Transnational and Borderland Studies in Mathematics Education

Richard S. Kitchen, Marta Civil

• Publisher: Routledge
• Year: 2012
• ISBN:
  9781136895197
• Edition: 1
• Topic: Éducation
• Subtopic: Éducation générale

Every year, significant numbers of immigrant children from Mexico enter classrooms in the United States. These immigrants comprise a heterogeneous group of students with diverse needs, abilities, and experiences. Transnational and Borderland Studies in Mathematics Education is the first collection to offer research studies across these communities. Providing invaluable research on both sending and receiving communities in Mexico and the US, this collection considers the multiple aspects of children's experiences with mathematics, including curriculum, classroom participation structures, mathematical reasoning and discourse – both in and out of school – and parents' perceptions and beliefs about mathematics instruction. An important treatment of an insufficiently documented subject, this collection brings together researchers on both sides of the border to foster and support an interest in documenting evidence that will set the stage for future studies in mathematics education.

1

ECOLOGICAL APPROACHES TO TRANSNATIONAL RESEARCH ON MATHEMATICAL REASONING

A Focus on Latino/a Mathematics Learners in the Borderlands

JUDIT MOSCHKOVICH

University of California, Santa Cruz

Introduction

Transnational experiences, especially with languages, are becoming increasingly relevant to understanding students’ lives in the United States. In 2001, 4.5 million of K-12 students in public schools (9.3%) were labeled as English learners (Tafoya, 2002). Between 1979 and 2006 the number of school-age children (ages 5–17) who spoke a language other than English at home more than doubled, increasing from 3.8 million, 9% of the population, to 10.8 million, 20% of the population (Planty et al., 2008). The majority of English learners in the United States are Latinos/as. In 2006, about 72% of school age children (ages 5–17) who spoke a language other than English at home spoke Spanish (Planty et al., 2008). In some states the numbers are even greater. For example, in California, 25% (1.5 million) of the children in public school in 2001 were labeled English learners and 83% of those children spoke Spanish as their primary language (Tafoya, 2002).

Not all Latino/a students in the United States are immigrants because many were born and raised in the United States, but Latino/a students are likely to have transnational experiences. Latino/a students who are immigrants and those who were born and raised in the United States may have transnational experiences interacting with or hearing about the experiences of their parents or family members born and raised in another country. Immigrant students will have arrived in the country at many different ages, thus bringing a variety of educational experiences from their country of origin that could inform their schooling. Some Latino/a students may also immigrate and emigrate more than once, creating transnational experiences that do not fit the assumption that they come once and stay. Some may continue to visit their country of origin or their parents’ country of origin over their childhood, adolescence, and young adulthood. Although not in school during those visits, they probably participate in transnational home, community, and peer cultural practices. Although students’ experiences in the two countries involve many different settings, contexts, and practices that may be relevant to schooling, this chapter will only focus on mathematical reasoning practices.

Transnational populations involve two (or more) cultures and are by definition cross-cultural. Cross-cultural research requires that we make our assumptions about the nature of cultural practices explicit. Definitions of culture¹ are contested and vary across academic disciplines. A definition of culture or an account of debates around its definition is beyond the scope of this chapter. However, educational anthropology and cultural psychology provide assumptions to ground transnational or cross-cultural studies.

First and foremost, we cannot assume “cultural uniformity or a set of harmonious and homogeneous shared practices” (González, 1995, p. 237) about any cultural group. To avoid essentializing cultural practices or describing culture as individual traits Gutiérrez and Rogoff (2003) propose that we focus not on individual reasoning but on what they call “repertoires of practice,” using the assumptions that individuals develop, communities change, and learners have access to multiple practices. In Lee's (2003) words in her introduction to this article, Gutiérrez and Rogoff argue that we should “neither attribute static qualities to cultural communities nor assume that each individual within such communities shares in similar ways those practices that have evolved over generations” (p. 4).

Researchers have also described pitfalls to avoid in cross-cultural research. One major pitfall to avoid is using deficit views. Learners from nondominant groups have been characterized by a deficit model in which their failures in schools are related to their home environment. In particular, research on Latino/a students has often focused on: “…language genres, behavior patterns, motivations, attitudes, and expectations that are either unacknowledged by the schools or seen as developmental deficits that must be ‘remediated’ or proscribed before learning can begin” (Garcia & González, 1995, p. 422).

In designing transnational research studies that focus on Latino/a students and their communities, we need to move from deficit models of these students’ home cultures to frameworks that value the resources that students bring to the mathematics classroom from their previous experiences and their home culture. Only then can instruction be designed that builds on these experiences.

Ecological Approaches

Researchers in education have recently called for ecological approaches that integrate a dynamic view of cultural practices into the study of learning and development and document the resources in everyday thinking (Gutiérrez & Rogoff, 2003; Lee, 2003). These ecological approaches are based on the ecological framework proposed by Bron-fenbrenner (1989), on seminal studies that examined cross-cultural learning and development and documented the complexity of reasoning in everyday settings (e.g., Lave, 1988; Saxe, 1991), and on studies of learning and development among youth from nondominant communities (e.g., Gutiérrez et al., 1999; Gutiérrez & Rogoff, 2003; Lee, 1993; Nasir, 2000).

Some ecological approaches (i.e., Brenner, 1998b; Nasir, 2000) focus on “people's participation in practices across varied contexts, including the family, the neighborhood, peer social networks, and participation within various institutional settings such as churches, schools, and other community organizations” (Lee, 2003, p. 4). Other ecological approaches focus on participation in one setting, such as a classroom, while using an ecological approach to examine student participation in classroom work (i.e., Brenner, 1998a; Moschkovich, 1999a, 2008; Moschkovich & Brenner, 2000). While ecological approaches are not uniform, they share some fundamental assumptions that shift the focus from individualistic views of thinking, reasoning, and learning to an ecological focus. As Lee (2008) explains, these approaches “share a number of fundamental propositions”:

• Context matters: Contexts help to shape people and people shape contexts.

• Routine practices count.

• The cognitive, social, physical, and biological dimensions of both individuals and contexts interact in important ways. Lee, 2008, p. 268)

The study of students’ transnational experiences with mathematical reasoning practices requires such ecological approaches not only because this work is cross-cultural, but also because it involves several interacting levels of analysis. These approaches provide theoretical notions and methods that simultaneously address the cognitive, domain specific, cross-cultural, and linguistic nature of mathematical reasoning in transnational settings. I will first describe three components of a conceptual framework that, in combination, simultaneously frame the multiple aspects and levels of analysis for the study of mathematical reasoning practices in transnational settings. In the second section of the chapter I illustrate how this ecological approach provides resources for meeting challenges specific to research with transnational populations who use two (or more) languages. In the closing section I describe strategies for avoiding deficit models of mathematics learners.

Three Components of an Ecological Approach

In this section I describe three components of a conceptual framework for framing transnational studies using an ecological approach: a naturalistic paradigm (Moschkovich & Brenner, 2000); a situated view of language and Discourse—following Gee I use the term Discourse with a capital “D” to mark a view of Discourse as more than utterances or text—(Gee, 1996, 1999; Moschkovich, 2002); and an ethnomathematical perspective on mathematical activity (D'Ambrosio, 1991). These descriptions clarify several theoretical notions—including culture, context, and mathematical activity—and make explicit the assumptions that each component brings to transnational studies. These three components provide an integrated ecological approach for analyzing mathematical reasoning practices in the following ways: (a) A naturalistic paradigm considers the ecological validity of problems, tasks, and questions used to explore mathematical reasoning. (b) Situated views of language and discourse provide an ecological approach to language, in particular to the meaning of utterances, texts, and inscriptions used during mathematical reasoning. (c) Ethnomathematics provides an ecological view of mathematical practices because it assumes that mathematical reasoning practices are multiple, heterogeneous, and connected to other cultural practices.

Naturalistic Paradigm²

A naturalistic paradigm frames studies in ways that are consistent with current approaches in anthropology and cultural psychology, two disciplines that can theoretically ground cross-cultural work. The principles for a naturalistic paradigm include considering multiple points of view and studying cognitive activity in context. Although this paradigm is not specific to the analysis of mathematical activity, it provides a general stance toward mathematical reasoning as a cultural practice. This paradigm provides three constructs: the ecological validity for cognitive tasks, a definition of context (Lave 1988), and an ethnographic stance. Using this paradigm implies that design considers the ecological validity (Bronfenbrenner, 1977; Cole, Hood, & McDermott, 1978) for mathematical tasks used in a study, that context is defined as a complex, multifaceted, and interactional phenomenon, and that analysis is conducted from an ethnographic stance. These three constructs can be applied to the study of transnational mathematical practices in several ways (I will focus on the first two; for more details on an ethnographic stance see Moschkovich & Brenner, 2000).

Ecological validity³ in psychological studies (Bronfenbrenner, 1977; Cole et al., 1978) ensures that participants’ reasoning is examined on cognitive tasks that are connected to regular cultural practices, in or out of school. Cultural psychology has shown that, when this is not the case, participants look less competent that they actually are (Cole et al., 1978). Bronfenbrenner (1977) suggests that for a study to be considered ecologically valid it should be designed to fulfill three conditions:

First, it must maintain the integrity of the real-life situations it is designed to investigate. Second, it must be faithful to the larger social and cultural contexts from which the subjects came. Third, the analysis must be consistent with the participants’ definition of the situation. (p. 35)

Cole et al. (1978) recommend that “the analysis of any behavior should begin with a descriptive analysis of at least one real world scene” (p. 4). This descriptive analysis informs the design of experiments (or quasi-experiments) that preserve some aspects of the real-world setting while modifying others. A study can start with observations in a setting where cognitive phenomena occur regularly without intervention. To explore further the cognitive phenomena observed in the natural setting originally, researchers then design interviews, quasi-experiments, tests, and interventions, based on those observations.

When considering whether a task is ecologically valid, context is not assumed to be a unilateral experience. Traditional task analyses of mathematics problems conducted by experts assume that learners know what experts know and thus use an impoverished notion of context, defined by how an expert sees and interprets a task. In contrast, a more complex notion of context adds complexity to how we see and interpret mathematical tasks. Lave (1988) provides a definition for context and distinguishes it from the setting. For Lave, a setting is the physical and social environment.⁴ A description of a setting includes the objects, people, and activities that are present. In contrast, Lave defines context as the relationship between a setting and how participants interpret the setting, including the meaning of practices. A description of context delves more deeply into the different meanings that a setting and the practices taking place in a setting have for different participants. Context is thus not a single entity, such as a place, nor is it experienced in the same ways by all participants. Instead, context is

…an identifiable, durable framework for activity, with properties that transcend the experience of individuals, exist prior to them, and are entirely beyond their control. On the other hand, context is experienced differently by different individuals. (Lave, 1988, p. 151)

Studying mathematical activity in context means not only considering the place where the activity occurs, but also considering how context, the meaning that the place and the practices have for the participants, is socially constructed. It is not sufficient to describe the setting in which learning takes place (classrooms, stores, and homes); rather, reasoning and learning need to be described within the larger set of sociocultural practices that happen to occur in particular physical settings.

One way to address the importance of context is to study cognition in the settings in which it naturally and regularly occurs without intervention. Naturalistic research methods were developed to study behavior within such “natural” settings and here they have much to offer transnational research. However, because mathematical reasoning may not always be visible in these “natural” settings, researchers may need to combine data from a “natural” setting with data from a more structured situation that includes an intervention or a design experiment. Nonetheless, to understand the process of learning using a naturalistic paradigm, it is essential to include at least some data from a “natural” setting, such as a classroom, home, store, playground, or other complex settings.

Lastly, but perhaps most importantly, a naturalistic paradigm draws on anthropology for notions of relativity that acknowledge the knowledge of the people we study (Spindler & Spindler, 1987). A relativistic stance means that we try to understand the knowledge of others in their own terms as much as possible prior to comparing it to other knowledge systems, including those of experts. Cognitive anthropology (Spradley, 1980) and grounded theory (Glaser & Strauss, 1967) prescribe specific research techniques that enable the systematic elicitation of participants’ knowledge. Relativism allows us to move from deficiency models of learners to exploring their reasoning in terms of its potential for progress, a move that is especially relevant to research with learners from nondominant communities. A relativistic stance toward culture avoids reducing cultural practices to essential or individual traits.

Situated View of Language and Discourses

Data collected in transnational studies focused on mathematical reasoning practices will involve some aspect of language. A situated view of language as it relates to mathematics learners (Moschkovich, 200...

Table of contents

• Front Cover
• Half Title
• STUDIES IN MATHEMATICAL THINKING AND LEARNING
• Title Page
• Copyright
• Dedication
• Contents
• ACKNOWLEDGEMENTS
• PREFACE
• 1 ECOLOGICAL APPROACHES TO TRANSNATIONAL RESEARCH ON MATHEMATICAL REASONING: A FOCUS ON LATINO/A MATHEMATICS LEARNERS IN THE BORDERLANDS
• 2 CROSSING THE BORDER BETWEEN HOME AND SCHOOL: DOMINICAN PARENTS’ PERSPECTIVES ON THE TEACHING AND LEARNING OF MATHEMATICS
• 3 IMPRESSIONS OF MEXICAN IMMIGRANT FAMILIES ON THEIR EARLY EXPERIENCES WITH SCHOOL MATHEMATICS IN ARIZONA
• 4 BECOMING A “LIBERAL” MATH LEARNER: EXPANDING SECONDARY SCHOOL MATHEMATICS TO SUPPORT CULTURAL CONNECTIONS, MULTIPLE MATHEMATICAL IDENTITIES, AND ENGAGEMENT
• 5 ENGAGING UNDERPRIVILEGED MEXICAN STUDENTS IN REFORMORIENTED MATHEMATICS INSTRUCTION
• 6 CONSIDERING MEXICAN AND U.S. TEACHERS’ VIEWS ON THE TEACHING AND LEARNING OF MATHEMATICS THROUGH A TEACHING FOR DIVERSITY LENS
• 7 TEACHERS’ TASK MANAGEMENT PRACTICES IN THE CONTEXT OF ROUTINE AND NONROUTINE MATHEMATICS PROBLEMS: A DESCRIPTIVE ANALYSIS
• 8 TEACHERS’ CONCEPTIONS OF MATHEMATICS AND MATHEMATICS TEACHING AND LEARNING: THE CASE OF TWO ELEMENTARY TEACHERS IN NORTHERN MEXICO
• 9 LOOKING FORWARD: ESTABLISHING A RESEARCH AGENDA FOR TRANSNATIONAL AND BORDERLAND STUDIES IN MATHEMATICS EDUCATION
• EPILOGUE
• INDEX