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A Teacher‐Researcher Perspective on DesigningMulticultural Mathematics Experiences for PreserviceTeachersJanet M. SharpPublished online: 09 Jul 2006.
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A Teacher-Researcher Perspective on DesigningMulticultural Mathematics Experiences for PreserviceTeachers
JANET M. SHARP
Many cultures come together to shape the learn-ing environment of an elementary classroom.A teacher has personal connections to one or
more cultures, the teacher's style of and skill at communi-cation is cultural, and, of course, the students bring theirown unique cultures to the classroom environment.Moreover, the various subject matters studied withinthose classroom walls also have cultural connections.Teachers who are knowledgeable about the culture-relatedactivity in their classrooms can use this knowledge to pro-vide effective instruction. Boyer's (1990) stage theory ofethnic growth can provide guidance to teachers who rec-ognize the need to develop culturally rich experiences fortheir classrooms. However, part of a teacher's multicul-tural awareness also must be in relation to the historicaland cultural aspects of the subject matter itself.
For instance, in mathematics, it is a fact that two plus twois four. Unless some nontraditional algebraic structures aredefined, such as an alternative base system, two plus two isalways four. Mathematics, in this skeletal sense, is a disci-pline free from cultural biases. However, the Indian nu-meral 2 (Menninger, 1969) and the German notation +(Smith, 1919) provide evidence of cultural artifacts inmathematics. In ancient Greece, a learner would have usedspatial thinking to mentally arrange two items and twoitems in order to imagine the sum of four as shown in Figure1. The Greek learner might also have additionally notedthat four is a square number, resulting from 2x2.
In addition, examples and contexts selected by teachersmay affect the cultural communication of the classroomenvironment. Consider a young boy who grew up on afarm being taught about "2+2" within the context of count-ing floors in a skyscraper. Next, visualize a young girl who
Figure 1A Mental Image of "2 Plus 2"
grew up in an urban neighborhood being taught about"2+2" within the context of counting cattle in a pasture. Therespective cultures of the students (and teachers) have cre-ated two different cultural experiences for the learning of2+2. Effective mathematics teachers know about the cul-ture of their subject area and about the cultures of their stu-dents and then adjust instruction accordingly.
Mathematics is a dynamic collection of mental ideasresulting from beliefs and values from a myriad of cul-tures. In fact, van Oers characterized mathematics as be-ing defined through "a long series of lively debatesamong mathematicians about what is to be accepted asvalid mathematics" (1996, p. 95). So, the cultures of thosemathematicians will impact the debate, and mathemat-ics cannot truly be considered a culture-free discipline.Moreover, the mental nature of mathematics precludesseparation from the learning and teaching of mathemat-ics, which are not culturally neutral activities (see Ander-son, 1990; Bishop, 1988; D'Ambrosio, 1990; Rauff, 1996;and Sleeter, 1989). Anderson (1990) outlines six "peda-gogical disasters" in mathematics teaching which arebased on the belief that there exist certain hidden realitiesa learner must experience before being able to learnmathematics. Acceptance of "the myth that mathematicsis pure abstraction and, therefore, antithetical to one'scultural and working environment" (p. 350) is amongthose pedagogical disasters because it reinforces the no-tion that mathematics is separate from culture. The pur-pose of this article is to discuss the appropriateness andimpact of some multicultural mathematics education as-signments for future elementary teachers, assignmentsthat were designed to counter the above myth. Thisstudy will discuss a teacher-researcher's efforts to useBoyer's stage theory to guide her students' developmentof cultural awareness with respect to mathematics andthe teaching of mathematics.
BACKGROUND
Multicultural education is first of all a complete edu-cational program with curriculum, pedagogy, and ac-
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tion in which students acquire academic content.Secondly, multicultural education guides students to-ward the development of an understanding of theirown backgrounds, backgrounds of others (Bennett,1990), and the ability to make decisions and engage insocial actions reflective of that knowledge (Nieto, 1996).In a multicultural education experience, students areencouraged to respect, appreciate, and celebrate diver-sity and to overcome ethnocentric and prejudicial atti-tudes and discriminatory behaviors.
The mathematics typically nurtured in the culture ofthe school classroom consists of a Western view ofmathematics, which is characterized by structure, de-duction, operations, and applicability (Nelson, Joseph,& Williams, 1993; Powell & Frankenstein, 1997; Rauff,1996). Many mathematics classroom environments donot recognize the multicultural nature of mathematicsand tend to initiate students into the Eurocentric viewof mathematics (Bishop, 1988; Frankenstein, 1997; Pinx-ten, 1994). However, one aspect of social constructivistlearning theory has provided an avenue to challengethis approach to teaching. Proponents of this theory de-scribe teaching as the creation of social classroom situa-tions to give students shared experiences to construct"shared" mathematical knowledge (Wood, Cobb, &Yackel, 1995). By definition, those social, albeitknowledge-gaining, experiences are inseparable fromstudents' cultures and histories (Bauersfeld, 1995), andthe teacher must learn to find appropriate mathemati-cal situations that match those students' cultures(Sleeter, 1989).
On the one hand, "teaching becomes a matter ofcreating situations in which children actively partici-pate in scientific, mathematical, or literary activitiesthat enable them to make their own individual con-structions" (Wood, 1995, p. 337). On the other hand,to consciously engage in such instruction, "it is nolonger sufficient to just look at cultural influences,with adjustment of teaching style and class atmos-phere to perceived student preference; rather there isa need for an understanding of the importance of aca-demic competence combined with an understandingof academia as culture" (Garaway, 1994, p. 109). Ifteachers are able to avoid narrow views that inadver-tently promote single-culture learning of mathemat-ics and instead are able to know their students,complete with history and culture, teachers will beable to provide better mathematical learning experi-ences for their students.
Mathematics learned from such a consciously multi-cultural perspective results in students' more completeunderstanding of the subject (Frankenstein, 1997; Katz,1994), as well as an understanding that mathematics isdeveloping from beliefs and values from many cultures(Joseph, 1993). This rich multicultural perspective in-cludes use of culturally relevant mathematics class-
room activities that celebrate the diversity that character-izes American society.
Such a view of the teaching and learning of mathemat-ics leads students toward an understanding of ethno-mathematics. Ethnomathematics is a global view ofmathematics as a field of study which developed througha unique interplay of cultural activity. Multiculturalmathematics is about the celebration of the many culturalaspects of the school curriculum and pedagogy associ-ated with the teaching and learning of mathematics.D'Ambrosio defines ethnomathematics as "a programmewhich looks into the generation, transmission, institu-tionalization and diffusion of knowledge with emphasison the socio-cultural environment" (1990, p. 369). It is themathematics born and nurtured within a cultural system.Therefore, ethnomathematics is shaped by the history,mores, and morals of that culture and the people's indi-vidual experiences while serving their society. Successfulteaching is the quest to enable students to value, access,and utilize their ethnomathematical understandings,without narrow constraints associated with singular cul-tural viewpoints.
This perspective also includes teacher/student discoursein which a diversity of children's culturally determined,problem-solving strategies are valued (Nelson, 1993). Whenthe learning environment enhances all students' understand-ings of mathematics, and as classroom activities come to re-flect all students' cultures, the cultural nature of mathematicsbecomes a natural part of the curriculum. There are multicul-tural resources providing examples of such lessons (e.g.,Grant & Sleeter, 1989; Joseph, 1990; Nelson, Joseph, & Wil-liams, 1993; and Zaslavsky, 1973), but, in the end, it is theteacher who knows the students and must learn to connectthe learning situation to individual students. Belief in learn-ing as a construction process built upon previous knowledgerequires teachers to know the ethnomathematics of each stu-dent, as well as the world in general, and to provide corre-sponding learning experiences.
Teachers who base teaching episodes on this multicul-tural constructivist belief consistently expose their viewsof teaching and learning of mathematics within lessonscelebrating diversity. These teachers view the mathemat-ics curriculum as a multicultural entity complete with arich history inclusive of all cultures (Nelson, 1993).Moreover, they know that mathematicians are a group ofpeople whose diversity is so complex that no culture,gender, economic level, or ethnic group is viewed as hav-ing superior innate capabilities. Their rich learning expe-riences are demonstrative of the belief that all childrencan learn mathematics (NCTM, 1989). From such class-rooms come the sounds of children's voices whosemathematical curiosity has been deliberately triggeredby a carefully created mathematical situation. Theseteachers act on the belief that all children do learn mathe-matics through a multicultural mathematics educationprogram.
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For instance, Moore (1988) taught iteration (a repetitiveprocess), tessellations (covering of a plane with repeatedtracings of a fundamental figure), and symmetry (both ax-ial and central) through views of petroglyphs (drawings"created and carved on surfaces of caves, cliffs, and stonesby the ancestors of the American Indians" (p. 30)). Taylor,Stevens, Peregoy, and Bath (1991) taught middle-gradestudents how Native American mathematicians used tes-sellations in creative work ranging from sand paintings torugs and jewelry. When one of the students at that NativeAmerican school created a tessellation of an eagle, it wasan obvious example of the student's real-world cominginto the mathematics classroom. "The tessellated eagle...is culturally relevant, as eagles hold a special place ofhonor in Indian cultures" (p. 17).
Unfortunately, such examples are not standard fare formathematics classrooms. Typically, teachers claim that in-clusion of diverse people in the mathematics curriculum isinapplicable (Sleeter, 1989) or is achieved by inserting cul-tural names into typical word problems (Trent & Gilman,1985). These teachers do not view mathematics as a multi-cultural discipline, and they do not think of mathemati-cians as being representatives from culturally diversepeople. In fact, many historical works have obscured theheritage of many mathematicians or implied that mathe-maticians of importance hailed only from Greece orEurope (Powell & Frankenstein, 1.997). For example, thereexists no evidence to suggest that Euclid was not a BlackEgyptian or African mathematician (Lumpkin, 1997), yethe continues to be portrayed as Greek in many works. Togain understanding of multicultural mathematics educa-tion and apply it in terms of curriculum, pedagogy, and ac-tion, future mathematics teachers must experience aprocess of personal and professional development.
STAGE THEORY
Boyer (1990) developed an eight-stage theory about in-dividual ethnic growth, in which he outlines a person'sprogressively sophisticated multicultural view of self andsociety. These stages can be generalized to the classroomenvironment:
Celebration — all learners experience a festive freedom for ac-tive participation in the larger society at a levelvoid of apprehension or negative responses
Appreciation— all learners feel a positive visualization of theirprofiles because they are clearly a legitimate partof all phases of the school experience
Respect — all learners feel that all cultural elements havesignificant worth in any environment
Acceptance — all learners hear an acknowledgment and discus-sion of cultural elements as a proper, normal, in-evitable part of the school experience
Recognition — all learners see formal acknowledgment and spe-cial mention of cultural elements in the school ex-perience
Tolerance — there are cultural discussions designed toacknowledge cultural elements
Existence — the limited inclusion of cultural elements inthe school experience is both insignificantand uninspiring
Non-existence —- there is a complete absence of cultural ele-ments in the school experience
These stages can be used to guide the creation of ex-periences designed to facilitate movement through thestages and can be used to examine students' responsesto multicultural situations. For example, when the pre-service mathematics teachers cited by Trent and Gilmanwere asked to create word problems that would be rele-vant to Native American students,
about half of these future teachers attempted to use In-dian names to make the problems more meaningful,while the other half made no attempt to complete theassignment and instead insisted that the same prob-lems be given to everyone as "there is no significantethnic content in mathematics." (1985, p. 43-44).
These preservice teachers considered multiculturalramifications of mathematics to be irrelevant. So, theyexhibited traits associated with the nonexistence stageof ethnic growth. To work toward enhancement ofsuch students' multicultural views, Suzuki claims that"multicultural education should start where peopleare at" (1979, p. 48). It therefore makes sense to usestage theory to discern "where people are at." Preserv-ice teachers, such as those studied by Trent and Gilman(1985), need to experience existence of mathematicalreality through a variety of cultural viewpoints. "Un-less all students develop the skill to see reality frommultiple perspectives, not only the perspective ofdominant groups, they will continue to think of [real-ity] as linear and fixed" (Nieto, 1996, p. 319).
In many nonwestern cultures, linear, analytic think-ing is not the preferred mode of thought. Rather, circu-larity and holistic thinking dominate the successful andvalued thought processes (Asante & Davis, 1985; Chi-ang, 1993; Howell, 1979). Mathematics has been discov-ered and invented by people of cultures who usenon-linear as well as linear styles of communication(Frankenstein, 1997; Pinxten, 1997). A more fulfillingand accurate knowledge of mathematics and its histori-cal development can be achieved by students who ex-perience mathematics from a variety of culturalviewpoints and thinking styles. To this end, whilestudying to become teachers, they need appropriatelyplanned learning experiences that begin at their levelsof ethnic growth and which deepen their understand-ing of mathematics as a cultural entity.
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THE STUDY: PROCEDURES
Subjects
Over the course of four semesters, five groups of pre-service elementary school teachers (n=140) at two mid-size, midwestem universities responded to three activi-ties designed to enhance their cultural understandingsabout mathematics education. The course in which theactivities were completed was the elementary mathe-matics teaching methods course.
Twenty-three (16%) males and 117 (84%) femalesparticipated in the study. Two African American fe-males (1-4%), 3 Hispanic females (2.1%), 110 EuropeanAmerican females (79%), 2 females who did not reportethnicity, 22 European American males (16%), and 1 Af-rican American male (.7%) completed the activities. Allstudents had completed one introductory course in so-cial and cultural foundations of education and wouldcomplete an upper level course in multicultural educa-tion prior to graduation.
Speaking from a European American female pro-fessor-researcher background, I found my emergingpersonal theories about multicultural mathematicsmodified after a review of the 140 students' work withthe three activities, and five students' privatelyshared, unsolicited personal thoughts. I had taughtthe course six times prior to teaching the four semes-ters described in this article, with continual revision.My efforts reflected Noffke's statement that, "whileteaching a course, it is natural to develop the same'questioning attitude' in ourselves as teacher educa-tors as we would have in teachers" (1992, p. 28). Ingeneral, when the teacher educator engages in reflec-tive thought about carefully planned, theoretically-based teaching activities, personal theories in thepractice of teaching develop.
In the following discussion, both quantitative andqualitative reporting are used as I describe my efforts tounderstand my students' work on the various assign-ments and activities. This dual approach of reflectionabout my students' accomplishments provided a morefull and complex view of their thoughts than wouldhave been achieved with sole use of either style of re-porting.
Activity 1
In a whole group setting, I told a story about a mem-ber of the mathematics department of my alma mater.At the conclusion of the story, students were asked tovisualize the professor's office and conversation. Afterthe students had a picture of the mathematics professorin their minds, they drew a picture of that mathemati-cian (see Figure 2).
After students finished their drawings, I asked,"Who does mathematics?" Being typical elementary
education majors, they dutifully replied, "everyone." Ofthe five classes, 82%, 74%, 76%, 75%, and 79% of the stu-dents had drawn older European American men. I asked iftheir drawings were inconsistent with their "everyone"statement. At first the students were quiet, but then anemotional, positive discussion ensued. As best I could, Iused this discussion to pull any students who were at thenonexistence stage of Boyer's stages of ethnic growth to-ward the existence stage.
Even though the majority of these students drew oldermales with European ancestry, one could argue that theysimply relayed their true collegiate experiences. Themathematics department at one university had no femalefaculty, at another university, the mathematics departmenthad only two female faculty. For both universities, thosefaculty who were not of European descent were not U.S.American. These preservice teachers have not experiencedthe existence of many different people who do mathemat-ics and so they do not visualize university mathematiciansas diverse. However, because they will teach children ofmany cultures, they must examine how their personallearning experiences may have influenced their beliefs.They must develop cultural consciousness (Bennett, 1990).This is an effective activity because it begins where the stu-dent was, and thus provides an opportunity to advance to-ward a higher stage.
After reflection on the students' drawings, I createdanother higher-stage activity. The students were to createfrom photographs a poster depicting two people whoregularly do mathematics in their daily activities. I re-quired that these two people should be different in somemanner, whether it be due to gender, race, or some otherattribute. Through this recognition level assignment, thestudents might formally acknowledge the many culturesthat are represented among people who do mathematics,or possibly, they might demonstrate respect or celebra-tion. By participating in both activities some students ap-peared to have advanced through stages. A notableadvancement came from one European American femalestudent:
"I've found a black female and a white male. I know I can'tuse the white male. But, I can't find anyone else. Whatshould I do?""Why do you think you can't use the white male?""Because this is supposed to be multicultural!"
Before the assignment, she believed participation in a multi-cultural education program required the exclusion of Euro-pean American males. "There is a widespread perceptionthat multicultural education is only for students of color, orfor urban students, or for so-called disadvantaged stu-dents" (Nieto, 1996, p. 312). Because it appears that the stu-dent has engaged in some discussions (and developed somemisconceptions) designed to acknowledge cultural ele-ments, she is at the tolerance stage of ethnic growth. This as-signment started at the recognition stage. Only through
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Figure 2Activity at the Existence Level and Tolerance Level: Draw a Mathematician
Once upon a time, there was a mathematician who loved to teach projective geometry. We studiedwhat seemed like everything there could have been to know about intersecting lines. The professorassigned homework on a daily basis. Between solving the problems in that book and studying newtheorems and corollaries, I came to know geometry. However, I'll never forget the day that myprofessor came into the room and asked what geometry could there be in railroad tracks? If a pair ofrailroad tracks were projected into infinity, what would happen? What does an artist see when drawinga pair of railroad tracks?
Draw a picture of the mathematician discussing the railroad tracks.
one-on-one discussions with me was she able to succeed.I observed that when the assignment was one step be-yond where the student was, she experienced difficulties.
Activity 2
Individually, each student selected a specific geometryobjective and then wrote a story, published a magazine,or produced a video about that geometry topic. The as-signment requirements articulated that the geometric oc-currences be presented in terms of the multiculturalworld around them. For example, a student might dem-onstrate congruence by collecting photographs of Arabictessellations, Chinese lattice patterns, or Celtic knot-works.
One European American female student created andwrote articles for a geometry magazine using no cultural,gender, or ethnic references. She said, "I addressed themulticultural aspect by not addressing it." She felt thebest course of action was simply to directly and antisepti-cally explain geometry ideas through writing, as thoughdifferent cultures did not exist. "In that way," she de-
fended, "no one feels left out." This student, positioneddistinctly at Boyer's first stage of ethnic growth, nonex-istence, felt if she left everyone out, this resulted in noone feeling left out. Because my original intent requiredstudents to create a mathematics teaching aid promot-ing one or more goals of multicultural education, the as-signment was at the acceptance stage, four steps fromthis student's stage of growth. Because she could not ac-knowledge diversity, she could not complete the assign-ment successfully.
Another European American female student wrote ageometry story about turtles and frogs playing to-gether. The characters in the story encountered differentshapes in their worlds and they classified them accord-ing to typical (Western) geometry words. She stated,"This would be interpreted as students of different cul-tures playing together." This student is at the secondstage of ethnic growth, existence. When people are atthis stage "there is inclusion as opposed to total absence"(Boyer, 1990, p. 40). The fact that the student broached themulticultural aspect through a metaphor may indicate thestudent's view of multicultural education as a controver-
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sial topic unsuitable for straight-forward discussion.Had she at least recognized that beings from differentcultures (frogs and turtles certainly have different "so-cietal" structures) might have different geometry wordsand experience difficulty in communicating with oneanother, she might have managed a more appropriateresponse to the assignment. "It is one thing to developknowledge and awareness of human similarities andanother to develop empathy" (Bennett, 1990, p. 294).But, this student had not grown to the stage of the as-signment and could not appropriately complete it sinceit was well beyond her stage of ethnic growth.
A third European American female student sharedher thoughts about the story she had written and illus-trated about her son's attempt to see mathematics in theworld around him. She included geometry along withthe other mathematics from which she believed shecould not separate geometry. Her response to the as-signment was accurate and appropriate. A paraphraseof her comments follows:
I am part of an entire generation of children whowere raised believing that we should avoid makingstatements identifying a person's ethnicity. I feel veryuncomfortable when stating a person's ethnicity. Wehave been taught that everyone is the same and that tonotice differences is bad and unfair to that person. Forexample, I hate it when someone says, "You know . . .the black guy" when trying to identify a particular per-son in a group of people. I have quietly fought what Ithought was the good fight by consciously avoidingstating a person's ethnicity.
My husband is Cuban, so my son is Hispanic. In mystory, I identified the auntie as Hispanic, but did notidentify the child's ethnicity. Am I supposed to makesure everyone knows the ethnicity of every characterin every story I ever write (mathematical or not)? Iwant to recognize the ethnicity of my characters.
I don't think that I should do so by coloring the peo-ple, it seems to me that if I were to color the people"people-colors" in my book, then that would be takenas a stereotypical response.
Clearly, the student is struggling with the intendedmulticultural essence of the activity. She seems to be atthe acceptance stage and is trying to get to the respectstage. Her primary concern was how to show respectwithout stereotypically showing recognition. Becauseshe had no difficulty recognizing mathematics as acomposition of multicultural factors, the assignmentwas appropriate for her. It started where she was. Eventhough I thought I had created an appropriate assign-ment for all students, I again found overwhelming evi-dence that when not following Suzuki's (1979)suggestion of starting where they are, assignments fail.Although it will be challenging to create assignments inline with each of my students' starting points, I believe itis possible. However, I have ceased to use Activity 2 be-
cause it requires students to be further along in their stagesof ethnic growth than they tend to be.
Instead, I have used my findings to create a new as-signment, which has been more successful. First, I pres-ent an in-class geometry lesson about symmetry based ona story about a Navajo weaver (Krensky, 1991). This les-son is at the recognition level. Then, I give an acceptancelevel assignment: Students must develop a K-6 geometryor measurement activity that could be used in conjunc-tion with a children's literature book. Each student mustbase a geometry lesson on a children's book or folk-talethat highlights a culture. I have found this approach to beeffective. An example of a student's response is shown inFigure 4.
Activity 3
Each student explored the geometric ideas within the"Ohio Star" quilt pattern used by U.S. pioneer women (Za-slavsky, 1994). In written form, the students justified usinga cultural approach when teaching about congruence (seeFigure 3). Ninety-seven percent of all students respondedfavorably to such an approach and communicated theirbelief(s) that such a lesson would enhance children's un-derstanding because it provided a meaningful context forlearning. "European American youths often feel that theydo not have a culture, at least not in the same sense thatclearly culturally identifiable youths do" (Nieto, 1996, p.313). Because 79 percent of the students were females ofEuropean descent, it may be the case that they foundmeaning in the quilt context: It offered recognition of a cul-ture to which they may have felt more closely connected.Twelve percent of the students mentioned extending thisactivity to quilts associated with the Southern populationor Native American nations. "People must possess a de-gree of self- and group-esteem, as well as personal security,before they can be empathetic in their interrelations withothers" (Bennett, 1990, p. 283).
Activity 3 seemed to match many students' culturalawareness. It was also somewhat open-ended, so thosestudents who were able to move along did so. In addition,the cultural focus of Activity 3 was relevant to the majorityof the students. I learned that deliberate celebration ofmathematics topics within our own culture helped thesepreservice teachers come closer to visualizing celebrationof all cultures. I now use this activity in my class each se-mester.
Decisions about continued use of these activities werebased on class discussions as well as on conversationswith individual students. The low number of students(five) who sought me out to discuss their thoughts may re-sult from a struggle with disparity between personal expe-riences and political correctness. Or they may findmulticultural mathematics a difficult subject to discuss.Three of these five students had completed the upper levelmulticultural course. Clearly, students were entering myclass at several stages of development according to Boyer's
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Figure 3Activity at the Recognition Level: Making a Star
This 3 x 3 quilt block is called "Ohio Star". The quiltmaker might use the same pattern for all theblocks in her quilt, or she might vary it with other designs. She thinks about the fact that when theblocks are placed one beside the other, the pattern within the block will also create an overall patternfor the entire quilt. She selects her fabric and colors according to how the overall pattern will look.
From the time of theAmerican colonies,women have organizedquilting bees. Thiswas one way thatwomen could easilyget together sociallyand do their work atthe same time.African-Americanwomen developedtheir own styles, someof which were basedon more traditionalAfrican patterns.
1. Analyze the 3 x 3 quilt block. Using the block, find:(a) 2 different pairs of congruent shapes.
Shade them in with a marker and describe them in words.(b) an example of symmetry. Describe this symmetry.
Be careful to distinguish it from among the three possible kinds of symmetry.
Adapted from C.Zaslavsky (1994).
theory. Therefore I felt a need to design additional activi-ties for students who were ready to experience higher lev-els of multicultural mathematics.
Additional activities
As a result of these experiences, I created three more in-dass activities aimed at the top three levels of Boyer's stagetheory.
At the respect level, students should see all cultural ele-ments having worth in the whole classroom as well as theoutside world. I sought to bring into my classroom an ideafrom several cultures of the outside world. Number wordsand numeration systems typically emerge together. In theU.S./English language/number words for 11 through 19
are inconsistent with number words for 21 through 99.This fact gives rise to the opportunity to discuss somemathematics that crosses cultures: base-ten numerationand the multicultural nature of the various numberwords. Figure 5 shows a place-value activity designedfor students at the respect level.
At the appreciation level, students must visualizeseveral ethnic profiles as clearly legitimate parts of thewhole school experience. One way to address studentsat this level is through study of the mathematics of an-other culture, in which the corresponding thinking pro-cesses directly challenge those pre-dominantlydisplayed and valued in the textbooks of the U.S. soci-ety. Often, students are unaware that an idea as funda-mental as an algorithm for subtracting two-digit
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Figure 4Activity at the Acceptance Level: Creating the Calendar
Read the book, Thirteen Moons on Turtle's Back by Joseph Bruchae and Jonathan London.
(In this book, the yearly 13 moons are described Although nearly all tribal nations of the nativeAmericas had different names for each of the 13 moons which appear each year, this book selects onemoon from each of 13 tribal nations and highlights it, describes it, and notes the corresponding tribalnation.)
Select one moon as described in the book, decide the season within wKich that moon would occur, andexplain your reasoning. ^
Now, suppose die accepted calendar for Earth had been created from the information in this book.Create the calendar in the space below, determine what today's date would be in this new calendar.Extension: Explain how this new calendar deals with leap year.
numbers can have cultural underpinnings (see Figure6). When a mathematical algorithm is the context tostudy the fact that various cultures develop unique al-gorithms, students can more completely appreciate thealgorithm and move along in their ethnic growth.
In contrast, an important cultural activity, such as Af-rican drumming, can be studied and applied as a con-text for learning mathematics. After the students learnsome history about drumming and the importance ofdancing in many African cultures, the U.S. students cancelebrate the mathematics by joining in and playingsome songs. In the activity described in Figure 7, stu-dents first learn to play a basic 6-count rhythm (osti-nato). They continually repeat this rhythm, while an ex-pert musician plays an 8-count rhythm (tumbao). In Af-rican drumming, many different rhythms areconcurrently played by different drummers. A poly-rhythmic song results as these rhythms of differentlengths are played simultaneously. As the rhythms be-come more and more out-of-sync, so to speak, dancersbecome more agitated and thus more inclined to"move." Because dancing is a valued and importantpiece of many African cultures (Grand, 1985), the abil-ity of a song to move a dancer is paramount. But, thenatural mathematical question is, how do the variousrhythms compare (ratios) and when will two givendrummers "meet back up" (least common multiple) at asimultaneous beginning. Figure 7 displays a recordsheet to accompany this experience.
CONCLUSIONS/IMPLICATIONS FORTEACHING
Future teachers find difficulty believing mathemat-ics is multicultural. They know 2+2 is four. But, in order
to think about and sensibly teach about 2+2, future teach-ers need to know how visualizations and contexts con-structed by people in various cultures actually create thecultural nature of mathematics. It is the responsibility ofthe teacher educator to create activities leading futureteachers toward a celebration of mathematics as a multi-cultural entity and toward the belief that all students canbecome mathematicians. Activities are most effectivewhen designed with the students' stages of ethnic growthin mind. In this study, I found that activities at the recogni-tion stage seemed to provide successful learning experi-ences only after the initial nonexistence stage activity anddiscussion. Assignments at the acceptance level (orhigher) were successful for only a few students. I intend tocontinue to study the results of using mathematics lessonsdesigned for each stage of ethnic development.
Collegiate mathematics activities and experiences mustbe designed to gently lead preservice teachers through asmany stages as possible. Teacher educators should planthese learning experiences carefully being certain to beginwith lower level activities. Once a common base has beenestablished, teacher educators can move closer to the goalof modeling the appropriate pedagogy to celebrate culturein the teaching and learning of mathematics.
The events described in this article involved college stu-dents studying to become teachers. However, uses ofculturally-rich mathematics activities, at various levels ofethnic growth, need much more research at all levels(K-12) of the typical school experience. In addition, vari-ous sub-sets of elements of the cultural experience oflearning mathematics, such as language usage in, commu-nication about, approaches to problem solving of, and pre-ferred visualizations of mathematical situations need to beexamined in far greater detail.
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Figure 5Activity at the Respect Level: Numeration Systems
The following number words are spoken regularly in Kuanyama, Africa:vali tatu tano-na-valine tano-na-mwe tano-na-tatutano mwe
C. Zaslavsky (1973) Africa Counts (p.39-41).
In the above list they are out of order. They represent the numbers values of one through eight (1,2,3,4,5,6,7, 8). Put the words in an order you can defend, "na" means "and".(There are several solutions to this question, but that should not be taken to mean that the Kuanyamanpeople do not have a consistent numeration system! Some necessary information has been withheld,thus preventing the correct solution. However, a "defensible " solution is quite possible.)
What do you suppose is their number word for nine (9)?
teacher note:FY1...
mwe= 1, vali =2,tatu = 3, ne = 4, tano = 5
(Zaslavsky, 1973. p. 39 - 41)
The Korean number word for 17 directly translates to "ten - and - seven".
Research and find the number words for 17 in 9 different languages.Record the literal translations for those words.
How do the translations compare to the U.S./English word "seventeen"?
What role does place-value play in a child's development of his or her number words?
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Figure 6Activity at the Appreciation Level: Mexican Subtraction
In many cities in Mexico, children are taught to complete subtraction problems by working left to right.For example, the exercise: 52 - 29 would be completed as follows:
52-29
Given problem
52-293
Subtract the "tens"
1252
-293-2
Want to subtract the
1252
-293-23
Subtract the "ones"ones", borrow fromthe answer (3 "tens")to get enough "ones".
Solve the following exercises using the Mexican method of subtraction:
63-46
41-18
38-24
146- 79
Why should a teacher be able to do computations with a variety of algorithms?
What does a teacher need to know about a child's background in order to plan effective instruction?
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Figure 7Activity at the Celebration Level: Using Drum Rhythms to Learn about Ratios
Use these symbols to record a bean-shaker rhythm* Shake — Rest
Adapted from Dworsky & Sansby (1994).
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Janet M. Sharp is Associate Professor of Mathematics Educa-tion at Iowa State University, Ames.
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