turn out. It has a critical bearing upon the practice of OR and upon the training of OR practitioners.