Volume 24 Number 3
July 1985

Duane E. Schindler and David M. Davison

Due to the international character of modern mathematics, its concepts may be transmitted, studied and developed in a variety of languages throughout the world. This development is as recent as this century. The language of modern mathematics is part of the continuum of technical language development that began in Europe in the seventeenth century. Modern mathematics also drew heavily on the ancient language stocks in Europe, Asia Minor, and North Africa and may be said to represent the cumulative technical language development of diverse peoples over thousands of years (Closs, 1977). Mathematics as a technical language is not, nor has it ever been, the exclusive domain of the English language.

Researchers Closs (1977) and Green (1978) concluded that historians of mathematics have concentrated on the mainstream development of mathematics and have largely ignored mathematics "in cultures not directly contributing to it" (Closs, 1977, p. 1). However, Gay and Cole (1967), and Closs, presented studies of mathematics in other cultures that show separate development of mathematics in the cultures. Both studies noted the technical language indices of separate development due to cultural uses and perceived need. Closs noted that there is no comprehensive study of pre-Columbian mathematical development in the American continents.

Research suggests that any classroom that may contain children whose dominant language is not English, needs to have a teacher who can analyze the dominant language of these children and create a second language mathematics descriptions that are meaningful based on their language's use of the mathematics concepts. Leap, et al. (1982), attributes some errors in American Indian students' mathematics problem-solving to use of American Indian language mathematics-based problem solving strategies and not to some inaccurate mastery of Western mathematical skills. An effective teaching methods class for teachers of children whose dominant language is not English should explore the cognitive relationships of the language in the development of mathematical concepts.

The review of the literature of mathematics learning among American Indians in higher education indicates that not only is there underrepresentation of American Indians employed in the field of mathematics, researchers are beginning to identify some causes of the underrepresentation.

According to Green (1978), factors related to the mathematics education of American Indians that must be considered include math avoidance, differences in perceived utility, distinguishing nuances of meaning in the English language, second language development, and mathematics vocabulary in the American Indian language. Thus, mathematics program development cannot strictly consist of remedial mathematics programs for American Indians, but must also address processing in the native language and cultural differences.

Green (1978) discounts the idea that American Indian students fear and avoid mathematics because they have experienced failure in a mathematics class or lack inherited skills. She found, rather, that many American Indian students dropped mathematics classes before failure occurred. A further observation relates to the lack of classroom use of historical material dealing with the mathematical contributions of American Indians. Green also points out that in addition to the omission of contributions of ancient American Indians from mathematics classes, the classes with American Indian students often saw mathematics as a form of punishment rather than a "career/life tool" or "an orderly, logical system which can bring pleasure in and of itself" (Green, 1978, p. 9).

A review of the literature highlights a problem for institutions of higher education charged with the preparation and in-service education of teachers of mathematics. To be able to expand a teacher education program effectively, faculty and teacher education students need to understand how mathematics concepts are learned in the language of selected American Indian tribes. For instance, the State of Montana contains seven American Indian Reservations representing at least ten different American Indian languages. Therefore, programs for the mathematics preparation and in-service education of teachers in the state of Montana cannot ignore the need to consider this variety of languages. A requirement of such a teacher education program should be an understanding of the American Indian languages and culture as they relate to school curriculum. One particular facet of this issue to explore is how mathematics instruction can be addressed in teacher education programs for teachers in schools with a significant enrollment of American Indian students.

In the cross-cultural teaching of mathematics at least two factors are critical. First, perceived utility of mathematics is important to all peoples. Second, mathematics learning is directly related to language development.

In a review of studies of mathematics taught in English among a variety of non-Western cultures, it has been found that indigenous peoples are often unable to solve mathematics problems that are not perceived as culturally relevant. For example, the abstract addition of thirty-seven and fourteen is meaningless to some non-Westerners. The addition of fourteen pigs to thirty-seven pigs in a world community where pigs are an important commodity may be an appropriate re-structuring of this problem in more meaningful terms. Cultural relevance is one of the more important factors to consider in the examination of mathematics operations and technical terms developed in American Indian languages. Leap, et al. (1982), writing of Picuris American Indian mathematics cautioned that it involves:

Language development is another critical factor for consideration. This involves specialized vocabulary and second language expansion.

Halliday lists seven techniques for developing language to serve the emerging needs of a society. Before a language is expanded to include new technical mathematics terms in its vocabulary, these processes need to be considered and an approach to development selected using one or more of the techniques.

Prior to development, the language to be developed should have its technical vocabulary compared to English language terms. Where the terms of the non-English language do not match the English language terms or are missing certain elements, expansion of the non-English language should utilize the techniques Halliday suggested.

Examination of mathematics concepts in English for counterparts in American Indian languages reveals factors related to the development of American Indian languages that differ from the contexts, constructs, and technical terms used in English language mathematics instruction. It is very difficult and perhaps impossible for native speakers of Navajo to construct an exactly parallel systematic analysis of the mathematics concepts in English.

Leap's account of his unsuccessful experience in translating an English language problem into Tewa suggests that such translation can result in a very practical problem becoming "re-written as a topic of senseless speculation" (Leap, et al., 1982, p. 30).

Gay and Cole (1967), in their study of the Kpelle of Liberia, suggested that a presentation of mathematics concepts in the English language without a consideration of Kpelle language development and cultural usage led to rote memorization and no comprehension of the English language concepts of mathematics presented to the Kpelle. In fact, the Kpelle exposed to the presentation of mathematics in the English language could not transfer this acquired knowledge of mathematics in English to solve new problems in similar settings. Yet, teachers of mathematics typically test students' mastery of concepts by variation of the problem presentations. In the case of the Kpelle, we know that no transfer of training could be said to have occurred based on such a test of mastery of concepts. It is reasonable to infer that a similar result would occur if native language speaking American Indians are taught mathematics concepts in the English language in an identical manner. One plausible but difficult alternative is the teaching of mathematics in the languages of American Indians.

The Eastern Montana College Title VH, Dean's Grant was awarded to address the needs of a multicultural/bilingual American Indian population by focusing on the expansion of selected teaching methods classes. A primary goal of the program is to extend the capacity of the institution to provide training relevant to bilingual educators in the innovative use of technology, mathematics, and science. Therefore, in the first year of the Dean's Grant, the Project Coordinator concentrated on research of English language mathematics concepts and terms used in the Crow American Indian language; one of the principal aboriginal language groups in Montana.

The authors selected the Crow American Indian language for the study of English language mathematics concepts and vocabulary based on several factors. First, the Crow language is supported through an extensive network of Title VII, Bilingual Education programs located in schools on the Crow Indian reservation. In addition, the Crow language spoken fluency rate is high among the adult Crow reservation population, and nearly eighty per cent of Crow Indian children typically are fluent speakers of the Crow language (Read, 1978).

In order to explore the relationship between the acquisition of mathematics concepts in the English language and in the Crow language and to document the existence and use of mathematics vocabulary in the Crow language the authors devised a structured interview study with Crow Indian adults in several reservation communities, and also with Crow Indian children in one reservation school as informants. The children, who were dominant Crow language speakers, were asked to name numbers in the Crow language and to indicate when they used the Crow language to count and when they used the English language. They were also asked, if they were able, to give the Crow language names for arithmetic operations, for fractions, and for particular geometric figures.

The adults included a wide range of English education backgrounds; however, all adults were fluent in the spoken use of the Crow language. In addition to questions similar to the ones the children were asked, the adults were asked whether numbers were used other than to count, and whether the mathematics children are taught has any relevance outside of schooling. The authors also explored the influence of the Crow culture on the acquisition of mathematics concepts.

Adult informants indicated that while number names exist up to one million, Crow Indians have little everyday use for numbers greater than one thousand. Closs recounts an analogous example in a story from the Copper Eskimo that concludes with the eventual death by starvation of two Eskimo men who set out to count the hairs of a wolf and a caribou to settle their argument. The storyteller adds: "That is what happens when one starts to do useless and idle things that can never lead to anything" (Closs, 1977, p. 8).

Only one of the children interviewed could count beyond twenty in the Crow language. The effect of years of schooling Crow Indian children in English appears to be that Crow language mathematics vocabulary is being lost. Even when the children knew the Crow number names (as in counting), they appeared to be thinking in English and translating into the Crow language. They reported that they used the English language number names when talking with other Crow language speaking children, but that they speak in the Crow language to Crow language speaking adults.

Another area of interest was the Crow language names for geometrical figures and their associated concepts in the language. For example, according to the adult informants, names for square and circle were well known, but the name for triangle (meaning in English literally three points) was of recent origin and less widely known. Names for other terms such as sphere were descriptions and recent inventions. Informants agreed that terms associated with geometric and English language related mathematics operations that are currently not used could be easily developed. Mathematics vocabulary currently in use reflects a functional use within the Crow culture.

Crow language terms exist for addition and subtraction because these operations have meaning in the Crow culture. Crow language terms for operations such as multiplication and division were not found in the survey. This may be due to the informants not typically using the terms. The adult informants noted that number names are important in ceremonies and have special significance in the Crow culture.

Crow language mathematics operations are not limited to addition and subtraction. The mathematics language development stage in the Crow language is comparable to William Leap's characterization of the Copper Eskimos counting vocabulary: "whatever 'limitations' may seem to be imposed upon the Copper Eskimo by their counting vocabulary, these 'limitations' in no way restrict the ability of individual members of the group to perform enumerative-related tasks when cultural context or social purpose requires it" (Leap, et al, 1982, p. 16). A mathematics operation such as division may not have a term in the American Indian language but this in no way hinders the description of the act or the performance of the act. A specific mathematics operation that is not typically performed in an American Indian language creates a conflict between what is linguistically possible and what is culturally real (Leap, et al., 1982, p. 26). Closs concluded that "It is . . . significant to know that the number systems of many tribes can be extended indefinitely, if required . . . [This] implies that a number system is structurally mature . . ." (Closs, 1977, p. 7). Using Closs' criteria Crow language mathematics is structurally mature.

Based on the interview data, it appears the Crow language with the Crow language geometric terms and Crow language uses of mathematics operations is not typically being used to aid mathematics taught in English in the Crow reservation schools. If the Crow language is to be used to teach mathematics to Crow language speaking children, teachers of Crow language speaking Indian children need to be aware of the operations of mathematics within the Crow language and to be able to use the logical constructs within the Crow language to assist Crow language speaking children in the accommodation of mathematics instruction in English. This means that specialized teacher preparation programs for the teachers of Crow language speaking children should include study of mathematics concepts in the Crow language. Crow bilingual education programs currently in the elementary schools may emphasize Crow language lessons on mathematics functions in the Crow language and thus assist non-Crow speaking teachers in facilitating Crow language speaking children's accommodation of the English language mathematics concepts. If Crow bilingual education programs do this, there is an indication that Crow language instruction in mathematics may enhance the English language mathematics achievement of Crow language speaking children. An experimental Crow language bilingual mathematics class which taught initial reading and mathematics in the Crow language to Crow language speaking first grade level children resulted in these first graders receiving higher scores on a standard English language achievement test of mathematics ability than a matching group of Crow language speaking first grade level children who were given initial mathematics instruction in English ("Final Evaluation Report," 1975).

More research is also needed to explore the impact of latent use of American Indian language in the processing of mathematics in the English language. In particular, "cultural code switching" (Leap, et al., 1982, p. 18) needs further investigation for its impact on the processing of mathematics in the English language. Many American Indian students do not speak their native language, but nevertheless retain cognitive structures of the native language with which they attempt to construct English language analyses. American Indian students may be in some cases composing sentences using the local American Indian language grammatical rules "while persons outside the tribe are interpreting those sentences using a second set of grammatical rules . . . " (Leap, 1982, p. 150). This process adversely influences the American Indian students' ability to comprehend and accommodate mathematics concepts which are presented in an English language format.

In this study, the authors investigated the role of the Crow language and its impact on mathematical concepts in the English language. The findings of this study suggest that in preparing teachers of mathematics more attention needs to be paid to the structure/thought processes of the native language when that language is not English.

School mathematics is typically presented using the English language and an English language method of processing. Students who are not native speakers of English go through a process of translation and accommodation in their language. This creates several problems for them. Because the process is different, the comprehension level and accommodation of these students may not match that of English language dominant students.

Teacher education programs for teachers of American Indians should include mathematics history relevant to American Indians. Courses should also include presentation of the historical contributions to the field of mathematics and traditional uses of mathematics among American Indian tribes. The authors believe Crow language bilingual education programs can minimize many of the problems discussed through emphasis on teaching Crow speaking children the interrelationships of the mathematics terms and concepts in English and Crow.

Bradley, Claudette (1984). Issues in Mathematics Education for Native Americans and Directions for Research. Journal for Research in Mathematics Education, 15: 96-106.

Closs, Michael P. (1977). A Survey of Mathematics Development in the New World (Report 410-77-0222). Ottawa, Canada: University of Ottawa.

Closs, Michael P. (1975). Final Evaluation Report 1974-1975 Crow Agency Bilingual Project (Evaluation Report). Crow Agency, Montana.

Gay, John and Cole, Michael (1967). The New Mathematics and an Old Culture: A Study of Learning Among the Kpelle of Liberia. New York: Holt, Rinehart and Winston.

Green, Rayna (1978). Math Avoidance: A Barrier to American Indian Science Education Science Careers (BIA Educational Resource Bulletin). Washington D.C.: U.S. Government Printing Office.

Halliday, M.A.K. (1978). Language as Social Semiotic: The Social Interpretation of Language and Meaning. Baltimore: University Park Press.

Leap, William (1982). Semilingualism as a Form of Linguistic Proficiency. In Robert St. Clair and William Leap (Eds.), Language Renewal Among American Indian Tribes: Issues, Problems and Prospects (pp. 149-159). Rosslyn, Virginia: National Clearinghouse for Bilingual Education.

Leap, William L., McNett, Charles, Jr., Cantor, Joel, Baker, Robert, Laylin, Laura, and Renker, Ann (1982). Dimensions of Math Avoidance Among American Indian Elementary School Students, (Final Report). Washington, D.C.: The American University.

Read, John Arthur Stanley (1978). A Sociolinguistic Study of Crow Language Maintenance. Unpublished Doctor's dissertation, University New Mexico, Albuquerque, New Mexico.