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Bringing Native American Perspectives to Mathematics and Science Teaching Author(s): Sharon Nelson-Barber and Elise Trumbull Estrin Source: Theory Into Practice, Vol. 34, No. 3, Culturally Relevant Teaching (Summer, 1995), pp.

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Sharon Nelson-Barber Elise Trumbull Estrin

Bringing Native American Perspectives to Mathematics and Science Teaching

EW WOULD QUARREL with the assertion that mathematics and science instruction for American

students needs to improve. Much of the public rhetoric agitating for reform centers around the allegation that such reform is necessary to ensure the economic security of the United States. That contention has been disputed by those who point out that there are not even enough jobs for the current crop of college educated students (Bracey, 1994; Rotberg, 1994). But setting this dispute aside, it is clear that students from certain ethnic groups are systematically under- represented in careers that require high-level mathematics and scientific knowledge. This is the case for American Indian students, perhaps more than any other group.

Despite a rich foundation of cultural values and experiences that could serve as the basis for success in mathematics and science courses that would prepare them for scientific and technical careers, Amer- ican Indian students do not successfully complete such courses. In fact, as a group, they do not successfully complete high school: In some communities, the dropout rate is greater than 60 percent.

Why should all of this be true? We argue that American Indian students have had inadequate opportunities for success in the existing educational system, particularly in the areas of mathematics and

Sharon Nelson-Barber and Elise Trumbull Estrin are senior research associates at the Far West Laboratory for Educational Research and Development, San Francisco.

science, because of the ways mathematics and science are typically taught. In this article, we identify some of the sources of the problem and make sug- gestions for improving mathematics and science instruction for American Indian students.

Many American Indian students have extensive knowledge of mathematics and science-knowledge that is rooted in naturalist traditions common to Native communities and arrived at through observation and direct experience. Because many Indian communities follow traditional subsistence lifestyles, parents routinely expose their offspring to survival routines, often im- mersing the children in decision-making situations in which they must interpret new experiences in light of previous ones (Kawagley, 1990; Pomeroy, 1988). Unfortunately, a majority of teachers recognize neither Indian students' knowledge nor their considerable learning strategies.

Thus, not only is potentially important content knowledge ignored but well-developed ways of knowing, learning, and problem solving also go un- recognized. The result is that critical opportunities to build on or draw from Indian students' existing knowledge are missed. At the same time, Indian students may feel both confused by classroom approaches to mathematics and science that are not grounded in experience, and denigrated by a system that appears to assume they know nothing about these realms.

As national, state, and local standards of instructional content and student performance continue

THEORY INTO PRACTICE, Volume 34, Number 3, Summer 1995 Copyright 1995 College of Education, The Ohio State University 0040-5841/95$1.25



Nelson-Barber and Estrin Native American Perspectives

to be developed, it will be crucially important for educators and parents from American Indian communities (as from other communities) to be involved in deciding how those standards can best be applied and assessed in their contexts. The usual approach of setting change in motion for the "mainstream" student and then considering how to adapt it to the needs of "non-mainstream" students cannot be expected to result in the success of those who are an afterthought.

Mathematics and Science Education Reforms On the whole, the reform efforts of groups such

as the National Council of Teachers of Mathematics (NCTM) and the National Committee on Science Education Standards and Assessment (NCSESA) represent a move toward a potentially improved pedagogy for all students. For example, the new philo- sophic thrust of mathematics education is toward encouraging students to explore math concepts and solve problems related to their everyday experiences and then build symbolic and axiomatic knowledge on that foundation (NCTM, 1989). Similarly, new science frameworks define understanding in terms of concepts rather than specific skills or knowledge of facts and formulas (NCSESA, 1994).

Methodologies such as thematic instruction provide for the integration of key concepts and move students into a more active mode of learning well beyond the passive memorization of facts. In the current vision of science learning, students are active, exploratory, hands-on learners who take a strong role in constructing their own knowledge.

The theory of learning reflected in the mathematics and science standards and associated reforms is "constructivist." That is, learning is viewed as something that takes place as a result of an interaction between what students bring to a task or setting and what they encounter (cf. Cobb, 1994). They "construct" knowledge, based on what they already know, what they are motivated to learn about, and how new experience and information are presented to them. The notion of a one-way transfer of knowledge from teacher to child is antithetical to the constructivist conceptualization of learning. Within a constructivist framework, the child's personal experience and knowledge base become keys to instruction and to understanding how children construe new information and experiences.

The issues involved are developmental as well as cross-cultural. Children enter school with intuitive understanding of many mathematical and scientific ideas. Part of the task of schooling is to help learners frame their intuitive knowledge more formally and in conventional symbolic terms. Kieren (1992) has characterized intuitive mathematics knowledge as "cores for technical symbolic mathematics" (p. 353). "This implies that everyday life or intuitive actions are not simply used to illustrate or provide a context for episodal memory of symbolic algorithms to be learned. Mathematics in its own right exists at these levels and can serve as a powerful base for more formal knowing" (Kieren, 1992, p. 353).

Children need to be able to connect formal symbolic representations to real objects, actions, and experiences-mapping back and forth between the real world and the representation of it (Kieren, 1992), constructing their own understandings rather than just learning procedures. Of course, these intuitive understandings are grounded in particular cultural experiences.

For many teachers, these new standards, methodologies, and ways of understanding student learning will require significant shifts in thinking along with upgrading of pedagogical skills. And, if teachers are to be supported to apply these new understandings successfully with all students, they will need opportunities to develop cross-cultural competencies as well. American Indian ways of teaching, such as modeling and providing for long periods of observation and practice by children, are quite harmonious with constructivist notions of learning. "Mainstream" teachers can look to their American Indian colleagues for such examples of culturally-responsive practice as well as insights into how to interpret student performance on classroom tasks.

Assumptions About Mathematics and Science Although both the NCTM and the draft science

standards stress the importance of students' linking concepts to their existing knowledge and experience, a philosophical, cultural, and historical framework that would put these standards in perspective is missing. Short of a brief reference to recognizing cultural diversity with the goal of ensuring that all students participate fully in science learning, the proposed science standards refer only to "the methods, attitudes, and habits of mind of scientific inquiry" (NCS- ESA, 1994, p. 6). Assumptions about the universality

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THEORY INTO PRACTICE / Summer 1995 Culturally Relevant Teaching

of these "methods, attitudes, and habits of mind" are not questioned. Were we to examine the relevance of the standards to a Navajo science curriculum and pedagogy, for example, we might discover both conflict and complementarity.

The current reforms represent a "Western"' approach to mathematical thinking, which is situated in a world-view that posits discrete entities, quantities, and states. Natural phenomena are often ana- lyzed out of context, "objectivized," and scientific "facts" are thought to be value-free, determinate through rigorous observation and/or experimentation. In addition, European Americans, and perhaps the academic culture in general, tend to view societies, science, mathematics, and the like as moving inexo- rably toward more complex ("advanced" or "better") states. What is to come will be better than what went before. Time is sequential, linear.

All of these features of Western thought run counter to the ways of knowing of many Indian cultures, where interrelationships, flux, observation and evaluation in context, and a more circular view of time prevail. Moreover, "progress" may be conceived of more in spiritual and ethical terms than in terms of decontextualized knowledge.

The Navajo way of viewing reality, for example, focuses on processes and phenomena in flux. "Among the Navajo, where the focus is on process, change is ever-present; interrelationship and motion are of primary significance. These incorporate and subsume space and time" (Ascher, 1991, p. 129). Space is not conceived of as separate from time and motion. Spatial boundaries are not independent of the processes of which they are a part, so that seg- menting space in an "arbitrary and static way" without accounting for flux over time (Ascher, 1991, p. 129) is senseless.

In addition, a Navajo approach to science entails elements that are missing from the current mathematics and science standards, that is, the ethical and historical dimensions that situate science knowledge in a context. A more complete approach to science instruction would include these dimensions, promot- ing an inquiring stance toward "science" itself. Ques- tions such as, "What knowledge is important to the survival of our society, our earth?" "To what use will knowledge be put?" "What might be the effects of using knowledge in this way?" would arise from taking ethical and historical standpoints.

The Western approach serves to obscure important social information in technical facts that purport to be value free. Perhaps proposed content standards having to do with "science and society" are meant to address these dimensions, and, at least hypothetically, standards relating to "scientific inquiry" and "scientific explanations" could entail a philosophy of science orientation. Still, a more explicitly cross-cultural perspective would better equip us to recognize the values embedded in Western science and to identify the information required to understand alternatives (Levidow, 1988).

Culture-Based Variations in Ways of Knowing All cultures generate mathematical and scien-

tific knowledge, but that knowledge need not look the same from one group to another. In fact, mathematics and science may be understood as kinds of cultural knowledge or cultural products rather than objective bodies of truth (Ascher, 1991; Bishop, 1991). With regard to mathematics, looking cross- culturally one sees that mathematical ideas are con- cerned with space, number, and logical relationships and with systems for organizing these elements. All cultures exhibit the inclination to impose order on the natural world-to create categories and classifi- cation systems to organize daily life in these terms. However, the ways in which any culture deals with mathematical ideas are dictated by local needs, such as the need to organize social and kinship systems or to devise effective approaches to navigation.

Mathematical ideas and culture are inseparable. Moreover, mathematical ideas and systems cannot be compared along any single linear, developmental scale-contrary to the way Western texts on mathematics have tended to represent "progress" in mathematical thinking (Ascher, 1991).

Cultures' approaches to problem solving and the ways in which they differentiate, classify, and interpret the natural phenomena they experience or observe are variable. Learning styles are also affect- ed by one's environment and closely linked to the demands of daily life. For example, "discovery learning" may be discouraged in an environment fraught with physical dangers (as is the case for many traditionally-raised Indian children) (Lancy, 1983). Chil- dren from such an environment may not immediately respond well to certain contemporary teaching/learning methodologies (Green, 1978).

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Obedience to parental constraints may be obliga- tory in a socially dangerous environment, as Gadsden (1994) has suggested with regard to African American children who live in a racist society where minori- ties cannot expect equal protection under the law. In fact, student competence might be said to depend upon the match between the demands of a task, the context in which it is embedded, the instructional approach, and the culturally-developed skills of the learner (Gallagher, 1993).

Unfortunately, systems of mathematics and science that deviate from the Western mode are often seen as incomplete or inferior rather than in terms of their adequacy to serve the purposes of the group that developed them. As suggested earlier, there is no necessary developmental path that mathematical ideas must take in every culture. And it is not possi- ble to construct a hierarchy of conceptual structures or strategies: Certain numeration systems do not nec- essarily come before others, and the Navajo concept of space-time is neither inferior nor superior to the Western concept of space and time as separate dimensions (Ascher, 1991).

American Indian philosopher Vine Deloria (1992) notes that dominant-culture interpretations of American Indian science do not recognize the "whole basis on which tribal peoples acquired their insights" (p. 17). For example, in tribal science, the observer is not separate from the observed; and Western science, itself, is slowly coming to recognize such separation as impossible. Another contrast is the way in which tribal peoples have tended to refrain from abstract- ing broad generalizations from experience and then using them to explain new experiences. Instead, they have understood new information and experience in terms of the full tapestry of continuous experience and contextualized knowledge. Instead of matching generalizations with new phenomena, tribal people match their more specific body of information with the immediate event or experience.

Children can be underestimated in the classroom when educators who are unfamiliar with vary- ing socialization patterns judge experiences that con- tribute to cognitive development against norms of interaction that do not reflect the norms of the communities to which children belong. Children learn not only the language and lore of their cultures, they also learn their cultures' theories of language and cognition (Lancy, 1983). They learn what knowledge

is important for what purposes and when and how it is acquired and displayed. Therefore, an understanding of children's classroom learning and participa- tion requires an understanding of community values and norms.

In an account that demonstrates the subjective, culturally-determined nature of "abstract" categories, Frankenstein and Powell (1994) retell the story of researchers working with a group of people whom "academic anthropologists" would label "primitive." The researchers presented five objects from each of four categories-food, clothing, tools, and cooking utensils-and asked the participants to sort them. Most of the respondents arranged items in groups of two, explaining practical connections such as, "The knife goes with the orange, because it cuts it" (p. 83). As cited in Rose (1988), "[the people] at times volunteered 'that a wise man would do things in the way this was done.' When an exasperated experi- menter asked finally, 'How would a fool do it?' he was given back groupings of the type . . . initially expected-four neat piles with foods in one, tools in another, etc." (p. 291).

The experience mentioned above is a reminder that when assessment of cognition or achievement does not take into account the ways of knowing of particular groups, it cannot pretend to have valid outcomes. New forms of assessment, such as portfo- lios and performance tasks, may turn out to be more equitable, if-and only if-they allow the flexibility and contextualization necessary to capture the meaning of student performances.

Differences between American Indian children and European American children have been noted and sometimes misunderstood, resulting in stereo- typing at times. However, according to many ac- counts, Indian children do tend more toward a "watch-then do," or "listen-then do" strategy than the "trial and error" approach favored in Western schools (cf. Green 1978; John-Steiner & Ostereich, 1975; More, 1989; Swisher & Deyhle, 1989). Ac- cording to a young Keres man cited by Tafoya (1989), "You don't ask questions when you grow up. You watch and listen and wait, and the answer will come to you. It's yours then, not like learning in school" (p. 32).

In some cultures, such as Navajo and Chipewyan, adults do not pay overdue attention to children or cor- rect them, because the adult would need to continually

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Culturally Relevant Teaching

monitor their obedience. By ignoring children's be- havior, adults protect their own authority (Scollon & Scollon, 1983). Indirect teaching occurs in other ways as well. For example, when it is time to learn certain things, a young boy may be taken out into the coun- tryside and left all day to learn for himself about the land and water-reflecting an approach of guiding, facilitating, and setting up a circumstance through which the child can learn. (Compare this with new

conceptualizations of teachers in constructivist class- rooms.)

In considering what would constitute a curriculum and an approach to instruction that is valid for

any cultural group, we must first consider the cus- tomary ways of knowing and acquiring knowledge of that group. We are faced with essential epistemological questions such as, What counts as important knowledge or knowing? How is that knowledge represented in language or other forms (visual models)? What counts as evidence for claiming something to be true? How and when should knowledge or understanding be expressed or shared? As we have noted, a blanket approach to teaching students is not likely to succeed.

When essentially different approaches to knowledge come in contact with one another, they may conflict or collide, or they may complement each other. What happens is partially dependent on how children are enculturated to new ways of thinking (as in school). For example, Hopi biologist Frank Dukepoo (1993) asserts: "Elements of cultural knowledge do not create barriers in the study of science; but rather can enhance, excite and stimulate scientific curiosity and inquiry. . . . [Teaching] can engender cultural appreciation, understanding, respect and acceptance" (p. 4).

In some ways, this case is similar to the experience of speaking more than one language: Learning a second language gives perspective on the first, ex- pands one's notion of language, and makes for greater cognitive flexibility, if one is supported to maintain the first language and if respect is accorded to that language (cf. Hakuta, 1990). As asserted by Ascher (1991), "whenever we increase our understanding of other cultures, we increase understanding of our own by seeing what is or is not distinctive about us and by shedding more light on assumptions that we make which could, in fact, be otherwise" (p. 186).

Bridging Cultures in the Classroom It appears that many elements must coalesce if

mathematics and science education are to become meaningful to American Indian students. If meaning is made through connections to "personal models of reality" (Kieren, 1992), then ways must be found to help American Indian students forge those connections. Using ethnomathematics and ethnoscience in instruction is one way to connect with students' home and community experiences. Ethnomathematics has been defined as forms of mathematics embedded in cultural activities, forms that vary from culture to culture (Nunes, 1992). Likewise, ethnoscience can be conceptualized as forms of scientific and techno-

logical activities in their social and cultural contexts. "Culture" in the sense used here is not restrict-

ed to ethnic groups. Rather, it is meant to include workplace and social groups as well. In real life (in contrast to school), mathematics and science are not used for their own sake but to carry out everyday tasks and solve everyday problems (Nunes, 1992). Builders attempting a new design, for example, may not "mathematize" their work by capturing it in formal mathematical terms, but they are using mathematics.

It is often the case with ethnomathematics and ethnoscience practices that they are not recorded, formalized, and passed on beyond the context of their immediate usefulness. For this reason, "the chain of historical development, which is the spine of a body of knowledge structured as a discipline, is not rec- ognizable. Consequently, ethnomathematics is not recognized as a structured body of knowledge, but rather as a set of ad hoc practices" (D'Ambrosio, 1991, p. 22). However, as Borba (1990) has observed, the practices developed by different groups are likely to be more efficient for solving problems related to their cultures than are academic mathematical practices, because the solutions are tailored to the particular obstacles encountered by that group.

For example, the nature of a map that will be useful to a traveler going by foot through Arctic territory, in which physical features (including trail contour and visibility) change frequently because of weather effects, will be quite different from one useful to a traveler going by car on highways through the Midwest. The mapmaker in the first case will need to select environmental benchmarks that resist destruction or disguise by weather.

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Some educators look directly to ethnomathematics and ethnoscience to serve as the sources of real-world connections that will make classroom theories and procedures meaningful (Bishop, 1988; Fran- kenstein, 1990; National Council of Teachers of Mathematics, 1989; Nunes, 1992; Wheeler, 1993). Others, while not mentioning ethnomathematics or ethnoscience per se, suggest classroom mathematical or scientific investigations that build in some

way on students' knowledge and lived experiences (Mathematical Sciences Education Board, 1989; Ovando, 1992).

As Ovando (1992) has said, "All children bring with them to school a base for scientific knowledge, skills, and experiences. And this base can be related to the school's curriculum" (p. 223). Using ethnomathmatics and ethnoscience activities from students' communities, teachers can help students see common conceptual underpinnings to everyday and school mathematics or to see where the mathematics and science of school diverge from their own everyday conceptions.

When ethnomathematics and ethnoscience are

transported from their natural environment into the classroom, they may lose some of their authenticity. That is, the immediate need to engage in such practices is lost; and an activity may seem contrived. In addition, understanding ethnomathmatics and ethnoscience practices in "classroom" terms may not be simple. For example, one may be able to identify the underlying conceptual similarity between adding the prices of goods in the marketplace mentally or oral- ly-where hundreds, tens, and ones maintain their value-and adding a column of figures, where "carry the one" can refer to any place value. However, it

may be difficult for students to make the leap from one realm to the other.

Ethnomathematics, classroom mathematics, and the mathematics of academic mathematicians have been associated not only with different institutions or settings but also with different functions. Aca- demic mathematicians (usually at universities) are expected to produce mathematical knowledge; schools are supposed to reproduce and transmit the knowledge; and people in the workplace and in everyday life are expected to use this mathematical knowledge as a tool to solve problems. This neat and tidy division of domains is open to question. In fact, mathematical understanding is spontaneously

created across many contexts. But such understand-

ing is apparently very context-bound. Lave (1988) has shown that workers often cannot

consistently apply their ethnomathematical practices out of the familiar contexts in which they customar-

ily use them. Moreover, many workers, interviewed about the mathematics required for their jobs, fail to see the mathematics in what they do (whether it is

measuring lengths of carpet, placing articles in tes- sellated patterns, or weighing fluids) (Dowling, 1991). Often, people are operating with what Dowling (1991) has called "frozen mathematics" (p. 96)- automatized, non-mathematized routines. Nor is mathematics of the classroom automatically carried into the real world of daily life. It is as though there are semi-permeable membranes among everyday, classroom, and academic realms of mathematics.

One way of thinking about the "elements of mathematical knowledge building" has been suggested by Kieren (1992). Figure 1 shows ethnomathematical knowledge and intuitive knowledge at the core of a set of elements, with technical symbolic and axiomatic deductive knowledge expanding around that core.

Figure 1. Elements of mathematical knowledge building (adapted from Kieren, 1992, p. 352; reprint- ed with permission).

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Examples of ethnomathematical knowledge have been given. Intuitive knowledge, as explained by Kieren (1992), arises from interactions with objects, people, and situations in the "real world" and entails imagery, premathematical thinking, and informal use of language (rather than formal mathematical lan-

guage). For example, intuitive knowledge of what students will come to understand more formally as "fractions" may first take the form of ability to par- tition quantities equally among friends (e.g., sharing cookies, marbles) and to talk in simpler mathematical terms about how quantities were divided up (e.g., "I gave each person one cookie, and then I gave each one another cookie until I used them all up.").

Intuitive knowledge can be built upon in the same ways as students' ethnomathematical knowledge, which may have been learned in the context of everyday experience through cultural activities such as cooking, rug weaving, or budgeting family funds. The implication of Figure 1 is that ethnomathematical and intuitive knowledge are not simply underdeveloped versions of "true mathematics" but that they are essential elements of mathematical knowledge around which other, more formal knowledge is developed.

Figure 2 shows an instructional sequence that would be compatible with the notion of building from students' experiences and helping them recognize and talk about their own ethnomathematical and intuitive knowledge. This basis for mathematical understanding would be used to introduce students to more formal symbolic representation and mathematical procedures in the classroom. Finally students would

Start here 3 Students' lived experiences

Axiomatic knowledge

Ethnomathe- matical knowledge .and intuitive understanding

Technical symbolic representation

Figure 2. Proposed instructional sequence.

learn how to express mathematical relationships in the most formal terms, i.e., in the form of axioms. (There are obvious parallels to science-with ethnoscience knowledge, intuitive knowledge, technical symbolic and procedural scientific knowledge, and abstract, theoretical knowledge serving as somewhat distinct realms.)

One can infer from cognitive development research that starting with experience and working toward understanding of formal concepts and procedures may be more appropriate than the reverse for many students. In other words, rather than introducing a new concept by asking students to read about the topic, a teacher might first ask students to describe their own experiences, observations, and interactions with the topic (Pomeroy, 1988). American Indian students, for example, are often knowledgeable about science as it relates to preserving and maintaining the environment- water pollution, toxic waste, agricultural and industrial chemicals, and wise use of technology (Preston, 1991, p. 50). Such knowledge should be drawn upon in the classroom.

Because ethnomathematics and ethnoscience offer the potential for an active classroom role for each student-by virtue of being more familiar to the students than to the teachers, in most cases, and of requiring the students' interpretation-they are an attractive route to acquisition of "school" mathematics and science. We must bear in mind, however, that apparent connections across the "membranes" of these different realms of theory and practice may not be evident to students. Teachers might want to ask students to talk about the connections they see and share how they would go about solving a particular kind of daily problem using mathematics they have learned in school and to contrast that process to one they would use intuitively. The point would not be to invalidate an ethnomathematical or ethnoscientific solution but to explore the pros and cons of such a solution versus a "classroom" solution. The following activities incorporate elements of ethnomathematics and ethnoscience, but they require school interpretations for activities whose procedures might otherwise be demathematized.

Ethnomathematics Activities Any activity should be chosen on the grounds

of meaningfulness to the students and utility for ad- dressing curricular objectives. One problem with

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books of such activities is the risk that they will be used in a way that is disconnected from "regular" instruction. That said, let us look at a couple of activities that illustrate both decontextualized and con- nected ways of applying ethnomathematics.

Designing and/or interpreting rug patterns are frequently-used ethnomathematical activities in class- rooms attended by American Indian students. An example related to Navajo culture can be found in Addison-Wesley's Multiculturalism in Mathematics, Science, and Technology: Readings and Activities (1993). In "Spider Woman Weaves a Rug," students read background information on Navajo weaving and answer questions about weavers' practices (including questions for critical thinking about mathematical principles involved). Next, they use colored pens and paper to design a rug, following instructions that will produce particular geometric patterns. Instruc- tions read:

In the following activity you will make your own burnt- water design. Begin with an 18" by 24" sheet of paper or newsprint. (p. 129)

Explicit directions for folding paper and using card- board "steps" to create a pattern follow, e.g.:

Make a vertical fold to divide your paper in half. With the paper folded, make a horizontal fold to divide the paper in half again. Open the folded paper and lay it flat on the table. Use a blue pen or marker to draw your vertical and horizontal axes by tracing the fold lines. (p. 129)

Symmetry and number patterns are explored in follow-up questions.

Compare the activity above to a somewhat similar activity-intended as an assessment task-"Designing and Weaving a Rug," developed by fifth grade teachers in the Chinle Public Schools on the Navajo Reservation in northeast Arizona (Koelsch, 1994). This exercise also requires students to design a Navajo rug; however, because it was devised for Chinle students by Chinle teachers, with reference to culturally congruent curriculum standards, it has a more contextualized char- acter. Instructions read:

You are spending the summer in the canyon with grandmother. In order to earn some money for new school clothes, grandmother has offered to weave a rug for you to sell, if you will create the design. 1. Design your rug in the grid below. [Depiction of graph paper-like grid "rug" hanging from Navajo loom.]

2. Grandmother says it will take her five minutes to weave each row. How many minutes will it take her to weave your rug? Count the number of rows and multiply by five. Show your work below. 3. A tourist to the canyon came through on a jeep tour and stopped to watch grandmother weave. She was so impressed with grandmother's weaving that she asked grandmother to weave her a rug using her favorite pattern. Use the pattern below to design a rug. Use only this pattern in as many ways as you want. [A simple pattern involving 4 squares is shown.] 4. Color in the tourist's rug using two or three colors.

The task has three more parts that call on the student to make further calculations based on the number of colors used and grandmother's daily available time.

The striking feature of the Chinle activity is its strong connection to local context, which could be expected to make the task more relevant and more inviting or motivating. As research suggests (cf. Lipka, 1990; Scollon & Scollon, 1983), the strong contextualization allows students to "enter" the task more effec- tively as, one hopes, they readily envision the situation presented. It also provides an opportunity to bridge the differences that often arise in cross-cultural contexts.

Ordinarily this grandmother would probably not calculate formally how long she would need to weave the rugs. In keeping with phenomena noted by Sapir (1957) and Whorf (1956), our experience tells us that "time" can be construed quite differently in Na- tive and non-Native contexts. For instance, on one occasion one of us observed a tourist asking a traditional Navajo grandmother, "How long have you been a potter?" At first the woman replied that she did not understand the question. However, after the tourist persisted asking many more questions such as, "May- be 15 years, 20 years?" she finally replied, "I have always been a potter."

To this grandmother, and others socialized in traditional ways, the notion of using "time" to bound one's vocation seemed a puzzlement. But in the Western classroom, there is rationale for wanting to know the time required; such skills are useful beyond the traditions of one's community when viewed as part of a larger repertoire of mathematics and science abilities. Also, both the child and the tourist might be eager to know when their rugs would be ready. One can readily see the potential this activity provides for exploring such differences by combin- ing cultural/ethnomathematical aspects with "school mathematics" procedures.

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On another level, one that challenges values associated with the use of technology and mathematical and scientific information to bring about "progress," Chinle teachers have asked students to take on controversial and difficult environmental issues such as the mining of minerals on the reservation or the deforestation of Navajo lands. Without prescribing what students should think, they have asked them to use high-level language and literacy skills to develop their understanding of complex scientific, social, economic, political, and cultural issues and to make decisions that would affect the environment and the financial status of their communities.

The task is a lengthy one: Students collect information from many kinds of sources on a subtopic that interests them. They do library research, interview tribal members (including elders), prepare a written report, and, if they wish, design a presenta- tion for either live or video performance. This activity has the potential to include ethnomathematical and ethnoscientific content, but, again, students' products are meant to reflect school curriculum content

aligned both with cultural values and with the Ari- zona curriculum frameworks. Processes of inquiry can be a hybrid of more culture-based and school- based approaches.

For example, interviewing elders about the background of the deforestation issue and about their beliefs and values regarding the land would have to take a culturally-accepted form. Students would likely have to translate the kind of data they gather into a form that fits the scientific inquiry approach of school. Still, by virtue of examining an issue that concerns their own community, they would not be using mathematics and science in a decontextualized or objectivized way as is so often the case in the Western instructional mode.

According to Benally (1988), even efforts that truly intend to recognize local beliefs and traditions remain essentially mainstream and are not consistent with American Indian realities. Benally explains that the Navajo educational focus is on preparing indi- viduals to reach a state of hozho-a state achieved through a balanced and harmonious life. All knowledge is viewed with respect to its ability to draw one closer to this spirit of harmony. "Traditionally," he writes, "the individual is taught the interrelationship and interdependence of all things and how we must

harmonize with them to maintain balance and har-

mony" (p. 10). By contrast, "Western organization of knowledge, with its fragmentation ... and lack of connectedness, does not promote hozho" (p. 12).

Faculty from the Navajo Community College- Shiprock campus have developed a framework that serves to reorient mainstream Western teaching and curricular practices to Navajo philosophy. Using the four directions as a cultural touchstone, they align their efforts to help students develop a balanced and harmonious life with four branches of knowledge: humanities and fine arts (knowledge associated with the east); professional and vocational studies (knowledge associated with the south); social sciences (knowledge associated with the west); and the natural sciences (knowledge associated with the north).

As an example, a course in the natural sciences taught in accordance with this philosophy would dem- onstrate the interconnectedness and interdependence of all things. In other words, it would place that science within the broader context of life values and show its relevance to making a living, to one's social well-being. It would also consider issues of particular impacts on the natural environment. The themes of knowledge and goals of learning woven throughout would be holistic and reflect the Navajo world-view.

Beyond Method and Content Bringing these non-school-based forms of know-

ing into the classroom entails not only content but ways of thinking about and solving problems. Sum- marizing a number of studies, Bartolome (1994) suggests that successful pedagogies that are culturally responsive may have succeeded not solely because of cultural content and processes but because they humanize the classroom and equalize relations between students and teacher and among different social groups of students. Perhaps when students are given choices, when their own experiences and ways of knowing and communicating are reflected in the classroom, the most powerful result is in the changed relations that ensue. Students can become "equal partners in their own learning" (p. 186).

A feature Bartolome sees as the most promis- ing is students' and teachers' joint negotiation of meaning in pursuit of a learning goal.

Teachers act as cultural mentors of sorts when they introduce students not only to the culture of the classroom,

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but to particular subjects and discourse styles as well. In the process, teachers assist the students in appropri- ating the skills (in an additive fashion) for themselves so as to enable them to behave as "insiders" in the particular subject or discipline. (p. 188)

Other researchers, among them McDermott (1977) and Nelson-Barber and Harrison (1995), as well as acclaimed math teacher Jaime Escalante, showcased in the motion picture Stand and Deliver (Warner Brothers, 1988), have shown a variety of methods to be effective when trust between teachers and students exists. In fact, Escalante asserts that even when students seem to be uninterested in education, it is the teacher who is responsible for giving students the desire to learn. In an interview, Escal- ante relates: "My skills are really to motivate these kids, to make them learn, to give them ganas-the desire to do something-to make them believe they can learn. . . . The teacher gives us the desire to learn, the desire to be Somebody" (Meek, 1989, p. 46).

Similarly, Dukepoo (1993) tells the story of a teacher who emphasized "culture" by requiring students to count and categorize "beads, buckskins and bolo ties," but did little to arouse their interest in mathematics. Clearly, Escalante and Dukepoo make an important point that goes beyond superficial elements of culture as an explanation for success.

Using mathematics and science content from students' lives will not compensate for lack of a real relationship between teacher and student. Reliance on method or content alone is an inadequate strategy. If teachers respect and value students, that stance will be communicated and likely have important pos- itive consequences-whether the curriculum and methodology parallel students' experience or not (A. Cook, personal communication, July 1994).

Conclusion Today's teachers are being called upon to

present modern mathematics and science not only in ways that result in learning particular content standards but also in ways that will actively engage our nation's increasingly diverse student body. There is a tremendous need for teachers who are able to en- courage American Indian students to become explorers and creatively develop their understanding of these subjects, as well as for teachers who can create connections among science, mathematics, technology, and social issues. In truth, many teachers simply do

not have the background or experience to take on these tasks, particularly in view of other needed skills: integrating math and science with other disciplines and with restructuring programs at the school site; replacing lectures and cookbook exercises with exercises that enlist all students; providing resources and techniques such as heterogeneous cooperative grouping (understanding that what counts as "coop- eration" is culturally defined); and developing skills to serve as mentors and leaders of educational reform.

We believe that the academic success rate of American Indian students in mathematics and science programs will continue to increase when more attention is paid to the unique perspectives of these students. In response to the call to "celebrate" diversity, we argue that this is a simple stance that risks trivializing diversity or prematurely attenuating an open-ended exploration of what it means in all its com- plexity. As long as we are all celebrating superficial ethnic differences, we avoid the threat of examining the assumptions that maintain existing inequities and limit our vision. Alternatively, the society-at-large stands to benefit from gaining a perspective on the ways in which values and practices based in mathematics, science, and technology affect human relations and the earth. The issues go beyond empowering single groups to opening up greater possibilities for the whole society.

Note This article is a shortened version of a monograph of the Native Education Initiative of the Regional Educational Laboratories (see Nelson-Barber & Estrin, 1995). 1. We put "Western" in quotation marks to indicate that what passes for Western European mathematics is in fact a result of historical influences from India, China, and Africa as well as Europe.

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