The Mathematics Educator Being A Mathematics Learner:
The Mathematics Educator2007, Vol. 17, No. 1, 7–14Being a Mathematics Learner: Four Faces of IdentityRick AndersonOne dimension of mathematics learning is developing an identity as a mathematics learner. The social learningtheories of Gee (2001) and Wenger (1998) serve as a basis for the discussion four “faces” of identity:engagement, imagination, alignment, and nature. A study conducted with 54 rural high school students, withhalf enrolled in a mathematics course, provides evidence for how these faces highlight different ways studentsdevelop their identity relative to their experiences with classroom mathematics. Using this identity frameworkseveral ways that student identities—relative to mathematics learning—can be developed, supported, andmaintained by teachers are provided.This paper is based on dissertation research completed at Portland State University under the direction of Dr. Karen Marrongelle. Theauthor wishes to thank Karen Marrongelle, Joyce Bishop, and the TME editors/reviewers for comments on earlier drafts of this paper.Learning mathematics is a complex endeavor thatinvolves developing new ideas while transformingone’s ways of doing, thinking, and being. Buildingskills, using algorithms, and following certainprocedures characterizes one view of mathematicslearning in schools. Another view focuses on students’construction or acquisition of mathematical concepts.These views are evident in many state and nationalstandards for school mathematics (e.g., NationalCouncil of Teachers of Mathematics [NCTM], 2000).A third view of learning mathematics in schoolsinvolves becoming a “certain type” of person withrespect to the practices of a community. That is,students become particular types of people—those whoview themselves and are recognized by others as a partof the community with some being more central to thepractice and others situated on the periphery (Boaler,2000; Lampert, 2001; Wenger, 1998).These three views of mathematics learning inschools, as listed above, correspond to Kirshner’s(2002) three metaphors of learning: habituation,conceptual construction, and enculturation. This paperfocuses on the third view of learning mathematics. Inthis view, learning occurs through “socialparticipation” (Wenger, 1998, p. 4). This participationincludes not only thoughts and actions but alsomembership within social communities. In this sense,learning “changes who we are by changing our abilityto participate, to belong, to negotiate meaning”(Wenger, 1998, p. 226). This article addresses howstudents’ practices within a mathematics classroomRick Anderson is an assistant professor in the Department ofMathematics & Computer Science at Eastern Illinois University.He teaches mathematics content and methods courses for futureelementary and secondary teachers.Rick Andersoncommunity shape, and are shaped by, students’ senseof themselves, their identities.Learning mathematics involves the development ofeach student’s identity as a member of the mathematicsclassroom community. Through relationships andexperiences with their peers, teachers, family, andcommunity, students come to know who they arerelative to mathematics. This article addresses thenotion of identity, drawn from social theories oflearning (e.g., Gee, 2001; Lave & Wenger, 1991;Wenger, 1998), as a way to view students as theydevelop as mathematics learners. Four “faces” ofidentity are discussed, illustrated with selectedquotations from students attending a small, rural highschool (approximately 225 students enrolled in grades9–12) in the U.S. Pacific Northwest.MethodThe students in this study were participants in alarger study of students’ enrollment in advancedmathematics classes (Anderson, 2006). All students inthe high school were invited to complete a survey andquestionnaire. Of those invited, 24% responded.Fourteen students in grades 11 and 12 were selected forsemi-structured interviews so that two groups wereformed: students enrolled in Precalculus or Calculus(the most advanced elective mathematics coursesoffered in the school) and students not taking amathematics course that year. These studentsrepresented the student body with respect to postsecondary intentions, as reported on the survey, andtheir interest and effort in mathematics classes, asreported by their teacher. All of the students had takenthe two required and any elective high schoolmathematics in the same high school. One teachertaught most of these courses. When interviewed, this7
teacher indicated the “traditional” nature of thecurriculum and pedagogy: “We’ve always stayedpretty traditional. We haven’t really changed it tothe really ‘out there’ hands-on type of programs.”Participant observation and interviews with studentscorroborated this statement. Calvin, a high schoolsenior, had enrolled in a mathematics class each yearof high school and planned to study mathematicseducation in college. During an interview, he describeda typical day:Just go in, have your work done. First the teacherexplains how to do it. Like for the PythagoreanTheorem, for example, she tells you the steps for it.She shows you the right triangle, the leg, thehypotenuse, that sort of thing. She makes us writeup notes so we can check back. And then after thatshe makes us do a couple [examples] and then ifwe all get it right, she shows us. She gives us timeto work. Do it and after that she shows us thecorrect way to do it. If we got it right, then weknow. She makes us move on and do anassignment.IdentityAs used here, identity refers to the way we defineourselves and how others define us (Sfard & Prusak,2005; Wenger, 1998). Our identity includes ourperception of our experiences with others as well asour aspirations. In this way, our identity—who weare—is formed in relationships with others, extendingfrom the past and stretching into the future. Identitiesare malleable and dynamic, an ongoing construction ofwho we are as a result of our participation with othersin the experience of life (Wenger, 1998). As studentsmove through school, they come to learn who they areas mathematics learners through their experiences inmathematics classrooms; in interactions with teachers,parents, and peers; and in relation to their anticipatedfutures.Mathematics has become a gatekeeper to manyeconomic, educational, and political opportunities foradults (D’Ambrosio, 1990; Moses & Cobb, 2001;NCTM, 2000). Students must become mathematicslearners—members of mathematical communities—ifthey are to have access to a full palette of futureopportunities. As learners of mathematics, they will notonly need to develop mathematical concepts and skills,but also the identity of a mathematics learner. That is,they must participate within mathematical communitiesin such a way as to see themselves and be viewed byothers as valuable members of those communities.8Identity as a Mathematics Learner: Four FacesThe four faces of identity of mathematics learningare engagement, imagination, alignment, and nature.Gee’s (2001) four perspectives of identity (nature,discursive, affinity, institutional) and Wenger’s (1998)discussion of three modes of belonging (engagement,imagination, alignment) influenced the development ofthese faces. Each of the four faces of identity as amathematics learner is described below.EngagementEngagement refers to our direct experience of theworld and our active involvement with others (Wenger,1998). Much of what students know about learningmathematics comes from their engagement inmathematics classrooms. Through varying degrees ofengagement with the mathematics, their teachers, andtheir peers, each student sees her or himself, and isseen by others, as one who has or has not learnedmathematics.Engaging in a particular mathematics learningenvironment aids students in their development of anidentity as capable mathematics learners. Otherstudents, however, may not identify with thisenvironment and may come to see themselves as onlymarginally part of the mathematics learningcommunity. In traditional mathematics classroomswhere students work independently on short, singleanswer exercises and an emphasis is placed on gettingright answers, students not only learn mathematicsconcepts and skills, but they also discover somethingabout themselves as learners (Anderson, 2006; Boaler,2000; Boaler & Greeno, 2000). Students may learn thatthey are capable of learning mathematics if they can fittogether the small pieces of the “mathematics puzzle”delivered by the teacher. For example, Calvin stated,“Precalculus is easy. It’s like a jigsaw puzzle waitingto be solved. I like puzzles.”Additionally, when correct answers on shortexercises are emphasized more than mathematicalprocesses or strategies, students come to learn thatdoing mathematics competently means getting correctanswers, often quickly. Students who adopt thepractice of quickly getting correct answers may viewthemselves as capable mathematics learners. Incontrast, students who may require more time to obtaincorrect answers may not see themselves as capable ofdoing mathematics, even though they may havedeveloped effective strategies for solving mathematicalproblems.Four Faces of Identity
One way students come to learn who they arerelative to mathematics is through their engagement inthe activities of the mathematics classroom:The thing I like about art is being able to becreative and make whatever I want But in maththere’s just kind of like procedures that you have towork through. (Abby, grade 11, Precalculus class,planning to attend college)Math is probably my least favorite subject I justdon’t like the process of it a lot— going through alot of problems, going through each step. I just getdragged down. (Thomas, grade 12, Precalculusclass, planning to attend college)Students who are asked to follow procedures onrepetitive exercises without being able to makemeaning on their own may not see themselves asmathematics learners but rather as those who do notlearn mathematics (Boaler & Greeno, 2000). Asubstantial portion of students’ direct experience withmathematics happens within the classroom, so thetypes of mathematical tasks and teaching and learningstructures used in the classroom contributesignificantly to the development of students’mathematical identities. In the quotation above, Abbyexpressed her dislike of working through proceduresthat she did not find meaningful. In mathematics class,she was not able to exercise her creativity as she did inart class. As a result, she may not consider herself to bea capable mathematics learner.On one hand, when students are able to developtheir own strategies and meanings for solvingmathematics problems, they learn to view themselvesas capable members of a community engaged inmathematics learning. When their ideas andexplanations are accepted in a classroom discussion,others also recognize them as members of thecommunity. On the other hand, students who do nothave the opportunity to connect with mathematics on apersonal level, or are not recognized as contributors tothe mathematics classroom, may fail to see themselvesas competent at learning mathematics (Boaler &Greeno, 2000; Wenger, 1998).ImaginationThe activities in which students choose to engageare often related to the way they envision thoseactivities fitting into their broader lives. This isparticularly true for high school students as theybecome more aware of their place in the world andbegin to make decisions for their future. In addition tolearning mathematical concepts and skills in school,students also learn how mathematics fits in with theirRick Andersonother activities in the present and the future. Studentswho engage in a mathematical activity in a similarmanner may have very different meanings for thatactivity (Wenger, 1998).Imagination is the second face of identity: theimages we have of ourselves and of how mathematicsfits into the broader experience of life (Wenger, 1998).For example, the images a student has of herself inrelation to mathematics in everyday life, the place ofmathematics in post-secondary education, and the useof mathematics in a future career all influenceimagination. The ways students see mathematics inrelation to the broader context can contribute eitherpositively or negatively to their identity as mathematicslearners.When asked to give reasons for their decisionsregarding enrollment in advanced mathematics classes,students’ responses revealed a few of the ways theysaw themselves in relation mathematics. For example,students had very different reason for taking advancedmathematics courses. One survey respondent stated, “Ineed math for everyday life,” while another claimed,“They will help prepare me for college classes.” Thesestudents see themselves as learners of mathematics andmembers of the community for mathematics learningbecause they need mathematics for their present orfuture lives. Others (e.g., Martin, 2000; Mendick,2003; Sfard & Prusack, 2005) have similarly noted thatstudents cite future education and careers as reasonsfor studying mathematics.Conversely, students’ images of the waymathematics fits into broader life can also causestudents to view their learning of further mathematicsas unnecessary. Student responses for why they chosenot to enroll in advanced mathematics classes included“the career I am hoping for, I know all the math for it”and “I don’t think I will need to use a pre-cal math inmy life.” Students who do not see themselves asneeding or using mathematics outside of the immediatecontext of the mathematics classroom may develop anidentity as one who is not a mathematics learner. Ifhigh school mathematics is promoted as somethinguseful only as preparation for college, students who donot intend to enroll in college may come to seethemselves as having no need to learn mathematics,especially advanced high school mathematics(Anderson, 2006).Students may pursue careers that are available intheir geographical locale or similar to those of theirparents or other community members. If these careersdo not require a formal mathematics education beyondhigh school mathematics, these students may limit their9
image of the mathematics needed for work toarithmetic and counting. In addition, due to the lack offormal mathematical training, those in the workplacemay not be able to identify the complex mathematicalthinking required for their work. For example, Smith(1999) noted the mathematical knowledge used byautomobile production workers, knowledge notidentified by the workers but nonetheless embedded inthe tasks of the job. When students are not able tomake connections between the mathematics they learnin school and its perceived utility in their lives, theymay construct an identity that does not include theneed for advanced mathematics courses in high school.The students cited in this paper lived in a rurallogging community. Their high school mathematicsteacher formally studied more mathematics than mostin the community. Few students indicated personallyknowing anyone for whom formal mathematics was anintegral part of their work. As a result, careersrequiring advanced mathematics were not part of theimages most students had for themselves and theirfutures.AlignmentA third face of identity is revealed when studentsalign their energies within institutional boundaries andrequirements. That is, students respond to theimagination face of identity (Nasir, 2002). Forexample, students who consider advanced mathematicsnecessary for post-secondary educational oroccupational opportunities direct their energy towardstudying the required high school mathematics. Highschool students must meet many requirements set byothers—teachers, school districts, state ofessional organizations. By simply followingrequirements and participating in the requiredactivities, students come to see themselves as certain“types of people” (Gee, 2001). For example a “collegeintending” student may take math classes required foradmission to college.As before, students’ anonymous survey responsesto the question of why they might choose to enroll inadvanced mathematics classes provide a glimpse intowhat they have learned about mathematicsrequirements and how they respond to theserequirements. Students were asked why they takeadvanced mathematics classes in high school. Onestudent responded, “Colleges look for them onapplications,” and another said, “Math plays a big partin mechanics.” Likewise, students provided reasons forwhy they did not take advanced mathematics courses10in high school, including “I have already taken two[required] math classes,” and “I might not take thoseclasses if the career I choose doesn’t have therequirement.” While some students come to seethemselves, and are recognized by others, asmathematics learners from the requirements theyfollow, the opposite is true for others. Students whofollow the minimal mathematics requirements, such asthose for graduation, may be less likely to seethemselves, or be recognized by others, as studentswho are mathematics learners.The three faces of identity discussed to this pointare not mutually exclusive but interact to form andmaintain a student’s identity. When beginning highschool, students are required to enroll in mathematicscourses. This contributes to students’ identity throughalignment. As they participate in mathematics classes,the activities may appeal to them, and their identity isfurther developed through engagement. Similarly,students—like the one mentioned above who isinterested in mechanics—may envision theirparticipation in high school mathematics class aspreparation for a career. Mathematics is both arequirement for entrance into the career and necessaryknowledge to pursue the career. Thus, identity inmathematics is maintained through both imaginationand alignment.NatureQ: Why are some people good at math and somepeople aren’t good at math?A: I think it’s just in your makeup genetic Iguess. (Barbara, grade 12, Precalculus, planning toattend vocational training after high school)The nature face of identity looks at who we arefrom what nature gave us at birth, those things overwhich we have no control (Gee, 2001). Typically,characteristics such as gender and skin color areviewed as part of our nature identity. The meanings wemake of our natural characteristics are not independentof our relationships with others in personal and broadersocial settings. That is, these characteristics compriseonly one part of the way we see ourselves and otherssee us. In Gee’s social theory of learning, the natureaspect of our identity must be maintained andreinforced through our engagement with others, in theimages we hold, or institutionalized in therequirements we must follow in the environmentswhere we interact.Mathematics teachers are in a unique position tohear students and parents report that their mathematicslearning has been influenced by the presence orFour Faces of Identity
absence of a “math gene”, often crediting nature fornot granting them the ability to learn mathematics. Theclaim of a lack of a math gene—and, therefore, theinability to do mathematics—contrasts with Devlin’s(2000b) belief that “everyone has the math gene” (p. 2)as well as with NCTM’s (2000) statement that“mathematics can and must be learned by all students”(p. 13). In fact, cognitive scientists report,“Mathematics is a natural part of being human. It arisesfrom our bodies, our brains, and our everydayexperiences in the world” (Lakoff & Núñez, 2000, p.377). Mathematics has been created by the humanbrain and its capabilities and can be recreated andlearned by other human brains. Yet, the fallacy persistsfor some students that learning mathematics requiresspecial natural talents possessed by only a few:I’m good at math. (Interview with Barbara, grade12, Precalculus class)I’m not a math guy. (Interview with Bill, grade 12,not enrolled in math, planning to join the militaryafter high school)Math just doesn’t work f
Mathematics has become a gatekeeper to many economic, educational, and political opportunities for . mathematics in post-secondary education, and the use of mathematics in a future career all influence imagination. The
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Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
