1:****W*W***********w · AUTHOR Preston, Vera TITLE Mathematics and Science Curricula in Elementary and. Secondary Education for American Indian and Alaska. Native

DOCUMENT RESUME

ED 343 767 RC 018 626

AUTHOR Preston, VeraTITLE Mathematics and Science Curricula in Elementary and

Secondary Education for American Indian and AlaskaNative Students.

SPONS AGENCY Department of Education, Washington, DC. IndianNations At Risk Task Force.

PUB DATE 91

NOTE 18p.; In: Indian Nations At Risk Task ForceCommissioned Papers. See RC 018 612.

PUB TYPE Information Analyses (070)

EDRS PRICE MF01/PC01 Plus Postage.DESCRIPTORS *American Indian Education; American Indians;

Cooperative Learning; *Demonstration Programs;*Educational Strategies; Elementary SecondaryEducation; Experiential Learning; Higher Education;Mathematics Education? *Mathematics Instruction;Science Education; *Science Instruction; TeacherEducation; *Teaching Methods

IDENTIFIERS Culturally Relevant Curriculum

ABSTRACTIssues related to the improvement of mathematics and

science education pertain to Native students as well as to thegeneral population. Native students are most successful at tasks thatuse visual and spatial abilities and that involve simultaneousprocessing. Instruction should build on Native students' strengths.Experiential learning and cooperative learning are two methods thatare particularly effective with Native students in improving studentattitudes and problem-solving abilities and reducing mathematicsanxiety. Storytelling techniques can be used to develop culturallyrelevant problems. Career days show students the uses of mathematicsin the real world. Curriculum development strategies includeestablishing the relationships and connections between mathematicsand other subjects, and incorporating culturally relevant materials,such as Maya or Inca mathematics and science. Strategies of exemplaryprograms include summer math camps for Native students, summerinstitutes to improve teacher instructional skills and methods,after-school and summer enrichment activities in science andengineering, instructional materials developed to accompany a scienceseries on public television, magnet schools, after-school collegepreparatory courses in mathematics, and parent resource centers.Recommendations are offered related to instructional methods, programdevelopment, and federal funding. This paper contains over 130references. (SV)

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from the original document.***********************.t***********************************************

Mathematics and Science Ctuvricula in Elementaryand Secondary Education for American Indian

and Alaska Native Students

Vera Preston

IntroductionWake up, America! Your child:en are at risk.Three of every four Americans stop studyingmathematics before completing career or Jobprerequisites. Most students leave schoolwithout sufficient preparation in mathe-matics to cope either with on-the-jobdemands for problem-solving or with collegerequirements for mathematical literacy...Since mathematics is the foundation ofscience and technology, it serves as a key toopportunity and careers.. Moreover, mathe-matics contributes to literacy certain distinc-tive habits of mind that are of increasingimportance to an informvd citizenry in atechnological age. ... Despite massive effort,relatively little is accomplished by remedia-tion programs No one not educators.mathematician*, or researchers knowshow to reverse a consistent early pattern oflow achievement and failure. Repetition rare-ly works; more often than not. it simplyreinforces previous failure. The best time tolearn mathematics is when it is first taught;the best way to teach mathematics is to teachit well the first time. (Everybody Counts.1989, pp. 1-3. 13)

Testimony from the Indian Nat:xis at Risk(INAR) Task Force meetings, regional hearings,and written testimony submitted to the INAR TaskForce represents the diversity of Native peoplesand the variety of schools attended by AmericanIndian and Alaska Native students. The commonthread existing throughout the extensive tes-timony is the need for American Indian and AlaskaNative students to have their culture acknow-ledged and to be taught in the ways they learn. Thetype of school, the type of class (language arts,social studies, science, mathematics), the area ofthe country does not change the need to berespected as a human being and to be taught in amanner that allows the individual to understandand internalize the material to be learned. Allchildren, regardless of race, gender, or ability,deserve to be taught in the ways they learn bybuilding on their strengths and overcoming theirweaknesses. Rather than emphasizing the dif-

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ferences, the message obtained from the testimonyaccentuates the common concern of Native peoplefor educating their children. They want theirchildren to be educated in a manner which enrichestheir horizons and guarantees a variety of optionsfor lifelong learning and for various vocations.

For years reports have stated the need forreforming mathematics and science education inthe United States. A Nation at Risk (NationalCommission on Excellence in Education 1983),Educating Americans for the 21st Century (Nation-al Science Board Commission on Precollege Educa-tion in Mathematics, Science and Technology1983), and Everybody Counts -A Report tu the Na-tion on the Future of Mathematics Education (Na-tional Research Council 1989) are representativeof the reports concerning mathematics and scienceeducation. Compared to students throughout theworld, students in the United States rank in abilitywith students from Third World countries. Nation-al Assessment of Educational Progress (NAEP)reportsaf results of 1986 assessments indicate thatability to answer problems requiring higher orderthinking is lacking for most students in the UnitedStates (Brown, Carpenter, Kouba, Lindquist, Sil-ver, and Swafford, 1988, pp. 911-248). Somereports have addressed the issue of what needs tobe done to solve the problems. Ciaims of easysolutions to difficult problems should be viewedwith great suspicion. Research indicates the waystudents learn is not always the way students aretaught. Since teachers tend to teach the way theywere taught and the way in which they learned, thevicious cycle ofmiseducatim continues to handicapthe lives of children of all ages, abilities, and races.

The National Council of Teachers of Mathe-matics (NCTM) has responded to the need forchange in mathematics teaching by producing Cur-riculum and Evaluation Standards for SchoolMathematics (NCTM, 1989). It is a set of stand-ards for mathematics curricula in kindergartenthrough high school developed by classroomteachers, supervisors, educational researchers,teacher educators, and university mathematiciansworking together. NCTM is very much aware ofthe

Indians N Alons At Risk; Solutions for the 1990s

skepticism of people toward change after the *newmath" of the sixties which imposed change from thetop down. The current recommendations stress theimportance of all stIdents learning mathematicsand the importance of evising mathematics in-struction. The National Science Teachers ofAmerica (NSTA) has developed a Statement ofInitiatives similar to the NCTM Standards. Em-phasis is on more unified, in-depth hands-onscience curricula for all students kindergartenthrough twelfth grade.

What can be done to improve the mathematicsand science education of all children includingAmerican Indian and Alaska Native children?Why is mathematics and science education so im-portant to all students? This paper will addressthe issues in mathematics and science education.

Mathematics and ScienceMazhematics and science influence the lives of

everyone. Scientific, mathematical, and tech-nological literacy is crucially important for citizensto understand the technological world in whichthey live. Mathematics has been called "the queeiiand servant of the sciences." It is the study ofpatterns and relationships and is the foundationon which understanding of many fields of study isbuilt. An understanding of mathematical conceptsand mathematical reasoning is necessary in al-most every occupation in today's technologicalsociety. Mathematics has been called a criticalfilter because so many occupations require a math-ematical background. Many people perceive math-ematics as an infinite set of meaningless ruleswhich are impossible to memorize. Mathematics isconstantly changing; more than half the mathe-matics known today has been developed since1940. Science and mathematics are interdepen-dent in our technological age. For example, under-standing of environmental issues such as wateravailability and water pollution require an under-standing of both mathematics and science.

Mathematics and ScienceLearning and Teaching

Fundamental change in the way mathematicsand science is taught is necessary (NCTM Stand-ards, 1989; Willoughby, 1990). Information frommathematics and science educational research andfrom cognitive psychologists should be used todevelop plans for organizing instruction for math-ematics and science classes. Research indicatesAmerican Indian and Alaska Native students aremost successful at tasks which use visual andspatial abilities. (More, 1986) Many studet.i,s, in-cluding Native students, use holistic, global

processing of knowledge which enables them toperceive various elements as a whole. Seeing thewhole picture, a holistic, global process is referredto as simultaneous processing. Processing intbr-mation sequentially and in an analytical manneris refened to as successive processing. (Das, Kirby,& Jarman, 1975. Quoted in More, 1986, pp. 8-9)Research results indicate Native students usesimultaneous processing more frequently and ef-fectively than successive processing. It also indi-cates that Native children are more fieldi,dependent (FI) than field dependent (FD) andthat they tend to think in relational styles ratherthan in analytic styles. (More, 1986, pp. 9, 12)Research on culturally based communicationstyles related to the classroom has been done byScollon and Scollon (1983) and Phillips (1972).Studies of teacher effectiveness which relate tostyles of communication of Native Indian studentshave been done by Kleinfeld (1970). TraditionalNative cultures teach children to watch, listen,think and then do the task to be learned. Manyclassrooms encourage trial-and -error learning.Many students become quite proficient at guessinguntil they get the correct answer. Trial-and-errorlearning is the antithesis of traditional Nativelearning styles.

Field independence is the degree to which anindividual can separate a figure from itsbackground, a part from the whole, oneselffrom the environment or other people. A fieldindependent (FI) person is more able to pro-vide an organizational structure to a disor-ganized set of facts or observations (e.g..making a mental map of the surroundingterrain). A field dependent (FD) person is lessable to separate a part of the whole, but ismore conscious of other people... . The Flperson will impose his or her own structureon a field (spatial or conceptual) more readi-ly. The FD person is mom socially aware andmore responsive to those around him or her.(More, 1986, pp. 10-11).There is considerable overlap between thelearning styles of Indian and Non-Indian stu-dents. Indeed the similarities are probablygreater than the differences. Furthermore, itis likely that the learning styles of somegroups of Non-Indians will be remarkablysimilar to those of Native Indians. This couldoccur when there are similar factors affectingthe way of life such as degree of remotenessof the community, economic level, and sig-nificant cultural differences from themajority. However in such situations cul-tural differences should not be downplayed.Even though the pattern of learning styles issimilar, the causes and the methods of work-

Mathematics and Science

Mg with the learning styles may not be."(More. 1986. p. 20)

It is important to realize that teaching stylesthat are appropriate for Native students are alsoappropriate for other students. Methods of teach-ing found in many schools are not appropriate tothe ways children learn. This is one of the majorreasons for the need ofremedial classes. If studentswere taught concepts in the way they learn, theneed for remedial classes would be greatlydiminished. Since Native students have strengthsin using visual/spatial/perceptual information,new material should be introduced using methodsbuilding on their strengths. The overall view ofnew information should be stated before the infor-mation is broken down into small segments. Ad-vance organizers should be used. Students'strengths should be used to develop skills in othermodes of learning. (More, 1986, p. 21) Studentsshould have the right to learn beginning materialin their native language to allow them to developa cognitive base on which to learn for theremainder of their lives. (McDonald, 1989, p. 16)

Children are born with a love of learning andan innate curiosity. The first three years of life achild learns a tremendous amount of information.Parents are the first and most important teachersof a child. How can a child's natural desire to learnabout his/her surroundings be nurtured and main-tained in academic situations? Bow can studentslearn the necessary material in the various clas-ses? Many theories exist regarding how peoplelearn. There is danger in stating that n specificgroup of individuals learn in a particular way. It isimportant to be sensitive to the needs of eachindividual and help that individual build on her/hisstrengths and increase ability in areas of weak-ness. (More, 1986) Many people have a preferredlearning style. Many researchers stress that stu-dents should be allowed to learn from theirstrengths. Research in mathematics anxiety indi-cates that students overcome their anxiety and areable to perform better in mathematics if they buildsuccess on success. (Tobias, 1978)

Manipulative, hands-on, Materials"Tell me, and I forget; Show me, and I remem-

ber; Involve me, and I understand." AncientChinese Saying.

Students who use hands-on materials, other-wise known as manipulatives, have better at-titudes toward mathematics and better ability toproblem solve (Suydam and Weaver,1975; Traf-ton,1975). Students who use manipulatives andare guided to bridge the gap from concrete to pic-torial to symbolic representations are better able

to understand concepts and to transfer their learn-ing to other situations (Surdam,1986). Teacherguidance ;i3 important in bridging the gaps fromthe concrete to the pictorial to the symbolic stage.(Kouba et al., 1988b) Educators from the earlypert of the century to the present have advocatedthe us,) of physical, or concrete, materials in theclassroom. (Brownell, 1935; Burns,1986; Montes-806,1965; Piaget,1952; Reys,1971; Suydam,1986)Recommendations from reports of the most recentNA EP results encourage the use of concretematerials and activities. NCTM has recommendedthe use of physical, concrete materials in the class-room. (NCTM 1980; NCTM 1987) An Agenda forAction: Recommendations for School Mathematicsof the 1980s (NCTM, 1980, p. 12) includes thefollowing: "Teachers should use diverse instruc-tional strategies, materials, and resources, such as... the use of manipulatives, where suited, to il-lustrate or develop a concept or skill."

There are many benefits which occur as a resultof using manipulatives. Mathematics anxiety isalmost non-existent in classes where manipula-tives are used frequently. The attitude towardlearning mathematics is consistently more positiveas a result of using manipulatives and of par-ticipating in cooperative learning. Student under-standing of concepts is greater among all abilitylevels. Parental involvement is easier to obtain byhaving some activities done at home using commonhousehold materials. Students need guidance withthe teacher as facilitator on the use of themanipulatives. Assistance and guidance in under-standing the concepts represented by themanipulatives and in bridging activities to go fromthe concrete to the pictorial to the symbolic orabstract are essential in using the materials to bestadvantage (Heddens,1986). Learning what to doand why enables students to retain informationand transfer the learning to other situations(Burns,1986). Fuson (1981) writes that there is ameta-cognitive benefit in the use of concrete ob-jects. Teachers' sensitivity to styles oflearning willhelp their students. A balanced view of variousmethods of teaching and learning will assist stu-dents to recognize their particular style of learn-ing. (More, 1986) The teaching/learning style isdetermined by the learning task. Learning how tolearn and recognizing the underlying concepts andfeatures of mathematical problems is very impor-tant.

Cooperative LearningCurrent issues in educational circles include

teaching thinking through the curriculum, prob-lem solving, cooperative learning, writing acrossthe curriculum, and learning how to learn

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Indians Nations At Risk: Solutions for the 1990s

(Simon,1986). Cooperative, or group, learning isbeing recognized as important in learning con-cepts. Since people work cooperatively in mostsituations outside school, studeuts should haveopportunities to work together in school. The manybenefits which occur as a result of cooperativelearning are an ability to communicate verbally theinformation and concepts being studied, an under-standing of other peoples' attitudes, a positive at-titude toward learning, recognizing there arevarious paths to the solution, and clarifying one'sown understanding. !Slavin, 1987)

Cooperative learning is a popular concept at themoment in education. The merits and disad-vantages of cooperative learning are discussed infour articles in the Febmary 1991 issue of Educa-tional Leadership (Kohn, 1991; Slavin, 1991). It isimportant to note that cooperative learning hasbeen used successfully by teachers over the yearswithout the label of "cooperative learning." Muchof the discussion in the articles relates to givingrewards for group effort. Rather than gettingbogged down in semantics or side issues, teachersneed to consider the advantages of cooperativelearning and apply the concept as appropriate totheir particular situation in their specific class-room. Teachers must acknowledge that coopera-tion in the world outside the classroom is expectedand necessary for success in projects throughoutthe business world. Mathematics and sciencelearning should involve learners working together.Mathematicians and scientists talk to each otherand try new ideas and check each other's reason-ing. They do not work in total isolation. Researchoverwhelming supports the concept of cooperativelearning and working together. It is particularlyappropriate to the learning style of many Nativestudents. There are many different forms ofcoope. ative learning. The most successful approaches have incorporated group goals and individual accountability (Slavin 1991). Many of theadvantages of cooperative learning are social aswell as academic. Mathematics anxiety is practi-cally non-existent when students are allowed towork together to discuss and learn new informa-tion.

Many teachers used cooperative learningbefore it was called cooperative learning. Studentsworked together and helped one another andrecorded the results of their experimentation, theirsearching for patterns, or their explanation of newdefinitions, concepts and material. Each groupturned in one paper with their results. Studentswere each responsible for the material and wroteindividual reports later. The method of havingstudents work together was successful with stu-

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dents in gifted classes and in remedial classes atall levels.

Storytelling and Problem Solving inMathematics and Science.

Storytelling as a means ofteaching is tradition-ally used by American Indians and Alaska Natives.Using stories to pose mathematical or scientificproblems is a natural way to relate situations ofthe student's home life and culture within themathematics classroom. Students arrive in schoolwith the ability to solve problems which havemeaning for them. School textbooks and mathe-matics lessons frequently convince students theyare unable to solve the problems which had pre-viously been easy for them. One of the importantskills in solving problems is drawing pictures rep-resenting the problem. (Davis & McKillip, 1980,pp. 80-91)

An ancient example of using storytelling to posea mathematical problem is the story of the persongoing to St. Ives. "As I was going to St. Ives, I meta man with seven wives. Each wife had seven cats,each cat had seven kits. How many were going toSt. Ives?" Only one person was going to St. Ives;the others were leaving St. Ives. The problem in-volves listening and reasoning. The mathematicsinvolved is rather straightforward.

Students enjoy making up story problems whenthe teacher provides an encouraging, accepting,respectful atmosphere in the classroom. The entirefamily could participate in posing story problemswhich incorporate mathematical and scientificproblems. For example each of the following math-ematical type of questions could be posed in a story.How much flour is used for making fry bread? Howdoes the volume of the fry bread change when it iscooked? When popcorn is popped, how much doesthe volume change? How many seeds are there ina watermelon? What are some strategies for deter-minitig how many seeds are in the watermelon?What are the differences in types of seeds? Whatare the shapes of leaves? How many leaves are onthe plant? How many steps does each person in thefamily take for walking the same distance? (Whatis the stride of each person?) How many sheep peracre can the land support without irrigation? Howmany cattle per acre can the land support? Howmany bales of hay can be put in a pickup? What isthe most efficient shape to build a house ifmaterials are scarce? (A circle is the most efficientshape to provide the most area for the sameamount of fencing. A square is the most efficientstraight-sided figure. Why?) How many days did itrain last week, last month, last year, ten years ago?


What is the difference in temperature from 5:00AM to 5:00 PM on the same day?

Mathemafics AnxietyMany students develop 6. fear of mathematics

during the elementary, middle school, junior highyears which cripples their ability to l3am mathe-matics in high school and later. Other students dowell in mathematics classes until they reach al-gebra or geometry. When they have difficulty inmathematica, they quit taking elective mathe-matics classes. They take the path of least resis-tance when taking required mathematics classes.It is important for students to be encouraged, pos-sibly required to take mathematics classes eachyear of high school. Students must realize thatmathematics is a critical filter. The more mathe-matics a student studies, the higher the expectedlifetime earnings, and the more opportunities foremployment are available. Students can choose ajob or career based on what they enjoy doing ratherthan on what requires no mathematics. Counselorsshould be required to insist that students stay inmathematics classes rather than allowing them todrop out when the going gets tough.

Mathematics anxiety and test anxiety is usual-ly found in students after the elementary schoolyears. When students are asked to relate theirexperiences leading to mathematics anxiety theyusually relate it to something which occurred inelementary school. Reasons students at the com-munity college level give for their mathematicsanxiety include moving a lot during elementaryschool, family problems, receiving beatings from aparent when multiplication facts were not learnedfast enough, getting behind when ill and nevercatching up, being told they were not smart enougl-,to do mathematics, and getting behind in mathe-matics becanse they did not understand a par-ticular concept. Many students who had difficultywith mathematics in elementary school havelearned they have a learning disability. With per-sistence and determination many students withlearning disabilities are successful in mathematicsclasses.

Acknowledging mathematics anxiety and sug-gesting ways to overcome it helps students. Stu-dents can be given a list of books which discussmathematics and test anxiety and how to deal withit. The most effective way for a student to overcomemathematics anxiety is to have opportunities to besuccessful in solving mathematical problems.(Langbort, 1985; Tobias, 1978) Many activities fordiscovering patterns and for solving logic problemsimprove attitudes toward mathematics. There aremany books available which provide interesting

mathematics and science activities for students ofall ages.

Career EducationCareer days for elementary, middle, and junior

high school students enable the students to learnabout different types of careers. Native studentsshould be encouraged to consider careers in fieldswhich will help their communities. Careers inhealth fields, environmental science, agriculturalscience, various type of engineering, attorneys,counselors, and business all have the potential forhelping their communities. Expanding YourHorizons Workshops and Career Days wereoriginally organized for females. The format isappropriate for American Indian and Alaska Na-tive children. Students should be encouraged tocontinue their education so they will be preparedto help their community in many ways.

Students generally lose their innate en-thusiasm for science and mathematics during thecrucial junior high years. Therefore, it is criticallyimportant to provide elementary, middle, andjunior high school students opportunities to ex-perience challenging and interesting projects andprograms in mathematics and science. It is alsocritically important for students to recognize thatmathematics is a critical filter for many interestingand exciting careers. Most professions require col-lege mathematics courses through calculus. Moststudents who are successful in their college math-ematics classes took a mathematics class theirsenior year in high school and then took a collegemathematics class their first semester in college.When mathematics classes are taken consecutive-ly they are much easier than when a number ofsemesters have gone by between classes.

After-school, weekend, and summer programsprovide fun, interesting, challenging opportunitiesfor students to learn about mathematics andscience. Programs which emphasize hands-on ex-periences, applications to real life situations, andfield trips are particularly successful in encourag-ing students to consider mathematics and scienceoriented careers. It is important to take advantageof resources within a community when planningprograms. Many people are willing to share infor-mation about their career when they are given theopportunity to do so.

Environmentalists are just realizing what Na-tives have known for centuries. Careers whichrelate to environmental issues are important forpeople throughout the world. Native studentsshould be informed of the opportunities to helptheir communities by studying various fields ofscience related to preserving and maintaining theenvironment. Water pollution, toxic waste, agricul-


tural and industrial chemicals are some of thetopics which can be addressed. The wise use oftechnology can best be monitored by people whounderstand the technology.

Medical pereonnel are needed in rural areasand small towns throughout the United States.Native people need to encourage their children toconsider pursuing a career in one of the manyhealth related professions. Dieticians, nurses,radiology technicians, physical therapists, occupa-ti on al therapists, family physicians, andphysicians who are specialists are some of thehealth related professions students can consider.

Each area where Natives live have specificneeds regarding the environment and theeconomic and business situations. People areneeded who have the vision to develop small busi-nesses to meet the needs of the area. Attorneys areneeded to provide necessary information and expertise for protecting the interests of the people.Biomedical engineers, chemical engineers, civil en-gineers, electrical engineers, biological engineers,and other engineers are needed to protect the en-vironment and help the community. Leaders withvision who are aware of long-term needs of theNative community are needed. Educators withsensitivity, ability, and knowledge of subject mat-ter are needed to provide role models for thechildren. Our children are precious. They are thefuture.

Parents and other members of the extendedfamily have a tremendous influence on children.Encouragement and belief in a child's ability topersevere through the challenging times on thepath to a career that will be fulfilling to the in-dividual and will provide thE opportunity to enrichand help the community are crucially important.Many students state that the reason they stayedin school was because someone believed in themand encouraged them when they were discouraged.

An activity which helps middle school studentsdiscover the importance of mathematics in careersis "mathenger hunt," based on a scavenger-huntidea. Teams of students are selected by the teacher.Students gather a minimum of twenty signaturesfrom people who state they use mathematics intheir career. The class decides on a point system.Additional points can be earned by obtaining addi-tional evidence from the people related to their useof mathematics. Some side effects of the projectwere offers by people to speak to the class, scrap-books of the signatures and other information wereshared with other classes, and the class compiledword problems relating to careers (Falba & Weiss,1991, pp. 88-90).

Instructional methods thatmotivate Native students, challengetheir minds, stimulate theircreativity and initiative.

The diversity of Native peoples providestremendous opportunity for developing materialsfor mathematics and science classes which can beshared with schools throughout the country. Avision where all children are given the opportunityto share in the joy of holistic lifelong learning isshared by people from all walks of life and frommany races. There is beauty in learning and recog-nizing that all of life is interrelated. One day thatvision will come true when all work together for thechildren. People throughout the country shoulddemand that students be taught in ways whichbuild on their learning strengths. Students,regardless of their ability level, learn new conceptsfrom the concrete (hands-on) to the pictorial to theabstract (Piaget, 1952; Brownell, 1935). Activitiesare needed to bridge the gap from the pictorial tothe abstract and need to be included in the learningactivities.

Learning is fun. Math is fun. Science is fun.Teaching needs to stimulate children's naturalcuriosity. Teachers do not necessarily have to rein-vent the wheel. They can adapt material to makeit appropriate for the area in which students live.There are many materials available which providechallenging, enriching, and relevant experiencesin mathematics and in science. Using manipula-tives throughout the school years as new conceptsare introduced will stimulate interest and assistunderstanding. When materials are provided in amathematics laboratory type of situation, studentsenjoy solving problems posed using variousmaterials. Consider the following example. Whenan apple is cut crosswise, a star is formed. Studentscan be asked how many points does the star have?How many seeds are there? Why is a star formed?How was the apple formed on the tree? In whatsoil and weather conditions do apples grow? Howmany types of apples are there? Which type ofapples make the best juice, pie, snacks? Wh,...tchemical is found in apples that is calming forpeople?

In-service training is helpful for teachers tolearn to use new manipulatives and other newmaterials. There are various programs availablefor providing elementary teachers with experiencein using mathematics and science materials. Someprograms teach specific mathematics and scienceclasses. Some programs combine both aspects.Open Math is a resource book for a set of videos,originally television programs, developed atArizona State University by Jonathan Knaupp and


Gary Kna.niller. Teachers can r:atch a video eachweek and try one new activity. The following weekthey can share their successes and their failuresand learn from one another. Littie by little theyexpand their repertoire of instructional methods.

Change occurs slowly in education. For changeto occur in the way children are taught, teachersmust be provided opportunities to learn how to usenew methods of teaching. A one-day workshop isnot sufficient to help a teacher change the way inwhich she/he teaches. Summer school classes orworkshops combined with regularly scheduled in-service during the academic year is a way in wiiichto effect change. It is difficult, frustrating, andslighly frightening to change teaching methods.Teachers need encouragement and support tochange the way they teach. An example of the typeof person who can inspire others to do their best isDr. Helen Neely Cheek. Dr. Cheek was a wonder-ful, inspiring teaeher and educator who taughtelementary and secondary teachers new methodsof teaching. She provided opportunities to learnnew methods of instruction little by little until eachteacher was comfortable using row materials andnew methods of instruction. She emphasized recog-nizing each student's strengths and weaknessesand building on their strengths to overcome theirweaknesses. (Preston & Skinner, 1986)

Curriculum Development StrategiesMathematical Connections is standard four of

the NCTM Standards for kindergarten throughhigh school. Relationships between mathematicaltopics and other curriculum -las are encouraged."Unless connections are made, children will seemathematics as a collection of isolated topics"(NCTM Standards, 1989, P. 33).

Mathematics as Communicadon is StandardTwo of the NCTM Standards. Students need to beable to clarify their thinking to fully understandmathematical concepts. Communication by repre-senting concepts using manipulatives, by usingdiagrams and drawings, by discussing ideas, bylistening, by reading, and by writing are all impor-tant. Knowledge must be able to be communicatedto oth ers before its usefulness and applicability canbe determined and utilized.

Culturally Televant mathematics and sciencematerials address both of t.he above-mentionedstandards. Geometric designs found in all Nativecultures can be used to develop geometric andalgebraic understanding. Art can be used todevelop concepts of estimation to know the amountof paint, clay, beads, etc. which are needed for theproject. Chemistry and mathematics can be usedto obtain the colors needed for art projects.Geometry is used to develop perspective in many

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paintings (Serra, 1989). When mathematics andscience units are developed using the intemats ofthe students, motivation and enthusiasm for learning are increased. Tesselations, Pascal's Triangle,and the Fibonacci Sequence are topics which relateeasily to Native mathematics and science lessons.Tesselations, otherwise known as til4ngs, can becreated using Native designs. Pascal's 'Manglecan be used to develop algebraic, probability, andnetwork concepts, and to create petterns, artwork,and beadwork designs. The Fibonacci Sequencecan be used to study botany, architectuT e, art, andpackaging and marketing topics in advertbing. Itcan be included in a study ofmedieval times sinceFibonacci lived in the thirteenth century. (Preston,1985)

Linda Skinner has developed a week-longprocess where teachers, paraprofessionals,Elders, parents, students, and other communitymembers collectively experience various aspects ofculture. From their experiences and uniqueperspectives they create and develop culturallyrelevant learning activities and materials. Theyreturn to their individual communities withrenewed enthusiasm strengthened by the focusedenergy on the common goal of creating the besteducation experiences for the children of their com-munities.

Mayan and Inca Scienceand Mathematics

Examples of culturally relevant mathematicsmaterial related to other disciplines are MayanMathematics and Inca Mathematics. A unit onMayan Mathematics or on Inca Mathematics canbe developed to include geography, social studies,astronomy, archaeology, agricultural methods,development of corn, development of potatoes, art,glyph translations, architecture, language arts,music, physical education, peobability, acousticalengineering, civil engineering, textiles, minerals.Various topics could be studied in different gradelevels.

Mayan mathematics was highly developed. Theconcept of zero was recognized by Mayans longbefore it. was recognized in European mathematics.A concept of zero is necessary before a mathemati-cal system can have a place value system. It al-


lowed Mayans to think about and writ,' numbersthat were very large. They had two separate butrelated mathematical systems. One was avigesimal, base twenty system, used to countthings and the other was an adjusted base twentysystem used to count time. Only three symbolswere used to represent all their numbers, a dot toreprescnt one, a bar to represent five, and a cacaobean to represent zero. (Most references state thatthe symbol for zero is a type of shell. Patrick Scott,(Jniversity of New Mexico, has been working withMayans in Guatemala who believe the symbol wasa cacao bean.)

"The Mayas thought of time as a never-endingflow into the future and back into the past. Theybelieved that history repeated itself when condi-tions were exactly as they had been in the past. Forthis reason, it was extremely important that thepriests continually study prevailing conditions inorder to compare them with past conditions" (Cal-lahan, 1969). Charles H. Smiley, Professor ofAstronomy, Brown University, Providence, RhodeIsland, in conversations with the author, claimedto have predicted a hurricane for the New Englandcoast using information from the Mayan DresdenCodex, one of the remaining books of the Maya. Heboarded up his house on the coast and had nodamage.

Mayans were able to construct temples,pyramids, and ball courts with admirableprecision. Acoustics allowed priests to whisperand be heard across the plaza. Astronomical obser-vatories allowed them to observe and accuratelyrecord information regarding planets, comets, andstars. As communication with Chinese mathe-maticians and astronomers increases, astronomi-cal information for studying the correlation ofMayan dates with current knowledge of historicaldr.tes improves. Agricultural accomplishments ofthe Mayas include development of many crops weuse today. Raised fields were used in swampy areasand water reservoirs were established in dry areas.

The Incas of South America were great or-ganizers. Their highway system was more exten-sive than the Roman highway system that isrecognized in history books. Parts of the Inca high-way system are still used. Irrigation methods usedterracing for maximum use of water. ThroughoutPeru there are Inca terraces on the sides of moun-tains that are still used. Surgeons were able toperform delicate, suftessful brain surgery. Theyhad an extensive knowledge of herbal medicine.The weaving with feathers and gold thread cannobe duplicated today. The artistry using gold isunmatched with today's knowledge of technology.It is still a mystery hlw the structures of the Incas

were built without the use 00 Itheels to transportthe gigantic stones and without mortar betweenthe stones. After centuries of earthquakes, manyof the Inca stmctures remain intact while modernstructures are destroyed in recent earthquakes.

Exemplary Schools, Programs, andProjects

The Colorado School of Mines in Golden,Colorado, has held summer math camps forAmerican Indian and Alaska Native middle schoolstudents since 1988. The first one was held inconjunction with the American Indian Science andEngineering Society (AISES) and was funded bythe Educational Foundation of America. Dr. Bar-bara Bath is the director and Dr. Ardel Boes, headof the mathematics department, is the ad-ministrator of the math camp each summer. Thepurpose is to expose Native students to mathe-matics in various ways and provide opportunitiesto learn about various mathematics and scienceoriented careers. One of the major goals is todevelop or reinforce positive attitudes towardmathematics. Results from pre and post tests ofattitude and cognitive reasoning indicate improve-ment is fairly consistent for all the students. Thetype of mathematics studied helps students intheir regular math classes during the year. Ac-tivities in the mathematics classes emphasize pat-terns, logical thinking, and spatial relations.Hands-on, manipulative activities are used exten-sively. For example, pattern blocks are used tomake designs and patterns in the mathematicsclasses. The designs can be duplicated using thecomputer. Computer classes include developingtesselations and other geometric designs. Logo isused because of its problem solving capabilities forstudents. Since the second year mathematiciansand computer scientists from Bell Laboratories ofAT&T have volunteered to be mentors to assist thestudents in their projects. Students' informalevaluations state one of the benefits of the mathcamp is learning to deal with new situations andadjust to them. Parents contacted Dr. Bath andinformed her that their child's attitude towardschool, achievement in school, and self-concept hadimproved after attending the math camp.

The second year a student program and ateacher education program was funded by theDepartment of Education. A Middle SchoolTeacher Enhancement Geometry Program wasfunded by the National Science Foundation (NSF).A Young Scholars Program and a Middle SchoolTeachers Algebra program was funded by NSF thethis 1 year. The summer of 1991 another YoungScholars Program and Middle School Teachers

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program will be funded by NSF. The program hasbeen designated a Program of Excellence by theColorado Commission for Higher Education andhas received five years of funding for the MiddleSchool American Indian and Alaska Native MathCamp and the Teacher Education program.

The University of Kansas, Haskell IndianJunior College, and the Bureau of Indian Affairstogether developed and presented two eight-weekmathematics and science elementary teacher in-stitutes in the summers of 1988 and 1989 fundedby the National Science Foundation. Funding hasbeen allocated by the National Science Foundationfor the summers of 1991 through 1993. The Math-ematics and Science Teachers for ReservationSchools (MASTERS) Project focuses on improvingclassroom instruction skills of teachers. Threethemes are emphasized: students' concretefirsthand (hands-on) experience with the conceptbeing taught; cultural relevancy of the curriculumobjectives, materials, and teaching technique, andpositive role models and community involvementin instruction.

Science is studied using hands-on experiences.Air and space topics are a focus of many of theactivities. Science and mathematics activities arerelated to all aspects of life using a whole languageapproach. Biographical information about NativeAmerican scientists has been compiled. Scienceand mathematics activities related to thebiographical information will be developed in fu-ture workshops. The vision of the program is todevelop mathematics and science activities withinall aspects of the curriculum. During the summerof 1991 there will be thirty-five new students andtwelve returning students. MASTERS' goals in-clude improving mathematics and science instruc-tion at the individual teacher level, within theschools of the participating teachers, and withinthe overall system of reservation schools.

"Change is not a quick process: and changeagents are not made overnight. MASIERS'goal is to produce a core of skilled mathe-matics and science teacher leaders who canwork with their fellow faculty and com-munity, not impose improved science andmathematics teaching programs. (Smith.W.S.. 1990, pp. 1-2)National Action Council for Minorities in En-

gineering, Inc. (NACME) launched TechForce2001, a program for students who will be earningengineering degrees at the beginning of the newcentury, in 1989. NACME, the South DakotaSchool of Mines and Technology, AISES, and thelocal school districts in Rapid City and Kyle formeda coalition to provide a Scientific Knowledge forIndian Learning and Leadership (SKILL) pro-

gram. Students in five elementary schools in RapidCity and Kyle will participate in after-schoolacademic year and in summer enrichment ac-tivities. Students will study science and engineer .ing as they relate to their everyday lives. ScienceMade Simple is a two week non-residential sum-mer camp. Since NACME requires communitypartnership and matching funds, Amoco, theSouth Dakota Community Foundation, U. S. WestCommunications, and Title II (United StatesDepartment of Education) now help support theprogram. (Campbell, G. Jr., 1990)

Porter Middle School in Austin, Texas, has avery innovative science teacher, Sharon Mitchell.She is the adviser of the Earthnauts, a groupstarted "by kids for kids." The students work in abiodome to test theories of growing food withoutpesticides or without otherwise harming the earth.They grow vegetables in water instead of soil. Thegroup has published a book of poetry, written aproposal to NASA to gather ground-level data tohelp the agency fine-tune readings from orbitingsatellites, and produced a series of comic booksabout the environment for elementary students.Students are in charge of the biodome and theplants grown there. Involvement in Earthnautshas caused many students to plan to go into variousbranches of science. (Smith, S., 1990, pp. Bl, B4)

The New Jersey Institute of Technology spon-sored an Experimental Mathematical Science andCommunications (EMSAC) program for seventhgrade students during the summers of 1983 and1984. The 1984 program focused on mapping andsurveying, observational astronomy, Apple Logo,communications, and probability and statistics.The prorain was for inner-city students. NCTM's1981 Yearbook, Teaching Statistics and Prob-ability, was used to develop the probability andstatistics concepts introduced. Students playedcoin toss games and games using dice. Games weredetermined tu be fair or unfair by playing thegames and keeping results. After discussions ofresults of the games, students determined whichgames were fair or unfair. A Casino Day was heldto study the concept of determining odds. One ofthe learning objectives was to have students real-ize that there is at least one best strategy forplaying each game. After Casino Day the gameswith obvious strategies were discussed. Pascal'striangle was used to help explain the odds.Geometric probability was introduced by discuss-ing variation s of th e Lady or the Tiger game. Deter-mining the winnings when a game endsprematurely while a person is ahead was studied.Monte Carlo techniques and simule ions wereused to discuss prizes in boxes of cookies. Student

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interest was highest when there was active par-ticipation. Students enjoyed playing games, toss-ing coins, and other activities to help themunderstand probability and statistics. (Scheinok,1988, pp. 310-314).

St. Paul, Minnesota, School District is develop-ing an American Indian Magnet School for gradesK-8. The 1990 school year is for planning. Theopening is scheduled for the 1991-92 school year.The priorities are to provide the opportunity foroptimum learning for each student and to helpeach student recognize her/his self-worth. Hands-on approaches and laboratory work will be used toteach mathematics and science. The University ofMinnesota's National Science Foundation Projectwill work closely with the school. EarlyChildhood/Family Education Programs will beavailable for the entire range of situations fromexpectant parents up to families of high schoolstudents. A parent resource center will be avail-able for parenting groups; it will offer activities tostrengthen families.

Chicago area teachers and scientists havedeveloped instructional materials to go with eachof the thirteen episodes of a new Public Broadcast-ing System series "The New Explorers." The scien-tists in the series include an ethnobotanist, anastronomer, a zoologist, a surgeon, a chemist, anornithologist, a microscopist, environmental scien-tists, and computer scientists. One of the results ofdeveloping the materials is teachers getting toknow the scientists. Students are able to visitscientists to learn more about where scientistswork and what they like about their jobs. Forinformation about the project and the teachers'guides, write to the Chicago Science Explore-sProgram, Division of Educational Programs, Ar-gonne National La5oratory, 9700 S. Cass Ave.,Argonne, IL 60439-4845. (Koepke, 1991, pp. 50-55)

Boothbay Harbor and Wiscasset, Maineteachers and scientists are working together in akindergarten through twelfth grade interdiscipli-nary program where students are developing mapsof the area. The students use computers to inter-pret information from Frances's SPOT satellite tocreate highly detailed maps of their area. Theproject, called the GAIA Crossroads Project,received technical support from an Apple Corn-p titer Crossroads Education Grant. (Wolcott, 1991)

The Houston Independent School District andBaylor College ofMedicine in Houston collaboratedto develop the Mathematics/Science/ComputerCenter at Fondren Middle School in the fall of1987. It exists as a special population within theschool. There are now classes for sixth througheighth grades. Multidisciplinary units relate all

10

subjects together. Units developed include: SpaceFrontiers, Mammals, Preservation of the Earth,Perception/Sense Organs, Chemistry of Life,Weather, Origins of People, Topology, Prejudice,Ecolcgy, Astronomy, Research, Topography, Ar-chitecture, and Music. Parente are involved andkept informed by attending an open house at thebeginning of the year, by receiving newsletters,and by attending a science night. A week-longsummer science camp for magnet students wassponsored by Baylor in 1988 and 1989. (Miller,LaVois, and Thomson, 1991)

Mercer Island, Washington, High School has aninnovative physics teacher, Jim Minstrel!. He usesresearch on how students learn and how tocounteract the preconceptions with which theyenter physics classes in planning his classes. Hereceived a grant to research how students learn."The project has three closely related goals: toidentify in more detail students' preconceptionsabout the physical world; to explore ways teacherscan use those preconceptions to teach physicallaws; and to prepare classroom materials thatother teachers can use to teach physics (Wolcott,1991)."

Professional Development Program (PDP) ofthe University of California at Berkeley hasdeveloped a PDP High School MathematicsWorkshop Model to encourage more minority stu-dents to do well in college-preparatory mathe-matics courses. The workshops are held at the highschools before or after school twice a weekthroughout the school year. Consistent attendanceis required; it is not a "drop-in" program. Theworkshops are designed to supplement specificcourses rather than to remediate. Difficult andnonroutine problems are included in the problemsets used. Workshops are led by college studentswho are majoring in mathematics, science, or en-gineering. For information contact Uri Treisman,PDP Program, 230B Stephens Hall, University ofCalifornia, Berkeley, CA 94720. (Stanley, 1991)

The American Association for thq Advance-ment of Science (AAAS) is sponsoring Project 2061,a long-range, multiphase project to determinewhat scientific literacy should be and how to ac-complish it. Project 2061 published Science for AllAmericans in 1989. The AAAS project "Linkagesto the Future" is assisting local Girl Scout councilsin developing hands-on activities and increasingconnections between scientists, engineers, andscientific organizations in the local areas. MakingMathematics and Science Work for Hispanics isanothei /OAS drogram. For informat,on contactAAAS, 1333 1' Street, N. W., Washington, DC20005. (Reeves, 1991)


Various programs could be used for Native stu-dents with slight or no modifications. Someprograms to consider in addition to previouslymentioned programs follow. Information regard-ing Math/Science Network, Family Math, FamilyScience, Math for Girls and Other Piliblem Solvers,How to Encourage Girls in Math and Science, andExpanding Your Horizons Workshops is availablefrom the Lawrence Hall of Science, University ofCalifornia-Berkeley, Berkeley, California 94720.Futures with Jaime Escalante is a mathematicsand science series created by the Ihiulic Broad-casting Service, 1320 Bi addock Place, Alexandria,VA 22314-1698. Phillips Petroleum Company,Bartlesville, Oklahoma, 74005, has sponsored twoseries of mathematics and science videos, Chal.lenge of the Unknown (mathematics) and Searchfor Solutions (science). The videos are available forrent free of charge. Teacher guides have beendeveloped to go with each video.

RecommendationsMathematics is a "critical filter" which provides

opportunities to pursue interests in many differentfields of study. Native students must receive a solidfoundation in mathematics in order to have optionsavailable to them. Mathematics and science mustbe taught in the way children learn from theconcrete to the pictorial to the abstract. Mathe-matics and science should be taught in an ex-periential manner. Activities should includeexperiments and the use of hands-on materials.Mathematicians and scientists in the real worldwork together and use models to demonstrate theirideas. Students should be expected to use hands-onmaterials at all levels, kindergarten through thetwelfth grade, to demonstrate their ideas. Studentsshould work together in groups in various types ofactivities. Mathematics and science should betaught in interdisciplinary units which includelanguage arts, social studies, physical education,art and music all aspects of the curriculum.

Computers, calculators, and graphing cal-culators should be used within the science andmathematics classes. The use of calculators andcomputers allows students to study problem solv-ing using higher level thinking skills. Studentsmust be instructed in the use of current technologyto prepare themselves to live in a technological age.

Culturally relevant materials should be used toteach mathematics and science. Materials areavailable and other materials can be developed. Anational clearinghouse or computer networkshould be established to assist teachers andparents in determining what materials are

presently available. Mathematics and science aronaturally related to the lives of Native students.

A core curriculum at the elementary and secon-dary levels that challenges all students andprepares them for the next stages in their educa-tion should be provided. Schools and school sys-tems should use the NCTM Standards to setspecific goals for student mathematics' learning.These goals should be used to develop culturallyrelevant multidisciplinary units for kindergartenthrough the eighth grade. At the junior high andsenior high level, mathematics and scienceteachers should communicate with teachers ofother disciplines to coordinate some topics. Highschools must ensure that graduates are preparedto continue into post-secondary education with theexpected skills in reading, writing, speaking/lis-tening, mathematics, reasoning and studying.Schools should not steer or 'track" Native studentsinto vocational or general curriculum in ways thatpreclude solid academic preparation necessary topursue a college education.

Native children should be expected to succeed!Make sure that this expectation of success isshared by students, parents, teachers, and ad-ministrators. Schools and school districts shouldmake the educational process more familiar toNative parents by holding school meetings in thecommunities or by meeting with parents regularlyto review their children's progress. Money, time,and effort should be invested in pre-school andelementary school programs to provide a solidfoundation of learning and self-esteem for thechildren. Wh sn mathematics and science aretaught in vv s students learn there will be no needfor remedial instruction. All children can learn!Summer programs for junior high and middleschool students are very important in determiningtheir classes for high school. Encourage and spon-sor students to attend pre-college summer enrich-ment programs in mathematics, science, andhealth science. Provide opportunities for studentsto learn about careers and opportunities in math-ematics, science, and health fields.

Establish programs which identify gifted Na-tive studer ts early in their school careers. Provideenriched and accelerated programs, credit by ex-amination, and other special arrangements to en-sure that these Native students are appropriatelychallenged rather than lost in the system. Inde-pendent learning should be encouraged. Parents ofgifted students should encourage their children tobe self-motivated and to study independently theirown area of interest. Help them find informationor write for references and material about theirinterests.

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Education needs to be more holistic to achievea joy of learning which will lead to life-long learn-ing. Parents and community leaders should insistthat children be taught in the way they learn.Parents and community leaders should insist uponthe purchase of hands-on materials to be used inthe classrooms and instruction of faculty in the useof same. Insist that teachers new to the systemattend in-service training which sensitizes them tothe Native culture of the area and the special needsof the Native children in that particular school.Native parents, grandparents, and relatives needto recognize the importance of motivating theirchildren to finish school and to help them succeedin iho educational process.

Thq Federal government must maintain thenational Gnmmitment made to the Native people.Improvement of the current situation requirescommitment of time and money. ACTION isneeded NOW. President Bush and the governorshave stated gods for education but have not ap-propriated fuks to meet those goals. Words withno action are meaningless. Inaction approachescriminal negligence of our children who are ourfuture! Money is better spent on education that iseffective and relevant now rather than on remedialprograms and criminal justice programs later.Funds are required to make salaries competitiveand to provide continuous training for ad-ministrators, faculty, and staff regarding learningstyles, instructional methods, including the use ofhands-on (manipulative) materials and the use ofculturally relevant material. Funds should beavailable to pay for administrators and faculty toattend conferences, summer institutes, and otherclasses to maintain current knowledge of activitiesand research in their field. Funds are requirod todevelop culturally relevant curricula within thepreviously determined curricula. Funds are re-quired to provide hands-on materials and con-sumable materials for science and mathematicsclasses, kindergarten through twelfth grade, in amanner similar to the way in which materials areprovided for art classes. Funds are required toimprove existing facilities, or build new facilities,which adhere to federal guidelines for the safety ofall students and which provide accessibility for thehandicapped. It should be noted that excellentfacilities just by their existence do not produceinnovative, effective instructional methods. Thefacilities provide an environment conducive forlearning and teaching. If funds are spent only onstructural renovations of the school and not onteacher support programs, teacher training, andclassroom materials, then the money has beenwasted. The entire situation needs to be considered

es a whole not in isolated parts. Time is requiredto accomplish the changes necessary. Neverthe-less, ACTION is required immediately. Thechildren are watching us to determine by our ac-tions how important education is to us? What arethey seeing?

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About the AuthorVera Preston grew up in Stillwater, Ok-

lahoma. Her BS in Education is from OklahomaState University; her MA in mathematics is fromthe University of Maine-Orono. Her thesis was

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Mathematics in the Aztec, Mayan, and Inca Cut.tures (written as V.P. Callahan). She was a doc-toral student of Dr. Helen Cheek, a Choctaweducator, at Oklahoma State University. She iscompleting her doctoral work at the University ofTexas at Austin with Dr. Charles Lamb.

She teaches mathematics at Austin Com-munity College (ACC) in Austin, Texas, where shehas developed a 'Building Success in Mathe-matics" course. She develops culturally relevantAmerican Indian and Alaska Native mathematicsmaterials. She developed and taught the mathe-matics curriculum for the American Indian andAlaska Native Math Camp at the Colorado Schoolof Mines the summer of 1988. She has taughtstudents of all ability levels in the seventh gradethrough college. She has taught at a predominatelyAmerican Indian school, Red Rock School, in RedRock, Oklahoma. She developed and taught "Mathis Fun" two-week workshops for four-year-olds tothirteen year olds in her home in Stillwater, Ok-lahoma. From 1962-1964 she served in the PeaceCorps in Ecuador. In 1982 she attended the EquityConference sponsored by the NCTM in Albuquer-que, New Mexico. She is past president of Womenand Mathematics in Education.

She speaks at National Council of Teachers ofMathematics (NCTM) regional and annual meet-ings on various topics which include: "Multicul-tural Activities in the Classroom," "MayanMathematics," and "Patterns in Mathematics."She presents workshops for teachers of kindergar-ten through college students on t'ue use ofmanipulatives and multicultural activities in theclassroom.

U.S. Dept. of Education

Office of EducationalResearch and Improvement (OEM)

Date FilmedAugust 8, 1992

Documents

1:**********W***********W***************w · AUTHOR Preston, Vera TITLE Mathematics and Science Curricula in Elementary and. Secondary Education for American Indian and Alaska. Native

1:****W*W***********w · AUTHOR Preston, Vera TITLE Mathematics and Science Curricula in Elementary and. Secondary Education for American Indian and Alaska. Native