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DOCUMENT RESUME RC 018 626 ED 343 767 Preston, Vera AUTHOR Mathematics and Science Curricula in Elementary and TITLE Secondary Education for American Indian and Alaska Native Students. Department of Education, Washington, DC. Indian SPONS AGENCY Nations At Risk Task Force. PUB DATE 91 18p.; In: Indian Nations At Risk Task Force NOTE Commissioned Papers. See RC 018 612. Information Analyses (070) PUB TYPE MF01/PC01 Plus Postage. EDRS PRICE *American Indian Education; American Indians; DESCRIPTORS Cooperative Learning; *Demonstration Programs; *Educational Strategies; Elementary Secondary Education; Experiential Learning; Higher Education; Mathematics Education? *Mathematics Instruction; Science Education; *Science Instruction; Teacher Education; *Teaching Methods Culturally Relevant Curriculum IDENTIFIERS ABSTRACT Issues related to the improvement of mathematics and science education pertain to Native students as well as to the general population. Native students are most successful at tasks that use visual and spatial abilities and that involve simultaneous processing. Instruction should build on Native students' strengths. Experiential learning and cooperative learning are two methods that are particularly effective with Native students in improving student attitudes and problem-solving abilities and reducing mathematics anxiety. Storytelling techniques can be used to develop culturally relevant problems. Career days show students the uses of mathematics in the real world. Curriculum development strategies include establishing the relationships and connections between mathematics and other subjects, and incorporating culturally relevant materials, such as Maya or Inca mathematics and science. Strategies of exemplary programs include summer math camps for Native students, summer institutes to improve teacher instructional skills and methods, after-school and summer enrichment activities in science and engineering, instructional materials developed to accompany a science series on public television, magnet schools, after-school college preparatory courses in mathematics, and parent resource centers. Recommendations are offered related to instructional methods, program development, and federal funding. This paper contains over 130 references. (SV) 1:**********W***********W***************w******************************* Reproductions supplied by EDRS are the best that can be made from the original document. ***********************.t*********************************************** Mathematics and Science Ctuvricula in Elementary and Secondary Education for American Indian and Alaska Native Students U.S. DEPARTMENT OP EDUCATION Once ot Educenonel Research end ['Mover/tem EDUCATIONAL RESOURCES INFORMATION CENTER IERICI document nes been reprOduced as received from Inn person or poem/anon originating it .ve boon made to improve Vera Preston C Minor changes reproduction au% Ay Points 0/ view os opinions steles) in tniedockr mint do not necessarily represent Octet OERI poettion or policy Introduction ferences, the message obtained from the testimony accentuates the common concern of Native people Wake up, America! Your child:en are at risk. for educating their children. They want their Three of every four Americans stop studying children to be educated in a manner which enriches mathematics before completing career or Job their horizons and guarantees a variety of options prerequisites. Most students leave school for lifelong learning and for various vocations. without sufficient preparation in mathe- For years reports have stated the need for matics to cope either with on-the-job reforming mathematics and science education in demands for problem-solving or with college the United States. A Nation at Risk (National requirements for mathematical literacy... Since mathematics is the foundation of Commission on Excellence in Education 1983), science and technology, it serves as a key to Educating Americans for the 21st Century (Nation- opportunity and careers.. Moreover, mathe- al Science Board Commission on Precollege Educa- matics contributes to literacy certain distinc- tion in Mathematics, Science and Technology tive habits of mind that are of increasing 1983), and Everybody Counts -A Report tu the Na- importance to an informvd citizenry in a tion on the Future of Mathematics Education (Na- Despite massive effort, technological age. ... tional Research Council 1989) are representative relatively little is accomplished by remedia- of the reports concerning mathematics and science not educators. tion programs No one education. Compared to students throughout the mathematician*, or researchers knows how to reverse a consistent early pattern of world, students in the United States rank in ability low achievement and failure. Repetition rare- with students from Third World countries. Nation- ly works; more often than not. it simply al Assessment of Educational Progress (NAEP) reinforces previous failure. The best time to reportsaf results of 1986 assessments indicate that learn mathematics is when it is first taught; ability to answer problems requiring higher order the best way to teach mathematics is to teach thinking is lacking for most students in the United it well the first time. (Everybody Counts. States (Brown, Carpenter, Kouba, Lindquist, Sil- 1989, pp. 1-3. 13) ver, and Swafford, 1988, pp. 911-248). Some Testimony from the Indian Nat:xis at Risk reports have addressed the issue of what needs to (INAR) Task Force meetings, regional hearings, be done to solve the problems. Ciaims of easy and written testimony submitted to the INAR Task solutions to difficult problems should be viewed Force represents the diversity of Native peoples with great suspicion. Research indicates the way and the variety of schools attended by American students learn is not always the way students are Indian and Alaska Native students. The common taught. Since teachers tend to teach the way they thread existing throughout the extensive tes- were taught and the way in which they learned, the timony is the need for American Indian and Alaska vicious cycle ofmiseducatim continues to handicap Native students to have their culture acknow- the lives of children of all ages, abilities, and races. ledged and to be taught in the ways they learn. The The National Council of Teachers of Mathe- type of school, the type of class (language arts, matics (NCTM) has responded to the need for social studies, science, mathematics), the area of change in mathematics teaching by producing Cur- the country does not change the need to be riculum and Evaluation Standards for School respected as a human being and to be taught in a Mathematics (NCTM, 1989). It is a set of stand- understand manner that allows the individual to ards for mathematics curricula in kindergarten and internalize the material to be learned. All through high school developed by classroom children, regardless of race, gender, or ability, teachers, supervisors, educational researchers, deserve to be taught in the ways they learn by teacher educators, and university mathematicians building on their strengths and overcoming their working together. NCTM is very much aware ofthe weaknesses. Rather than emphasizing the dif- 1 Indians N Alons At Risk; Solutions for the 1990s processing of knowledge which enables them to skepticism of people toward change after the *new perceive various elements as a whole. Seeing the math" of the sixties which imposed change from the whole picture, a holistic, global process is referred top down. The current recommendations stress the to as simultaneous processing. Processing intbr- importance of all stIdents learning mathematics mation sequentially and in an analytical manner and the importance of evising mathematics in- struction. The National Science Teachers of is refened to as successive processing. (Das, Kirby, & Jarman, 1975. Quoted in More, 1986, pp. 8-9) America (NSTA) has developed a Statement of Research results indicate Native students use Initiatives similar to the NCTM Standards. Em- simultaneous processing more frequently and ef- phasis is on more unified, in-depth hands-on fectively than successive processing. It also indi- science curricula for all students kindergarten cates that Native children are more field through twelfth grade. i,dependent (FI) than field dependent (FD) and What can be done to improve the mathematics that they tend to think in relational styles rather and science education of all children including than in analytic styles. (More, 1986, pp. 9, 12) American Indian and Alaska Native children? Research on culturally based communication Why is mathematics and science education so im- styles related to the classroom has been done by portant to all students? This paper will address Scollon and Scollon (1983) and Phillips (1972). the issues in mathematics and science education. Studies of teacher effectiveness which relate to Mathematics and Science styles of communication of Native Indian students have been done by Kleinfeld (1970). Traditional Mazhematics and science influence the lives of Native cultures teach children to watch, listen, everyone. Scientific, mathematical, and tech- think and then do the task to be learned. Many nological literacy is crucially important for citizens classrooms encourage trial-and -error learning. to understand the technological world in which Many students become quite proficient at guessing they live. Mathematics has been called "the queeii It is the study of until they get the correct answer. Trial-and-error and servant of the sciences." learning is the antithesis of traditional Native patterns and relationships and is the foundation learning styles. on which understanding of many fields of study is built. An understanding of mathematical concepts Field independence is the degree to which an and mathematical reasoning is necessary in al- individual can separate a figure from its background, a part from the whole, oneself most every occupation in today's technological from the environment or other people. A field society. Mathematics has been called a critical independent (FI) person is more able to pro- filter because so many occupations require a math- vide an organizational structure to a disor- ematical background. Many people perceive math- ganized set of facts or observations (e.g.. ematics as an infinite set of meaningless rules making a mental map of the surrounding which are impossible to memorize. Mathematics is terrain). A field dependent (FD) person is less constantly changing; more than half the mathe- able to separate a part of the whole, but is matics known today has been developed since The Fl more conscious of other people... . 1940. Science and mathematics are interdepen- person will impose his or her own structure on a field (spatial or conceptual) more readi- dent in our technological age. For example, under- ly. The FD person is mom socially aware and standing of environmental issues such as water more responsive to those around him or her. availability and water pollution require an under- (More, 1986, pp. 10-11). standing of both mathematics and science. There is considerable overlap between the learning styles of Indian and Non-Indian stu- Mathematics and Science dents. Indeed the similarities are probably Learning and Teaching greater than the differences. Furthermore, it is likely that the learning styles of some Fundamental change in the way mathematics groups of Non-Indians will be remarkably and science is taught is necessary (NCTM Stand- similar to those of Native Indians. This could ards, 1989; Willoughby, 1990). Information from occur when there are similar factors affecting mathematics and science educational research and the way of life such as degree of remoteness from cognitive psychologists should be used to of the community, economic level, and sig- develop plans for organizing instruction for math- nificant cultural differences from the ematics and science classes. Research indicates majority. However in such situations cul- tural differences should not be downplayed. American Indian and Alaska Native students are Even though the pattern of learning styles is most successful at tasks which use visual and similar, the causes and the methods of work- spatial abilities. (More, 1986) Many studet.i,s, in- use holistic, global cluding Native students, Mathematics and Science Mg with the learning styles may not be." to understand concepts and to transfer their learn- (More. 1986. p. 20) ing to other situations (Surdam,1986). Teacher guidance ;i3 important in bridging the gaps from It is important to realize that teaching styles the concrete to the pictorial to the symbolic stage. that are appropriate for Native students are also (Kouba et al., 1988b) Educators from the early appropriate for other students. Methods of teach- pert of the century to the present have advocated ing found in many schools are not appropriate to the us,) of physical, or concrete, materials in the the ways children learn. This is one of the major classroom. (Brownell, 1935; Burns,1986; Montes- reasons for the need ofremedial classes. If students 806,1965; Piaget,1952; Reys,1971; Suydam,1986) the were taught concepts in the way they learn, Recommendations from reports of the most recent need for remedial classes would be greatly NA EP results encourage the use of concrete diminished. Since Native students have strengths materials and activities. NCTM has recommended in using visual/spatial/perceptual information, the use of physical, concrete materials in the class- new material should be introduced using methods room. (NCTM 1980; NCTM 1987) An Agenda for building on their strengths. The overall view of Action: Recommendations for School Mathematics new information should be stated before the infor- of the 1980s (NCTM, 1980, p. 12) includes the mation is broken down into small segments. Ad- following: "Teachers should use diverse instruc- vance organizers should be used. Students' tional strategies, materials, and resources, such as strengths should be used to develop skills in other the use of manipulatives, where suited, to il- ... modes of learning. (More, 1986, p. 21) Students lustrate or develop a concept or skill." should have the right to learn beginning material There are many benefits which occur as a result in their native language to allow them to develop of using manipulatives. Mathematics anxiety is a cognitive base on which to learn for the almost non-existent in classes where manipula- remainder of their lives. (McDonald, 1989, p. 16) tives are used frequently. The attitude toward Children are born with a love of learning and learning mathematics is consistently more positive an innate curiosity. The first three years of life a as a result of using manipulatives and of par- child learns a tremendous amount of information. ticipating in cooperative learning. Student under- Parents are the first and most important teachers standing of concepts is greater among all ability of a child. How can a child's natural desire to learn levels. Parental involvement is easier to obtain by about his/her surroundings be nurtured and main- having some activities done at home using common tained in academic situations? Bow can students household materials. Students need guidance with learn the necessary material in the various clas- the teacher as facilitator on the use of the ses? Many theories exist regarding how people manipulatives. Assistance and guidance in under- learn. There is danger in stating that n specific standing the concepts represented by the group of individuals learn in a particular way. It is manipulatives and in bridging activities to go from important to be sensitive to the needs of each the concrete to the pictorial to the symbolic or individual and help that individual build on her/his abstract are essential in using the materials to best strengths and increase ability in areas of weak- advantage (Heddens,1986). Learning what to do ness. (More, 1986) Many people have a preferred and why enables students to retain information learning style. Many researchers stress that stu- and transfer the learning to other situations dents should be allowed to learn from their (Burns,1986). Fuson (1981) writes that there is a strengths. Research in mathematics anxiety indi- meta-cognitive benefit in the use of concrete ob- cates that students overcome their anxiety and are jects. Teachers' sensitivity to styles oflearning will able to perform better in mathematics if they build help their students. A balanced view of various success on success. (Tobias, 1978) methods of teaching and learning will assist stu- Manipulative, hands-on, Materials dents to recognize their particular style of learn- ing. (More, 1986) The teaching/learning style is "Tell me, and I forget; Show me, and I remem- determined by the learning task. Learning how to ber; Involve me, and I understand." Ancient learn and recognizing the underlying concepts and Chinese Saying. features of mathematical problems is very impor- Students who use hands-on materials, other- tant. better at- wise known as manipulatives, have titudes toward mathematics and better ability to Cooperative Learning problem solve (Suydam and Weaver,1975; Traf- Current issues in educational circles include ton,1975). Students who use manipulatives and teaching thinking through the curriculum, prob- are guided to bridge the gap from concrete to pic- lem solving, cooperative learning, writing across torial to symbolic representations are better able the curriculum, and learning how to learn 3 4 Indians Nations At Risk: Solutions for the 1990s dents in gifted classes and in remedial classes at (Simon,1986). Cooperative, or group, learning is being recognized as important in learning con- all levels. cepts. Since people work cooperatively in most Storytelling and Problem Solving in situations outside school, studeuts should have Mathematics and Science. opportunities to work together in school. The many benefits which occur as a result of cooperative Storytelling as a means ofteaching is tradition- learning are an ability to communicate verbally the ally used by American Indians and Alaska Natives. information and concepts being studied, an under- Using stories to pose mathematical or scientific standing of other peoples' attitudes, a positive at- problems is a natural way to relate situations of titude toward learning, recognizing there are the student's home life and culture within the various paths to the solution, and clarifying one's mathematics classroom. Students arrive in school own understanding. !Slavin, 1987) with the ability to solve problems which have Cooperative learning is a popular concept at the meaning for them. School textbooks and mathe- moment in education. The merits and disad- matics lessons frequently convince students they vantages of cooperative learning are discussed in are unable to solve the problems which had pre- four articles in the Febmary 1991 issue of Educa- viously been easy for them. One of the important tional Leadership (Kohn, 1991; Slavin, 1991). It is skills in solving problems is drawing pictures rep- important to note that cooperative learning has resenting the problem. (Davis & McKillip, 1980, been used successfully by teachers over the years pp. 80-91) without the label of "cooperative learning." Much An ancient example of using storytelling to pose of the discussion in the articles relates to giving a mathematical problem is the story of the person rewards for group effort. Rather than getting going to St. Ives. "As I was going to St. Ives, I met bogged down in semantics or side issues, teachers a man with seven wives. Each wife had seven cats, need to consider the advantages of cooperative each cat had seven kits. How many were going to learning and apply the concept as appropriate to St. Ives?" Only one person was going to St. Ives; their particular situation in their specific class- the others were leaving St. Ives. The problem in- room. Teachers must acknowledge that coopera- volves listening and reasoning. The mathematics tion in the world outside the classroom is expected involved is rather straightforward. and necessary for success in projects throughout Students enjoy making up story problems when the business world. Mathematics and science the teacher provides an encouraging, accepting, learning should involve learners working together. respectful atmosphere in the classroom. The entire Mathematicians and scientists talk to each other family could participate in posing story problems and try new ideas and check each other's reason- which incorporate mathematical and scientific ing. They do not work in total isolation. Research problems. For example each of the following math- overwhelming supports the concept of cooperative ematical type of questions could be posed in a story. learning and working together. It is particularly How much flour is used for making fry bread? How appropriate to the learning style of many Native does the volume of the fry bread change when it is students. There are many different forms of cooked? When popcorn is popped, how much does coope. ative learning. The most successful ap the volume change? How many seeds are there in proaches have incorporated group goals and in a watermelon? What are some strategies for deter- dividual accountability (Slavin 1991). Many of the minitig how many seeds are in the watermelon? advantages of cooperative learning are social as What are the differences in types of seeds? What well as academic. Mathematics anxiety is practi- are the shapes of leaves? How many leaves are on cally non-existent when students are allowed to the plant? How many steps does each person in the work together to discuss and learn new informa- family take for walking the same distance? (What is the stride of each person?) How many sheep per tion. Many teachers used cooperative learning acre can the land support without irrigation? How before it was called cooperative learning. Students many cattle per acre can the land support? How worked together and helped one another and many bales of hay can be put in a pickup? What is the most efficient shape to build a house if recorded the results of their experimentation, their searching for patterns, or their explanation of new materials are scarce? (A circle is the most efficient shape to provide the most area for the same definitions, concepts and material. Each group turned in one paper with their results. Students amount of fencing. A square is the most efficient and wrote were each responsible for the material straight-sided figure. Why?) How many days did it individual reports later. The method of having rain last week, last month, last year, ten years ago? students work together was successful with stu- 4 Mathematics and Science What is the difference in temperature from 5:00 mathematics and science activities for students of AM to 5:00 PM on the same day? all ages. Career Education Mathemafics Anxiety Many students develop 6. fear of mathematics Career days for elementary, middle, and junior high school students enable the students to learn during the elementary, middle school, junior high about different types of careers. Native students years which cripples their ability to l3am mathe- matics in high school and later. Other students do should be encouraged to consider careers in fields which will help their communities. Careers in well in mathematics classes until they reach al- health fields, environmental science, agricultural gebra or geometry. When they have difficulty in mathematica, they quit taking elective mathe- science, various type of engineering, attorneys, matics classes. They take the path of least resis- counselors, and business all have the potential for helping their communities. Expanding Your tance when taking required mathematics classes. Horizons Workshops and Career Days were It is important for students to be encouraged, pos- originally organized for females. The format is sibly required to take mathematics classes each appropriate for American Indian and Alaska Na- year of high school. Students must realize that tive children. Students should be encouraged to mathematics is a critical filter. The more mathe- continue their education so they will be prepared matics a student studies, the higher the expected to help their community in many ways. lifetime earnings, and the more opportunities for Students generally lose their innate en- employment are available. Students can choose a thusiasm for science and mathematics during the job or career based on what they enjoy doing rather crucial junior high years. Therefore, it is critically than on what requires no mathematics. Counselors important to provide elementary, middle, and should be required to insist that students stay in junior high school students opportunities to ex- mathematics classes rather than allowing them to perience challenging and interesting projects and drop out when the going gets tough. programs in mathematics and science. It is also Mathematics anxiety and test anxiety is usual- critically important for students to recognize that ly found in students after the elementary school years. When students are asked to relate their mathematics is a critical filter for many interesting and exciting careers. Most professions require col- experiences leading to mathematics anxiety they lege mathematics courses through calculus. Most usually relate it to something which occurred in students who are successful in their college math- elementary school. Reasons students at the com- ematics classes took a mathematics class their munity college level give for their mathematics senior year in high school and then took a college anxiety include moving a lot during elementary mathematics class their first semester in college. school, family problems, receiving beatings from a When mathematics classes are taken consecutive- parent when multiplication facts were not learned ly they are much easier than when a number of fast enough, getting behind when ill and never semesters have gone by between classes. catching up, being told they were not smart enougl-, After-school, weekend, and summer programs to do mathematics, and getting behind in mathe- matics becanse they did not understand a par- provide fun, interesting, challenging opportunities for students to learn about mathematics and ticular concept. Many students who had difficulty with mathematics in elementary school have science. Programs which emphasize hands-on ex- periences, applications to real life situations, and learned they have a learning disability. With per- field trips are particularly successful in encourag- sistence and determination many students with ing students to consider mathematics and science learning disabilities are successful in mathematics oriented careers. It is important to take advantage classes. of resources within a community when planning Acknowledging mathematics anxiety and sug- programs. Many people are willing to share infor- gesting ways to overcome it helps students. Stu- mation about their career when they are given the dents can be given a list of books which discuss opportunity to do so. mathematics and test anxiety and how to deal with Environmentalists are just realizing what Na- it. The most effective way for a student to overcome tives have known for centuries. Careers which mathematics anxiety is to have opportunities to be relate to environmental issues are important for successful in solving mathematical problems. people throughout the world. Native students (Langbort, 1985; Tobias, 1978) Many activities for should be informed of the opportunities to help discovering patterns and for solving logic problems their communities by studying various fields of improve attitudes toward mathematics. There are science related to preserving and maintaining the many books available which provide interesting environment. Water pollution, toxic waste, agricul- Indians Nations At Risk: Solutions for the 1990s Instructional methods that tural and industrial chemicals are some of the topics which can be addressed. The wise use of motivate Native students, challenge technology can best be monitored by people who their minds, stimulate their understand the technology. creativity and initiative. Medical pereonnel are needed in rural areas The diversity of Native peoples provides and small towns throughout the United States. tremendous opportunity for developing materials Native people need to encourage their children to for mathematics and science classes which can be consider pursuing a career in one of the many shared with schools throughout the country. A health related professions. Dieticians, nurses, vision where all children are given the opportunity radiology technicians, physical therapists, occupa- to share in the joy of holistic lifelong learning is ti on al therapists, family physicians, and shared by people from all walks of life and from physicians who are specialists are some of the many races. There is beauty in learning and recog- health related professions students can consider. nizing that all of life is interrelated. One day that Each area where Natives live have specific vision will come true when all work together for the needs regarding the environment and the People throughout the country should children. economic and business situations. People are demand that students be taught in ways which needed who have the vision to develop small busi- build on their learning strengths. Students, nesses to meet the needs of the area. Attorneys are regardless of their ability level, learn new concepts needed to provide necessary information and ex from the concrete (hands-on) to the pictorial to the pertise for protecting the interests of the people. abstract (Piaget, 1952; Brownell, 1935). Activities Biomedical engineers, chemical engineers, civil en- are needed to bridge the gap from the pictorial to gineers, electrical engineers, biological engineers, the abstract and need to be included in the learning and other engineers are needed to protect the en- activities. vironment and help the community. Leaders with Learning is fun. Math is fun. Science is fun. vision who are aware of long-term needs of the Teaching needs to stimulate children's natural Native community are needed. Educators with curiosity. Teachers do not necessarily have to rein- sensitivity, ability, and knowledge of subject mat- vent the wheel. They can adapt material to make ter are needed to provide role models for the it appropriate for the area in which students live. children. Our children are precious. They are the There are many materials available which provide future. challenging, enriching, and relevant experiences Parents and other members of the extended in mathematics and in science. Using manipula- family have a tremendous influence on children. tives throughout the school years as new concepts Encouragement and belief in a child's ability to and assist are introduced will stimulate interest persevere through the challenging times on the understanding. When materials are provided in a path to a career that will be fulfilling to the in- mathematics laboratory type of situation, students dividual and will provide thE opportunity to enrich enjoy solving problems posed using various and help the community are crucially important. materials. Consider the following example. When Many students state that the reason they stayed Students an apple is cut crosswise, a star is formed. in school was because someone believed in them does the star have? can be asked how many points and encouraged them when they were discouraged. How many seeds are there? Why is a star formed? An activity which helps middle school students How was the apple formed on the tree? In what discover the importance of mathematics in careers soil and weather conditions do apples grow? How is "mathenger hunt," based on a scavenger-hunt of many types of apples are there? Which type idea. Teams of students are selected by the teacher. apples make the best juice, pie, snacks? Wh,...t Students gather a minimum of twenty signatures chemical is found in apples that is calming for from people who state they use mathematics in people? their career. The class decides on a point system. In-service training is helpful for teachers to Additional points can be earned by obtaining addi- learn to use new manipulatives and other new tional evidence from the people related to their use materials. There are various programs available of mathematics. Some side effects of the project for providing elementary teachers with experience were offers by people to speak to the class, scrap- in using mathematics and science materials. Some books of the signatures and other information were programs teach specific mathematics and science shared with other classes, and the class compiled classes. Some programs combine both aspects. word problems relating to careers (Falba & Weiss, Open Math is a resource book for a set of videos, 1991, pp. 88-90). originally television programs, developed at Arizona State University by Jonathan Knaupp and Mathematics and Science paintings (Serra, 1989). When mathematics and Gary Kna.niller. Teachers can r:atch a video each science units are developed using the intemats of week and try one new activity. The following week the students, motivation and enthusiasm for learn they can share their successes and their failures ing are increased. Tesselations, Pascal's Triangle, and learn from one another. Littie by little they and the Fibonacci Sequence are topics which relate expand their repertoire of instructional methods. easily to Native mathematics and science lessons. Change occurs slowly in education. For change Tesselations, otherwise known as til4ngs, can be taught, teachers to occur in the way children are created using Native designs. Pascal's 'Mangle how to use must be provided opportunities to learn algebraic, probability, and can be used to develop teaching. A one-day workshop is new methods of network concepts, and to create petterns, artwork, change the way in not sufficient to help a teacher and beadwork designs. The Fibonacci Sequence which she/he teaches. Summer school classes or botany, architectuT e, art, and can be used to study workshops combined with regularly scheduled in- packaging and marketing topics in advertbing. It wiiich service during the academic year is a way in medieval times since can be included in a study of and to effect change. It is difficult, frustrating, Fibonacci lived in the thirteenth century. (Preston, slighly frightening to change teaching methods. Teachers need encouragement and support to 1985) change the way they teach. An example of the type of person who can inspire others to do their best is Dr. Helen Neely Cheek. Dr. Cheek was a wonder- ful, inspiring teaeher and educator who taught elementary and secondary teachers new methods Linda Skinner has developed a week-long of teaching. She provided opportunities to learn teachers, paraprofessionals, process where little by little until each new methods of instruction Elders, parents, students, and other community teacher was comfortable using row materials and members collectively experience various aspects of She emphasized recog- new methods of instruction. culture. From their experiences and unique nizing each student's strengths and weaknesses perspectives they create and develop culturally and building on their strengths to overcome their relevant learning activities and materials. They weaknesses. (Preston & Skinner, 1986) with return to their individual communities Curriculum Development Strategies renewed enthusiasm strengthened by the focused creating the best energy on the common goal of Mathematical Connections is standard four of education experiences for the children of their com- the NCTM Standards for kindergarten through munities. high school. Relationships between mathematical -las are encouraged. topics and other curriculum Mayan and Inca Science "Unless connections are made, children will see mathematics as a collection of isolated topics" and Mathematics (NCTM Standards, 1989, P. 33). Examples of culturally relevant mathematics Mathematics as Communicadon is Standard material related to other disciplines are Mayan be Two of the NCTM Standards. Students need to Mathematics and Inca Mathematics. A unit on able to clarify their thinking to fully understand Mayan Mathematics or on Inca Mathematics can mathematical concepts. Communication by repre- be developed to include geography, social studies, senting concepts using manipulatives, by using methods, astronomy, archaeology, agricultural diagrams and drawings, by discussing ideas, by development of corn, development of potatoes, art, listening, by reading, and by writing are all impor- glyph translations, architecture, language arts, communicated tant. Knowledge must be able to be acoustical music, physical education, peobability, to oth ers before its usefulness and applicability can engineering, civil engineering, textiles, minerals. be determined and utilized. Various topics could be studied in different grade Culturally Televant mathematics and science levels. materials address both of t.he above-mentioned standards. Geometric designs found in all Native cultures can be used to develop geometric and algebraic understanding. Art can be used to Mayan mathematics was highly developed. The develop concepts of estimation to know the amount Mayans long concept of zero was recognized by of paint, clay, beads, etc. which are needed for the before it. was recognized in European mathematics. project. Chemistry and mathematics can be used A concept of zero is necessary before a mathemati- to obtain the colors needed for art projects. cal system can have a place value system. It al- Geometry is used to develop perspective in many 7 Indians Nations At Risk: Solutions for the 1990s were built without the use 00 Itheels to transport lowed Mayans to think about and writ,' numbers the gigantic stones and without mortar between that were very large. They had two separate but related mathematical systems. One was a the stones. After centuries of earthquakes, many vigesimal, base twenty system, used to count of the Inca stmctures remain intact while modern structures are destroyed in recent earthquakes. things and the other was an adjusted base twenty system used to count time. Only three symbols Exemplary Schools, Programs, and numbers, a dot to were used to represent all their Projects represcnt one, a bar to represent five, and a cacao bean to represent zero. (Most references state that The Colorado School of Mines in Golden, Colorado, has held summer math camps for the symbol for zero is a type of shell. Patrick Scott, (Jniversity of New Mexico, has been working with American Indian and Alaska Native middle school Mayans in Guatemala who believe the symbol was students since 1988. The first one was held in a cacao bean.) conjunction with the American Indian Science and "The Mayas thought of time as a never-ending Engineering Society (AISES) and was funded by flow into the future and back into the past. They the Educational Foundation of America. Dr. Bar- believed that history repeated itself when condi- bara Bath is the director and Dr. Ardel Boes, head of the mathematics department, is the ad- tions were exactly as they had been in the past. For this reason, it was extremely important that the ministrator of the math camp each summer. The priests continually study prevailing conditions in purpose is to expose Native students to mathe- order to compare them with past conditions" (Cal- matics in various ways and provide opportunities lahan, 1969). Charles H. Smiley, Professor of to learn about various mathematics and science oriented careers. One of the major goals is to Astronomy, Brown University, Providence, Rhode Island, in conversations with the author, claimed develop or reinforce positive attitudes toward to have predicted a hurricane for the New England mathematics. Results from pre and post tests of coast using information from the Mayan Dresden attitude and cognitive reasoning indicate improve- Codex, one of the remaining books of the Maya. He ment is fairly consistent for all the students. The boarded up his house on the coast and had no type of mathematics studied helps students in their regular math classes during the year. Ac- damage. Mayans were able to construct temples, tivities in the mathematics classes emphasize pat- pyramids, and ball courts with admirable terns, logical thinking, and spatial relations. precision. Acoustics allowed priests to whisper Hands-on, manipulative activities are used exten- sively. For example, pattern blocks are used to and be heard across the plaza. Astronomical obser- vatories allowed them to observe and accurately make designs and patterns in the mathematics record information regarding planets, comets, and classes. The designs can be duplicated using the stars. As communication with Chinese mathe- computer. Computer classes include developing maticians and astronomers increases, astronomi- tesselations and other geometric designs. Logo is cal information for studying the correlation of used because of its problem solving capabilities for Mayan dates with current knowledge of historical students. Since the second year mathematicians dr.tes improves. Agricultural accomplishments of and computer scientists from Bell Laboratories of the Mayas include development of many crops we AT&T have volunteered to be mentors to assist the students in their projects. Students' informal use today. Raised fields were used in swampy areas and water reservoirs were established in dry areas. evaluations state one of the benefits of the math The Incas of South America were great or- and camp is learning to deal with new situations ganizers. Their highway system was more exten- adjust to them. Parents contacted Dr. Bath and sive than the Roman highway system that is informed her that their child's attitude toward recognized in history books. Parts of the Inca high- school, achievement in school, and self-concept had way system are still used. Irrigation methods used improved after attending the math camp. The second year a student program and a terracing for maximum use of water. Throughout teacher education program was funded by the Peru there are Inca terraces on the sides of moun- Department of Education. A Middle School tains that are still used. Surgeons were able to perform delicate, suftessful brain surgery. They Teacher Enhancement Geometry Program was had an extensive knowledge of herbal medicine. funded by the National Science Foundation (NSF). The weaving with feathers and gold thread canno A Young Scholars Program and a Middle School be duplicated today. The artistry using gold is Teachers Algebra program was funded by NSF the unmatched with today's knowledge of technology. this 1 year. The summer of 1991 another Young It is still a mystery hlw the structures of the Incas Scholars Program and Middle School Teachers 8 9 Mathematics and Science program will be funded by NSF. The program has gram. Students in five elementary schools in Rapid City and Kyle will participate in after-school been designated a Program of Excellence by the academic year and in summer enrichment ac- Colorado Commission for Higher Education and has received five years of funding for the Middle tivities. Students will study science and engineer . ing as they relate to their everyday lives. Science School American Indian and Alaska Native Math Made Simple is a two week non-residential sum- Camp and the Teacher Education program. The University of Kansas, Haskell Indian mer camp. Since NACME requires community partnership and matching funds, Amoco, the Junior College, and the Bureau of Indian Affairs South Dakota Community Foundation, U. S. West together developed and presented two eight-week Communications, and Title II (United States mathematics and science elementary teacher in- Department of Education) now help support the stitutes in the summers of 1988 and 1989 funded program. (Campbell, G. Jr., 1990) by the National Science Foundation. Funding has Porter Middle School in Austin, Texas, has a been allocated by the National Science Foundation very innovative science teacher, Sharon Mitchell. for the summers of 1991 through 1993. The Math- She is the adviser of the Earthnauts, a group ematics and Science Teachers for Reservation started "by kids for kids." The students work in a Schools (MASTERS) Project focuses on improving biodome to test theories of growing food without classroom instruction skills of teachers. Three themes are emphasized: students' concrete pesticides or without otherwise harming the earth. They grow vegetables in water instead of soil. The firsthand (hands-on) experience with the concept group has published a book of poetry, written a being taught; cultural relevancy of the curriculum proposal to NASA to gather ground-level data to objectives, materials, and teaching technique, and help the agency fine-tune readings from orbiting positive role models and community involvement satellites, and produced a series of comic books in instruction. about the environment for elementary students. Science is studied using hands-on experiences. Students are in charge of the biodome and the Air and space topics are a focus of many of the plants grown there. Involvement in Earthnauts activities. Science and mathematics activities are has caused many students to plan to go into various related to all aspects of life using a whole language branches of science. (Smith, S., 1990, pp. Bl, B4) approach. Biographical information about Native The New Jersey Institute of Technology spon- American scientists has been compiled. Science and mathematics activities related to the sored an Experimental Mathematical Science and Communications (EMSAC) program for seventh biographical information will be developed in fu- grade students during the summers of 1983 and ture workshops. The vision of the program is to 1984. The 1984 program focused on mapping and develop mathematics and science activities within surveying, observational astronomy, Apple Logo, all aspects of the curriculum. During the summer communications, and probability and statistics. of 1991 there will be thirty-five new students and The prorain was for inner-city students. NCTM's twelve returning students. MASTERS' goals in- 1981 Yearbook, Teaching Statistics and Prob- clude improving mathematics and science instruc- ability, was used to develop the probability and within the tion at the individual teacher level, statistics concepts introduced. Students played schools of the participating teachers, and within coin toss games and games using dice. Games were the overall system of reservation schools. determined tu be fair or unfair by playing the "Change is not a quick process: and change games and keeping results. After discussions of agents are not made overnight. MASIERS' results of the games, students determined which goal is to produce a core of skilled mathe- games were fair or unfair. A Casino Day was held matics and science teacher leaders who can work with their fellow faculty and com- to study the concept of determining odds. One of munity, not impose improved science and the learning objectives was to have students real- mathematics teaching programs. (Smith. ize that there is at least one best strategy for W.S.. 1990, pp. 1-2) playing each game. After Casino Day the games National Action Council for Minorities in En- with obvious strategies were discussed. Pascal's gineering, Inc. (NACME) launched TechForce triangle was used to help explain the odds. 2001, a program for students who will be earning Geometric probability was introduced by discuss- engineering degrees at the beginning of the new ing variation s of th e Lady or the Tiger game. Deter- mining the winnings when a game ends century, in 1989. NACME, the South Dakota School of Mines and Technology, AISES, and the prematurely while a person is ahead was studied. Monte Carlo techniques and simule ions were local school districts in Rapid City and Kyle formed a coalition to provide a Scientific Knowledge for used to discuss prizes in boxes of cookies. Student Indian Learning and Leadership (SKILL) pro- 9 1 0

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DOCUMENT RESUME RC 018 626 ED 343 767 Preston, Vera AUTHOR Mathematics and Science Curricula in Elementary and TITLE Secondary Education for American Indian and Alaska Native Students. Department of Education, Washington, DC. Indian SPONS AGENCY Nations At Risk Task Force. PUB DATE 91 18p.; In: Indian Nations At Risk Task Force NOTE Commissioned Papers. See RC 018 612. Information Analyses (070) PUB TYPE MF01/PC01 Plus Postage. EDRS PRICE *American Indian Education; American Indians; DESCRIPTORS Cooperative Learning; *Demonstration Programs; *Educational Strategies; Elementary Secondary Education; Experiential Learning; Higher Education; Mathematics Education? *Mathematics Instruction; Science Education; *Science Instruction; Teacher Education; *Teaching Methods Culturally Relevant Curriculum IDENTIFIERS ABSTRACT Issues related to the improvement of mathematics and science education pertain to Native students as well as to the general population. Native students are most successful at tasks that use visual and spatial abilities and that involve simultaneous processing. Instruction should build on Native students' strengths. Experiential learning and cooperative learning are two methods that are particularly effective with Native students in improving student attitudes and problem-solving abilities and reducing mathematics anxiety. Storytelling techniques can be used to develop culturally relevant problems. Career days show students the uses of mathematics in the real world. Curriculum development strategies include establishing the relationships and connections between mathematics and other subjects, and incorporating culturally relevant materials, such as Maya or Inca mathematics and science. Strategies of exemplary programs include summer math camps for Native students, summer institutes to improve teacher instructional skills and methods, after-school and summer enrichment activities in science and engineering, instructional materials developed to accompany a science series on public television, magnet schools, after-school college preparatory courses in mathematics, and parent resource centers. Recommendations are offered related to instructional methods, program development, and federal funding. This paper contains over 130 references. (SV) 1:**********W***********W***************w******************************* Reproductions supplied by EDRS are the best that can be made from the original document. ***********************.t*********************************************** Mathematics and Science Ctuvricula in Elementary and Secondary Education for American Indian and Alaska Native Students U.S. DEPARTMENT OP EDUCATION Once ot Educenonel Research end ['Mover/tem EDUCATIONAL RESOURCES INFORMATION CENTER IERICI document nes been reprOduced as received from Inn person or poem/anon originating it .ve boon made to improve Vera Preston C Minor changes reproduction au% Ay Points 0/ view os opinions steles) in tniedockr mint do not necessarily represent Octet OERI poettion or policy Introduction ferences, the message obtained from the testimony accentuates the common concern of Native people Wake up, America! Your child:en are at risk. for educating their children. They want their Three of every four Americans stop studying children to be educated in a manner which enriches mathematics before completing career or Job their horizons and guarantees a variety of options prerequisites. Most students leave school for lifelong learning and for various vocations. without sufficient preparation in mathe- For years reports have stated the need for matics to cope either with on-the-job reforming mathematics and science education in demands for problem-solving or with college the United States. A Nation at Risk (National requirements for mathematical literacy... Since mathematics is the foundation of Commission on Excellence in Education 1983), science and technology, it serves as a key to Educating Americans for the 21st Century (Nation- opportunity and careers.. Moreover, mathe- al Science Board Commission on Precollege Educa- matics contributes to literacy certain distinc- tion in Mathematics, Science and Technology tive habits of mind that are of increasing 1983), and Everybody Counts -A Report tu the Na- importance to an informvd citizenry in a tion on the Future of Mathematics Education (Na- Despite massive effort, technological age. ... tional Research Council 1989) are representative relatively little is accomplished by remedia- of the reports concerning mathematics and science not educators. tion programs No one education. Compared to students throughout the mathematician*, or researchers knows how to reverse a consistent early pattern of world, students in the United States rank in ability low achievement and failure. Repetition rare- with students from Third World countries. Nation- ly works; more often than not. it simply al Assessment of Educational Progress (NAEP) reinforces previous failure. The best time to reportsaf results of 1986 assessments indicate that learn mathematics is when it is first taught; ability to answer problems requiring higher order the best way to teach mathematics is to teach thinking is lacking for most students in the United it well the first time. (Everybody Counts. States (Brown, Carpenter, Kouba, Lindquist, Sil- 1989, pp. 1-3. 13) ver, and Swafford, 1988, pp. 911-248). Some Testimony from the Indian Nat:xis at Risk reports have addressed the issue of what needs to (INAR) Task Force meetings, regional hearings, be done to solve the problems. Ciaims of easy and written testimony submitted to the INAR Task solutions to difficult problems should be viewed Force represents the diversity of Native peoples with great suspicion. Research indicates the way and the variety of schools attended by American students learn is not always the way students are Indian and Alaska Native students. The common taught. Since teachers tend to teach the way they thread existing throughout the extensive tes- were taught and the way in which they learned, the timony is the need for American Indian and Alaska vicious cycle ofmiseducatim continues to handicap Native students to have their culture acknow- the lives of children of all ages, abilities, and races. ledged and to be taught in the ways they learn. The The National Council of Teachers of Mathe- type of school, the type of class (language arts, matics (NCTM) has responded to the need for social studies, science, mathematics), the area of change in mathematics teaching by producing Cur- the country does not change the need to be riculum and Evaluation Standards for School respected as a human being and to be taught in a Mathematics (NCTM, 1989). It is a set of stand- understand manner that allows the individual to ards for mathematics curricula in kindergarten and internalize the material to be learned. All through high school developed by classroom children, regardless of race, gender, or ability, teachers, supervisors, educational researchers, deserve to be taught in the ways they learn by teacher educators, and university mathematicians building on their strengths and overcoming their working together. NCTM is very much aware ofthe weaknesses. Rather than emphasizing the dif- 1 Indians N Alons At Risk; Solutions for the 1990s processing of knowledge which enables them to skepticism of people toward change after the *new perceive various elements as a whole. Seeing the math" of the sixties which imposed change from the whole picture, a holistic, global process is referred top down. The current recommendations stress the to as simultaneous processing. Processing intbr- importance of all stIdents learning mathematics mation sequentially and in an analytical manner and the importance of evising mathematics in- struction. The National Science Teachers of is refened to as successive processing. (Das, Kirby, & Jarman, 1975. Quoted in More, 1986, pp. 8-9) America (NSTA) has developed a Statement of Research results indicate Native students use Initiatives similar to the NCTM Standards. Em- simultaneous processing more frequently and ef- phasis is on more unified, in-depth hands-on fectively than successive processing. It also indi- science curricula for all students kindergarten cates that Native children are more field through twelfth grade. i,dependent (FI) than field dependent (FD) and What can be done to improve the mathematics that they tend to think in relational styles rather and science education of all children including than in analytic styles. (More, 1986, pp. 9, 12) American Indian and Alaska Native children? Research on culturally based communication Why is mathematics and science education so im- styles related to the classroom has been done by portant to all students? This paper will address Scollon and Scollon (1983) and Phillips (1972). the issues in mathematics and science education. Studies of teacher effectiveness which relate to Mathematics and Science styles of communication of Native Indian students have been done by Kleinfeld (1970). Traditional Mazhematics and science influence the lives of Native cultures teach children to watch, listen, everyone. Scientific, mathematical, and tech- think and then do the task to be learned. Many nological literacy is crucially important for citizens classrooms encourage trial-and -error learning. to understand the technological world in which Many students become quite proficient at guessing they live. Mathematics has been called "the queeii It is the study of until they get the correct answer. Trial-and-error and servant of the sciences." learning is the antithesis of traditional Native patterns and relationships and is the foundation learning styles. on which understanding of many fields of study is built. An understanding of mathematical concepts Field independence is the degree to which an and mathematical reasoning is necessary in al- individual can separate a figure from its background, a part from the whole, oneself most every occupation in today's technological from the environment or other people. A field society. Mathematics has been called a critical independent (FI) person is more able to pro- filter because so many occupations require a math- vide an organizational structure to a disor- ematical background. Many people perceive math- ganized set of facts or observations (e.g.. ematics as an infinite set of meaningless rules making a mental map of the surrounding which are impossible to memorize. Mathematics is terrain). A field dependent (FD) person is less constantly changing; more than half the mathe- able to separate a part of the whole, but is matics known today has been developed since The Fl more conscious of other people... . 1940. Science and mathematics are interdepen- person will impose his or her own structure on a field (spatial or conceptual) more readi- dent in our technological age. For example, under- ly. The FD person is mom socially aware and standing of environmental issues such as water more responsive to those around him or her. availability and water pollution require an under- (More, 1986, pp. 10-11). standing of both mathematics and science. There is considerable overlap between the learning styles of Indian and Non-Indian stu- Mathematics and Science dents. Indeed the similarities are probably Learning and Teaching greater than the differences. Furthermore, it is likely that the learning styles of some Fundamental change in the way mathematics groups of Non-Indians will be remarkably and science is taught is necessary (NCTM Stand- similar to those of Native Indians. This could ards, 1989; Willoughby, 1990). Information from occur when there are similar factors affecting mathematics and science educational research and the way of life such as degree of remoteness from cognitive psychologists should be used to of the community, economic level, and sig- develop plans for organizing instruction for math- nificant cultural differences from the ematics and science classes. Research indicates majority. However in such situations cul- tural differences should not be downplayed. American Indian and Alaska Native students are Even though the pattern of learning styles is most successful at tasks which use visual and similar, the causes and the methods of work- spatial abilities. (More, 1986) Many studet.i,s, in- use holistic, global cluding Native students, Mathematics and Science Mg with the learning styles may not be." to understand concepts and to transfer their learn- (More. 1986. p. 20) ing to other situations (Surdam,1986). Teacher guidance ;i3 important in bridging the gaps from It is important to realize that teaching styles the concrete to the pictorial to the symbolic stage. that are appropriate for Native students are also (Kouba et al., 1988b) Educators from the early appropriate for other students. Methods of teach- pert of the century to the present have advocated ing found in many schools are not appropriate to the us,) of physical, or concrete, materials in the the ways children learn. This is one of the major classroom. (Brownell, 1935; Burns,1986; Montes- reasons for the need ofremedial classes. If students 806,1965; Piaget,1952; Reys,1971; Suydam,1986) the were taught concepts in the way they learn, Recommendations from reports of the most recent need for remedial classes would be greatly NA EP results encourage the use of concrete diminished. Since Native students have strengths materials and activities. NCTM has recommended in using visual/spatial/perceptual information, the use of physical, concrete materials in the class- new material should be introduced using methods room. (NCTM 1980; NCTM 1987) An Agenda for building on their strengths. The overall view of Action: Recommendations for School Mathematics new information should be stated before the infor- of the 1980s (NCTM, 1980, p. 12) includes the mation is broken down into small segments. Ad- following: "Teachers should use diverse instruc- vance organizers should be used. Students' tional strategies, materials, and resources, such as strengths should be used to develop skills in other the use of manipulatives, where suited, to il- ... modes of learning. (More, 1986, p. 21) Students lustrate or develop a concept or skill." should have the right to learn beginning material There are many benefits which occur as a result in their native language to allow them to develop of using manipulatives. Mathematics anxiety is a cognitive base on which to learn for the almost non-existent in classes where manipula- remainder of their lives. (McDonald, 1989, p. 16) tives are used frequently. The attitude toward Children are born with a love of learning and learning mathematics is consistently more positive an innate curiosity. The first three years of life a as a result of using manipulatives and of par- child learns a tremendous amount of information. ticipating in cooperative learning. Student under- Parents are the first and most important teachers standing of concepts is greater among all ability of a child. How can a child's natural desire to learn levels. Parental involvement is easier to obtain by about his/her surroundings be nurtured and main- having some activities done at home using common tained in academic situations? Bow can students household materials. Students need guidance with learn the necessary material in the various clas- the teacher as facilitator on the use of the ses? Many theories exist regarding how people manipulatives. Assistance and guidance in under- learn. There is danger in stating that n specific standing the concepts represented by the group of individuals learn in a particular way. It is manipulatives and in bridging activities to go from important to be sensitive to the needs of each the concrete to the pictorial to the symbolic or individual and help that individual build on her/his abstract are essential in using the materials to best strengths and increase ability in areas of weak- advantage (Heddens,1986). Learning what to do ness. (More, 1986) Many people have a preferred and why enables students to retain information learning style. Many researchers stress that stu- and transfer the learning to other situations dents should be allowed to learn from their (Burns,1986). Fuson (1981) writes that there is a strengths. Research in mathematics anxiety indi- meta-cognitive benefit in the use of concrete ob- cates that students overcome their anxiety and are jects. Teachers' sensitivity to styles oflearning will able to perform better in mathematics if they build help their students. A balanced view of various success on success. (Tobias, 1978) methods of teaching and learning will assist stu- Manipulative, hands-on, Materials dents to recognize their particular style of learn- ing. (More, 1986) The teaching/learning style is "Tell me, and I forget; Show me, and I remem- determined by the learning task. Learning how to ber; Involve me, and I understand." Ancient learn and recognizing the underlying concepts and Chinese Saying. features of mathematical problems is very impor- Students who use hands-on materials, other- tant. better at- wise known as manipulatives, have titudes toward mathematics and better ability to Cooperative Learning problem solve (Suydam and Weaver,1975; Traf- Current issues in educational circles include ton,1975). Students who use manipulatives and teaching thinking through the curriculum, prob- are guided to bridge the gap from concrete to pic- lem solving, cooperative learning, writing across torial to symbolic representations are better able the curriculum, and learning how to learn 3 4 Indians Nations At Risk: Solutions for the 1990s dents in gifted classes and in remedial classes at (Simon,1986). Cooperative, or group, learning is being recognized as important in learning con- all levels. cepts. Since people work cooperatively in most Storytelling and Problem Solving in situations outside school, studeuts should have Mathematics and Science. opportunities to work together in school. The many benefits which occur as a result of cooperative Storytelling as a means ofteaching is tradition- learning are an ability to communicate verbally the ally used by American Indians and Alaska Natives. information and concepts being studied, an under- Using stories to pose mathematical or scientific standing of other peoples' attitudes, a positive at- problems is a natural way to relate situations of titude toward learning, recognizing there are the student's home life and culture within the various paths to the solution, and clarifying one's mathematics classroom. Students arrive in school own understanding. !Slavin, 1987) with the ability to solve problems which have Cooperative learning is a popular concept at the meaning for them. School textbooks and mathe- moment in education. The merits and disad- matics lessons frequently convince students they vantages of cooperative learning are discussed in are unable to solve the problems which had pre- four articles in the Febmary 1991 issue of Educa- viously been easy for them. One of the important tional Leadership (Kohn, 1991; Slavin, 1991). It is skills in solving problems is drawing pictures rep- important to note that cooperative learning has resenting the problem. (Davis & McKillip, 1980, been used successfully by teachers over the years pp. 80-91) without the label of "cooperative learning." Much An ancient example of using storytelling to pose of the discussion in the articles relates to giving a mathematical problem is the story of the person rewards for group effort. Rather than getting going to St. Ives. "As I was going to St. Ives, I met bogged down in semantics or side issues, teachers a man with seven wives. Each wife had seven cats, need to consider the advantages of cooperative each cat had seven kits. How many were going to learning and apply the concept as appropriate to St. Ives?" Only one person was going to St. Ives; their particular situation in their specific class- the others were leaving St. Ives. The problem in- room. Teachers must acknowledge that coopera- volves listening and reasoning. The mathematics tion in the world outside the classroom is expected involved is rather straightforward. and necessary for success in projects throughout Students enjoy making up story problems when the business world. Mathematics and science the teacher provides an encouraging, accepting, learning should involve learners working together. respectful atmosphere in the classroom. The entire Mathematicians and scientists talk to each other family could participate in posing story problems and try new ideas and check each other's reason- which incorporate mathematical and scientific ing. They do not work in total isolation. Research problems. For example each of the following math- overwhelming supports the concept of cooperative ematical type of questions could be posed in a story. learning and working together. It is particularly How much flour is used for making fry bread? How appropriate to the learning style of many Native does the volume of the fry bread change when it is students. There are many different forms of cooked? When popcorn is popped, how much does coope. ative learning. The most successful ap the volume change? How many seeds are there in proaches have incorporated group goals and in a watermelon? What are some strategies for deter- dividual accountability (Slavin 1991). Many of the minitig how many seeds are in the watermelon? advantages of cooperative learning are social as What are the differences in types of seeds? What well as academic. Mathematics anxiety is practi- are the shapes of leaves? How many leaves are on cally non-existent when students are allowed to the plant? How many steps does each person in the work together to discuss and learn new informa- family take for walking the same distance? (What is the stride of each person?) How many sheep per tion. Many teachers used cooperative learning acre can the land support without irrigation? How before it was called cooperative learning. Students many cattle per acre can the land support? How worked together and helped one another and many bales of hay can be put in a pickup? What is the most efficient shape to build a house if recorded the results of their experimentation, their searching for patterns, or their explanation of new materials are scarce? (A circle is the most efficient shape to provide the most area for the same definitions, concepts and material. Each group turned in one paper with their results. Students amount of fencing. A square is the most efficient and wrote were each responsible for the material straight-sided figure. Why?) How many days did it individual reports later. The method of having rain last week, last month, last year, ten years ago? students work together was successful with stu- 4 Mathematics and Science What is the difference in temperature from 5:00 mathematics and science activities for students of AM to 5:00 PM on the same day? all ages. Career Education Mathemafics Anxiety Many students develop 6. fear of mathematics Career days for elementary, middle, and junior high school students enable the students to learn during the elementary, middle school, junior high about different types of careers. Native students years which cripples their ability to l3am mathe- matics in high school and later. Other students do should be encouraged to consider careers in fields which will help their communities. Careers in well in mathematics classes until they reach al- health fields, environmental science, agricultural gebra or geometry. When they have difficulty in mathematica, they quit taking elective mathe- science, various type of engineering, attorneys, matics classes. They take the path of least resis- counselors, and business all have the potential for helping their communities. Expanding Your tance when taking required mathematics classes. Horizons Workshops and Career Days were It is important for students to be encouraged, pos- originally organized for females. The format is sibly required to take mathematics classes each appropriate for American Indian and Alaska Na- year of high school. Students must realize that tive children. Students should be encouraged to mathematics is a critical filter. The more mathe- continue their education so they will be prepared matics a student studies, the higher the expected to help their community in many ways. lifetime earnings, and the more opportunities for Students generally lose their innate en- employment are available. Students can choose a thusiasm for science and mathematics during the job or career based on what they enjoy doing rather crucial junior high years. Therefore, it is critically than on what requires no mathematics. Counselors important to provide elementary, middle, and should be required to insist that students stay in junior high school students opportunities to ex- mathematics classes rather than allowing them to perience challenging and interesting projects and drop out when the going gets tough. programs in mathematics and science. It is also Mathematics anxiety and test anxiety is usual- critically important for students to recognize that ly found in students after the elementary school years. When students are asked to relate their mathematics is a critical filter for many interesting and exciting careers. Most professions require col- experiences leading to mathematics anxiety they lege mathematics courses through calculus. Most usually relate it to something which occurred in students who are successful in their college math- elementary school. Reasons students at the com- ematics classes took a mathematics class their munity college level give for their mathematics senior year in high school and then took a college anxiety include moving a lot during elementary mathematics class their first semester in college. school, family problems, receiving beatings from a When mathematics classes are taken consecutive- parent when multiplication facts were not learned ly they are much easier than when a number of fast enough, getting behind when ill and never semesters have gone by between classes. catching up, being told they were not smart enougl-, After-school, weekend, and summer programs to do mathematics, and getting behind in mathe- matics becanse they did not understand a par- provide fun, interesting, challenging opportunities for students to learn about mathematics and ticular concept. Many students who had difficulty with mathematics in elementary school have science. Programs which emphasize hands-on ex- periences, applications to real life situations, and learned they have a learning disability. With per- field trips are particularly successful in encourag- sistence and determination many students with ing students to consider mathematics and science learning disabilities are successful in mathematics oriented careers. It is important to take advantage classes. of resources within a community when planning Acknowledging mathematics anxiety and sug- programs. Many people are willing to share infor- gesting ways to overcome it helps students. Stu- mation about their career when they are given the dents can be given a list of books which discuss opportunity to do so. mathematics and test anxiety and how to deal with Environmentalists are just realizing what Na- it. The most effective way for a student to overcome tives have known for centuries. Careers which mathematics anxiety is to have opportunities to be relate to environmental issues are important for successful in solving mathematical problems. people throughout the world. Native students (Langbort, 1985; Tobias, 1978) Many activities for should be informed of the opportunities to help discovering patterns and for solving logic problems their communities by studying various fields of improve attitudes toward mathematics. There are science related to preserving and maintaining the many books available which provide interesting environment. Water pollution, toxic waste, agricul- Indians Nations At Risk: Solutions for the 1990s Instructional methods that tural and industrial chemicals are some of the topics which can be addressed. The wise use of motivate Native students, challenge technology can best be monitored by people who their minds, stimulate their understand the technology. creativity and initiative. Medical pereonnel are needed in rural areas The diversity of Native peoples provides and small towns throughout the United States. tremendous opportunity for developing materials Native people need to encourage their children to for mathematics and science classes which can be consider pursuing a career in one of the many shared with schools throughout the country. A health related professions. Dieticians, nurses, vision where all children are given the opportunity radiology technicians, physical therapists, occupa- to share in the joy of holistic lifelong learning is ti on al therapists, family physicians, and shared by people from all walks of life and from physicians who are specialists are some of the many races. There is beauty in learning and recog- health related professions students can consider. nizing that all of life is interrelated. One day that Each area where Natives live have specific vision will come true when all work together for the needs regarding the environment and the People throughout the country should children. economic and business situations. People are demand that students be taught in ways which needed who have the vision to develop small busi- build on their learning strengths. Students, nesses to meet the needs of the area. Attorneys are regardless of their ability level, learn new concepts needed to provide necessary information and ex from the concrete (hands-on) to the pictorial to the pertise for protecting the interests of the people. abstract (Piaget, 1952; Brownell, 1935). Activities Biomedical engineers, chemical engineers, civil en- are needed to bridge the gap from the pictorial to gineers, electrical engineers, biological engineers, the abstract and need to be included in the learning and other engineers are needed to protect the en- activities. vironment and help the community. Leaders with Learning is fun. Math is fun. Science is fun. vision who are aware of long-term needs of the Teaching needs to stimulate children's natural Native community are needed. Educators with curiosity. Teachers do not necessarily have to rein- sensitivity, ability, and knowledge of subject mat- vent the wheel. They can adapt material to make ter are needed to provide role models for the it appropriate for the area in which students live. children. Our children are precious. They are the There are many materials available which provide future. challenging, enriching, and relevant experiences Parents and other members of the extended in mathematics and in science. Using manipula- family have a tremendous influence on children. tives throughout the school years as new concepts Encouragement and belief in a child's ability to and assist are introduced will stimulate interest persevere through the challenging times on the understanding. When materials are provided in a path to a career that will be fulfilling to the in- mathematics laboratory type of situation, students dividual and will provide thE opportunity to enrich enjoy solving problems posed using various and help the community are crucially important. materials. Consider the following example. When Many students state that the reason they stayed Students an apple is cut crosswise, a star is formed. in school was because someone believed in them does the star have? can be asked how many points and encouraged them when they were discouraged. How many seeds are there? Why is a star formed? An activity which helps middle school students How was the apple formed on the tree? In what discover the importance of mathematics in careers soil and weather conditions do apples grow? How is "mathenger hunt," based on a scavenger-hunt of many types of apples are there? Which type idea. Teams of students are selected by the teacher. apples make the best juice, pie, snacks? Wh,...t Students gather a minimum of twenty signatures chemical is found in apples that is calming for from people who state they use mathematics in people? their career. The class decides on a point system. In-service training is helpful for teachers to Additional points can be earned by obtaining addi- learn to use new manipulatives and other new tional evidence from the people related to their use materials. There are various programs available of mathematics. Some side effects of the project for providing elementary teachers with experience were offers by people to speak to the class, scrap- in using mathematics and science materials. Some books of the signatures and other information were programs teach specific mathematics and science shared with other classes, and the class compiled classes. Some programs combine both aspects. word problems relating to careers (Falba & Weiss, Open Math is a resource book for a set of videos, 1991, pp. 88-90). originally television programs, developed at Arizona State University by Jonathan Knaupp and Mathematics and Science paintings (Serra, 1989). When mathematics and Gary Kna.niller. Teachers can r:atch a video each science units are developed using the intemats of week and try one new activity. The following week the students, motivation and enthusiasm for learn they can share their successes and their failures ing are increased. Tesselations, Pascal's Triangle, and learn from one another. Littie by little they and the Fibonacci Sequence are topics which relate expand their repertoire of instructional methods. easily to Native mathematics and science lessons. Change occurs slowly in education. For change Tesselations, otherwise known as til4ngs, can be taught, teachers to occur in the way children are created using Native designs. Pascal's 'Mangle how to use must be provided opportunities to learn algebraic, probability, and can be used to develop teaching. A one-day workshop is new methods of network concepts, and to create petterns, artwork, change the way in not sufficient to help a teacher and beadwork designs. The Fibonacci Sequence which she/he teaches. Summer school classes or botany, architectuT e, art, and can be used to study workshops combined with regularly scheduled in- packaging and marketing topics in advertbing. It wiiich service during the academic year is a way in medieval times since can be included in a study of and to effect change. It is difficult, frustrating, Fibonacci lived in the thirteenth century. (Preston, slighly frightening to change teaching methods. Teachers need encouragement and support to 1985) change the way they teach. An example of the type of person who can inspire others to do their best is Dr. Helen Neely Cheek. Dr. Cheek was a wonder- ful, inspiring teaeher and educator who taught elementary and secondary teachers new methods Linda Skinner has developed a week-long of teaching. She provided opportunities to learn teachers, paraprofessionals, process where little by little until each new methods of instruction Elders, parents, students, and other community teacher was comfortable using row materials and members collectively experience various aspects of She emphasized recog- new methods of instruction. culture. From their experiences and unique nizing each student's strengths and weaknesses perspectives they create and develop culturally and building on their strengths to overcome their relevant learning activities and materials. They weaknesses. (Preston & Skinner, 1986) with return to their individual communities Curriculum Development Strategies renewed enthusiasm strengthened by the focused creating the best energy on the common goal of Mathematical Connections is standard four of education experiences for the children of their com- the NCTM Standards for kindergarten through munities. high school. Relationships between mathematical -las are encouraged. topics and other curriculum Mayan and Inca Science "Unless connections are made, children will see mathematics as a collection of isolated topics" and Mathematics (NCTM Standards, 1989, P. 33). Examples of culturally relevant mathematics Mathematics as Communicadon is Standard material related to other disciplines are Mayan be Two of the NCTM Standards. Students need to Mathematics and Inca Mathematics. A unit on able to clarify their thinking to fully understand Mayan Mathematics or on Inca Mathematics can mathematical concepts. Communication by repre- be developed to include geography, social studies, senting concepts using manipulatives, by using methods, astronomy, archaeology, agricultural diagrams and drawings, by discussing ideas, by development of corn, development of potatoes, art, listening, by reading, and by writing are all impor- glyph translations, architecture, language arts, communicated tant. Knowledge must be able to be acoustical music, physical education, peobability, to oth ers before its usefulness and applicability can engineering, civil engineering, textiles, minerals. be determined and utilized. Various topics could be studied in different grade Culturally Televant mathematics and science levels. materials address both of t.he above-mentioned standards. Geometric designs found in all Native cultures can be used to develop geometric and algebraic understanding. Art can be used to Mayan mathematics was highly developed. The develop concepts of estimation to know the amount Mayans long concept of zero was recognized by of paint, clay, beads, etc. which are needed for the before it. was recognized in European mathematics. project. Chemistry and mathematics can be used A concept of zero is necessary before a mathemati- to obtain the colors needed for art projects. cal system can have a place value system. It al- Geometry is used to develop perspective in many 7 Indians Nations At Risk: Solutions for the 1990s were built without the use 00 Itheels to transport lowed Mayans to think about and writ,' numbers the gigantic stones and without mortar between that were very large. They had two separate but related mathematical systems. One was a the stones. After centuries of earthquakes, many vigesimal, base twenty system, used to count of the Inca stmctures remain intact while modern structures are destroyed in recent earthquakes. things and the other was an adjusted base twenty system used to count time. Only three symbols Exemplary Schools, Programs, and numbers, a dot to were used to represent all their Projects represcnt one, a bar to represent five, and a cacao bean to represent zero. (Most references state that The Colorado School of Mines in Golden, Colorado, has held summer math camps for the symbol for zero is a type of shell. Patrick Scott, (Jniversity of New Mexico, has been working with American Indian and Alaska Native middle school Mayans in Guatemala who believe the symbol was students since 1988. The first one was held in a cacao bean.) conjunction with the American Indian Science and "The Mayas thought of time as a never-ending Engineering Society (AISES) and was funded by flow into the future and back into the past. They the Educational Foundation of America. Dr. Bar- believed that history repeated itself when condi- bara Bath is the director and Dr. Ardel Boes, head of the mathematics department, is the ad- tions were exactly as they had been in the past. For this reason, it was extremely important that the ministrator of the math camp each summer. The priests continually study prevailing conditions in purpose is to expose Native students to mathe- order to compare them with past conditions" (Cal- matics in various ways and provide opportunities lahan, 1969). Charles H. Smiley, Professor of to learn about various mathematics and science oriented careers. One of the major goals is to Astronomy, Brown University, Providence, Rhode Island, in conversations with the author, claimed develop or reinforce positive attitudes toward to have predicted a hurricane for the New England mathematics. Results from pre and post tests of coast using information from the Mayan Dresden attitude and cognitive reasoning indicate improve- Codex, one of the remaining books of the Maya. He ment is fairly consistent for all the students. The boarded up his house on the coast and had no type of mathematics studied helps students in their regular math classes during the year. Ac- damage. Mayans were able to construct temples, tivities in the mathematics classes emphasize pat- pyramids, and ball courts with admirable terns, logical thinking, and spatial relations. precision. Acoustics allowed priests to whisper Hands-on, manipulative activities are used exten- sively. For example, pattern blocks are used to and be heard across the plaza. Astronomical obser- vatories allowed them to observe and accurately make designs and patterns in the mathematics record information regarding planets, comets, and classes. The designs can be duplicated using the stars. As communication with Chinese mathe- computer. Computer classes include developing maticians and astronomers increases, astronomi- tesselations and other geometric designs. Logo is cal information for studying the correlation of used because of its problem solving capabilities for Mayan dates with current knowledge of historical students. Since the second year mathematicians dr.tes improves. Agricultural accomplishments of and computer scientists from Bell Laboratories of the Mayas include development of many crops we AT&T have volunteered to be mentors to assist the students in their projects. Students' informal use today. Raised fields were used in swampy areas and water reservoirs were established in dry areas. evaluations state one of the benefits of the math The Incas of South America were great or- and camp is learning to deal with new situations ganizers. Their highway system was more exten- adjust to them. Parents contacted Dr. Bath and sive than the Roman highway system that is informed her that their child's attitude toward recognized in history books. Parts of the Inca high- school, achievement in school, and self-concept had way system are still used. Irrigation methods used improved after attending the math camp. The second year a student program and a terracing for maximum use of water. Throughout teacher education program was funded by the Peru there are Inca terraces on the sides of moun- Department of Education. A Middle School tains that are still used. Surgeons were able to perform delicate, suftessful brain surgery. They Teacher Enhancement Geometry Program was had an extensive knowledge of herbal medicine. funded by the National Science Foundation (NSF). The weaving with feathers and gold thread canno A Young Scholars Program and a Middle School be duplicated today. The artistry using gold is Teachers Algebra program was funded by NSF the unmatched with today's knowledge of technology. this 1 year. The summer of 1991 another Young It is still a mystery hlw the structures of the Incas Scholars Program and Middle School Teachers 8 9 Mathematics and Science program will be funded by NSF. The program has gram. Students in five elementary schools in Rapid City and Kyle will participate in after-school been designated a Program of Excellence by the academic year and in summer enrichment ac- Colorado Commission for Higher Education and has received five years of funding for the Middle tivities. Students will study science and engineer . ing as they relate to their everyday lives. Science School American Indian and Alaska Native Math Made Simple is a two week non-residential sum- Camp and the Teacher Education program. The University of Kansas, Haskell Indian mer camp. Since NACME requires community partnership and matching funds, Amoco, the Junior College, and the Bureau of Indian Affairs South Dakota Community Foundation, U. S. West together developed and presented two eight-week Communications, and Title II (United States mathematics and science elementary teacher in- Department of Education) now help support the stitutes in the summers of 1988 and 1989 funded program. (Campbell, G. Jr., 1990) by the National Science Foundation. Funding has Porter Middle School in Austin, Texas, has a been allocated by the National Science Foundation very innovative science teacher, Sharon Mitchell. for the summers of 1991 through 1993. The Math- She is the adviser of the Earthnauts, a group ematics and Science Teachers for Reservation started "by kids for kids." The students work in a Schools (MASTERS) Project focuses on improving biodome to test theories of growing food without classroom instruction skills of teachers. Three themes are emphasized: students' concrete pesticides or without otherwise harming the earth. They grow vegetables in water instead of soil. The firsthand (hands-on) experience with the concept group has published a book of poetry, written a being taught; cultural relevancy of the curriculum proposal to NASA to gather ground-level data to objectives, materials, and teaching technique, and help the agency fine-tune readings from orbiting positive role models and community involvement satellites, and produced a series of comic books in instruction. about the environment for elementary students. Science is studied using hands-on experiences. Students are in charge of the biodome and the Air and space topics are a focus of many of the plants grown there. Involvement in Earthnauts activities. Science and mathematics activities are has caused many students to plan to go into various related to all aspects of life using a whole language branches of science. (Smith, S., 1990, pp. Bl, B4) approach. Biographical information about Native The New Jersey Institute of Technology spon- American scientists has been compiled. Science and mathematics activities related to the sored an Experimental Mathematical Science and Communications (EMSAC) program for seventh biographical information will be developed in fu- grade students during the summers of 1983 and ture workshops. The vision of the program is to 1984. The 1984 program focused on mapping and develop mathematics and science activities within surveying, observational astronomy, Apple Logo, all aspects of the curriculum. During the summer communications, and probability and statistics. of 1991 there will be thirty-five new students and The prorain was for inner-city students. NCTM's twelve returning students. MASTERS' goals in- 1981 Yearbook, Teaching Statistics and Prob- clude improving mathematics and science instruc- ability, was used to develop the probability and within the tion at the individual teacher level, statistics concepts introduced. Students played schools of the participating teachers, and within coin toss games and games using dice. Games were the overall system of reservation schools. determined tu be fair or unfair by playing the "Change is not a quick process: and change games and keeping results. After discussions of agents are not made overnight. MASIERS' results of the games, students determined which goal is to produce a core of skilled mathe- games were fair or unfair. A Casino Day was held matics and science teacher leaders who can work with their fellow faculty and com- to study the concept of determining odds. One of munity, not impose improved science and the learning objectives was to have students real- mathematics teaching programs. (Smith. ize that there is at least one best strategy for W.S.. 1990, pp. 1-2) playing each game. After Casino Day the games National Action Council for Minorities in En- with obvious strategies were discussed. Pascal's gineering, Inc. (NACME) launched TechForce triangle was used to help explain the odds. 2001, a program for students who will be earning Geometric probability was introduced by discuss- engineering degrees at the beginning of the new ing variation s of th e Lady or the Tiger game. Deter- mining the winnings when a game ends century, in 1989. NACME, the South Dakota School of Mines and Technology, AISES, and the prematurely while a person is ahead was studied. Monte Carlo techniques and simule ions were local school districts in Rapid City and Kyle formed a coalition to provide a Scientific Knowledge for used to discuss prizes in boxes of cookies. Student Indian Learning and Leadership (SKILL) pro- 9 1 0

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