Words, Sums, and Differences

Allen Klinger and Pamela Munro, UCLA

 

C. Project Description

 

Introduction.             Language is a support system for culture that can nonetheless be a barrier to achievement and reward. Computers have changed the way that business is done and individuals communicate for many, but there are significant groups unable to participate. More generally, many populations have less economic opportunity than the majority group in their country. This effort is designed to impact a linguistic minority in California and Mexico and to derive knowledge about what works best to support making access to modern technology possible for disadvantaged communities.

 

      Conducting conversations about computer and communications technology and mathematics is at the core of this proposal. The conversations will be between graduate students at UCLA and recent immigrants, native speakers of Zapotec. Most Zapotec immigrants speak no English when they reach the United States. They usually arrive with little preparation for work, coming with at most a sixth grade education in an imperfectly learned second language, Spanish, whose structure is radically different from that of their mother tongue. Our conversations will be conducted in English, Spanish, and Zapotec, both in Los Angeles and later in Oaxaca, Mexico, and will have three purposes:

 

1.     To provide a window on, and simple learning in, basic ideas of science, engineering and mathematics;

2.     To support the speaking of Zapotec while affirming its co-equal role with English (or Spanish) as a vehicle for discussing logical concepts, visual relationships, and probability; and,

3.     To develop a core vocabulary relevant to computing, with definitions of its component words in simple, easy-to-understand, terms.

 

            Our research tests the following hypothesis: if people learn new concepts on the basis of discussion and questioning in their own language, they will understand more than if the ideas are presented in a second or third language. We focus on a single language community, speakers of Valley Zapotec on the West Side of Los Angeles and in the state of Oaxaca, Mexico. This focus will accelerate adoption of the technical terms developed during the project, both in Los Angeles and in an underdeveloped region of our neighbor, Mexico. Prior linguistic research on Valley Zapotec at UCLA provides needed background, and existing contacts in the community will facilitate recruitment of project participants. Ascher (1991) and Joseph (1991) provide persuasive evidence that a structure about mathematical concepts exists in many cultures.

 

      This study addresses the basic research question of what fundamental concepts must be transmitted to enable success in working with contemporary computer and communication technologies, and how presenting these concepts in the student's native language aids learning and retention of knowledge. Our effort will introduce basic ideas in science, engineering and computing to individuals who are particularly able to benefit, Zapotec speakers in Los Angeles and Mexico, a minority population at significant educational and economic disadvantage. We will thus be able to observe how populations adapt to technology and use formal education. This is a seminal research activity with important potential benefit to society. In particular it has potential to extend to similar situations where including a minority in the activities of the majority culture can have great value (e.g., the Kurds in Turkey and Iraq).

 

The materials. We will employ, add to, and extend existing web-based material on mathematics now in the site of one principal investigator, We will bring UCLA graduate students into on-campus use of those materials, by qualifying them for stipends based on their success on preference-oriented multiple choice quizzes, such as those presented in

Klinger ("Training and ThinkingÓ http://www.cs.ucla.edu/~klinger/training.html) March 2002, using topics among those surveyed in Steen (ed. 1990), On the Shoulders of Giants, and other sources. The web materials now in the site of one principal investigator are highly visual and conceptual in nature (see Figure 1, where three representative examples appear), The references list gives more information in three forms: locators (Web Files), figures, and resources. (Even that list is only a small amount compared to what is readily available. Many other items are available just in the site of the co-Principal Investigator; expansion of them could be part of this work.) In all, the web references provide a basis for covering a broad range of topics relating to science, engineering and mathematics, particularly from the viewpoint of working with computers. They will be the basis of preparatory meetings at UCLA to involve graduate students in off-campus short presentations of mathematical ideas for discussion-stimulation.

 

      Each of the web files will be used in training sessions with UCLA graduate student members of the research team, and will then serve as source and reference materials for Zapotec conversations. The grant research will involve bringing basic mathematical concepts into these conversations. These conversations will take either of two paths:

 

1. Helping non-native-English-speakers towards acquisition of science, engineering, and technology skill, and familiarity with some regular curriculum mathematical topic.

2. Supporting UCLA participants' understanding of traditional Zapotec culture as it is today.

 

Sample Visual Resources for use in Stimulating Mathematics-Related Conversations

a)     Division Example Ð Raise to a Power, Exponents, Large Numbers

 

b)    Grouping Example Ð Inclusion, Parity, Hierarchy

 

c)     Start a Quantitative Discussion from a Game Ð Modulo, Tempo, Stacks, Queues

 

Figure 1. Some mathematical web resources [(a) and b) drawn by Gavin Wu.]

 

      An informal approach to mathematics can make use of games (rules of many games are available from [wr19]), kinship relations, and space partitioning. All these items lend themselves to linguistic as well as analytical description. Other aspects of Zapotec culture including the division of activities into men's, women's, and children's work and pastimes; agricultural practices; holidays; relationships to the natural world; religious/civic/justice administration; and kinship constraints on social activity will be examined for their connection to the mathematical curriculum. We will explore how basic logical relationships in these cultural areas connect with abstract principles in mathematical and scientific reasoning.

 

Mathematical structures.  The mathematician Sonya Kovalesky wrote, "Many who have never had the occasion to discover more about mathematics, confuse it with arithmetic and consider it a dry and arid science. In reality, however, it is a science which demands the greatest imagination" (see Osen 1974). "Beauty and insight" (Hoffman 1998, p. 44) play important roles in mathematical creation. Sometimes both come from an amateur (see Bell 1990 on Fermat). The research will seek to stimulate thought and inquiry by all the participants, university- and community-based. That should result as questions arise prompted by introductory talks. Some may be based on games. Others will concern key words from the language of mathematics.

 

      Simple statements can be beautiful, as in the title of a paper by Erdos and Selfridge (1975): "The Product of Consecutive Integers is Never a Power". Richard Feynmann, the Nobel prize winning physicist, put it this way: "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in" (1967). The statement echoes the book by Dantzig (1930): Number, The Language of Science.

 

      But to many people, the window that mathematical thought opens on science and engineering is tightly closed. Even simplest aspects of mathematics involve symbols, abbreviation and conventions. These may form an awkward barrier to understanding. Even more important, the conventions aren't always the best way to indicate the ideas involved. A beautiful structure has become an obstacle.

 

      Further, operations (iteration, recursion, approximation) and notations for mathematical expression (mod/modulo, rho-theta-phi coordinates, sgn/sin) now take on basic and major roles in applications. The result is a bewildering array of technical language, one that can block access and involvement. Historically, mathematics has adjusted its terms and boundaries to deal with practical needs. For example gambling studies led to probability. A similar mathematical expansion came from the need to harness electricity. Electrical phenomena led to the operational (Heaviside) calculus, delta "functions," and Z-transforms. These led, in turn, to extending mathematical ideas to distributions. Similarly, long ago imaginary and complex numbers became part of the accepted structure of mathematics. Those who seek to enter the technical world encounter both unfamiliar vocabulary and a more confusing thing, ordinary words used in special ways. This causes acute difficulty for people working through a second or third language.

 

      Due to changes in available tools mathematics education is in ferment. Computers and handheld calculators have had a great impact on the curriculum. Several calculator products became revolutionary tools for educators (see the Texas Instruments TI-89 or Voyage 200 - http://education.ti.com/). Other changes have caused great turmoil concerning mathematics curricula. (A meeting in Vienna 2002 on these subjects involved over two hundred forty participants from forty-three different countries.)

 

      While we do not have the focus of mathematics researchers or teachers (as represented by Steen, On the Shoulders of Giants and the National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics), the outcome of our conversations could be principles and approaches that may impact the multi-sided evolution of mathematicsÕ structure. In the second year of this project we will disseminate a select set of questions from the Zapotec participants to a national panel of mathematics educators for review. These reviews will judge whether (and if yes, where) a discussion question is a useful fit with current K-12 or K-16 offerings.

 

Discussion topics. One Zapotec speaker told us, ÒWe need a window on technology.Ó Well established mathematical curricula and many useful suggestions in National Council of Teachers of Mathematics (1989), Osen (1974), and Steen (1990) will guide the choice of technical introductory talks. The central principle of this project will be to follow participantsÕ interests while helping them to become successful computer-users.

 

      UCLA graduate students and Zapotec speakers could have different valid reasons for wanting to explore a subject. People are willing to express themselves only when they are actively engaged. For example, such engagement occurs when someone seeks knowledge to support development of needed skills. We expect that employees of food service businesses (where many Zapotecs in Los Angeles work) will need metric conversion ability. They could benefit from dimensional analysis, multiplication and division on handheld calculators, and material on partitioning [wf2] or tallying [wf16] Space layout is likely to suit those working in the building trades, leading to Alexander (1977), Klinger and Salingaros (2000) [w15], and Steen (1990). Exploration of these practical subjects could lead to generalities along the lines of items discussed by Temple (1986), Ascher (1991), Bell (1990), and Dantzig (1930/1954). At the same time the existing structure of mathematics makes some topics intellectually appealing on their own, for example to university students, who may benefit from viewing video and web materials created through Project Mathematics! [wr8] and could there find visual inspiration on topics such as the Pythagorean theorem. Other sources of visual material range from Adams (1974) to Nelsen (1993, 2000) and [w15], [w20-22], [wf2], [wf4, [wf7], [wf14], and [wf16].

 

      Steen and colleagues active in [w9] partitioned the accepted structure into Òdeep mathematical ideasÓ (Steen 1990, p. 5): dimension, quantity, uncertainty (Figure 2 at end), shape (Figure 3 at end), and change. We seek to develop in some Zapotec speaker participants enough knowledge of portions of that structure to support future growth and learning. We will produce feedback mechanisms based on material published in Klinger (March 2002), Klinger (July 2002), and, choice among alternatives facilitated by [wf11], and a rating method described in Klinger (1997), [w16-17]. One of the deliverables of this research will be extended sets of examination material based on this methodology.

 

      Technology causes substantial modifications to educational curricula. At the elementary level there is teaching about sets and even efforts to add programming (via  ÒLogoÓ, see e.g., [wr6] and [wr7]). However among all the changes over recent years, one stands out. It is emphasis that students should acquire ability at problem solving.

 

      The complex nature of some problem areas and their relatively close connection to computing gives precedence (at least to professional information technology workers) to areas such as discrete mathematics, combinatorial problems, and network issues. Notwithstanding the practical problems they solve, the structure of mathematics treats some fields as ÒrecreationalÓ, such as the work of Golomb (1994), despite its importance for successful image transmission from space to earth. GolombÕs treatment of combinatorial issues could make his book, or the work of Riordan (1958), important sources for our project's graduate student discussion leaders.

 

      But there are linguistic issues as well. One highly successful computer utility, Mathematica, draws on Iverson (1980). (Iverson saw ways to overcome limitations of the first higher level computer language. This was in regard to mathematical  expressions : things  people  frequently  need. Examples include solving several simultaneous linear equations in more than one unknown.) Both  [wr1]  and [wr2], extend the insights of IversonÕs paper. Some graduate students may seek to steer the discussion work to concepts expressed by the accepted structure of mathematics (recursion/iteration, trigonometry, and algebra are a few examples). Those works are embedded in many mathematical software systems. The earliest, APL, Mathematica, and J are just a few of those that are available today, and there are parallel resources in display software and handheld devices. The mathematical education communities have responded. Texas Instruments marketing of software and handheld calculators [w26] both the value of the new technology, and the need for acquainting those left back by culture, to these modern mathematical tools.

 

      Finally, simple visually related problems such as [w20] can initiate discussions.  This resource poses the questions: Can one arrange nine apples in four baskets so there are an odd number of apples in each basket? and Can one arrange eight coins so that they lie in two straight lines having five in each? Both put individuals into problem solving mode. Answering the first question could introduce the concept of a hierarchy. Answering the second requires understanding of dimension, as well as at least connectivity and at most topology. See e.g., [w27].

 

From education to employment opportunities. Some potential participants in this research will have only an elementary education at best. Others may have a high school background. A few may have advanced academic training, at the community college or university level (in either the humanities or science).

 

      Economic development in California depends on the flow of people across borders. The state of CaliforniaÕs Community College system enables learning for working adults, people with no or poor high school records, and generally anyone interested and committed to knowledge-acquisition. The research seeks to honor the role of that system in helping individuals move to higher economic strata by aiding one instructor and two students from two-year college institutions whose students often transfer to UCLA. We will offer participation in the research to such qualified individuals at West Los Angeles, Los Angeles, or Santa Monica Community Colleges.

 

      Opportunities to participate will begin with the evaluation of questions from the discussions by such participants and review panels. They will be asked to grade each re its assisting people to know more about computers. They will be asked to input the impact of economic opportunities on the questionsÕ topics. There is a rapid pace of change in memory, processor, display, communication, and software technologies. That increases the importance of human needs re information training. Likewise it adds to  the value of the community college and other review results. The selection of topics and development of substantive and enduring instructional material is a research goal. To achieve it, many need to agree on basic principles to engage, challenge, and develop all kinds of students who want to study science, engineering, and mathematics. The means proposed here is in particular to find new ways to support students' involvement in exploratory learning using information systems and digital computer technology.      

 

The people and their language.  In Oaxaca, Valley Zapotec is spoken around the market town of Tlacolula, in the pueblos of San BartolomŽ Quialana, San Juan Guelav’a, San Lucas Quiavin’, San Marcos Tlapazola, Santa Ana del Valle, Teotitlan del Valle, and Tlacolula (Figure 4, at end of proposal). All schooling is in Spanish, with non-Zapotec-speaking teachers, though most children know no Spanish before entering school. When technical subjects such as mathematics are presented in an unfamiliar language, students face difficulties far beyond those of English speakers in the United States encountering the separate mathematical structures of arithmetic, algebra and geometry. Drop-out rates are high.

 

      Zapotecs from this region have been immigrating to the Los Angeles area in large numbers for three decades (Lopez and Munro 1999); there are now many thousands of Valley Zapotec speakers living on the West Side of Los Angeles (Figure 5). The Valley Zapotecs of Los Angeles are a cohesive community who constitute an ethnic minority within an ethnic minority Ñ to most U.S. citizens, they appear to be generic "Mexicans", but they are representative of many indigenous people who have suffered discrimination by Spanish speakers. Many live at or below the poverty line, yet aspire to move up to better jobs and to use and benefit from education. They speak a complex language whose grammar is very different from English and Spanish, both of which they may have to learn on arrival in the United States.

 

      Modern study and writing of Valley Zapotec began only a few decades ago with work on a Bible translation into San Juan Guelav’a Zapotec (Liga B’blica 1995). Since 1993, academic study of San Lucas Quiavin’ Zapotec at UCLA has produced two dissertations (Galant 1998, Lee 1999), a master's thesis (MŽndez 2000), a dictionary (Munro and Lopez, et al., 1999; currently being revised), and many scholarly articles.  Research on several other varieties of Valley Zapotec is in progress.

 

      The Zapotecs descend from a powerful ethnic group whose members ruled Southern Mexico from about 500 BC to 750 AD, leaving behind archeological sites like Monte Alb‡n and a hieroglyphic writing system that has not yet been deciphered. Their ancient language developed over the centuries into over fifty modern languages, all still known as "Zapotec" (Grimes 2002). Our project participants will be speakers of one of these languages, which we refer to as Valley Zapotec (Ethnologue Code ZAB, Grimes 2002). People in the Valley Zapotec region speak a variety of closely related dialects that are strongly differentiated from each other but still mutually intelligible. After 400 years of contact with Spanish, Valley Zapotec contains many Spanish loanwords. This is similar to the situation in English, which contains many words borrowed from other languages. As with the Spanish loans in Zapotec, many English loan words Ñ including such borrowings from French as beef, chair, music, and army Ñ are often unrecognized as loanwords by native speakers, and are used along with the core vocabulary.

 

      Our project will help participants view their native language not as a barrier to social mobility, but as a vehicle to enhance it, through discussion and questioning of technical concepts and vocabulary. Each group's own language provides the best possible record of its heritage and culture; when an ethnic group loses the native speakers of its language, some of its culture dies too. The astronomical expertise and many other accomplishments of classical Zapotec civilization are now forgotten, but the modern Zapotec languages remain the natural expression of the modern culture of this people. The opportunity to use Valley Zapotec for technical discussions will reinforce the usefulness and vitality of their language, and may help to ensure that the language is passed on to further generations.

 

Conversations and technical vocabulary. The basis of our research will be practical explorations of the underlying ideas behind mathematical formalisms. (As one mathematician put it ÒMathematics is a talky subjectÓ.) Interactive sessions at UCLA will bring our team of research assistants into an understanding of those concepts that are basic to workers who use information technology. We will establish a working mode that involves monthly, bimonthly, and weekly meetings with small groups of native speakers of Zapotec from the West Side of Los Angeles (Figure 3). The meetings will be held at UCLA or (more likely) in community rooms at shopping centers. Participants will be recruited from Oaxacan community organizations through previously established contacts initiated by research team member Felipe H, Lopez (a native speaker of Valley Zapotec).

 

      Each meeting will include one or more members of the project research staff. Sessions at the meeting will be predominantly in Zapotec, with only brief facilitation in English or Spanish. A meeting may begin with a research team member speaking for less than ten minutes on some computer-related aspect of mathematics. Next a volunteer participant will describe his or her understanding of that talk, as much as possible in Zapotec. The group will then discuss the concepts in Zapotec and seek a common agreed upon question to ask of the UCLA staff. The resulting sequence of give and take, question and answer will lead not only to the acquisition of knowledge, but to development of a vocabulary with which to discuss concepts like exponent, factorial, memory, storage, and links/pointers-locators, that are relevant to working with computers or to current science and engineering. Some sessions will also include consideration of elements of traditional Zapotec culture that reveal mathematical thinking and logical reasoning (see Ascher 1991). Sessions will be recorded (with the participants' permission) to aid in later retrieval of key terms and vocabulary. Items retrieved from recordings will initiate later sessions. They will also motivate development of additional web-based material.

 

      Eventually, the approaches we develop for exploring mathematics and computing through language will be used in the field. In later stages of the project we will visit villages in Mexico where Zapotec is spoken using materials and extending experiences begun in Los Angeles.

 

      The research process will result in a new Zapotec dictionary of the language developed to talk about, and work effectively with, computers. Some of the words we will start with are (in English) storage, memory, hierarchy, path, processing, data, and pointing or linking. The majority of words for introduced cultural items and concepts have been borrowed into Zapotec from Spanish, and we expect that many terms for technical concepts will be Spanish (or perhaps English) words adapted to Zapotec pronunciation. But core mathematical and computational concepts will often be expressed with existing native vocabulary, as speakers learn new senses for familiar words, just as English speakers have done with the examples just given. Since composing the definitions follows selection of the terms to include, this is a more general research issue and transcends the Zapotec-English relationship to encompass many other groups underrepresented in mathematics, including women, African-Americans, and Chicanos.

 

From specific to general. Zapotec is the mother tongue of more non-Spanish-speakers in Mexico than any other group except those who speak Nahuatl and Mayan. Here in the United States this language is the basic speech of tens of thousands of immigrants to Los Angeles, all living within a short bus ride of the UCLA campus. The combination of economic forces, geography, and group cohesion make this subculture ideal for investigations of how to help minorities accomplish adaptation to technology.

 

      The research project will begin with the production of units on topics that deliver a window into the existing mathematical structures of mainstream educational institutions. We will develop materials suitable for artisans and nurturing individuals, with two goals:

1. To encourage steps toward economic growth; and,

2. To foster transmission of the language to future generations.

 

      Developing the language as a usable tool for the modern world supports two things:

1. Passing on the linguistic heritage along with the minority group culture; and,

2. Ensuring that the technical ideas and concepts become available to future generations.

 

      Achieving these goals will lead to economic participation in the information economy by present and future generations of this minority community

 

      The proximity to UCLA of a substantial and significant linguistic group and the issues the investigators seek to address enables obtaining answers to these questions:

1. What discussion-topics are successful for stimulating subsequent mathematical or computer learning (measured by books read, courses taken, computing skills acquired)?

2. What is the impact of new learner adults on their childrenÕs progress in science, engineering, mathematics education (measured by courses taken and degrees received)?

 

Detailed plan. The continued existence of a language depends on its assimilating the changes civilizations experience over time. By working with a non-national-language such as Valley Zapotec, and examining its adaptation to modern industrial civilization, we seek to develop general knowledge transferable to working with overlooked and important cultures in an era dominated by terrorism. The project acknowledges the central role of technology as a force causing change, and seeks to use that as a way to gain understanding of the associated fundamental concepts,  particularly mathematics.

 

In the first year both PIs will contribute to the research, Allen Klinger on the technology materials and organization, Pamela Munro in posing means for linguistic recording and decoding of new constructs. Prof. Munro will continue research during the nine-month academic year under the support of the University but will be able to devote sustained time to organizing, writing, and deriving research principles from data obtained during the two summer months. Prof. Klinger will provide administrative and research activities during the eleven months listed, at a half-time level of effort. Graduate students including the two listed individuals  Felipe Lopez and Olivia Martinez will be chosen to interact with Valley Zapotec speakers and Profs. Munro and Klinger (Lopez and Martinez speak Valley Zapotec: it is LopezÕ native language), in the latter case frequently in order to stimulate means to initiate discussions that include mathematical and computer content. Each student will be involved in weekly meetings with Valley Zapotec speakers, with ten sessions required over a calendar year. Community college participants will join the effort at mid-year.

 

The second  year will repeat the work described in the preceding paragraph and will add organization and execution of two added things. The first is a national review of experiences derived from discussion questions. (A preliminary effort to be expanded in the third year.) The second involves extension of this work to locations in Mexico where the language is spoken. There will be two student researchers at a six-week on-site version of the West Los Angeles discussion sessions.

 

A third year will be devoted to producing two written items. The first is a Valley Zapotec technical dictionary. The second is a set of mathematical issues distilled from questions that were outcomes of discussion sessions with Valley Zapotec speakers.

 

Throughout all three years the Principal Investigators will work together and with student researchers to describe the activity and disseminate the insights gained to appropriate journals and conferences, including but not limited to Institute of Electrical and Electronics Engineers (IEEE) and its Computer Society (CS), American Society for Engineering Education (ASEE),  Association for Computing Machinery,(ACM), The Association for Computational Linguistics (ACL). Frontiers in Education (IEEE CS, ASEE and ACM sponsored), the journal  Computational Linguistics published by The MIT Press for ACL.

 

Travel funds will be used to bring researchers into the Valley in Mexico where Zapotec is spoken by many West Los Angeles immigrants. They will also be used to assemble a core group of technologists interested in improving access to Mathematics for a workshop meeting at UCLA.

 

Deliverable items and research conclusions. We will report on researchersÕ experiences in:

1.     Organizing expositions;

2.     Deriving understanding about conceptual structures in the non-English language used in conversations; and,

3.     Designing the technical vocabulary product (dictionary).

 

We will report on the aspects of the research that support extending this process to other minority cultures (both within the United States and in other countries). We will report on the aspects of computer and communications technology that require adaptation of the contemporary mathematics curriculum.

 

Prior NSF Support (Munro)

 

a. NSF award number, amount, and period of support

National Science Foundation grant 97-09415 to Pamela Munro,  $190,259 (including $5000 REU supplement, summer 1998), July 1997-December 1999.

 

b. the title of the project

San Lucas Quiavin’ Zapotec: Dictionary, Grammar, and Texts

 

c. summary of the results of the completed work, including, for a research project, any contribution to the development of human resources in science and engineering

            This grant supported a study of the Zapotec language spoken in San Lucas Quiavin’, Oaxaca, Mexico, and by numerous immigrants to the Los Angeles area. The project produced an extensive trilingual dictionary of San Lucas Quiavin’ Zapotec (Zapotec-English-Spanish), as well as theoretical accounts of aspects of the language's syntax, morphology and phonetics, supplemented by a text collection (final publication of which is still in progress; for the other resultant publications see d below). Development of this material provided an important resource for comparativists (especially those interested in the complex relationships among the different Zapotecan languages) and for the San Lucas Quiavin’ Zapotec people, whose language was not previously written. The procedures developed during the dictionary project and the analysis of the language's grammar have contributed more generally to broader literature on dictionary making and other topics in theoretical linguistics. The grant also supported completion of two dissertations (Galant 1998, Lee 1999) and one thesis (MŽndez 2000), listed in the References; another dissertation whose research benefited from the grant is in progress.

 

d. publications resulting the NSF award or drawing on its results (in reverse chronological order; see References for complete citations for starred entries)

Munro, Pamela. In press (2002-03). "Dictionary Entries for Verbs". In William J. Frawley, Kenneth C. Hill, and Pamela Munro, eds. Making Dictionaries: Preserving Indigenous Languages of the Americas. Berkeley Ð London: University of California Press.

Munro, Pamela. In press (2002). "Hierarchical Pronouns in Discourse: Third Person Pronouns in San Lucas Quiavin’ Zapotec Narratives". Southwest Journal of Linguistics.

Munro, Pamela. In press (2002). "'Peculiar to Themselves': Idioms in the Dictionary". Proceedings of the 28th Annual Meeting of the Berkeley Linguistics Society. Julie Larson and Mary Paster, eds. Berkeley: Berkeley Linguistics Society.

Munro, Pamela. 2000. "Field Linguistics." In The Handbook of Linguistics, ed. Mark Aronoff and Janie Rees-Miller. Malden, MA, and Oxford: Blackwell Publishers Inc., pp. 130-149.

*Lopez, Felipe H., and Pamela Munro. 1999.

*Munro, Pamela, and Felipe H. Lopez, et al. 1999.

 

e. brief description of the available data, samples, physical collections, and other related research products not described elsewhere

            Data recorded during the project is preserved in notebooks (following standard linguistic procedure) which are retained by the P.I. Analysis of this material is ongoing.

 

f. if the proposal is for renewed support, a description of the relation of the completed work to the proposed work

            The current proposal is not directly related to this one.

 

 

Conclusion. The research process we will describe here has a number of related but distinct goals.

 

      Languages like Valley Zapotec are often treated as cultural relics, by scholars and even by their speakers. Our research will demonstrate that such minority languages can be a viable medium for conversations about mathematics and computing, and will result in a new dictionary of terms for this enterprise.

 

      The project will result in the development of new materials and techniques for leading individuals speaking a minority language toward economic independence through increasing their ability to work with computer technology. Such abilities will be developed through these speakers' participation as members of a group conversing about new concepts in their own native language. The work will thus develop individualsÕ greater sense of the value of higher education while aiding a disadvantaged ethnic and linguistic minority to enter the modern world of information technology.

 

      We will also be able to observe how well this approach to mathematical, scientific, and computing education works. An advantage of our approach is that the materials we develop during the grant period can be easily adapted for use with Ð and by Ð members of other disadvantaged groups.

 

 

 

 

 

Figure 2. Demonstrating Probability Ð Easily-Obtained Platonic Solids Used For Games

 

 

 

 

 

Figure 3. Looking at Shape Ð Images Led to Using an Unconventional Solid Shape

 


Maps (Figures 4 and 5)

 

 



Figure 4.  The Northwest Tlacolula District, Oaxaca State, Mexico.

The state capital, Oaxaca City, is to the northwest. (From Garc’a Garc’a et al. [1995].)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Created with The GIMP

 

 


           

Figure 5. The West Side of Los Angeles,

home to most Valley Zapotec speaking immigrants to California.

The star marks UCLA. (From http://maps.yahoo.com.)