Department of Mathematics Education

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Annotated Bibliography of
Multicultural Issues in Mathematics Education

Patricia S. Wilson, Julio C. Mosquera P.,
Marilyn E. Strutchens, and Annicia J. Thomas

The Annotated Bibliography of Multicultural Issues in Mathematics Education is the product of work at the University of Georgia from June 1990- June 1994. We appreciate the advice and contributions of scholars throughout the world who have critiqued the contents and offered entries. We sincerely hope this work will contribute to the international effort that is being made to relate theoretical frameworks and research in diverse fields such as mathematics, history, psychology, sociology, and anthropology to work in mathematics education. The bibliography focuses on both the contributions of many cultures to mathematics and the ways in which culture may affect mathematics teaching and learning.

The bibliography contains journal articles, books, monographs, popular press, and conference papers related to multicultural issues in mathematics education. We have compiled a bibliography that addresses issues that mathematics educators need to consider for research and for practices. Most articles are not written by mathematics educators and many articles do not directly refer to mathematics or mathematics education, but the collection does offer relevant studies and theories for the mathematics education community.

The bibliography is multicultural, representing work about a variety of cultures, ethnic groups, geographic regions, and ages, as well as a variety of philosophical perspectives. While the collection is quite diverse, some individual entries are based on only one culture. A broad reading of the bibliography should help the reader develop a sense of how diverse cultural groups have constructed and continue to construct mathematical ideas, techniques, and structures, and have contributed to the development of mathematics. The literature also documents the under representation of some groups in both the study and practice of mathematics education as well as "traditional" accounts of the history of mathematics. Possible causes are suggested.

The bibliography is organized into three major sections: General (1), Mathematics (2), Science (3). Each major section is subdivided into three groups: Theory (1), Practice (2), and Research (3). For example, articles in section 2.1 are related to mathematics and mathematics education and are theoretical in nature. The categories are neither discrete nor exhaustive, but we did classify each entry so that it appears only once in the bibliography.

We would like to thank following graduate students at The University of Georgia for their advice, discussions, and annotations: Karen Brooks (US), Simeon Hau (Milawi), Daire Hubert (US), Steve Jackson (US), Cindy Jones (US), Julio Mosquera (Venezuela), Nicholas Oppong (Ghana), Shannon Primm (US), Jeneri Sagnia (Gambia), Marilyn Strutchens (US), Tingyao Zheng (Peoples Republic of China). We are also grateful to professors Dr. James Deegan (Ireland), Dr. Toshiko Kajji (Japan), and Dr. Mary Atwater (US) for their participation.

The project has been funded by the University of Georgia and the Eisenhower Program for the Improvement of Mathematics and Science. Although the grant has ended, we hope to continue to update the bibliography periodically and we welcome further contributions. Please send additional bibliographic entries and annotations, or annotations of current unannotated entries to :

Dr. Patricia S. Wilson
Mathematics Education
University of Georgia
105 Aderhold Hall
Athens, GA 30602
USA

1. GENERAL

1.1. Theory
1.2. Practice
1.3. Research

2.1. Theory
2.2. Practice
2.3. Research

3.1. Theory
3.2. Practice
3.3. Research

1. GENERAL

1.1. Theory

In this article James Anderson discussed the need for teachers to consider the cognitive styles or learning styles of students. He argued that white middle class students have a cognitive style (analytical) that is in line with the type of teaching which goes on in the classroom and are thus more apt to succeed than minority students who usually possess relational cognitive styles which do not correlate well with the type of teaching that goes on in the schools. The article contained tables which compare the characteristics of analytical versus relational learners and some fundamental differences between Non­p;Western versus Western World views. It also contained an extensive reference list.

This article was a literature review of conflicting explanations and paradigms that have emerged since the civil rights movement of the 1960s to explain the low academic achievement of ethnic youths. The author discussd ethnicity, socioeconomic status, cognitive styles and motivational styles. He concluded by stating that equity will exist for all students when teachers become sensitive to the cultural diversity in their classrooms, vary their teaching styles so as to appeal to diverse student population, and modify their curricula to include ethnic content. The article contained a large reference list.

In this article, Bracey argued that research that was done in Western cultures with a view to illustrate general laws of behavior did not usually produce similar results in other cultures. He based his arguments on the research findings of Sandra Marshall et al. of San Diego University and Paul Brandon et al. of University of Hawaii. Each of these studies examined gender in mathematics performance. While it was generally true that girls performed better than boys in the California Achievement program, in Hawaii, especially among the Caucasians, boys outperformed their female counterparts in mathematics. These were the issues he raised and examined in much detail in this article.

This article addressed the conflict between the institutional language of mathematics and personal mathematising for students. The author suggested that rather than focusing on the differences in cultures in the teaching of mathematics we may more profitably seek a common core of mathematics experience through personal mathematising. The institutional language of mathematics often conflicted with this personal mathematics.

Cohen discussed two conceptual styles: relational and analytic. In the article, the author compared the two styles across racial, socioeconomic status, and cultural groups. She discussed how different socialization practices fostered the development of one style over the other. Moreover, she reported that different environments were more compatible with one style versus the other.

This chapter was about the effect declaring English as an official language has on bilingual education within the United States. The author discussed agencies formed to advocate restrictions on immigration and their goals. He also compared English versus bilingual education.

In this book section, Diop attempts to answer the question, "How to define cultural identity?" The author writes that one must analyze the components of the collective personality. He discusses three factors which contribte to its formation-historical, linguistic, and psychological. The historical factor unifies the different elements of a people to make them into a whole. The linguistic factor implies that language has a major influence on cultural personality. Identifying a people by physical traits is a component of the psychological factor. The author feels that the difficulties and the failures in intercultural relations can be dealt with if the process in which two given cultures are born, developed, and make contact with each other should be evaluated.

This article was about the life of Thomas Fuller, an African shipped to America as a slave in 1724. He had never learned to read nor write, yet he had remarkable powers of calculation. The authors examined Fuller's story in three perspectives: the liberatory, the psychologistic, and the mathematical.

The purpose of this chapter was to evaluate the past 10 to 15 years' of psychological research on the intellectual development and education of poor children's minds. Ginsburg pointed out that school failure of poor children cannot be attributed to cognitive deficits which had been a popular theory in the past. The author also suggested new directions for research related to the school failure of poor children.

This part of the book reviewed the studies of environmental developments of the complex behavior of organisms. The issues which were indentified by psychologists and discussed here were crucial development stage, mother-child relaltionship, peers, environment, and exposure to complex situations. The author believed that both genetic and environmental contributions mold one's intelligence and the determination of the quantity of the contribution is impossible.

In this chapter, Jencks discussed the heredity versus environment issues that influence test scores. He examined the heritability of intelligence and/or the ability to perform well on IQ and achievement tests. He also looked at the possible environmental influences on these tests such as family background, economic background, and race. Jenks mentioned studies conducted in the United States relative to the controversy between genetic and environmental influences on intelligence. He supported the idea that the combination of genetic and environmental influences determined intelligence.

In this article, the author discussed the influence varying political ideologies have and should have in the different aspects of educational research. The author described a "Reality Principle" which embodied the knowledge and reality germane to the most fundamental process of education. This reality existed separate and distinct from the various political ideologies in which educational research may be conducted. The article included a discussion of similar findings in educational research which have come from different ideological backgrounds (U.S., Soviet Union, etc.).

The article discussed the existence of racism in the British educational system. He stated: "An antiracist process of education and socialization should enable pupils to develop a critical view of life and society. Furthermore, the process could enable pupils to operate across cultures, projecting a multiple presentation of self, without losing one's particular ethnicity or identity; maintaining, developing and exploring vertical and horizontal forms of communication, to negotiate a meaningful position in society with responsibility, status and access to power." (p. 280)

In this chapter Neisser presented an overview of the book. He compared and contrasted the different points of view of the authors featured in the book. The major focus of the chapter was the comparison of cognitive deficit views on intellectual differences versus cognitive conflict views.

In this chapter, the authors argued that tracking is embedded in a schooling context and a societal context. Together, the contexts help to better understand why tracking works to the disadvantage of most students. The purpose of these contexts is to appreciate what school reformers may be facing if they attempt to change tracking practices without considering strong assumptions and traditions that underlie tracking.

The author described and discussed the American caste system in terms of minorities. He listed three types of minorities: autonomous, immigrant, and caste-like. His major focus was on caste-like minorities with an emphasis on Black Americans. He discussed the negative effects of being a member of a caste-like minority in the United States.

This book contained a collection of articles on gender-related issues in the classroom. It was divided into three sections: important issues in education related to gender and race; interviews with girls about their experiences in school; and school accounts and actions related to sexism in the school. The book mentioned two views on improving girls' education: equality of opportunities, and the anti-sexist approach characterised as female-centered education. The latter view was focus of the majority of the papers in the book. Contained an extensive reference list.

The article was primarily a discussion of ethnography. The author compared the sociology on educational ethnography in the United Kingdom to anthropology education in the United States.

1.2. Practice

1.3. Research

Burton and Jones interpreted data collected by the National Assessment of Educational Progress. The data reflected trends in the levels of achievement of black and white students 9- and 13-years old in the United States from 1970 to 1980. There was a noticeable decrease in the differences in achievement between black and white students during this decade. The steady decline may be attributed to the increase of opportunities available to black youth in the past twenty-five years. The article contained several graphs to illustrate the shrinking differences in achievement levels.

Fulton and Calvin reported a study of three elementary school programs non-English-proficient Hispanic children: one bilingual multicultural, one integrated English as a second language (ESL), and one nonintegrated (ESL). They compared test scores in math, reading, and language achievement of first and sixth grade students. Their findings showed that bilingual multicultural students scored higher, on the average, than the rest of the students on most criteria.

In this chapter, the author observed high school students of different races interact with one another. He investigated the impact of the school's model of multicultural mainstream education on the students. The author discussed kinds of friendships, student cultural knowledge, race, handicap, and gender.

The article discussed the major differences between obtaining equity versus equality in the classroom. Grant stated that educational equity meant providing fairness and justice in the classroom life for students of color, poor students, and white female students. It required establishing a classroom environment that was not colorblind and teaching in a manner that accepted and affirmed the learning style differences based on culture and gender socialization.

This easy-to-read article profiled eight teachers judged to be effective by African-American parents and principals in teaching African-American students. The author used two very different teachers to illustrate the importance of culturally relevant teaching where teachers work within the dimensions of their conceptions of themselves and others, and their classrooms' social structure. Examples of each of these types of conceptions were provided.

Luttrell presented findings from qualitative research challenging feminist claims of a single or universal mode of knowing for women. She argued that what shapes how women think about learning and knowing is a complex combination of gender, racial, and class relations variables. The context of this research was adult education.

The research reported in this paper was based on activity theory. According to that theory culturally organized action guide the acquisition and organization of knowledge. The particular research reported here dealt with how worker in a milk processing plant organized their knowledge. The results showed that the activities in the plant were organized by social knowledge. Individuals, however, creatively synthesized several domains of knowledge in order to organize their own activities.

The article discussed the importance of defining equity and equality as two different terms. Secada stated that the heart of equity lies in our ability to acknowledge that, even though our actions might be in accord with a set of rules, their results may be unjust. Moreover, he believed that equality and the recognition that group inequalities may be unjust is one of the most powerful constructs of equity. He also pointed out that equality explores quantitative differences while equity addresses qualitative issues.

In this review, the author used studies which were based on the standardized test scores of Black children between the ages of 5 and 18. Those children who had obtained acceptable scores on standardized tests were used to identify the factors that seemed to influence the academic success of Black children in elementary and secondary schools. Shade used the following variables in her study: family status, structure, and interaction, sex differences, teacher­p;pupil interactions, personality characteristics, and intellectual performance patterns.

Shade examined the effects of ethnicity with a culturally induced lifestyle and perspective in the academic performances of Afro­p;Americans. She discussed the cultural foundations of Afro­p;American thought, social cognition, style of knowing, perceptual style, conceptual style, personality style, and cognitive and cultural styles. This article contained an extensive reference list.

2. MATHEMATICS

2.1. Theory

In this article, the author explored the history of mathematics. His intentions were to show that Europe should not be considered the only "civilized center" of the world. Anderson's objectives were to help students understand non-European founders and innovators of science and mathematics, Europe's affiliation with third world mathematics and science, and the basis of European capitalism.

In this beautifully written and illustrated book, the author analyzed the mathematical ideas in traditional cultures involving numbers, logic, spatial configuration, and the organization of these ideas into structures and systems. (K.S.)

Ascher and Ascher presented a definition and examples of what they considered as ethnomathematics. For them, it was the mathematics of non-literate people, people that had not developed a written system for their language. They presented an argument against the outdated view of non-literate peoples as primitive. They also argued that what we see as ethnomathematics in a given culture is always colored by our current view of mathematics.

This article was a reaction to an article written by James Tooley, published in the same journal in 1990, arguing against multicultural mathematics education. Part of Tooley's argument is that multiculturalist prescriptions were irrelevant to levels of achievement in mathematics. The context for the discussion was the design and implementation of a national curriculum in England. Bailey and Shan claim that Tooley misunderstand what multucuralist say and has a narrow view of achievement. The article explained that opponents of multiculturalism tend to ignore that mathematics teachers are instrumental in the transmission of values, attitudes, and beliefs. The authors say that mathematics educators should ask: What is the nature of math? Whose maths are we teaching? They conclude asking for culturally unbiased teaching and standardized assesment.

This article studied the relationship between learning mathematics and the cognitive process influenced by one's mother tongue. Two types of problems, A and B, were identified. "A" referred to the occurance when the instructional language was not the student's mother tongue. While "B" referred to the "distance" between the cognitive structures natural to the student and those assumed by the teacher, curriculum designer or teaching strategies and was believed to be more crucial and required urgent awareness. The new model of curriculum began from a starting point of assumptions about the learner's cognitive structures and took the adoption of traditional mathematics as a long term goal.

The author presented the results of a series of analysis of educational situations involving cultural issues. The author believed: 1) that mathematics was a pan-cultural phenomenon; 2) the identification of the associated value and its explanation relied on the mathematics educators in the certain culture; and 3) the most significant aspects of mathematics education in these issues were teacher education aspects because teachers bore the task of both enculturation and acculturation, i.e. cultural preservation and development.

Mathematics, like many other school subjects, was imposed on indigenous pupils in the colonial schools. According to Bishop, mathematics continues to have the status of a culture-free phenomenon in the otherwise turbulent waters of education and imperialism. Bishop identified three levels of response to the cultural imperialism of Western mathematics: 1) increasing interest in the study of ethnomathematics, 2) creating a greater awareness of one's own culture, 3) re-examining the whole history of Western mathematics itself. Bishop concluded his article claiming the resistance to Western mathematics is growing, critical debate is informing theoretical development, and research is increasing, in particular in those situations in which cultural conflict is recognized.

The author complained about the lack attention to cultural issues in a previously published special issue of Mathematics Teaching about geometry and the national curriculum in England. He claimed that mathematics educators should address the issues of the mathematics curriculum and diversity. Bishop complained that geometry was portrayed in the National Curriculum as culture-blind knowledge. He presented a number of recommendations for curriuclum developers. Among them were the following: "show that no one culture or country had, or has, a monopoly of mathematical ideas" and "show that many cultures and societies have contributed to the mathematical knowledge which the world now knows." Bishop complained that geometry is portrayed in the National Curriculum as culture blind knowledge.

The basic theme of this book was the exploration of the social context in which the teaching and learning of mathematics takes place. It was concerned primarily with the research findings of many studies, done both in the United States and in the United Kingdom, which were directly or indirectly concerned with the issues and problems which surround mathematics teaching today. The authors viewed these problems as constraints that were both external and internal to the teacher. External constraints are those imposed on the teacher by the institution, the pupils, parents, and society; while the internal constraints related more to the teacher's own attitude and knowledge and how he or she viewed the aims of education. Both constraints have been discussed bringing out how they affect or influence mathematics teaching and learning in the school.

The author explored current research in the areas of mainstream, cross-cultural and Aboriginal mathematics education and schooling and reflected on these findings in relation to the provision of more meaningful mathematics education for Aboriginal children. The review highlighted several features that should be inherited in any approach to the teaching of the mathematical-tecnoogical culture (or MT culture) in Aboriginal schools. They were the issues of: aboriginality, time, spatial awareness, experiences, language, bilingualism, ethnomathematics, and negotiation. The key question for educators to address was: "If Aboriginal people really want a mathematical education for their children have we the knowledge and flexibility to work with them to achieve that goal?" One hundred research papers and books were listed in the bibliography, giving an extensive reference list for further exploration of this topic.

The school mathematics of Western societies are a component of what Alan Bishop described as the widely accepted mathematico-technological (MT) culture. The author pointed out how Aboriginal cultural features often conflict with current approaches to teaching of the MT culture. Key factors which should be included in any attempts to teach MT culture to Aboriginal children. These were more purposeful experiences, maintaining their Aboriginality, providing more time for learning, making use of their existing spatial orientation, allowing talk in their native language, considering the mathematical knowledge they bring with them, and negotiating with students, parents and teachers the role of MT culture in the mathematical education of Aboriginal children. An extensive reference list provided.

The author used the situation of students in remote Aboriginal communities in the Northern territory of Australia to exemplify how historical, socio-political, and linguistic, as well as cultural and philosophical contexts of the classroom may inhibit communication and development of mathematical ideas. Harris emphasized the differences in teaching mathematics in English to children of another Indo-European language and teaching mathematics in English to children with a radically different language from English. The author offered "six pointers" related to cross-cultural teaching for consideration by teachers and others interested in mathematics education.

Hartz examimed the concept of democratic competence and the role that mathematics and its teaching might play in the development of such competence. He pointed out that there are two different arguments for democratisation: the social and the pedagogical. The discussion included a critique of the structural mathematics teaching of the sixties. Hartz ended his article asking teachers: "Can we build a mathematics curriculum which gives real democracy in materials and situations and real democracy in classrooms where we are responsible for so much of the lives of our pupils?"

The paper was about mathematics for all and the author was convinced that it is possible. She felt that mathematics learning cannot be different for students with different professional perspectives. It may differ, but only in its extension and in individual inclinations. She concluded her paper by pointing out that besides considering that mathematics should be learned as an applied discipline it should also be viewed in an applied context.

This document is the No. 35 of the Science and Technology Education Document Series published by UNESCO. It contains reports and papers presented in the Fifth Day Special Programme on "Mathematics, Education, and Society" celebrated at ICME-6 in Budapest, Hungary, in 1988. We included this document in this section because most of the articles presented theoretical perspectives and frameworks, but some of them are research reports delaing with issues such as mathematics education and bilingualism, ethnomathematics, power relations in the mathematics classroom, and so on. This document presented a complete overview of the different perspectives within the field of mathematics education concerned with the teaching and learning of mathematics and their connections with culture, language, ethnicity, and social class.

For thousands of years people did mathematics knowingly or unknowingly, and people in Botswana were no exception. Lea gave a clear idea of how traditional mathematics operated in the remote rural areas in Botswana. The article drew on the investigation carried out by graduate students on the people in rural areas. It examined the concepts of number and counting, the ways in which these rural people went about their day-to-day activities of addition, subtraction, multiplication, and division of numbers. Concepts of measurement, weight, and time were all highlighted in detail as well.

The purpose of this paper was to present the author's view on the learning process as it generates discrepancies and interruptions­p;mainly in the production of errors and unexpected successes, as well as in values (the evaluative reactions) that may arise in the course of the learning process. Mandler stated his view by presenting a brief outline of his constructivist view of emotion, and discussing some possible applications of his notions about emotion to problem solving and learning. Within his discussion he stressed a microanalytic approach and asked questions about the uses of affect and the specific effect of human error.

The purpose of this paper was to propose a theoretical framework for investigating the affective factors that help or hinder performance in mathematical problem solving. In motivating a need for this framework, McLeod summarized how affect influences several major categories of the mathematical problem solving process. These processes included the ability to retrieve information from the long­p;term memory, representational styles of solvers, the roles of the solvers' conscious and unconscious mental processes, the role of metacognition (knowledge about cognition and the regulation of cognition), and the role of automaticity. In addition, McLeod stated that affective influences on problem solving would vary according to the kind of heuristic strategy that the problem required and according to the phases through which the problem solver moved in addressing the problem.

The author looks at various papers that explore the problems inherent in universal mathematics education programs. He subscribes to views such as: the canonical school mathematics for mathematics were designed for a European elite and so there are serious adjustment problems when it is introduced into the mass educational system of a developing country. He contends that the relationship between mathematics and culture is the first and maybe the most general question which arises when mathematics for all is taken as a program. He concludes his paper by proposing two tasks: One is in the area of bringing about a cultural shift in developing countries. The second is a more specific task of working towards a better integration between universal mathematical education and the outside world to which students from developing countries will go.

Pool argued that mathematics educators should present exactly what they are looking for with multicultural mathematics education before they rush into a celebration of mathematical diversity. The author questioned the idea of treating the pupils as undifferentiated members of a cultural group ignoring their individuality. Pool used Whorf's hypothesis to criticize some of the claims about the development of mathematical ideas in non-Western cultures.

This article reports the results of a review of current literature realted to the perceived utility of mathematics and technical language development in the Crow Indian language. The authors state that many Indian languages have no counterparts to common mathematical words in English, such as multiplication and division. They suggest that teachers of Crow speaking children need to emphasize the interrelationship of mathematics teams and concepts in English and Crow. The article inlcuded mathematics history related to American Indians, and traditional uses of mathematics among American Indian tribes.

The author discussed the role and form mathematics education can and should take as a tool of democratization. A social argument of democratization was given which focused on mathematics applications which may have a "society-shaping" function. He also described a pedagogical argument of democratization which stated that our teaching of mathematics may implant servile attitudes in students to technological questions in our society, and that the teaching-learning situation should be based on democratic dialogue between student and teacher. Mellin-Olsen pointed out that these arguments may be in conflict, however, and asked the question about whether a mathematics curriculum could have been developed that is both open and empowering, and instill democratic competence.

The authors, an anthropologist and a mathematician, show how patterns from many cultures can be classified according to the symmetries which generate them. Flow charts enable one to determine the specific symmetry class of a pattern. Lavish black and white illustrations and explanatory diagrams accompany the text. (K.S.)

An interpretation of Mellin-Olsen's book as well as a call for a strategy that would increase the likelihood of school mathematics engaging students who reject school.

The author discusses the dissonance between school mathematics and the mathematical practices that students encounter in their real lives, both in the United States and in Africa. She discusses the significance of cultural differences in attempting to introduce a uniform curriculum in vastly different societies, and suggests ways of integrating cultural practices into mathematics curriculum at several levels. (K.S.)

2.2 .Practice

Ascher and Ascher pointed out that not enough attention is devoted to developments in mathematics in ancient America. They claim we need to overcome this restrictive frame and bias in order to appreciate the background of human intellectual accomplishments. Specifically, the authors were interested in the quipu, an artifact invented by the Incas in Perú. Quipus were colored cords with knots tied in them for recording numerical and relational information. Ascher and Ascher did not address educational issues in this paper, but the information provided could be helpful in designing mathematical activities and a history of mathematics course that includes groups or people who are traditionally excluded.

Frankenstein's mathematics textbook differs a great deal from traditional mathematics texts since it includes not only mathematical content but also approaches to learning mathematics, a social and political context for learning mathematics, and numerous historical insights. The style of the book provides strong support for the idea that mathematics is a human endeavor and mathematics can be a powerful tool for all people. The mathematical topics included integers, rational numbers, numerical operations, and variables. The author "situates the teaching of mathemaics within a rationale that links schooling to the wider considerations of citizenship and social responsability."

This is a paper on research on classroom environment, focusing on how mathematics teachers might apply ideas from research in guiding practical improvements in mathematics classrooms. In their study, use was made of a new short form of My Class Inventory (MCI). Which was found to be valid instrument. They then asked a teacher to use the MCI in a systematics attempts to improve a mathematics class. The results were promising. The authors conclude their paper with optimism and they quote Fraser and Fisher and write, "In recent studies of person-environment fit, students were found to achieve better when there was a higher congruence between the actual classroom environment and that preferred by the students".

This article was in part the result of research in which researchers developed some materials, then trial-tested and evaluated these materials for the teaching of mathematics from a global and multicultural perspective. The thesis used in developing the materials was that the issue of global inequality could be explored while also involving meaningful mathematical activities.

Moore presented evidence to support his hypothesis that preliterate tribes people were capable of mathematical thought as exhibited thorugh their invention and mastery of string art figures. This common activity did possessed elements of mathematical thought, namely, logic and intuition, analysis and synthesis, and generality and individuality, in accord with a definition of mathematics by Courant and Robbins. This information may impact American Indian students' conception of being mathematically disadvantaged when among Anglo students.

Moore indentified iteration, recursion, similitude, tiling, and symmetry as principals of mathematics-like thought used by petroglyph carvers. He supported his claim with examples of carvings which illustrate each principal. Concluded with several suggestions for classroom activities.

Materials in this handbook are the result of work of supervisors, administrators, teachers, counselors, and teacher-educators who attended 5 conferences organized by the NCTM in Florida, New Mexico, Maryland, and Minnesota. They included suggestions for conducting mathematics equity surveys, designing and organizing equity conferences and other teacher in-service activities, developinhg networking strategies, and developing curriculum and instructional strategies which deal with equity issues in mathematics. Also included is a resource list of mathematics equity materials and an appendix with papers that were presented at the conferences on underepresented groups in mathematics.

The authors outlined their work in revising the lower school science curriculum of a school system to take into account today's multicultural society. Their sources for this project were the current curriculum, suggestions from students, and units from the Third World Science Project. The authors' rough draft of the revised curriculum attempted to eliminate gender and ethnic biases and stereotypes by including illustrations from various cultures, not just the European and North American cultures. The units described in this article allowed students to read and/or write about the topic being studied in real world situations in order to make the material more relevant.

This is a copy of an Association of Teachers of Mathematics, in England, closing lecture in 1988 by the author. Using many anecdotes and examples, the author makes the point that mathematics is created in social settings and the directions of its development is socially determined. The claim is made that the interpretation of mathematics development (history books) was laden with ideological stances, political influences, and racial prejudices.

In this book, Tobias has examined the myths surrounding mathematics. She reported on intervention techniques that she tried out in an experimental clinic at her university. It is primarily a discussion of how intimidation, myth, misunderstanding, and missed opportunities have affected a large proportion of the population. The principal purpose for writing the book was to convince women and men that their fear of mathematics is the result and not the cause of their negative experiences with mathematics, and to encourage them to give themselves one more chance.

In this short article, the author explored some mathematical ideas developed in Africa outside of ancient Egypt. She claimed that history-of-mathematics books do not inlcude African mathematics leaving the impression that nothing had been accomplished in that part of the World. The main purpose of the article was to present some suggestions for the incorporation of mathematical ideas in the study of African culture, e.g. as a part of a total learning experience. Mathematical ideas related to weaving, knots, networks, divination, gambling, measuring, currency, and gaming were presented.

The author is concerned with Blacks and their relationship with mathematics. She reported the accomplishments of several Black mathematicians, as well as the prejudicies they had experienced within the field of mathematics.

The author presented ways of integrating the real-world as well as other school subjects into the mathematics curriculum. Investigating the various numeration systems, the unique styles of housing, and games from different cultures encouraged students to analyze their own concepts of mathematics. Each of these activities helped students to make meaningful connections between the mathematics taught in the classroom and real-life situations, in addition to exposing students to other cultures.

In this paper the author offers a series of activities with symmetrical designs and repeated patterns for the mathematics classroom. Ideas are taken from quilt patterns and Navajo rugs, and historical notes are included. The author tries to help students to become aware of the role of mathematics in society, realize that mathematics is a dynamic, growing, and changing human activity, and to learn to appreciate other cultures.

The author argued for the importance of incorporating a cultural perspective into the curriculum. She discussed topics such as numbers and numeration, design and pattern, architecture, and games of chance and skill.

2.3. Research

This paper presents results from the National Assessment of Educational Progress (mathematics assessment) involving 70,000 9-, 13- and 17-year-olds during the 1977-1978 school year. This article focuses on the assessment results for Blacks and Hispanics. Results indicated their performance was significantly below the national average for each age group assessed and they took less mathematics in high school, but most Black students liked mathematics, thought it was important, and indicated a greater desire than their peers to take more mathematics.

The author examined in great detail the significance of continous figure tracing among the peoples of the Malekula island in Oceania. She has noted that, within various traditions in Oceania, figures drawn depict different cultural meanings--myths that explain the origin of death, flora, and fauna--but implicit in them are remarkable mathematical ideas in geometry, topology, and algebraic algorithms. She gave examples of simple closed curves and regular graphs--graphs having all vertices of the same degree. With the regular graphs they developed algebraic skills based on succient statements of the drawing procedures. On the whole, the study provided a clear understanding of some graph theoretic considerations of other peoples whose culture may be regarded as different from the Western culture.

The author presented the very popular puzzle in which a person must ferry across a river a wolf, a goat, and a head of cabbage. The person has a boat that can carry only him/her and one other thing. African versions and Western versions of this puzzle are presented and analyzed. The author claimed that the existence and enjoyment of this puzzle in different cultures showed that interest in logic was not the exclusive province of any one culture or subculture, and that there was a pan-human concern for mathematical ideas. She concluded by pointing out that the case presented in this paper "is but one of the many examples that demonstrate that mathematical ideas are of concern in traditional non-Western cultures as well as in the Western cultures.

Awartani and Gray discussed the results of mathematics testing of 14 year olds and college freshmen in the West Bank. In testing the 14 year olds they found more substantial differences among students from different socioeconomic backgrounds than between males and females from the same background. With the college students there was no significant difference in the test scores of men and women. They discussed possible reasons for their findings and lastly encouraged further research specifically on how sex differences in mathematics achievement depend on cultural background and socioeconomic status.

In this article, the author raised a concept of "social construction" aimed at the better understanding of teaching and learning in the classroom. The author believed that every classroom was unique in its identity, people, atmosphere, events, pleasure, crisis, and history. Every person constructs his/her own mathematics knowledge through this uniqueness. He proposed a new orientation for mathematics education which viewed mathematics classroom teaching as controlling the organization and dynamics of the classroom for the purpose of sharing and developing mathematics meaning. The key was sharing. Therefore, the analysis of "social construction" focused on: 1) mathematics activities; 2) communication (pupil to pupil, pupil to teacher, and teacher to pupil); and 3) negotiation (goal-directed interaction guided by the teacher), and thus offered mathematics educators rich avenues to explore.

This book included the major findings of a three-year longitudinal study of 1,500 students in 6th through 12th grades in three New England schools. The findings concerned changes in students' ideas about mathematics, and their plans for mathematical involvement, negative attitudes of girls, and students' perceptions of the usefulness of mathematics. The author proposed interesting remedial strategies for the problems identified in the study.

This is a research report on the hypothesis that students reorganize their beliefs about mathematics to resolve problems that are primarily social rather than mathematical in origin. Cobb's contention is that cognition is necessarily contextually bounded. He concludes that many of the problematic situations that precipitate children's reorganization of their beliefs about mathematics are social rather than mathematical in origin.

The authors of this book were particularly concerned with the nature of cultural and linguistic influences on mathematics learning. Understanding the nature of mathematics performance in the schools requires much more than what cognitive researcher offers. Together with the explanations offered by cognitive research are important factors which affect mathematics performance. In addition cognitive issues, mathematics performance is influenced by bilingualism, gender, culture, class, affect, motivation, teacher competence, the availability of sound educational opportunities and the implemented curriculum (as opposed to the intended curriculum). Each of these factors may affect mathematics learning singly or as a part of a collection of all the factors.

The author reported a cross-cultural study of gender differences in performance on various mathematics items. Engelhard described the subjects, how they were selected, the test administered to them, and the results of the study. The data suggested that as the cognitive complexity of the item increased and the content moved from arithmetic to geometry, the male subjects performed better than the female subjects. A comparison of the results of the study in the United States and Thailand supported the findings of earlier studies that these gender differences were consistent across cultures. Several tables and an extensive reference list are provided.

This easy-to-read article dealt with a study of the mathematics vocabulary of Navajo Indians in the intermediate grades of elementary school. Several specific problems were identified and suggestions were included for teachers of not only Navajo students, but teachers of any students whose second language was English. The article included good ideas for dealing with "sound like" words such as angle and ankle, sum and sun, etc.

The purpose of this paper was to describe the various meanings people ascribe to the words attitude, affect, affective domain, belief system, emotion, and anxiety and to summarize some of the consistencies and inconsistencies among the meanings.The rationale for this paper was based on the difficulty that psychologists, mathematics educators interested in research on problem solving, and mathematics educators interested in research on attitudes toward mathematics have in communicating to one another using the aforementioned words due to the different meanings that each group imposed on each of the terms. The author justified the significance of clarifying the terms across the three groups by referring to ongoing research in areas related to attitudes, belief systems, emotions and other affective variables.

The authors reviewed findings from the National Assessment of Educational Progress for 1973 and 1978. At ages 9 and 13 blacks improved while whites declined in levels of mathematics achievement and yet substantial differences are found between average mathematics achievement scores of white and black youth. About one half of the white-black mean difference was accounted for by regression and school differences in background variables which played a more prominent role than individual differences within schools. The best single predictor of mathematics achievement was the number of high school algebra and geometry courses taken. Marked differences were found between predominantly black and predominantly white high schools in the average number of such courses taken. The adoption of policies that reduced those differences would be expected to result in relatively higher levels of mathematics achievement for black students.

In this book, Lancy presented the theoretical framework, research methodologies, and findings from a large mathematics education research project in Papua New Guinea. The project followed the lines of Piagetian research with elements from Vygotsky's socio-historical psychology. This position lead the researcher to postulate that societies rather than individual subjects passed through the developmental stages formulated by Piaget.

The Board provided a rational for all Americans to change because America was changing. Statistics were provided to show that nearly 40% of Americans under eighteen were minorities, and by the year 2020, these minorities would become the majority of students in the United States schools. The Board emphasized that a major part of the national effort in the 1990's would be to re-educate parents, principals, teachers, and the public, whose deeply-entrenched beliefs about who can learn mathematics appeared to limit Black, Hispanic, and American Indian children in developing their talents in mathematics. The Board listed six regional workshops and a national convocation as a first phase of making mathematics work for minorities.

Matthews reviewed twenty-four articles in order to organize variables (according to parent, student, and school) that influence the performance and participation of minorities in school mathematics. She found that few of the parental variables have been studied directly and that several school characteristics that appear to be influential have not been quantified. A variety of students' characteristics have been identified and their influence examined. Matthews listed and discussed the identified variables and suggested directions for further research.

This article presented and reviewed data from the Third National Assessment of Educational Progress in Mathematics that was conducted in 1982. Samples from White, Black, and Hispanic 9-, 13-, and 17-year-olds showed that although Black and Hispanic students continued to score below the national level of performance, they had made greater progress than whites since the 1978 assessment. The authors also reported greater gains by schools with heavy minority enrollment, and that the more mathematics courses taken increased scores for both blacks and whites.

The article contained an extensive review of the literature surrounding factors which may lead to the underrepresentation and underachievement of women and minorities in mathematics and science. The main categories for these factors were: cognitive, affective, schooling, and societal. The authors also discussed the mathematics and science pipeline, and gave several research implications and topics. The article contained an extensive bibliography.

This study attempted to replicate Piaget's investigation of the development of geometric concepts among children in the Ankole district of Uganda where there are no traditional precision measurement instruments either geometric or otherwise. Researchers studied the extent to which schooling experiences affected development in geometry. While the findings of the study indicated some evidence of developmental lag among non-Western school children relative to their Western counterparts, it was also revealed that when non-school cultural experiences were not inhibited or were facilitated, the influence of schooling on the children's understanding of conservation concepts was tremendous. Therefore, this investigation suggested that certain concepts of geometry may depend heavily on schooling rather than on the biologically based maturing of the logical structures of the child.

The article discusses an ex post facto analysis of the interrelationships of divergent thinking, general reasoning, field-dependence, mathematics achievement, reading of mathematical prose, syllogistic reasoning and mathematical word problem solving among Hispanic students. The findings of this study touch on two issues: educational policy and mathematics education research. The researchers think that the Spanish version of the reading, mathematics achievement, and mathematical problem-solving tests can be used by school personnel to assess these academic abilities among foreign born Spanish-speaking students at the prealgebra level. The research illustrated the complexity that exists between language, culture, and thought, on one hand, and the new avenues which the psychometric tradition can open for cross-cultural research in mathematics education in the psychometric tradition on the other.

Pea reviewed the two books: Cognition in Practice: Mind, Pathematics, and Culture in Everyday Life by Jean Lave, and Culture and Cognitive Development: Studies in Mathematical Understanding by Geofrey Saxe. Both authors attempted to provide empirical links between cultural practices and cognition for mathematical activities. Lave's book documented ten years of research concerning "the occurrence, organization and results of arithmetic practice in everyday situation." Saxe, in examining the interplay of culture and cognition, looked at the transfer of learning of school-linked cognitive forms to everyday practice-linked problems and vice versa. Pea believed the authors' views that problems emerged out of dilemmas and that learning arose when means were sought to resolve these dilemmas. He praised their focus on learning competencies rather than failures and proposed that such a focus will contribute more to advancing effective learning practices than repeated diagnoses of failures.

The authors reported their findings from an anthropological study of the Navajo's concept of space. To conduct this study, they elaborated a framework for investigating that concept. They claimed that it could be used to study the concept of space in any ethnic or cultural group. They also presented some possible consequences of their findings for mathematics education. According to them, mathematics educators of Navajo children had three choices: to develop the Western conceptualization, to develop the Navajo conceptualization, or to try to integrate them in a coherent way. The authors were in favor of the third option.

Porter's article was based on his premise that strengthening the teaching and learning of academic content would solve the larger problems of society. In studying, understanding, and/or defining mathematics education of disadvantaged students he discussed the need to consider the learners, teacher, curriculum,and milieu. Porter also discussed the idea that worthwhile mathematics content enabled students to apply their conceptual knowledge to novel problems and that good teaching was a rational, goal-oriented process. Though he tested and discussed characteristics of good teaching, he did note that there was a difference in deciding what constitutes good teaching and creating good teachers. Porter lastly discussed curriculum materials, texts that focused on skills versus problem solving, and the redundancy that occured through textbook series. Porter called for a reductionary change, a reformulation of policies, a redesign of textbooks, and the understanding of the fact that real and lastly changes lie ultimately with the individual teacher. His focus was that disadvantaged students are most in need and that is where the limited resources should be invested.

Rosin called his research a study on ethnoarithmetic. In this particular study, he investigated the mental arithmetic of an illiterate peasant in India in the context of buying golden medallions. Rosin showed that literacy is not a necessary condition for doing mathematics.

The objectives of this report were 1) to examine the concept of mathematics anxiety as measured by the Mathematics Anxiety Rating Scale (MARS) and the Math Anxiety Scale (MAS) with respect to their relationship to a measure of attitudes toward mathematics and arithmetic performance and 2) to provide data describing attitudes toward mathematics of participants in a math-anxiety treatment program. On the basis of the correlation between the Mathematics Anxiety Scale and the Math Anxiety Scale and correlations among the Fennma-Sherman Mathematics Attitude Scales and these measures, the researchers were unable to conclude for this sample of math-anxious individuals that the MARS and MAS measure the same construct. Mathematics anxiety is a unique affective variable that appears to be distinct from other affective variables.

The author addressed the issue of relationships between culture and cognition in the context solving arithmetic problems. The research was conducted with street vendors and non-vendors, largely unschooled, between 10- and 12-year-old kids in northeast Brazil. The findings supported a constructivist model of cognitive development. This model states that children create novel procedures and understandings in copying with their every day cultural practices.

Lucy Sells stated that mathematics courses in high school often served as a "critical filter" which hindered many female and minority students from pursuing mathematical related careers. She also discussed several programs such as SEED which were designed to help increase the enrollment level and achievement level of female and minority students.

The purpose of this study was to investigate the achievement­p;related behaviors of sixteen students in a seventh grade mathematics classroom. In particular, the study focused on the differences and similarities of the attitudes and achievement related behaviors of black and white girls and boys in the class. The attitudes and behavior on which the authors focused were confidence in learning mathematics, perceived usefulness of mathematics, enjoyment of mathematics, and the achievement­p;related behavior of persistence. The findings were clearer when race and gender were examined simultaneously than when either race or gender were examined alone.

The above three articles by Turner discuss the value and approach of teaching mathematics in Primary School through culturally motivated games, songs, and movement activities of Bhutanese children. The articles tie this approach ot existing literature on brain hemisphericity, the role of play and ethnomathematics. Examples of Bhutanese cultural/mathematical activites are presented.

The authors discussed the results of their study on the proportions of variance in mathematics achievement attributable to differences in the number of semesters of mathematics studied after taking into account other background influences. The study was conducted with a national random sample of 2,216 17-year old students. Eight background variables, representing the home, community, and individual factors which have been found to be related to student learning, were used. An extensive reference list was provided.

3. Science

3.1. Theory

This book contained a collection of articles about the teaching and learning of sciences. Some of the chapters in this book are individually listed in this bibliography.

3.2. Practice

3.3. Research

The purpose of this study was to learn more about how Black freshmen fare in science and engineering at large, historically white state universities, and the variables that tend to be related to their success and nonsuccess. Another goal of the study was to determine which factors among black students were significantly related to their success in science and engineering, and which ones were not. The authors concluded that if Black students come with realistic expectations of the university experience and if the university provides help to Black students who have problems, then more Black students will be successful. They also suggested a need for further study in this area.