Linguistic and Cultural Influences on Learning Mathematics

Rodney R. Cocking, Jose P. Mestre

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781136562631
• Edition: 1
• Topic: Didattica
• Subtopic: Didattica generale

The combined impact of linguistic, cultural, educational and cognitive factors on mathematics learning is considered in this unique book. By uniting the diverse research models and perspectives of these fields, the contributors describe how language and cognitive factors can influence mathematical learning, thinking and problem solving. The authors contend that cognitive skills are heavily dependent upon linguistic skills and both are critical to the representational knowledge intimately linked to school achievement in mathematics.

Chapter 1

Introduction:

Considerations
of Language Mediators
of Mathematics Learning

Rodney R. Cocking

National Institute of Mental Health

Jose P. Mestre

University of Massachusetts

Classical language studies have long been regarded as important components of good academic training, especially during elementary and high school years. This philosophy guided individuals like John Mill in how they educated their own children, as well as providing the cornerstones of many early educational institutions (e.g., Boston Latin). The educational philosophy cast language in a central role for at least three reasons: (a) Language studies involve discipline in learning formal structures, thus promoting the development of structural organization to thought expression; (b) language studies provide an historical perspective that has implications both for language (knowledge of word classes and word origins) and for historical time perspective on the evolution of academic content; and (c) finally, language studies were thought to be important because, by learning the language of the original literature, learners had direct access to others’ thinking. Thus, discipline, historical perspective, and access to knowledge were guiding rationales for making language studies a cornerstone to education.

Of these three reasons access, particularly as it pertains to educational opportunity, continues to enter into current debates over the role of language in learning. Battle lines have been drawn over political issues that are fueled by a plethora of legislation aimed at the access question. The controversy is over who deserves the largest percentage of a limited pool of school funds.

A brief look at the motivation behind access legislation will illustrate that none of the raging debates have anything to do directly with learning, understanding, or achievement. However, the effect of such debates greatly promotes or limits access to information and for this reason the policy debates are important to consider in the present context of language studies and school achievement.

The Civil Rights decision of 1974 stated that everyone deserves equal access to an education. The definition of equal access was unstated, but language skills were seen by many educators as limiting factors to some students’ ability to benefit fully from classroom instruction. The causes of the language limitations were varied and the remedies provided by numerous pieces of legislation that were drafted to ensure equal access were also varied. Thus, although equal access was the overarching Civil Rights concern, how to apply the “equal access” criterion in the case of language skills generated internal debates among educators. Who deserves special language programs and what constitutes a language deficit drew, and continues to draw, the major attention in debates over the role of language skills in academic success. Language limitations have been defined legislatively by the following: Title VII—Elementary and Secondary Education Act that contained a bilingual education provision; Title IV of the Indian Education Act recognizes that students may need assistance in improving school language skills that differ from their tribal languages; Headstart antipoverty programs of Title I (Chapter I) recognize that language skills training may be necessary for a wide segment of the population in preparing them for later academic work; the Indo-Chinese Refugee Act contains language assistance provisions for non-native speakers of English for helping students enter public schools; and P.L. 94.142—Education for All Handicapped specifies that there is a wide variety of language performance areas that can constitute handicapping conditions and that can limit students’ access to education.

These pieces of legislation recognize that students with below-norm language skills may be at risk educationally. This assumption is borne out of national educational statistics on who are underachievers in school. However, the relationship between language skills and academic performance is not a perfect one: some academic areas are affected more than others, some language skills appear to be more directly responsible for school success than others, and some language-minority groups are more adversely affected than others by the presence of more than one language in their environments. One academic area of current interest that appears to have a large language-related component is mathematics. The relationship between language skills and mathematics is not well understood, even though the same low math achievement statistics among certain groups of learners are reported year after year.

The legislative-related controversies and the educational statistics, like the older educational philosophies, continue to cast language in a central learning role. The purpose of this volume was to draw upon a wide array of experts from diverse fields to look at this important educational research problem. The intent was to pool the knowledge and methodologies of researchers in cognitive development, education, language studies, and mathematics to look at the problem of poor mathematics skills among language minority learners. The available literature indicated clearly that we were not dealing solely with mathematics learning or solely with language issues. Rather, the consensus was that cognitive processes of learning, comprehension, and symbolic thinking were all involved. That is, the topic was one of cognitive processes, rather than one singly determined by math or language skills.

RESEARCH QUESTIONS AND APPROACHES

We began by asking a series of questions that had answers or potential answers in differing sources. First, we asked what we know about the mathematics-language relationship that is potentially helpful in improving achievement for students experiencing difficulties in mathematics. Second, we asked where the research gaps exist pertaining to what we need to know about mathematical understanding and mathematics learning among language minority students. Finally, we asked what data banks and data sources exist that might be mined to address questions without collecting new data or that might be useful for pilot work before formulating new, expensive data-gathering projects.

A two-pronged approach was devised to look at these questions. One prong consisted of formulating the important research questions that bear on the topic. This activity included asking experts to write conceptual papers representing a variety of viewpoints, and tapping recent research projects for analyses that would address some of the relevant questions. Because we were dealing with a multiply determined problem, several substantive areas were identified as important in analyzing the problem. Specifically, the following areas were included: knowledge of large data bases on mathematics achievement, how language facilitates and/or limits one’s ability to access one’s knowledge, school performance among low-socioeconomic status students, the interaction of language and conceptual development, development of early number concepts, issues of intellectual development, and cultural background perspectives on development.

The second prong of the activity was simultaneously to identify and document existing data banks that might be appropriate for conducting secondary analyses. Secondary analyses, it was thought, could be useful in formulating and answering new research questions, as well as in fleshing-out a research model. The larger tasks of identifying and documenting available data bases had to be accomplished before secondary analyses could be performed. Well-known national data bases were easy to survey for what they had to offer, for the adequacy of documentation, and the overlap of previous reports with our current objectives. Federally funded research projects and state-level data bases were eventually identified but were usually problematic because of an inability to compare data across sites for pooling samples, because of the lack of information regarding the sample sizes themselves, or because of inadequate documentation. Data from individual researchers were identifiable only to the extent that projects were well-known through the published literature or by personal contact, and these data bases were small in scope and designed for such specific purposes as to contain few of the variables that overlapped across projects or with larger data bases. From over 160 data bases that we identified, only 18 potentially could be used in the secondary analyses.

Table 1.1 is a summary matrix of the relevant characteristics of data bases considered for possible inclusion in the secondary analyses of mathematics achievement of language minority students. A number of these data bases had potential for usefulness, but some general points had to be considered.

Generally, independent variables such as the student’s home language and language proficiency, as well as parental socioeconomic status (SES) were inconsistent across various data sets. Sometimes the non-English language variable was the language the student usually uses and sometimes it was the language used by others in the home. The extent to which various languages were used was often missing and definitions varied. Presence of a non-English language for children was variously reported by students, teachers, or parents. Proficiency data were almost totally lacking across all data sets. Socioeconomic indicators were almost as inconsistent as language indicators: Some data bases used parental educational attainment or incomes, whereas others used the types of indicators that are often used by school districts to allocate various sources of funds (eligibility for free or reduced-price lunches, for example). These kinds of variability made it difficult to equate data bases.

It should be noted that there was also a fair degree of variability across the mathematics achievement variable. Some data bases contained standardized test scores whereas others used criterion-referenced tests. Further, not all data bases had scores for both mathematics and reading achievement, so comparisons across subject matter fields was often difficult.

Data from states or large school districts often presented two types of problems: First, many state data bases are available only in aggregate form—for school districts or for entire schools. Second, access to state data bases is often difficult and time-consuming. States have various procedures that are designed to protect the confidentiality of their data and one may have to gain approval from various levels and pay to have personal identifiers stripped from the records. All of this takes time and increases the cost of secondary analyses.

Finally, the data bases that offer greatest utility in measuring the language and SES variables (i.e., the nationally representative studies) are generally lacking some of the finer grained educational process variables, such as teacher characteristics and time-on-task. Smaller studies that had in-depth measures of mathematics learning processes and problems often contained no measures of mathematics achievement.

Despite these problems, several large data bases were appropriate and a number of important questions could be addressed with a sample size large enough to yield meaningful information. The chapter by Myers and Milne summarizes several sets of secondary analyses as well as a research model that guided much of our thinking.

The chapters in the present volume represent a convergence of the two prongs outlined. Various analyses of data contributed to revisions and rethinking of concept issues; conceptual papers that were circulated among the chapter authors influenced the kinds of analyses that were suggested for the secondary analyses and what writers ultimately identified as areas needing more and/or better information.

In addition to these fine-grained analyses and attempts to tease apart the issues, exchanges among the contributors of this volume often involved an interplay between a search for detail and attempts to get a clearer focus on the larger picture of the educational concerns. The outcome of such interplay and dynamics among the contributors represented here is an emphasis on some subtle issues that, we think, provide valuable insight into how research should proceed. For instance, it was agreed among the writers that both mathematics and language were symbol systems, and therefore symbolic thinking was a relevant cross-over area to consider. Likewise, some writers argued that math itself is a language and therefore language-learning issues were important in thinking about the various strands of math representation. One of the contributors to this volume explores the subtleties of these various issues: Joan Stone discusses language and math issues from the perspective of a researcher who studies mathematics learning among deaf students. In such an approach, language-symbol system issues can be studied for similarities and differences with language-variation issues among both hearing and nonhearing learners of mathematics.

This volume includes discussions of cognitive-developmental changes in mathematics learning; influences of learning contexts, such as the opportunities students have to learn math and exposure to the material or the availability of learning aids and resource materials; teaching contexts, such as the expertise of teachers who are trying to deliver information about math subject matter and make it interesting to the students; the role of language in both teaching and learning, including effects of home language on learning. Discussions are often aimed at pinpointing special learning issues associated with language minorities and with competing language systems. The reader will see that research models and research designs are also points of confluence and divergence in writers’ interpretations of research findings. Secondary analyses that are reported emphasize the effects of primary language, home language, and language proficiency. Differences in opportunities to take advanced math courses and encouragement from background cultures to take courses in certain subject matter areas are the focal points for some authors in this volume. In summary, variety in emphasis and variety in perspective characterize what follows. What the authors unanimously state is the need to study the issues together and to infuse them with our collective expertise.

TABLE 1.1
Summary Characteristics of Relevant Databases

OVERVIEW OF THIS VOLUME

This volume is organized into three sections, each comprised of five chapters. The focus of the first section is...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Contributors
• Foreword
• Chapter 1 Introduction: Considerations of Language Mediators of Mathematics Learning
• Chapter 2 Conceptual Issues Related to Mathematics Achievement of Language Minority Children
• Chapter 3 Linking Language with Mathematics Achievement: Problems and Prospects
• Chapter 4 Intention and Convention in Mathematics Instruction: Reflections on the Learning of Deaf Students
• Chapter 5 Why Should Developmental Psychologists Be Interested in Studying the Acquisition of Arithmetic?
• Chapter 6 Patterns of Experience and the Language of Mathematics
• Chapter 7 Bilingualism, Cognitive Function, and Language Minority Group Membership
• Chapter 8 The Mathematics Achievement Characteristics of Asian-American Students
• Chapter 9 Mexican-American Women and Mathematics: Participation, Aspirations, and Achievement
• Chapter 10 Assumptions and Strategies Guiding Mathematics Problem Solving by Ute Indian Students
• Chapter 11 Opportunity to Learn Mathematics in Eighth-Grade Classrooms in the United States: Some Findings from the Second International Mathematics Study
• Chapter 12 The Role of Language Comprehension in Mathematics and Problem Solving
• Chapter 13 Linguistic Features of Mathematical Problem Solving: Insights and Applications
• Chapter 14 Bilinguals’ Logical Reasoning Aptitude: A Construct Validity Study
• Chapter 15 Effects of Home Language and Primary Language on Mathematics Achievement: A Model and Results for Secondary Analysis
• A Final Note …
• Epilogue: And Then I Went to School
• Author Index
• Subject Index