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The primary purpose of this chapter is to clarify the basic tenets of activity theory and constructivism, and to compare and contras instructional approaches developed within these global theoretical perspectives. This issue is worthy of discussion in that research and development programs derived from these two perspectives are both vigorous. For example, the work of sociocultural theorists conducted within the activity theory tradition has become increasingly influential in the United States in recent years. One paradigmatic group of studies conducted by Lave (1988), Newman, Griffin, and Cole (1089). and Scribner (1984) has related arithmetical computation to more encompassing social activities such as shopping in a supermarket, packing crates in a dairy, and completing worksheets in school. Taken together, these analyses demonstrate powerfully the need to consider broader social and cultural processes when accounting for children’s development of mathematic cal competeuce.
Mathematics for All began as a programme in the early 1980s when concerns about pupils’ access to mathematics education heightened due to the many issues surrounding the mathematics classroom and the mathematics student. The following chapter highlights these important issues by appropriately discussing the contexts within which these issues arise and may be resolved. The issues include curriculum content and assessment practices, equity among subgroups classified by gender, race, and socioeconomic status, the use of mathematics as a selection device, democracy in the mathematics classroom, and the value of culture in the teaching of mathematics. The chapter, likewise, echoes the voices of marginalized groups in the mathematics classroom that are products of undemocratic pedagogical practices, societal perceptions, and cultural realities. Prerequisites for a successful Mathematics for All programme are put forward and directions for further research are offered.
Previous studies have demonstrated that children use oral calculation procedures not taught in school. The present study provided evidence for situational variables that strongly influence the tendency to use such procedures. It also provided a qualitative analysis of the oral mathematics used by Brazilian third graders. Concrete problem situations were powerful elicitors of oral computation procedures, whereas computation exercises tended to elicit school-learned computation algorithms. Oral computation procedures involved the use of two reliably identifiable routines, decomposition and repeated grouping, that revealed the children's solid understanding of the decimal system. In general, the children were far more successful in using oral mathematics than written mathematics. An understanding of children's oral procedures may be useful in developing more successful programs for elementary mathematics instruction.
The educational sciences are generally construed around concerns of providing research that informs practices of learning and teaching in educational institutions. This research emphasizes questions of how to and has led to a "technification" of educational research, as primarily concerned with providing solutions to practical problems. In this paper we will show how mathematics education as a research field is not an exception, by analysing how theory is understood and used in the field, to address questions of how. We suggest that, although important, this research leave some important areas unaddressed, namely the ones which can emerge from posing questions of why. We argue that making this move implies rethinking and enlarging definitions and views of mathematics education research.
"New Perspectives on Mathematics Pedagogy" represents a serious attempt to understand pedagogy within mathematics classrooms. To that end, this symposium will address the key questions and issues surrounding mathematics pedagogy presently confronting vast numbers of researchers, as well as educators, and policy makers. Organised around presentations, responses, discussion and debate, the symposium is intended not only to enhance understanding but also to stimulate fresh thinking and initiate ongoing critical dialogue about the practice of mathematics pedagogy within teaching and learning settings. AIMS OF SYMPOSIUM This symposium aims to engage the audience in a critical discussion on a new and provocative book in the field of mathematics education. Unpacking Pedagogy: New Perspectives for Mathematics is a forthcoming (December 2009) publication by Information Age Publishing. Based on the chapters of this edited collection, this symposium will address the key questions and issues surrounding mathematics pedagogy presently confronting vast numbers of researchers, as well as educators, and policy makers. By pedagogy we mean the elements of practice characterised not only by the regularities of teaching but also the uncertainties of practice. If pedagogy is about the production of mathematical knowledge and the construction of mathematical identities, it is also about social relations and values. Pedagogy takes into account ways of knowing and thinking, language, emotion, and the discourses made available and generated within the physical, social, cultural, historical, and economic community of practice in which mathematics teaching is embedded. The symposium directly involves nine chapter authors who will present their research and/or act as respondents. Their presentations are not intended to provide analytic consensus in their attempts to understand what it is that structures the pedagogical experience. Highly influential in informing the analyses will be Foucault's understanding of how practices are produced within discourses and within power configurations; Lacan's notion of subjectivity; evolutionary frameworks of complexity science to rethink mathematics pedagogy; and Bourdieu's notion of habitus to explain the teaching/learning nexus.
The ethnographic experience is an indelible venture that continuously redefines one's life. Bringing together important cross-currents in the national debate on education, this book introduces the student or practitioner to the challenges, resources, and skills informing ethnographic research today. From the first chapter describing the cultural foundations of ethnographic research, by George Spindler, the book traces both traditional and new approaches to the study of schools and their communities.
This article shows how equity research in mathematics education can be decentered by reporting the “voices” of mathematically successful African American male students as they recount their experiences with school mathematics, illustrating, in essence, how they negotiated the White male math myth. Using post-structural theory, the concepts discourse, person/identity, and power/agency are reinscribed or redefined. The article also shows that using a post-structural reinscription of these concepts, a more complex analysis of the multiplicitous and fragmented robust mathematics identities of African American male students is possible—an analysis that refutes simple explanations of effort. The article concludes, not with “answers,” but with questions to facilitate dialogue among those who are interested in the mathematics achievement and persistence of African American male students—and equity and justice in the mathematics classroom for all students.
Over the past decade, the mathematics education research community has incorporated more sociocultural perspectives into its ways of understanding and examining teaching and learning. However, researchers who have a long history of addressing anti-racism and social justice issues in mathematics have moved beyond this sociocultural view to espouse sociopolitical concepts and theories, highlighting identity and power at play. This article highlights some promising conceptual tools from critical theory and post-structuralism and makes an argument for why taking the sociopolitical turn is important for both researchers and practitioners.
This article reports on a 2-year study about teaching and learning mathematics for social justice in an urban, Latino classroom and about the role of an NCTM Standards- based curriculum. I was the teacher in the study and moved with the class from seventh to eighth grade. Using qualitative, practitioner-research methodology, I learned that students began to read the world (understand complex issues involving justice and equity) using mathematics, to develop mathematical power, and to change their orientation toward mathematics. A series of real-world projects was fundamental to this change, but the Standards-based curriculum was also important; such curricula can theoretically promote equity, but certain conditions may need to exist. Social justice pedagogy broadens the concept of equity work in mathematics classrooms and may help promote a more just society.
I chose my critics well. All of their comments are helpful. From Lenore Langsdorf, I get a clearer sense of what I did. From Rachel Falmagne, I get a sense of where I might go next in terms of questions that remain. And especially from Adele Clarke, perhaps most difficultly, I get a sense of what I might have done differently to make this book stronger.
Getting Lost is an experiment in and of method against the normative critical framework of much feminist methodology in order to ask: if it is what it does, in a nominalist vein, what then is feminist methodology? The answers the book puts forward include: effaced, abjected, uncertain, engaged, reflex-ive (perhaps to a fault), and deeply invested in a sustained ethical engagement with those we study, particularly those with less power, while troubling what Adele terms “confession, testimonial and the intrusiveness of much research.” Situated as an index of more general tensions in the human sciences, I focus on how feminist methodology engages with a problematic of loss in taking fuller account of the fall into language and the loss of pure presence.
The book’s sensibility is toward that which shakes any assured ontology of the “real,” of presence and absence, a post-critical logic of haunting and undecidables. In this, it is important to remember that my methodological musings collected in the book are grounded in Troubling the Angels: Women Living with HIV/AIDS,1 a study that preceded the “new” anti-retroviral treatments of the mid-1990s. Hence this was, in many senses, a study of living with dying. Not-knowing was not difficult in such a space and I felt keenly how not wanting to not know is a violence that subsumes the Other into the Same. Abstracting a philosophy of inquiry from an archive of such work set me up well to explore the enablements that might be imagined from loss.
The genealogical period of Getting Lost, the questions that permeate it, its location in a political history of truth, is surely part of broader and deeper shifts in the doing of science, informed by arguments across a host of disciplines and interdisciplines, including my own position in educational research with its repositivization.2 Here many “qualitative researchers” are caught in paradigm wars we had dared to dream long over.
Getting Lost is informed also by the fieldwork of Troubling the Angels and the responsibility of being invited in to tell other people’s stories. Perhaps too clever by far in refusing to tell the “tidy tale” of these women’s struggles, I was in over my head. “This work is beyond me” was my mantra throughout the writing of Troubling the Angels.3
Part of this dizziness in writing Getting Lost was that it is of the future pluperfect tense of “what will have been said.”4 In one of the endless books on Nietzsche, I read of his “striking discovery” that in Sunrise and The Gay Science he may have “‘already provided the commentary [to Zarathustra]—before writing the text.’”5 Nietzsche viewed it as both a masterstroke and an act of folly on his part to compose the commentaries before the actual text. And so it was with the somewhat strange time of Getting Lost. Written before, during, and after Troubling the Angels, a sort of “folding forward” into a-book-that-was-not-yet inhabited the first text, Troubling the Angels. This strange time resulted in a poly-temporal dialogue across texts, time, and researching selves where I functioned as both author and (auto)critic of books that did not yet exist. Getting Lost, then, is a palimpsest where primary and secondary texts collapse into trace structures of one another that fold both backward and forward into books full of concealments and not knowings in an uncanny time of what “will always have already taken place.”
In sum, putting what Spivak terms an “identification crisis” around decol-onization front and center,6Getting Lost situates feminist ethnography as a seismograph of cultural shifts and intellectual movements in asking how research based knowledge remains possible after so much...
In this article, I address the need for a more clearly articulated research agenda around equity issues by proposing a working definition of equity and a focal point for research. More specifically, I assert that rather than pitting them against each other, we must coordinate (a) efforts to get marginalized students to master what currently counts as "dominant" mathematics with (b) efforts to develop a critical perspective among all students about knowledge and society in ways that ultimately facilitate (c) a positive relationship between mathematics, people, and equity on the planet. I make this argument partly by reviewing the literature on (school) contexts that engage marginalized students in mathematics. Then, I argue that the place that holds the most promise for addressing equity is a research agenda that emphasizes enabling the practice of teachers and that draws more heavily on design-based and action research, thereby redefining what the practice of mathematics means along the way. Specific research questions are offered.
Investigated the effectiveness of an experimental mathematics teaching program. The treatment program was primarily based on a naturalistic study of 40 relatively effective 4th-grade mathematics teachers. Students were tested before and after with a standardized test and a content test (posttest only), which had been designed to approximate the actual instructional content that students had received during the treatment. Observational measures revealed that teachers generally implemented the treatment, and analyses of product data showed that students of treatment teachers generally outperformed those of control teachers on both the standardized and content tests. Since strong efforts were made to control for Hawthorne effects, it seems reasonable to conclude that teachers and/or teaching methods can exert a significant difference on student progress in mathematics. (4 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
When a doctoral student plans to conduct qualitative education research, the aspect of the dissertation that often becomes problematic is determining which theoretical paradigm(s) might frame the study. In this article, the author discusses how he resolved the quandary through eclecticism. The author begins by describing briefly the purpose of his dissertation study, providing a justification for eclecticism in the selection of theories. He follows with a description of the three theories- poststructural theory, critical race theory, and critical theory-that framed his study and discusses briefly the methodology employed. The author concludes with a discussion of likely objections of his study and with an explanation of why his study was positioned within a critical postmodern paradigm.
This paper situates paradigm talk with its insistence on multiplicities and proliferations in tension with a resurgent positivism and governmental imposition of experimental design as the gold stan-dard in research methods. Using the concept of 'coloring epistemologies' as an index of such tensions, the essay argues for proliferation as an ontological and historical claim. What all of this might mean in the teaching of research in education is dealt with in a delineation of five aporias that are fruitful in helping students work against technical thought and method: aporias of objectivity, complicity, difference, interpretation, and legitimization. The essay concludes with a 'disjunctive affirmation' of multiple ways of going about educational research in terms of finding our way into a less comfortable social science full of stuck places and difficult philosophical issues of truth, interpretation and responsibility.
This paper engages withpoststructural ideas for a discussion on whatit means to engage in pedagogical work in thecontext of elementary/primary schoolmathematics classrooms. Central to the analysisare the pre-service student and the part thatthe teaching practicum plays in the `making'' ofa teacher. Drawing on insights from the work ofFoucault on power and subjectivity, instancesof teaching knowledge in production, asinterpreted by pre-service teachers, areexamined. The view is towards developing theoryfrom readings of specific regulatory strategiesthat impact powerfully on pre-service teachers''constructions of themselves as mathematicsteachers.
From a sociocultural perspective an object of research on mathematics teaching and learning can be seen as a particular moment
in the zoom of a lens. Researchers focus on a specific part of a complex process whilst taking account of the other views
that would be obtained by pulling back or zooming in. Researching teaching and learning mathematics must be seen in the same
way. Thus in zooming out researchers address the practices and meanings within which students become school-mathematical actors,
whilst zooming in enables a study of mediation and of individual trajectories within the classroom. In each choice of object
of research the range of other settings have to be incorporated into the analysis. Such analyses aim to embrace the complexity
of the teaching-learning process. This article will present a cultural, discursive psychology for mathematics education that
takes language and discursive practices as central in that meanings precede us and we are constituted within language and
the associated practices, in the multiple settings within which we grow up and participate.
In this paper we present ananalysis of the articles in EducationalStudies in Mathematics since 1990. It ispart of a larger
project looking at theproduction and use of theories of teachingand learning mathematics. We outline thetheoretical framework
of our tool ofanalysis and discuss briefly some of themethodological difficulties we face. Wethen present our findings from
the analysisof the journal and we also give one exampleof how we `read' an article, illustratingthe rules whereby criteria
are applied.
Our approach to emotion in school mathematics draws on social semiotics, pedagogic discourse theory and psychoanalysis. Emotions are considered as socially organised and shaped by power relations; we portray emotion as a charge (of energy) attached to ideas or signifiers. We analyse transcripts from a small group solving problems in mathematics class, and from an individual student. The structural phase of analysis identifies positions available to subjects in the specific setting, using Bernstein's sociological approach to pedagogic discourse. The textual phase examines the use of language and other signs in interaction and describes the positionings taken up by particular pupils. We then focus on indicators of emotion, and find indications of excitement and anxiety, linked to participants' positionings. Finally we consider implications of our approach.
Paulo Freire's critical education theory is “re-invented” in the context of a mathematics curriculum for urban working-class adults. The problems Freire poses for teachers in that context are explored, and work of other theorists which deepens or questions aspects of Freire's theory is discussed. Next, Freire's theory is applied to teaching basic mathematics and statistics for the social sciences. It is argued that such mathematical literacy is vital in the struggle for liberatory social change in our advanced technological society. Finally, this reflection on practice is used to pose further problems to be explored in the creation and re-creation of the “pedagogy of the oppressed.”
In this article, the author examines whether disparities in mathematics performance might be exacerbated by the track placement of native and non-native Latina/o English speakers in the Education Longitudinal Study of 2002. The effect of track placement on the mathematics performance of English Learners (EL) differed as a function of their level of English proficiency. The scores of Latinas/os with low levels of English proficiency in the general track were similar to the scores of students in the college track with comparable levels of English proficiency. The scores of non-native English speakers in the college track with high levels of English proficiency, however, were higher than those of their peers in the general track and nearly as high as those of native English speakers in the college track. Implications for the potential development of the mathematics language register of ELs are discussed.
"[Note: This abstract applies to all articles within this issue."] The chapters in this monograph describe qualitative research methods used to investigate students' and teachers' interactions with school mathematics. Each contributing author uses data from his or her own research to illustrate a particular technique or aspect of research design. The different chapters present a wide range of methods, representing a variety of goals and perspectives. Rather than a comprehensive reference manual, this monograph illustrates the diversity of methods available for qualitative research in mathematics education. The monograph begins with a discussion of key elements that contribute to the dynamic and evolving domain of mathematics education research. Background information is then provided that relates to the philosophical and epistemological assumptions underlying all qualitative research. In the chapters that follow, actual studies present the contexts for discussions of research design and techniques. Issues of research design include the importance of making explicit the underlying theoretical assumptions; the selection of an appropriate methodology; the interpretative, intersubjective nature of analysis; and the establishment of reliability and validity. Specific data collection techniques include clinical interviews, stimulated recall interviews, open-ended survey questions, and field notes and video or audio taping to record classroom events. Methods of analysis include participant validation, the categorization of data through constant comparison and software indexing and retrieval, phenomemographic analysis, and the identification of empirical examples of theoretical constructs. The monograph ends with a discussion of general issues, including the role of theory and the establishment of criteria for judging the goodness of qualitative research.
This article, the focus of which is on girls in mathematics, engages poststructural debates over knowledge and power to explore how female subjectivity is lived within the classroom, and the first section looks at some recent feminist reconstructionists' proposals developed from the idea of "different experience." The second section is set within the context of the poststructuralists' undermining of the "light" of progressive development, central to the Enlightenment project. Foucauldian ideas are introduced for a theoretical discussion about the ways in which the girl becomes gendered through available discourses and practices. Building on this discussion, the third section provides an analysis of some moments of classroom life and offers a different story about girls in school mathematics.
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Background/Context
School tracking practices have been documented repeatedly as having negative effects on students’ identity development and attainment, particularly for those students placed in lower tracks. Despite this documentation, tracking persists as a normative practice in American high schools, perhaps in part because we have few models of how departments and teachers can successfully organize instruction in heterogeneous, high school mathematics classes. This paper offers one such model through a qualitative and quantitative analysis.
Focus of Study
In an effort to better the field's understanding of equitable and successful teaching, we conducted a longitudinal study of three high schools. At one school, Railside, students demonstrated greater gains in achievement than students at the other two schools and higher overall achievement on a number of measures. Furthermore, achievement gaps among various ethnic groups at Railside that were present on incoming assessments disappeared in nearly all cases by the end of the second year. This paper provides an analysis of Railside's success and identifies factors that contributed to this success.
Participants
Participants included approximately 700 students as they progressed through three California high schools. Railside was an urban high school with an ethnically, linguistically, and economically diverse student body. Greendale was situated in a coastal community with a more homogeneous, primarily White student body. Hilltop was a rural high school with primarily White and Latino/a students.
Research Design
This longitudinal, multiple case study employed mixed methods. Three schools were chosen to offer a range of curricular programs and varied student populations. Student achievement and attitudinal data were evaluated using statistical techniques, whereas teacher and student practices were documented using qualitative analytic techniques such as coding.
Findings/Results
One of the findings of the study was the success of Railside school, where the mathematics department taught heterogeneous classes using a reform-oriented approach. Compared with the other two schools in the study, Railside students learned more, enjoyed mathematics more and progressed to higher mathematics levels. This paper presents large-scale evidence of these important achievements and provides detailed analyses of the ways that the Railside teachers brought them about, with a focus on the teaching and learning interactions within the classrooms.
Thomas Kuhn developed the construct of research paradigms to make sense of the history of conceptual change in the physical sciences. The construct has since been appropriated by a number of academic fields and by non‐academics as well. This paper traces the use of the construct in the educational research field. The bulk of the paper is organized around two questions: (a) Was it ever appropriate to characterize the educational research field’s acceptance of qualitative methods as equivalent to one of Kuhn’s paradigm revolutions? (b) Is paradigm talk appropriate today?
In this article we propose the beginnings of a constructivist model of mathematical learning that supersedes Piaget's and Vygotsky's views on learning. First, we analyze aspects of Piaget's and Vygotsky's grand theories of learning and development. Then, we formulate our superseding model, which is based on the interrelations between two types of interaction in constructivism—the basic sequence of action and perturbation, and the interaction of constructs in the course of re-presentation or other previously constructed items. When teaching children, we base our interactions with them on the schemes we infer by observing the children's interactions in a medium. This emphasis makes contact with both Piaget's and Vygotsky's ideas of spontaneous development. In our model, learning is understood as being spontaneous rather than provoked. To maintain our emphasis on spontaneity, we separate the unintentionality of the learner from the intentionality of the teacher. To ground our model, we describe how two ten-year-old children, Arthur and Nathan, used their partitioning and iterating operations while they worked with a teacher in a computer microworld. We conclude by stressing the teacher's responsibilty to induce perturbations in children that are emotionally bearable and in the child's zone of potential construction.
This paper reports on 3-year case studies of 2 schools with alternative mathematical teaching approaches. One school used a traditional, textbook approach; the other used open-ended activities at all times. Using various forms of case study data, including observations, questionnaires, interviews, and quantitative assessments, I will show the ways in which the 2 approaches encouraged different forms of knowledge. Students who followed a traditional approach developed a procedural knowledge that was of limited use to them in unfamiliar situations. Students who learned mathematics in an open, project-based environment developed a conceptual understanding that provided them with advantages in a range of assessments and situations. The project students had been "apprenticed" into a system of thinking and using mathematics that helped them in both school and nonschool settings.
In this article we examine the nature, scope, and significance of basic philosophical issues in the preparation of researchers. Following a brief review of the history of educational research and a discussion of the philosophy of science supporting much of this research, we present and discuss two central assertions in the context of the growing prominence of paradigmatic and methodological pluralism in education and the human sciences. The first assertion is that the curriculum for preparing researchers in education continues to be dominated by the epistemology of logical empiricism, the philosophy of science undergirding the quantitative research tradition. The second assertion is that research education tends to place a disproportionate emphasis on technical methods and procedures, with little attention given to the philosophical, moral, and political values that underpin procedural practices and that frame, however tacitly, the context for knowledge production. We argue that the hegemony of quantitative science and the narrow preoccupation with methodological rigor as the singular yardstick for judging good science are serious problems requiring immediate attention in research education programs. We present a case for expanding the research education curriculum to include a strong and broad foundation in the history, philosophy, sociology, and ethics of inquiry. To illustrate how this can be achieved, we propose two features of an intellectual culture that may be developed: (a) course work on philosophical issues in inquiry, which doctoral students should be required to take in preparation for—or as a supplement to—technical courses on statistical methods and research design; and (b) an atmosphere of interdisciplinary and multiparadigmatic collaborative research that provides an informal context for students to experience and practice the values fostered by such course work.
In March 2008, the National Mathematics Advisory Panel final report was released. This report was produced in response to an executive order from President George W. Bush. The report is important because of its subject matter—improving mathematics teaching and learning—its historically significant genesis, and the strong position that the report takes on the primacy of quantitative methods in education research. The author briefly introduces the report and then draws attention to some of the main points in the commentaries offered in this special issue of Educational Researcher. The special issue ends with a rejoinder from the chair and co-chair of the report.
Feminists in education increasingly use poststructuralism to trouble both discursive and material structures that limit the ways we think about our work. This overview of poststructural feminism presents several key philosophical concepts – language; discourse; rationality; power, resistance, and freedom; knowledge and truth; and the subject – as they are typically understood in humanism and then as they have been reinscribed in poststructuralism, paying special attention to how they have been used in education.
This article is about 8 African American middle school boys who have experienced success in mathematics. Working within a phenomenological methodological framework, the researcher investigated the limitations these students encounter and the compensating factors they experience. Critical race theory was the theoretical framework for this study; counter-storytelling was utilized to capture the boys' experiences, which is in stark contrast to the dominant literature concerning African American males and mathematics. Five themes emerged from the data: (a) early educational experiences, (b) recognition of abilities and how it was achieved, (c) support systems, (d) positive mathematical and academic identity, and (e) alternative identities.
A basic element of postmodernist theorizing in education is the critique of essentialistic modes of social analysis, including reproductionist theories that seek to specify how schooling is structurally linked to the capitalist socio-economic system. Critical postmodernists, notably Henry Giroux, Peter McLaren, and Stanley Aronowitz, agree that reproductionist theories are problematic and contend that an anti-essentialistic approach is more fruitful. I argue, however, that their version of anti-essentialism pre-supposes certain false dualisms, and is thus fundamentally flawed. Furthermore, their notion of postmodernity - which is based in part on anti-essentialism - fails to adequately characterize key developments in contemporary capitalism. In summary, critical postmodernism does not help us 'name the system' as part of the effort to develop a more viable theory of radical education. A structural, reproductionist theory that takes into account local struggles and 'non-class' phenomena remains an essential part of an emancipatory project to transform existing institutions.
the history of research in mathematics education is part of the history of a field—mathematics education—that has developed over the last two centuries as mathematicians and educators have turned their attention to how and what mathematics is, or might be, taught and learned in school / from the outset, research in mathematics education has also been shaped by forces within the larger arena of educational research / like mathematics education itself, research in mathematics education has struggled to achieve its own identity / it has tried to formulate its own issues and its own ways of addressing them / look back at some of the people and events that have given form, direction, and substance to the field of research in mathematics education
roots in mathematics / roots in psychology [research on thinking, studies of teaching and learning] / emergence of a profession (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Mathematics educators and researchers generally recognise the importance of students' active involvement in learning processes; this active engagement is positively thought of in relation to the development of understanding and the construction of mathematical knowledge. However, it is often not well understood, nor perhaps even considered, how engagement in these learning processes also produces the learner and his/her feelings of competence and confidence in/with mathematics. In this paper I undertake, from a post structuralist perspective, a meta-analysis of two short episodes from a paper by Manouchehri and Goodman (2000). I use these interactions to explore (a) how mathematical knowledge and identities are produced in teaching/learning interactions in the classroom and (b) the wider practical implications of this productive power ofprocess for mathematics education and research. Through analyses such as these it may be possible to breathe new life and purpose into mathematics education in the twenty-first century; not by negating what has come before, but in moving on to explore the exciting possibilities that present themselves when taken-for-granted assumptions of a rational, freely choosing proficient (NCTM, 2000) humanist subject are interrupted and revised.
This paper concentrates upon the relationship between mathematical education (ME) and critical education (CE) connected with the Frankfurt School and Critical Theory. To make the discussion as precise as possible a distinction is made between three alternatives in ME: Structuralism, pragmatism, and process-orientation. These alternatives are related to the key terms of CE in order to show the extent to which ME and CE contradict each other. The conclusion is that there does not exist any integration — nor even any close relationship-between ME and CE.Finally, this result is discussed in the light of the following two theses:(A)
It is necessary to increase the interaction between ME and CE, if ME is not to degenerate into one of the most important ways of socializing students into the technological society and at the same time destroying the possibilities for developing a critical attitude towards precisely this technological society.
(B)
It is important for CE to interact with subjects from the technical sciences, and among these ME, if CE is not to be taken over by the technological development and fade away into an unimportant and uncritical educational theory.
To illustrate aspects of critical mathematics education a project involving 14–15 years old students is described. Mathematics education can be organized so as to develop different types of knowing: mathematical knowing, which can be associated with skills developed in traditional teaching; technological knowing, which can be associated with a competence in mathematical model building; and reflective knowing, which can be seen as a competence in evaluating applications of mathematics. The thesis discussed says that if mathemacy should be developed as a competence of importance in a critical education, it must integrate mathematical, technological as well as reflective knowing. Via the description of the project, a possible educational meaning is given to this thesis. Especially, it is discussed what it could mean to involve students in reflections about mathematics as a tool for technological design.
Case studies were conducted to investigate the conceptions of mathematics and mathematics teaching held by three junior high school teachers. Examination of the relationship between conceptions and practice showed that the teachers' beliefs, views, and preferences about mathematics and its teaching played a significant, albeit subtle, role in shaping their instructional behavior. Differences among the teachers in their conceptions and practices are explained followed by a discussion of properties of their conceptual systems.