Mathematics Education Reform andDisadvantaged Preservlce Teachers:

A Case Study of a First Nations Studentby Ann Kajander

Pro sp ective elementary teachers facemultiple challenges with the implementa­tion of new curriculum, especially withchanges both in content and pedagogy. Forthis reason and others , there has been in­crea sed interest in the mathematical capa­bilities of preservice and in-service teach­ers (Ma 1999). Preservice teachers may notha ve experienced learning mathematicsusing instructional strategies consistentwi th reform practices.

The research evidence on achievementstrongly favors reform-based learningover traditional mathematics-instructionmethods (McDougall, Lawson, Ross,MacLellan, Kajander, and Scane 2000;Kamii 1994; National Council of Teachersof Mathematics [NCTM] 2000). If teachers'own learning in mathematics is of a tradi­tional nature, then the transition to reform­based teaching practices also will be achallenge for them as both learners andteachers.

Teacher beliefs and attitudes aboutteaching and learning mathematics maybe another barrier to success . Though at ­titudes toward mathematics have oftenbeen reported separately from achieve­ment (Franz 2000), Op't Eynde, DeCourte,and Verschaffel (2000, 3) have argued,"Mathematics related beliefs are situatedat the intersection of the cognitive ... and

the affective domain." Research in teach­ers' beliefs about mathematics shows thattheir attitudes and beliefs are strongly in­fluenced b y earlier school experiences(Wood, Cobb, and Yackel 1991; McDougallet al. 2000). However, targeting preserviceteachers' prior beliefs during instructiondoes have a significant impact on their be­liefs about teaching and learning (Fos s2000). Biagetti (2000, 3) has suggested aneed for "a better understanding of howteachers become engaged in self-sustain­ing, generative change, and the contextswhich support and advance this type ofteacher learning."

Any minority group of teachers ma yexperience social, cultural, economic, andinstitutional barriers they must overcome(Quiocho and Rios 2000), First Nations stu­dents may face special challenges due tosocial practices in their cultural heritage aswell as restrictions in their language. I hopeto illuminate some of these challengesthrough the eyes of a female preserviceAboriginal" teacher. Though this study re­fers to a Canadian student, it is possible thatsome of her experiences are common toother First Nations students throughoutNorth America.

• The term "Aborig ina l" is the accep ted term for referring tothe First Nat ions or Na tive American peopl e in Canada andwill be used thr ou ghout this essay.

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METHOD

This study took place in a mathemat­ics course for preservice elementary teach­ers in a concurrent (four-year) educationprogram, in a relatively small university inNorthern Canada. The university itself ca­ters to Aboriginal students by having spe­cial services provided to them-a lounge,support staff, tutoring, and so on. Thecourse itself is a lecture format and gener­ally has about 100 students per year. Thestudent in the discussion took the coursein the 1999-2000 school year. I have beenteaching the course for about 10 years, at­tempting to do so in the spirit of The Stan­dards (NCTM 2000) and other reform-baseddocuments. However, other than a gradingassistant, no other services, such as labora­tory periods, are available outside of thelectures.

The students complete 10 assignments,6-10 journal questions, 3 quizzes, a test, anexam, and a term project in each of the twoterms of the course. I collected data fromsamples of the student's work, particularlyjournal questions and projects, notes frominformal interviews, tape-recorded discus­sions, whole-class survey data, and studentgrades.

FINDINGS

Oshkiwahpikonese (a pseudonym cho­sen by the student meaning "new flower")is a soft-spoken Aboriginal woman. Whenshe came to see me before taking the math-

Ann Ka/ander teachesmathematics to preserviceeducation students atLakehead University inThunder Bay, Ontario. Shealso teaches mathematics inthe classroom. As part ofher

research , Dr. Kajander has developed anafter-school elementary-enrichmentprogram called Kindermath.

ematics course to express grave misgivingsabout her mathematical ability, it was noth­ing unusual. In fact, many students havesimilar feelings . In that year, out of morethan 100 students registered for the (full­year) course, 46 percent said they were ner­vous or unhappy about having to takemathematics, 20 percent noted it was diffi­cult for them but they were willing to workhard at it, 24 percent had positive feelingsabout mathematics, and 10 percent did notrespond to a question on the initial studentsurvey about their attitudes and feelingstoward mathematics. This situation wassimilar to that of previous years.

Oshkiwahpikonese's situation waseven more challenging than many otherstudents ' . As a Native Canadian womanwho had had many family problems grow­ing up, her technical skills in mathematicswere severely limited. In one of her initialjournal entries, in response to a questionabout calculator use in the classroom, shewrote, "The times I do not have a calcula­tor .. . I become so fearful that I might bewrong in my figures . However, I am begin­ning to go without a calculator for every­thing. Perhaps sometime in the future I willbe able to add, subtract, multiply, and di­vide in my head."

The first few times Oshkiwahpikonesetalked to me, she continually emphasizedhow nervous and scared she was. She wrotein her journal, "To be honest, I really don'tknow if I can do it, because it's hard forme." At this point, Oshkiwahpikonese'sconception of mathematics was fairly tra­ditional. When asked to describe in herjournal what mathematics was to her (justafter working on the first big problem ofthe course, the "checkerboard" problem ofhow many squares could be found on an 8X 8 grid), she wrote, "What mathematicsmeans to me is knowing or learning aboutnumbers, formulas, equations, problemsolving, and understanding the question.

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For example, the question about the check­erboard-it's frustrating."

A little later in the first term,Oshkiwahpikonese began to tell me moreabout herself. She told me she had beenworking and saving for several years tocome to the university. She also told methat, as a child, she had been abused by heruncle and then packed off to an arrangedmarriage when she was still in her teens.The marriage was also abusive. She has a16-year-old daughter who functionscognitively at about a Grade 2 level andwho had recently come home to her froman institution. She told me, "1 said to my­self, it's Oshkiwahpikonese's turn now."She had decided to come to the university.She chose education because she wantedto help the children on the reserve. Copingwith disaster seemed to be a normal wayof life for Oshkiwahpikonese. In the secondmonth of the course, she had to travel about1,000 kilometers to assist her brother, whohad had heart surgery-and, a few monthslater, she spent an extended time at inten­sive care with a cousin who had tried tocommit suicide.

In my mathematics course, I establishlearning groups at the beginning of thecourse for small-group activities .Oshkiwahpikonese came to me at one pointbecause she wanted to switch to a groupthat was in terested in meeting more regu­larly than did hers. She wrote, "My groupis not always available, and 1feel 1cannotdepend on asking simple questions whichare difficult for me to understand. There­fore, 1feel 1need more time to work on mymath! God, it's hard! Reading and reallyunderstanding!"

After a class discussion in which I men­tioned Kamii's (1994) work, Oshkiwah­pikonese took one of her books out of thelibrary and began reading it. She was stillstruggling with the class work. As she de ­scribed it, "1 was stuck throughout.. . .

I read and reread over and over again totry to understand what I am to do ."

Many of Oshkiwahpikonese's difficul­ties came from lack of knowledge of basicconventions. For example, the notations

2

2 •2(_)

left her completely confused as to when toadd and when to multiply.

When we began discussing integers, Iasked students to discuss models for sub­tracting in tegers in their journals. This wasthe first written evidence of Oshki­wahpikonese's cultural heritage. As shewrote, "The best I've come upon is the carmodel, [for] which 1would use a snowmo­bile or a boat, because, if I teach youngAboriginal children, they would under­stand more, and perhaps [I would] saythings in native language so they can un­derstand it."

I began to understand more about thedeep challenges Oshkiwahpikonese wasfacing after asking the class in the first termto read "Beyond Being Told Not to Tell"(Chazan and Ball 1999), a description ofreform-based mathematics classrooms andteacher learning. She wrote:

I found it very different to learnmath in this way. Because of the dif­ference-where a student, no matterwhatgrade theyare, they havetochal­lenge and prove their statements .That'swhy it's different! Furthermore,as a First Nations person or commu­nity, we are told not to create frictionof any sort-it really is something toknow. . . . In order to besuccessful [ina reform-based environment] andprove to an instructor or teacher youknowyourstuff, astudent hastoprovetheirpointandchallenge theiranswer.That's why I say-it's different beinga FirstNations person, very different .

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Oshkiwahpikonese was

now not just learning

new mathematics,

but rethinking its

relationship with

her culture.

How can we change what has beentold, taught, heldsacredforalong timewhen to show this side of learning isto be assertive or different? However,I should be aware that assimilation isalways in the minds of nonnativepeoples to always change the ways ofAnishnawbepeople.

This was my first experi-ence with the concept thatmathematics reformmight be cul turally inap­propriate for somepeople. Oshkiwahpiko­nese was now not justlearning new mathemat­ics, but rethinking its re­lationship with her cul­ture. She was askingwhe ther or not shewanted to make changesto her belief system about appropriate be­haviors for learners and ways to show re­spect for teachers and elders. As she put it,"It' s something I must think about-andyes , if it's a way to better learning, sobe it."

Oshkiwahpikonese was beginning tobe intrigued by some of the new ideas. Af­ter reading about the Van Hi ele levels(Shaughnessy and Burger 1985), I askedeach student to describe his or her ownlevel and jus ti fy the dec ision.Oshkiwahpikonese wrote, "That' s why Ithink I'm a level ' 1.' Because I do not re­member or know any of what is being ex­plained in class.Therefore, it's an enormousload for me to do each week. But, like I said,'I do the best I can and give all I have tolearn. ' From the little I've picked up-it'sneat!"

Toward the end of the first term,Oshkiwahpikonese confided in me that shehad started to use some of the ideas we haddone in class wi th young children on the

reserve, and that even her daughter hadshown some interest in what they weredoing. After working on a kaleidoscopereflection and rotation pattern using acetatetransparencies (Kajander 2003 ), Osh­kiwahpikonese started to show real anima­tion when she described to me her work

with the children and howgood it made her feel tounderstand it and be ableto show them:

I usedthe90-degreeanglefrom the templatethat was handed to us[she initially called it45 degrees by mis­take]. I traced the 90degrees into four sec­tions, colored by rotat-ing each piece. Prior tofinishing the coloring, I

took two mirrorsanddid thereflection.What I sawwasidentical [to thedraw­ing]. So I continued tof inish it, and itcame out as I saw it in the mirror.

I showed my 6-year-old nephewthis weekend when I baby-sat him. Hereallyenjoyed himself. We also madetwo small kaleidoscopeswhich he tookhome. My sister was impressed. Yes,it's true, children pickupandare verywilling to learn . That was myfirst ex­perience with the rotation andreflection.

It made me wonder [whether] thepatterns I enjoy admiring would behandled the same. . . . Just do a rota­tion and reflection to complete abeadwork project.

Oshkiwahpikonese began to look at herNative patterns, choosing for her termproject ancient Native patterns, called"petroglyphs," that contain iterative pat­terns. When I provided Oshkiwahpikonese

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with some books on fractal geometry thatillustrated similarly constructed images,she was thrilled. Later, as we began to lookat some geometry software in class, shebecame excited by the recursive potentialin Logo, and spent hours reproducing thesepatterns on the computer. This connectionto her culture seemed to be an importantturning point for her. Her whole mannerbegan to change when she talked to me anddescribed her work: "I get it now! I under­stand-and I enjoy it!"

When we did the "bouncing ball" ex­periment in class and recorded the linearrelation of the rebound height versus thedrop height of a ball, she tried it again athome. She later reported: "The bouncingball-I tried it. It really works like that! Myfather, an Anishnawabe, he knew thosethings too. How did he know?"

Unlike some other Native Canadianstudents with whom I have worked,Oshkiwahpikonese was willing and able toself-reflect on her learning: "The verbal­we 're not so good at. I need to try it. .. . Ihave to see it." For example, though shecould not recall how to solve a linear rela­tionship, the process was made clear to herby building and working with a balanceusing unit chips and unknown weights. Yetphrasing in textbook problems such as"express the given variable as a functionof ... " was completely unintelligible for her.Not only had she not learned many of thesewords in her Native language, in fact manyof the words did not even exist in her lan­guage. As well, the abstract nature of manyof the concepts seemed to elude her; as shenoted, she needed to "see" it "visually."

As we neared the end of the course,Oshkiwahpikonese became engrossed withcreating some geometric drawings of Ab­original designs and art on the computer.She continued to work with Aboriginalchildren, with whom she could practice hernew-found mathematical skills: "I'm al-

ways thinking, what am I going to do . . .what am I going to teach them?"

Oshkiwahpikonese's approach toteaching children seemed to be to provideenvironments in which children could playand explore, mixed with showing them"neat" things like the kaleidoscope. Basedon Oshkiwahpikonese's earlier commentsabout her cultural norm being not to chal­lenge others, it is unlikely that the sort ofverbal discussion often associated with so­cial constructivism was possible for her tofacilitate at this stage. However, she re­ported that the children enjoyed their dis­coveries when she worked with them.Oshkiwahpikonese also described repeat­edly how good it made her feel to be ableto share new ideas and how the teachingexperience validated her own new-foundunderstanding.

Oshkiwahpikonese's final grade in thecourse was a B. Though the grade did in­clude assignments, projects, and a journalassessment, 45 percent of the grade wasbased on formal written tests of a reason­ably traditional nature . Clearly, Oshki­wahpikonese had made huge progress intechnical skills as well as problem solving.In her final course evaluation, she wrote:"I was very scared when I enrolled in Sep­tember. All I can say is it feels like the firsttime I've ever taken math." It is importantto determine which aspects of this course,and of Oshkiwahpikonese herself, madethis experience so valuable for her, as anAboriginal, as a disadvantaged student,and possibly as a woman.

DISCUSSION

A turning point for Oshkiwahpikoneseseemed to be the extensive work she didwith Logo to create fractals and designsinspired from her Native art. This workexcited her, increased her self-confidencemathematically, and seemed to give thework cultural meaning for her. As Taylor

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(1997) noted, culturally relevant math­ematical experiences facilitated middleschool Native Americans' development ofpositive attitudes. Interestingly, Taylor(1997) also suggested using designs fromNative artwork for investigation and thestrategy of reproducing them using Logo.Oshkiwahpikonese had been able to comeup with this idea on her own, showinggreat excitement in doing so , a pursuitwhich lasted for the latter half of the courseand even afterward. She regularly told mehow wonderful this experience was for herand how successful it made her feel to beable to explain the ideas to young Aborigi­nal children. In fact, the validation of hernew ideas by the children in her own cul­ture seemed to be what gave her work themost meaning . She regularly spoke inglowing terms about her work with Ab­original children, exclaiming that it madeher feel so good to help them understandit and enjoy it. This work seemed to be theultimate validation of her efforts.

Writing in both her journal and on as­signments was also important toOshkiwahpikonese, as well as working ingroups with other students. Jacobs andBecker (1997) suggested that the four prin­ciples of feminist pedagogy can be used tobuild a gender-equitable, multiculturalmathematics classroom: using the student'sown experience, writing, and cooperativelearning; and de veloping a community oflearners. All of these elements were presentat least to some extent in Oshkiwah­pikonese's experiences during this year.She made extensive use of the writing pro­cess and actively sought more work ingroups. She used her own cultural experi­ence for inspiration and sought to enrichthe community by working with the chil­dren to practice and share her new ideas.

Preservice teachers face many obstaclesin their development, but the situation iseven more challenging for students with

relatively weak mathematical backgroundsand different cultural backgrounds(Quiocho and Rios 2000), including NativeNorth American students. The differencesin cultural backgrounds ma y become ap­parent pedagogically as well. Is reform­based learning inherently culturally biased,relying as it does on interaction and ques­tioning-behaviors that may be in opposi­tion to some cultural norms? Yet, as thiscase study illustrates, even disadvantagedstudents can make significant progress inan appropriate environment. Indeed, theyhave the potential to make a substantialcontribution to their own culture. In fact,the desire to contribute to one's own cul­ture may itself be a prime motivator forlearning, as seemed to be the case forOshkiwahpikonese.

Oshkiwahpikonese made greatprogress in improving technical skills aswell as problem-solving and investigativeability. Her confidence in her beliefs aboutlearning and about her own ability to domathematics also increased. Though shewould not have passed a formal entranceexam to enter an education program, hereducation will continue to be of great ben­efit to her people as well as to herself.

Oshkiwahpikonese's own determina­tion was certainly a factor in her success.However, her use of writing and reflection,as well as cooperative group work, oppor­tunities for hands-on experiences such aswith the balance or bouncing ball, and herefforts to try these ideas out with otherswere also valuable for her. She began as anexceptionally disadvantaged student interms of technical skills , but she made in­credible progress in a short time .

Further research is needed to investi­gate which of the student's experiences inthis case study are generalizable. However,I belie ve it is important that mathematicseducators give enough open-ended andrich investigative opportunities to all stu-

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dents to allow them the best chance pos­sible to find and create meaningful experi­ences for themselves. Reaching a disadvan-

taged student like Oshkiwahpikonese canbe one of the most rewarding experiencesof all.

The author is grateful to Peter Taylor for his sensitive and insightfulideas on the draft of this article-and to Doug McDougall, who sup­plied extensive editorial comments. As well, the student described inthe article provided detailed feedback on descriptions of her progress.Thanks also for the support of Lakehead University.

REFERENCESBiagetti, S. C. 2000. Teachers creating framew orks for their

stu de n ts ' algebraic thinking. Paper presented at theAnnual Meet ing of the Ame rica n Edu cati onal Resear chAssociation, New Orl eans, 24-28 April.

Cha zan, D., an d D. Ball. 1999. Beyond being told not to tell .For the Learning of Mathematics 19(2): 2- 10.

Foss , D. H . 2000. Conce p tions of mathematics teaching andlea rni ng : Middle leve l and seconda ry pre serv ice teach­ers. Pap er p resented at the Annual Meeting of th eAmeric an Educational Research Association, New Or­lean s, 24-28 April.

Franz, E. 2000. Mathematics an xiety: More than an emo tion .Paper presented at the Annual Meeting of the Ameri­can Ed ucational Research Association, Ne w Orleans,24-28 April.

[acobs.} . E., and ). R. Becker. 1997. Cre ating a gender-equi­table mult icultural classroom usin g feminist pedagogy.Multi culturaland gender equity in the mathematics class­room: The gift of diversity, ed . ). Tentacosta and M. ).Kenney, 107-1 4. Reston , Va.: Na tional Council of Teach ­ers of Math em at ics.

Kajander, A. 2003. Big ideas for small mathematicians: Kids dis­covering the beauty of math with 22 ready-to-go activities.Tucson , Ar iz.: Zephyr Press.

Karni i, C. 1994. Youngchildrencontinue to re-invent arithmetic,3rd grade: Implications of Piaget's theory. New York:Teachers Coll ege Press.

Ma , L. 1999. Knowing and teaching elementary mathematics.

Mah wah , N. ). : L. Erlbaum.McDou gall , D., A. Lawson .] , Ross.]. MacLellan , A. Kajande r,

and) . Scane . 2000. Research report: A study of the impactmath implementation strategy for the Ontario mathematicscurriculum, grades 7 and 8. Toronto, Ontar io: Un iver­sity of Toronto .

National Council of Teach ers of Mathematics. 2000. Principlesand standards ofschool mathematics. Reston . Va.: NC TM.

Op 't Eynde , P., E. DeCourte, and L. Vers chaffel. 2000. A socio­cons tru ctivist per sp ective on the origi ns and the roleof em ot ion s in math em ati cal problem so lving . Pap erpresented at the Annual Meeting of the Amer ican Ed u­cat ion al Research Association, New Orlean s, 24-28April.

Quiocho, A., and F.Rios. 2000. The pow er of their presence:minority group teacher s and their schooling . Review ofEducational Research 70(4): 485-528.

Shau ghnessy, )., and W. F.Burger. 1985. Spa dework p rio r tode d uction in geometry. Mathematics Teacher 78(6): 419­28.

Taylor, L. 1997. Int egrat ing mathem at ics and Ame rican In­d ian cultures. In Multicult ural and gender equity in themathematics classroom: The gif t of diversity, ed . ) .Tentacosta and M. ). Kenney, 169-76. Reston , Va.: Na­tional Council of Teachers of Mathematic s .

Wood , T., P. Cobb, and E. Yackel. 1991. Change in teachingmathematics: A case study. American Educational Re­search 'ournaI28( 3):587-616.

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© Kappa Delta Pi

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