Journal of American Indian EducationVolume 36 Number 3
|
|
An Exploration of American Indian Students' Perceptions of Patterning, Symmetry and Geometry Claudia Giamati and Marion Weiland Attempting to understand and interpret the perceptions of individuals from another culture using our own cultural lens can be misleading. Our experienced with Navajo students as they reacted to mathematics curriculum materials indicated that difficulties in performance were in fact a result of cultural influences on perception rather than a lack of ability. As we further explored the students' mathematical performance we became convinced that their primary language and cultural exposure greatly influenced their perceptions. Language has a deep influence on a person's view of mathematical relationships (Ascher, 1992). The Navajo language is one of the few languages that has more verbs than nouns (Pinxten, 1983; Witherspoon, 1977). The process of creating or doing something is often more important than the finished product (Rhodes, 1994). Past research (Pinxten,1983, 1985, 1991) has identified the cultural differences that exist and the dynamic action oriented language that influences the Navajo's way of acquiring knowledge. Pinxten states: There have been superficial attempts at designing curricula for culturally diverse populations such as American Indians. For example, Davison (1992) would have us believe that simply changing apples and oranges to horses and sheep in a primary school student's "word" problem will accommodate their cultural differences and make the math problem relevant to the American Indian student. However, since the notion of counting sheep in the herd is not culturally relevant (Smallcanyon, personal communication, March 12, 1995), how can it then help the student learn mathematics? Rather than focus on the contrived, why not uncover aspects of mathematics that are culturally relevant to Native American students? All indications pointed to the existence of rich, intuitive geometrical and visual capabilities within the Navajo students. Pinxten designed curricula based on intuitive geometry (Pinxten, 1991) in the spirit of Freudenthal's point of view which holds that "true mathematics is a meaningful activity in an open domain, rather that a haphazard one in an a priori fixed reference set" (Freudenthal, 1983, p. 39-40). However, the curriculum was meant for elementary school students, which does not aid the high school math teacher. Thus, it was not obvious just how a high school student's intuitive perceptions manifest in the classroom. One of the focuses of the geometry of the Navajo is the art of rug-making. The rug patterns have been widely used as a basis for judgement about the Navajo's intuitive understanding of geometry and symmetry (Zaslavsky, 1991). While this is an appropriate topic as a model for discussing symmetry, it only scratches the surface of the Navajo students' intuitions about and exposure to geometry based on their cultural experiences. The practice of weaving rugs and the patterns the rugs contain have been learned from the Pueblo Indians and the Spanish Conquistadors, respectively (Zaslavsky, 1991; Begaye, personal com-munication, March 22, 1995). In essence, the beautiful Navajo rugs are Navajo commercial art. The art that can illuminate more deeply what we might expect students to intuitively understand about geometry is sand-painting and basket weaving. The examination of such art reveals much more complex patterns that are not usually symmetrical in the geometric sense (Congdon-Martin, 1990; Orban-Szontagh, 1992). Considering the nature of space generally, concepts integral to ordering emerge. In speaking of the notion of succession as it relates to ordering, Pinxten explains: To the Navajo, centering is also an important concept in ordering. A rough translation of the verb "to center" is described by Pinxten: "this verb (to become the middle on both sides) defines a particular spatial unit, a very general order, or a division of some bigger unit in two, or all of these in one" (1983, p. 81). The center is a very general notion rather than a metrically established point. Navajo designs in rugs, baskets, and sand paintings are typically centered. Even as ordering is accomplished, people differ in the number of spatial dimensions they consider to be relevant. Further, they differ in the non-spatial dimensions they choose to incorporate, including, for example, temporal ones (Pinxten, 1983). The Navajo designs create a complex, dynamic world that is rarely, if ever, unidimensional. Our explorations sought to determine and describe just how these cultural differences and attributes might manifest themselves in a geometric sense. Background The project began when Dr. Weiland, a mathematics teacher at Greyhills Academy High School in Tuba City, Arizona, noticed recurrent anomalies in the patterns her Navajo students created while completing exercises included in the AlgebridgeTM curriculum. AlgebridgeTM, produced by the College Entrance Examination Board and the Educational Testing Service (Beck et al., 1989), is a supplementary curriculum aimed at bridging the gap between arithmetic and algebra. Unexpected patterns were produced by enough students to convince her that these perceptions were not simply an aberration of a few creative minds. Instead, it became apparent that a very fundamental difference in the way these Navajo students per-ceive certain mathematical relationships was revealing itself. These items were linear pattern sequence exercises which gave the beginning terms of a pattern, then left a blank space for the student to continue the pattern. The students often continued patterns in a circular rather than linear way as shown in Figure 1. Moreover, they treated sets of objects as an individual element in a pattern rather than as the individual objects intended by the creators of the AlgebridgeTM materials. Had the National Council of Teachers of Mathematics' (NCTM) Curriculum Standards (1989) not produced a climate demanding the production of more exploration oriented and open ended curriculum materials, educators may have not been encouraged to examine a phenomenon that anthropologists have long been aware of, knowledge as a cultural construct. When Navajo students took the RavenTM Standard Progressive Matrices test another surprising difference surfaced. The Raven is a pattern-completion test designed to assess "a person's capacity for intellectual activity" (Raven, 1988, p. 2). The students at Greyhills were administered this test to identify gifted students. From a mathematical perspective, its sections can be separated into part-whole patterns, transformational patterns, a combination of these two, and linear patterns. Navajo students usually perform well in comparison to their counterparts on all sections of the test except one, the linear patterns (Cox, per-sonal communication, October 11, 1994). Professionals who concern themselves with measures of giftedness have traditionally recognized the lack of success that Navajo students have on the section dealing with linear patterns, but their solution to this apparent deficiency has been to standardize the tests for Navajos without asking why these differences occur (Cox, personal communication, October 11, 1994). Furthermore, we noticed that on the results of the Comprehensive Test of Basic Skills for the ninth and tenth grades, the students scored best on Geometry & Measurement even though they had little or no formal training in geometry. Their poorest scores were on Ratio & Proportion, to which they had been greatly exposed in elementary and middle school. Instruction: Draw a figure that would come next in the sequence below:
Response intended by Algebridge TM:
Typical responses by the students:
Most placed the next hexagon so that the pattern would eventually form a circle. Some students would explain the response showing how they intended to complete the pattern
Figure 1. Student responses to an open-ended patterning exercise. We became interested in these differences in perception and patterning which we hypothesize are directly linked to the Navajo culture. Consequently, we set out to discover exactly what these differences are. We conducted interviews with some students who had completed the RavenTM. We encouraged them to try to reveal to us how they had come to their decisions when selecting a piece to complete the patterns. From these pilot interviews we learned how these Navajo students viewed mathematical relations in some key areas. They perceived patterning as a process rather than a sequence of objects. They viewed visual representations holistically rather than as a collection of parts. The students' methods of examining patterns were more circular than linear, as is traditional to Western culture. They more often worked from the outside in toward the center. They rarely used the linear grid approach of working left to right and top to bottom. The combination of these predispositions and other Navajo ways of seeing and patterning can be stumbling blocks to the students when they are, faced with several types of standardized tests that require them to identify patterns. The choices that come naturally to the Navajo student are marked wrong or are disregarded altogether. Method Fifty-six students in grades nine and ten were involved in the study. Fifty-four were Navajo and two were Hopi. The students had been tested for language fluency earlier in the year for English placement. Of the fifty-six, thirty-four were primarily Navajo speakers, two were primarily Hopi speakers, eleven were bilingual (English and Navajo), and nine were primarily English speakers. One child spoke both Navajo and Apache, but these languages are so similar as to not warrant separation from the Navajo speakers. (About 16% of our sample did not speak a Native language, which is similar to the 18% non-native language speakers found on the reservation generally.) All the students were enrolled at a Bureau of Indian Affairs grant school, Greyhills High School, which is also a member of the National Association of Laboratory Schools. As we began to talk to students and to examine their work, several tools that we had planned to use were deemed inappropriate. We discovered that the segmenting of a whole pattern into small boxes, so that a pattern could be completed by choosing the appropriate piece for the lower right-hand corner as is done in the Raven' test, was not the way most Navajo students would choose to look at a visual diagram. We chose to try an experimental approach. The decision was made to narrow our exploration to geometry and patterning. Our initial experiences with students had encouraged us that a structured but relatively open-ended patterning activity that engaged their interest would reveal some important cultural differences. We chose a patterning exercise along with interviews as our main methods of collecting data. The exercise was developed by Gilliland (1991) from her work with the Maori of New Zealand and grew out of her observation of the "frieze symmetry" used in the decoration of their rafters. She calls attention to the reflections, rotations, and translations which occur in frieze, or strip, decoration. The activity we chose was the creation of frieze strips with organic objects as design elements. The general expectation is that to create a frieze strip decoration is to create frieze symmetry. In speaking of frieze symmetry found in other cultures, Ascher similarly observes that the emphasis of the strip patterns is on "figures that are constrained to remain on a strip and are repeated along it with no change in size or shape . . ." (Ascher, 1992, p. 157). The mathematical definition of frieze symmetry requires that not only are the figures repeated with no change in size or shape, but that the transformation to obtain one copy from the other is always the same. Not only is the object repeated but also the procedure. In order to make frieze strips we provided the students with cut strips of paper and a variety of organic material, leaves, flowers, tangelos, cereal, etc. They were then given the following directions:
The word labor as part of the word elaborate was underlined to clarify the word's meaning. The students were told that this meant to work on the design, to add detail. The students were not shown any type of frieze patterns or any other type of pattern because there was no specific desired outcome. Our main hypothesis was that students would construct patterns that displayed symmetry, specifically frieze symmetry. Our secondary hypotheses were that the students' patterns would demonstrate aspects of transformational geometry and contexts for geometry that are culturally different from our own. Data Analyses The observational data are presented as mini-case studies describing the approaches of several students to the patterning exercise. They frequently reveal reflections of Navajo culture and natural philosophy incorporated into the students' designs.
Another student LB proceeded in a similar way: These designs are highly elaborated and hardly resemble the objects which inspired them. They are usually centered on the strip. As the geranium flowers are drawn into an elaborate design, the student sees this as a pattern which contains repetition, something used over and over again, but it does not include the lining up of a sequence of congruent shapes. In resisting the request to repeat the pattern, LB was not avoiding work. As he had originally created it, the design was aesthetically satisfying. He continued with the creation of another flower in order to please the teacher, but incorporated change in size and development. Other students created designs that incorporated the notion of context. This was sometimes added after the object was drawn, but in other cases the object was situated within its context from the beginning: In Figures 5 and 6 "repeating" can be seen to include viewing an object from a variety of perspectives. These examples incorporate changes in size and orientation of the elements and clearly demonstrate a preference to connect the objects to their larger natural context. Students who selected a tangelo for use in their patterning exercises rarely displayed only parts of the object in their designs. The tangelo was approached as a singular whole entity. This was true even when they had acted upon the object by removing the peel or cutting it.
JK drew his tangelo in the center of the strip. He had cut it to be viewed from the top as a cross-section. Around this he had arranged the peel he had removed and colored it. He wanted to leave it exactly as it was. When urged to repeat, he finally agreed, adding two more repetitions of the original design to the right and the left of center. These were not symmetrical, however, as one was larger and the other smaller than the original design in the center. When asked, X said that he did consider the orange with its rind to be the pattern. HG also peeled a tangelo. He wanted to include both the peel and a cross- section in his design but thought it was too large to fit on the strip. He then sectioned part of the tangelo and used a cross-section and nine separated sections lined up in a row. He said that this set was a pattern. He repeated that set. Both of these designs based on the tangelos show that some action has been taken. The tangelos have been peeled, cut, or sectioned. Their parts are arranged artistically. Time and action are indicated. Still, whenever possible, the students have retained the wholeness of the object. Expansive movement is important in Navajo natural philosophy (Pinxten, 1983) and many of their designs displayed this. This can include shrinking as well as expanding, unfolding, and growing as displayed in many of the students' designs. In addition to this expansive movement students' also depicted temporal change. TB's work is an example of this: TB began by copying a leaf at the bottom of the strip aligned vertically. She sat for some time after being asked to repeat the pattern on the strip of paper. Eventually, she began to draw the leaf as it would change over time. Another student, when asked to describe the pattern in TB's drawing, said it was the "emerging" leaf, the three taken together. Asked what he would do with that pattern if the strip was extended or doubled in length, he said he would continue with its stages until its "death." The idea of stages of development as movement in time is also present in MD's work, shown in Figure 5, wherein she presented the changing shape of the leaf through its stages of development. Further examples of the students' notion of movement in time and space can be seen in the work of PT, RT, and BK on Figure 9. Notions of expansion and shrinking, as related to the concept of expansive movement, are obvious in the patterns of RY and MR in Figure 10. In coding the data, we decided that wherever possible we would use loosely formal geometrical notions and definitions as a basis for categorization. Since the students depicted their patterns in both two and three dimensions, we separated these groups into two charts. We chose to represent the frequency chart in the form of a matrix with the students' primary language as rows: Primarily Navajo (Dine) speakers (D), Primarily Bilingual Speakers (B), Primarily English speakers (E), and Primarily Hopi Speakers (H). These assignments were based on assessments by the school’s English department for placement purposes. We examined the designs in the students’ patterns for indications of isometries (copying objects exactly) and similarities (the expanding or shrinking of copies of the object). A code was developed using the four standard isometries: translations or slides, rotations, reflections and glide reflections - a slide followed by a reflection . We also coded the patterns for similarities which can be loosely thought of as successive shrinking or expansion from one object to the next. Because some students chose to combine. two or more of these five transformations, we coded for that occurrence as well. Other modes of design that required recognition in code were the depiction of temporal change (T), symmetry (S), asymmetry (AS), an existing center (C), depiction of an action on the elements in the pattern (A), the incorporation of context (W), and color used as part of the pattern (H). Instances of students depicting the last image as different were also coded (LD), although it could arguably be included as intentional asymmetry.
We found that of the fifty-six students, twenty-four (42.9%) chose to draw two-dimensional designs (2D). Thirty-two (57. 1 %) chose to draw three dimensional designs (3D). Twenty-seven (48%) of all students combined two or more of the five standard transformations in one design (X). The primary combinations were translations with rotations and translations with similarities. Nineteen (33.9%) students used similarities and twenty-two (39.2%) used rotations as part of their design (see Figure 9). A mere five (8.9%) of the students used symmetry (S). The other 51 (9 1. 1%) depicted asymmetrical patterns (AS). Fourteen (25%) depicted some type of action (A) on a figure, including several who used words to describe the action. Seven (12.5%) students depicted a change in time in their patterns (T), while nine (16%) used color (H) as an essential feature of their patterns. Reflections (13) or glide reflections were rarely employed. As indicated earlier, thirty-four (60%) were primarily Navajo speakers (D), eleven (19.6%) were bilingual (B), nine (16%) were primarily English (E) speak- ers, and two (3.5%) were primarily Hopi (H) speakers. Nineteen (55.9%) of the primarily Navajo speakers (D) combined two or more of the standard transfor- translation mations in one design (X). Only four (36%) of the bilingual students (B) and three (33%) of the primarily English speakers (E) combined several transformations reflection glide reflection . Only three of the primarily Navajo (D) speakers (5.3 %) depicted any symmetry (S) in their pattern at all. None of the bilingual students (B) and only two multiple transformations (22%) of the primarily English speakers (E) depicted symmetry (S). Fourteen of color used in pattern the primarily Navajo (D) speakers (41 %) used similarities (a), whereas only two action on the pattern depicted (18%) of the bilingual (B) students and three (33%) of the primarily English speakers (E) used similarities (a) in their patterns. Seven (20 %) of the primarily Navajo (D) speakers had an identifiable center (C) to their patterns, only one (9%) bilingual (B) student had a center (C) and no primarily English (E) speakers depicted a center(C). Twenty two (64.7%) of the primarily Navajo (D) speakers chose to depict a pattern in three dimensions (3D), as did four (36%) of the bilingual (B) students, and five (55%) of the primarily English (E) speakers (see Figure 9). Ten (27%) of the primarily Navajo (D) speakers depicted action (A) in their patterns, but only two (18%) bilingual (B) students and two (22%) of the primarily English (E) speakers did. When comparing the data to our hypotheses we found some interesting results. Our main hypothesis was that students would construct patterns that displayed symmetry, specifically frieze symmetry. It was clear that the students knew what symmetry was, but they expressed their understanding by intentionally creating asymmetrical patterns. The students often added something to cause asymmetry in a seemingly symmetrical pattern. This was contrary to our expectations, largely influenced by viewing the rug patterns, which are highly symmetrical. We were further influenced by Gilliland's results (1991) and the outcomes we received when we tried the exercise with colleagues and other non Native people. Nothing in the literature prepared us for these outcomes, nor did our own cultural lenses. One of our secondary hypotheses was that the students' patterns would demonstrate aspects of transformational geometry. We found that this did happen. The students not only demonstrated their understanding of isometries, but of similarities as well. Recall that, loosely speaking, isometries are the act of copying an object exactly, while similarities are the act of expanding or shrinking copies of an object. The students rarely made equal copies of their diagrams. More often, the students chose to shrink or enlarge the diagrams of objects along the strip. A completely unexpected outcome that demonstrates a very deep command of transformational geometry was that diagrams were drawn in three dimensions. Several students depicted objects being transformed in space, both by position and by size. Students did not feel it necessary to hold the transformations constant in the same way that the size was not preserved. For example, rather than rotate an object by ninety degrees, then rotate the next one by one hundred eighty degrees, the angles the students chose for rotations were completely arbitrary. The other part of our secondary hypothesis was that the students' patterning exercise would demonstrate perceptions of elements of geometry that were culturally different from our own. There is no question that this occurred. In many instances, action of some kind was depicted. The patterns were dynamic, changing from frame to frame, rather than depicting static linear repetition. This corresponds to their verb laden language where action is the focus in description. Essentially, the attention that was given to action and process rather than the objects was an important revelation. The literature supports the outcomes we obtained but did not wholly prepare us for them (Pinxten, 1983, 1985, 1991; Ascher, 1992). Symmetry was often illustrated in a different way than Western culture views it. This was especially true when the intentional asymmetry was depicted. This actually contradicted the literature (Davison, 1992; Zaslavsky, 1991) and our own cultural perceptions of the Navajo culture. In addition, the ideas of spatial and temporal relations were also revealed by the task. As mentioned earlier, little in these patterns was unidimensional. The notion of placing an object in its context (space and time) was clearly demonstrated by the students. The lack of linear progression, while expected (Pinxten, 1983, 1985, 1991; Ascher, 1992), was also made clear. What was important from the perspective of an educator, is that the students demonstrated their general notions of symmetry, transformational geometry, and transformations. This afforded us a place to begin to visualize, and thus, understand, their ways of conceptualizing these notions. There are several implications that these discoveries have on the teaching of mathematics. The broadest implication is that teachers of mathematics who have Navajo students need to examine their methods of introducing mathematics and how the students perceive it. Careful attention to the actions and processes of doing mathematics and how mathematical knowledge evolves and progresses is needed since these are the aspects of mathematics the Navajo students will focus on. Creating situations for discovering these actions is essential. For example, the students were struggling with a unit on ratio and proportion where the use of similarity was employed. The focus of this lesson was to compare objects that were enlargements of each other and to record the scale factors, thereby setting up proportions. Using the notions that the students had demonstrated to us about their clear understanding of these ideas, the teacher began to use verbs such as growing, enlarging, and shrinking when discussing the similar figures. The students responded very well to these examples and were eventually able to demonstrate a much higher success rate on the problems in question. There are many implications for assessment. The patterning tests for giftedness and other standardized tests do not, as presently constructed, allow for cultural differences. There have been several very recent attempts to recruit teachers on the Navajo reservation to assist in discussing cultural differences that affect testing. At present these efforts have not changed the tests in any way. The implications for research are quite numerous. Our work toward further understanding the Navajo students' way of thinking and how it aids and impedes their learning of mathematics continues. We are interested in ways to assist Navajo students in calculus. Given that the Navajo students in our study demonstrated a natural understanding of dynamic and expansive movement, and given the dynamic nature of calculus, one might predict a higher level of success. This is not the case however. These questions remain open for other Native American tribes as well. The opportunities for further research are challenging. References Achenbach, T. M., & Edlebrock, C. (1986). Manual for the Teacher's Report Form and Teacher Version of the Child Behavior Profile. Burlington: University of Vermont, Department of Psychiatry. Brant, C. (1990). Native ethics and rules of behavior. Canadian Journal of Psychiatry, 32, 1385-1391. Crawford, J. (1987). The special case of bilingual education for Indian students. Education Week, 6, 16. Dale, E. E. (1949). The Indians of the Southwest. Norman: The University of Oklahoma Press. Deyhle, D. (1986). Break dancing and breaking out: Anglos, Utes and Navajos in a border reservation high school. Anthropology and Education Quarterly, 17, 111-127. DuBray, W. H. (1985). American Indian values: Critical factors in casework. Social Casework, 66(l), 30-37. Ebbott, E. (1985). Indians in Minnesota: Fourth Edition. Minneapolis: University of Minnesota Press. French, L., & Hornbuckle, J. (1980). Alcoholism among Native Americans: An analysis. Social Work, 25, 275-280. Hodgkinson, H. (1990). The demographics of American Indians. Washington, DC: Institute for Educational Leadership, Inc. Hollingshead, A. B. (1975). Four-factor index of social status. Available from the Department of Sociology, Yale University, New Haven, CT 06520. Hornett, D. (1991). Elementary-age tasks, cultural identity, and the academic performance of young American Indian children. Action in Teacher Education, 12, 43-49. Houser, S. (1991). Building institutions across cultural boundaries. Tribal College, 2, 11-17. Huffman, T. E., Sill, M. L., & Brokenleg, M. (1986). College achievement among Sioux and white South Dakota students. Journal of American Indian Education, 25(2), 32-38. Josephy, A. M. (1989). Now that the buffalo's gone. Norman: University of Oklahoma Press. Kayser, J. (1963). Scholastic performance and ethnicity: A preliminary study of seven school classes. Journal of American Indian Education, 3(l), 27-30. Kelso, D. & Attneave, C. (1981). Bibliography of North American Indian mental health. Westport, CT: Greenwood Press. Lazarus, P. J. (1982). Counseling the Native American child: A question of values. Elementary School Guidance & Counseling, 17(2), 83?88. Luria, A. R. (1971). Towards the problem of the historical nature of psychological processes. International Journal of Psychology, 6, 259?272. Luria, A. R. (1976). Cognitive development: Its cultural and social foundations. Cambridge, MA: Harvard University Press. Plas, J. M., & Bellet, W. (1983). Assessment of the value-attitude orientations of American Indian children. Journal of School Psychology, 21, 57-64. Mindel, C. H., & Habenstein, R. W. (1976). Ethnic families in America. New York: Elsevier North Holland, Inc. Radin, N., Williams, E., & Coggins, K. (1993). Paternal involvement in childrearing and the school performance of Native American children: An exploratory study. Family Perspective, 27(4), 375-391. Red Horse, J. G., Lewis, R., Feit, M., & Decker, J. (1978). Family behavior of urban American Indians. Social Casework, 59, 67-72. Rogoff, B. (1982). Integrating context and cognitive development. In M. E. Lamb & A. L. Brown (Eds.), Advances in developmental psychology, Vol. 2, 125-170. Hillsdale, NJ: Erlbaum. Rogoff, B. (1990). Apprenticeship in thinking. New York: Oxford University Press. Sack, W. H., Beiser, M., Clarke, G., & Redshirt, R. (1987). The high achieving Sioux Indian child: Some preliminary findings from the flower of two soils project. American Indian and Alaska Native Mental Health Research, 1, 37-51. Sobeck, J. L. (1990). Lifestyle and Behavior Risk Factors of Health: A Report on a Mail Survey Conducted in the Bay Mills Indian Community. Detroit, MI: Department of Community Medicine, Wayne State University. Sue, D. W., & Sue, D. (1990). Counseling the culturally different. New York: John Wiley & Sons. Thornton, R. (1987). American Indian holocaust and survival: A population history since 1492. Norman: University of Oklahoma Press. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Ward, C. J. (1994). Explaining gender differences in Native American high school dropout rates: A case study of Northern Cheyenne schooling patterns. Family Perspective, 27(4), 415-444. Wax, W. K. (1967). The warrior dropouts. Lawrence: The University of Kansas Press. Weeks, G. (1992). Mem-ka-weh: Dawning of the Grand Traverse Band of the Ottawa and Chippewa Indians. Traverse City, MI: Village Press, Inc. Weibel-Orlando, J. (1991). Indian country, LA.: Maintaining ethnic community in a complex society. Chicago: University of Illinois Press. Williams, E. (1995). Father and grandfather involvement in childrearing and the school performance of Ojibwa children: An exploratory study. Doctoral dissertation, University of Michigan. Williams, E., Radin, N., & Coggins, K. (1996a). Paternal involvement in childrearing and the school performance of Ojibwa children: An exploratory study. Merrill-Palmer Quarterly, 42(4), 578-595. Williams, E., Radin, N., & Coggins, K. (1996b). Grandfather involvement in childrearing and the school performance of Ojibwa children. Family Perspective, 30(2), 161-183. Wise, F., & Miller, N.B. (1983). The mental health of Native American children. In G. J. Powell, J. Yamamoto, A. Romero, & A. Morales (Eds.), The psychosocial development of minority group children, 344-361. New York: Harper & Row. |
