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The four faces of the identity of mathematics learning are 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 of these faces. Each of the four faces of identity as a mathematics learner is described in this pdf.
1.
The Mathematics Educator
2007, Vol. 17, No. 1, 7–14
Being a Mathematics Learner: Four Faces of Identity
Rick Anderson
One dimension of mathematics learning is developing an identity as a mathematics learner. The social learning
theories 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, with
half enrolled in a mathematics course, provides evidence for how these faces highlight different ways students
develop their identity relative to their experiences with classroom mathematics. Using this identity framework
several ways that student identities—relative to mathematics learning—can be developed, supported, and
maintained by teachers are provided.
This paper is based on dissertation research completed at Portland State University under the direction of Dr. Karen Marrongelle. The
author 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 that community shape, and are shaped by, students’ sense
involves developing new ideas while transforming of themselves, their identities.
one’s ways of doing, thinking, and being. Building Learning mathematics involves the development of
skills, using algorithms, and following certain each student’s identity as a member of the mathematics
procedures characterizes one view of mathematics classroom community. Through relationships and
learning in schools. Another view focuses on students’ experiences with their peers, teachers, family, and
construction or acquisition of mathematical concepts. community, students come to know who they are
These views are evident in many state and national relative to mathematics. This article addresses the
standards for school mathematics (e.g., National notion of identity, drawn from social theories of
Council of Teachers of Mathematics [NCTM], 2000). learning (e.g., Gee, 2001; Lave & Wenger, 1991;
A third view of learning mathematics in schools Wenger, 1998), as a way to view students as they
involves becoming a “certain type” of person with develop as mathematics learners. Four “faces” of
respect to the practices of a community. That is, identity are discussed, illustrated with selected
students become particular types of people—those who quotations from students attending a small, rural high
view themselves and are recognized by others as a part school (approximately 225 students enrolled in grades
of the community with some being more central to the 9–12) in the U.S. Pacific Northwest.
practice and others situated on the periphery (Boaler,
Method
2000; Lampert, 2001; Wenger, 1998).
These three views of mathematics learning in The students in this study were participants in a
schools, as listed above, correspond to Kirshner’s larger study of students’ enrollment in advanced
(2002) three metaphors of learning: habituation, mathematics classes (Anderson, 2006). All students in
conceptual construction, and enculturation. This paper the high school were invited to complete a survey and
focuses on the third view of learning mathematics. In questionnaire. Of those invited, 24% responded.
this view, learning occurs through “social Fourteen students in grades 11 and 12 were selected for
participation” (Wenger, 1998, p. 4). This participation semi-structured interviews so that two groups were
includes not only thoughts and actions but also formed: students enrolled in Precalculus or Calculus
membership within social communities. In this sense, (the most advanced elective mathematics courses
learning “changes who we are by changing our ability offered in the school) and students not taking a
to participate, to belong, to negotiate meaning” mathematics course that year. These students
(Wenger, 1998, p. 226). This article addresses how represented the student body with respect to post-
students’ practices within a mathematics classroom secondary intentions, as reported on the survey, and
their interest and effort in mathematics classes, as
reported by their teacher. All of the students had taken
Rick Anderson is an assistant professor in the Department of
Mathematics & Computer Science at Eastern Illinois University. the two required and any elective high school
He teaches mathematics content and methods courses for future mathematics in the same high school. One teacher
elementary and secondary teachers. taught most of these courses. When interviewed, this
Rick Anderson 7
2.
teacher indicated the “traditional” nature of the Identity as a Mathematics Learner: Four Faces
curriculum and pedagogy: “We’ve always stayed The four faces of identity of mathematics learning
pretty traditional. … We haven’t really changed it to are engagement, imagination, alignment, and nature.
the really ‘out there’ hands-on type of programs.” Gee’s (2001) four perspectives of identity (nature,
Participant observation and interviews with students discursive, affinity, institutional) and Wenger’s (1998)
corroborated this statement. Calvin, a high school discussion of three modes of belonging (engagement,
senior, had enrolled in a mathematics class each year imagination, alignment) influenced the development of
of high school and planned to study mathematics these faces. Each of the four faces of identity as a
education in college. During an interview, he described mathematics learner is described below.
a typical day:
Engagement
Just go in, have your work done. First the teacher
explains how to do it. Like for the Pythagorean Engagement refers to our direct experience of the
Theorem, for example, she tells you the steps for it. world and our active involvement with others (Wenger,
She shows you the right triangle, the leg, the 1998). Much of what students know about learning
hypotenuse, that sort of thing. She makes us write mathematics comes from their engagement in
up notes so we can check back. And then after that mathematics classrooms. Through varying degrees of
she makes us do a couple [examples] and then if
engagement with the mathematics, their teachers, and
we all get it right, she shows us. She gives us time
to work. Do it and after that she shows us the
their peers, each student sees her or himself, and is
correct way to do it. If we got it right, then we seen by others, as one who has or has not learned
know. She makes us move on and do an mathematics.
assignment. Engaging in a particular mathematics learning
environment aids students in their development of an
Identity identity as capable mathematics learners. Other
As used here, identity refers to the way we define students, however, may not identify with this
ourselves and how others define us (Sfard & Prusak, environment and may come to see themselves as only
2005; Wenger, 1998). Our identity includes our marginally part of the mathematics learning
perception of our experiences with others as well as community. In traditional mathematics classrooms
our aspirations. In this way, our identity—who we where students work independently on short, single-
are—is formed in relationships with others, extending answer exercises and an emphasis is placed on getting
from the past and stretching into the future. Identities right answers, students not only learn mathematics
are malleable and dynamic, an ongoing construction of concepts and skills, but they also discover something
who we are as a result of our participation with others about themselves as learners (Anderson, 2006; Boaler,
in the experience of life (Wenger, 1998). As students 2000; Boaler & Greeno, 2000). Students may learn that
move through school, they come to learn who they are they are capable of learning mathematics if they can fit
as mathematics learners through their experiences in together the small pieces of the “mathematics puzzle”
mathematics classrooms; in interactions with teachers, delivered by the teacher. For example, Calvin stated,
parents, and peers; and in relation to their anticipated “Precalculus is easy. It’s like a jigsaw puzzle waiting
futures. to be solved. I like puzzles.”
Mathematics has become a gatekeeper to many Additionally, when correct answers on short
economic, educational, and political opportunities for exercises are emphasized more than mathematical
adults (D’Ambrosio, 1990; Moses & Cobb, 2001; processes or strategies, students come to learn that
NCTM, 2000). Students must become mathematics doing mathematics competently means getting correct
learners—members of mathematical communities—if answers, often quickly. Students who adopt the
they are to have access to a full palette of future practice of quickly getting correct answers may view
opportunities. As learners of mathematics, they will not themselves as capable mathematics learners. In
only need to develop mathematical concepts and skills, contrast, students who may require more time to obtain
but also the identity of a mathematics learner. That is, correct answers may not see themselves as capable of
they must participate within mathematical communities doing mathematics, even though they may have
in such a way as to see themselves and be viewed by developed effective strategies for solving mathematical
others as valuable members of those communities. problems.
8 Four Faces of Identity
3.
One way students come to learn who they are other activities in the present and the future. Students
relative to mathematics is through their engagement in who engage in a mathematical activity in a similar
the activities of the mathematics classroom: manner may have very different meanings for that
The thing I like about art is being able to be
activity (Wenger, 1998).
creative and make whatever I want… But in math Imagination is the second face of identity: the
there’s just kind of like procedures that you have to images we have of ourselves and of how mathematics
work through. (Abby, grade 11, Precalculus class, fits into the broader experience of life (Wenger, 1998).
planning to attend college) For example, the images a student has of herself in
relation to mathematics in everyday life, the place of
Math is probably my least favorite subject… I just
don’t like the process of it a lot— going through a mathematics in post-secondary education, and the use
lot of problems, going through each step. I just get of mathematics in a future career all influence
dragged down. (Thomas, grade 12, Precalculus imagination. The ways students see mathematics in
class, planning to attend college) relation to the broader context can contribute either
positively or negatively to their identity as mathematics
Students who are asked to follow procedures on
learners.
repetitive exercises without being able to make
When asked to give reasons for their decisions
meaning on their own may not see themselves as
regarding enrollment in advanced mathematics classes,
mathematics learners but rather as those who do not
learn mathematics (Boaler & Greeno, 2000). A students’ responses revealed a few of the ways they
substantial portion of students’ direct experience with saw themselves in relation mathematics. For example,
mathematics happens within the classroom, so the students had very different reason for taking advanced
types of mathematical tasks and teaching and learning mathematics courses. One survey respondent stated, “I
need math for everyday life,” while another claimed,
structures used in the classroom contribute
“They will help prepare me for college classes.” These
significantly to the development of students’
students see themselves as learners of mathematics and
mathematical identities. In the quotation above, Abby
members of the community for mathematics learning
expressed her dislike of working through procedures
because they need mathematics for their present or
that she did not find meaningful. In mathematics class,
future lives. Others (e.g., Martin, 2000; Mendick,
she was not able to exercise her creativity as she did in
2003; Sfard & Prusack, 2005) have similarly noted that
art class. As a result, she may not consider herself to be
students cite future education and careers as reasons
a capable mathematics learner.
for studying mathematics.
On one hand, when students are able to develop
Conversely, students’ images of the way
their own strategies and meanings for solving
mathematics problems, they learn to view themselves mathematics fits into broader life can also cause
as capable members of a community engaged in students to view their learning of further mathematics
mathematics learning. When their ideas and as unnecessary. Student responses for why they chose
explanations are accepted in a classroom discussion, not to enroll in advanced mathematics classes included
“the career I am hoping for, I know all the math for it”
others also recognize them as members of the
and “I don’t think I will need to use a pre-cal math in
community. On the other hand, students who do not
my life.” Students who do not see themselves as
have the opportunity to connect with mathematics on a
needing or using mathematics outside of the immediate
personal level, or are not recognized as contributors to
context of the mathematics classroom may develop an
the mathematics classroom, may fail to see themselves
identity as one who is not a mathematics learner. If
as competent at learning mathematics (Boaler &
high school mathematics is promoted as something
Greeno, 2000; Wenger, 1998).
useful only as preparation for college, students who do
Imagination not intend to enroll in college may come to see
The activities in which students choose to engage themselves as having no need to learn mathematics,
are often related to the way they envision those especially advanced high school mathematics
activities fitting into their broader lives. This is (Anderson, 2006).
particularly true for high school students as they Students may pursue careers that are available in
become more aware of their place in the world and their geographical locale or similar to those of their
begin to make decisions for their future. In addition to parents or other community members. If these careers
learning mathematical concepts and skills in school, do not require a formal mathematics education beyond
students also learn how mathematics fits in with their high school mathematics, these students may limit their
Rick Anderson 9
4.
image of the mathematics needed for work to in high school, including “I have already taken two
arithmetic and counting. In addition, due to the lack of [required] math classes,” and “I might not take those
formal mathematical training, those in the workplace classes if the career I choose doesn’t have the
may not be able to identify the complex mathematical requirement.” While some students come to see
thinking required for their work. For example, Smith themselves, and are recognized by others, as
(1999) noted the mathematical knowledge used by mathematics learners from the requirements they
automobile production workers, knowledge not follow, the opposite is true for others. Students who
identified by the workers but nonetheless embedded in follow the minimal mathematics requirements, such as
the tasks of the job. When students are not able to those for graduation, may be less likely to see
make connections between the mathematics they learn themselves, or be recognized by others, as students
in school and its perceived utility in their lives, they who are mathematics learners.
may construct an identity that does not include the The three faces of identity discussed to this point
need for advanced mathematics courses in high school. are not mutually exclusive but interact to form and
The students cited in this paper lived in a rural maintain a student’s identity. When beginning high
logging community. Their high school mathematics school, students are required to enroll in mathematics
teacher formally studied more mathematics than most courses. This contributes to students’ identity through
in the community. Few students indicated personally alignment. As they participate in mathematics classes,
knowing anyone for whom formal mathematics was an the activities may appeal to them, and their identity is
integral part of their work. As a result, careers further developed through engagement. Similarly,
requiring advanced mathematics were not part of the students—like the one mentioned above who is
images most students had for themselves and their interested in mechanics—may envision their
futures. participation in high school mathematics class as
preparation for a career. Mathematics is both a
requirement for entrance into the career and necessary
A third face of identity is revealed when students knowledge to pursue the career. Thus, identity in
align their energies within institutional boundaries and mathematics is maintained through both imagination
requirements. That is, students respond to the and alignment.
imagination face of identity (Nasir, 2002). For
example, students who consider advanced mathematics Nature
necessary for post-secondary educational or Q: Why are some people good at math and some
occupational opportunities direct their energy toward people aren’t good at math?
studying the required high school mathematics. High A: I think it’s just in your makeup… genetic I
school students must meet many requirements set by guess. (Barbara, grade 12, Precalculus, planning to
others—teachers, school districts, state education attend vocational training after high school)
departments, colleges and universities, and
The nature face of identity looks at who we are
professional organizations. By simply following
from what nature gave us at birth, those things over
requirements and participating in the required
which we have no control (Gee, 2001). Typically,
activities, students come to see themselves as certain
characteristics such as gender and skin color are
“types of people” (Gee, 2001). For example a “college-
viewed as part of our nature identity. The meanings we
intending” student may take math classes required for
make of our natural characteristics are not independent
admission to college.
of our relationships with others in personal and broader
As before, students’ anonymous survey responses
social settings. That is, these characteristics comprise
to the question of why they might choose to enroll in
only one part of the way we see ourselves and others
advanced mathematics classes provide a glimpse into
see us. In Gee’s social theory of learning, the nature
what they have learned about mathematics
aspect of our identity must be maintained and
requirements and how they respond to these
reinforced through our engagement with others, in the
requirements. Students were asked why they take
images we hold, or institutionalized in the
advanced mathematics classes in high school. One
requirements we must follow in the environments
student responded, “Colleges look for them on
where we interact.
applications,” and another said, “Math plays a big part
Mathematics teachers are in a unique position to
in mechanics.” Likewise, students provided reasons for
hear students and parents report that their mathematics
why they did not take advanced mathematics courses
learning has been influenced by the presence or
10 Four Faces of Identity
5.
absence of a “math gene”, often crediting nature for mathematics. They do not engage in practices that are
not granting them the ability to learn mathematics. The recognized, in this case, to be the accepted practices of
claim of a lack of a math gene—and, therefore, the the community. As a result, they view themselves, and
inability to do mathematics—contrasts with Devlin’s are viewed by others, to be peripheral members of the
(2000b) belief that “everyone has the math gene” (p. 2) community of mathematics learners.
as well as with NCTM’s (2000) statement that As shown by the provided responses from students,
“mathematics can and must be learned by all students” each of the four faces of identity exists as a way that
(p. 13). In fact, cognitive scientists report, students come to understand their practices and
“Mathematics is a natural part of being human. It arises membership within the community of mathematics
from our bodies, our brains, and our everyday learners. I have chosen to represent these faces of
experiences in the world” (Lakoff & Núñez, 2000, p. identity as the four faces of a tetrahedron1 (Figure 1). If
377). Mathematics has been created by the human we rotate a particular face to the front, certain features
brain and its capabilities and can be recreated and of identity are highlighted while others are diminished.
learned by other human brains. Yet, the fallacy persists Each face suggests different ways to describe how we
for some students that learning mathematics requires see ourselves as mathematics learners although they
special natural talents possessed by only a few: are all part of the one whole. This representation of
I’m good at math. (Interview with Barbara, grade
identity maintains the idea that, as Gee (2001) wrote,
12, Precalculus class) “They are four strands that may very well all be
present and woven together as a given person acts
I’m not a math guy. (Interview with Bill, grade 12, within a given context” (p. 101). When considering the
not enrolled in math, planning to join the military
four faces of identity as a mathematics learner, this
after high school)
context is a traditional high school mathematics
Math just doesn’t work for me. I can’t get it classroom.
through my head. (Interview with Jackie, grade 12, While all four faces contribute to the formation of
not enrolled in math, planning to enroll in a students’ identities as mathematics learners, the nature
vocational program after high school) face provides the most unsound and unfounded
Although scientific evidence does not support the explanations for students’ participation in the
idea that mathematics learning is related to genetics, mathematics community. To allow for the development
some students attribute their mathematics learning to of all students to identify as mathematics learners,
nature. The high school student who says “I’m not a students and teachers must discount the nature face and
math guy” may feel that he is lacking a natural ability build on the other three faces of identity.
for mathematics. He is likely as capable as any other
Developing an Identity as a Mathematics Learner
student but has come to the above conclusion based on
his experience with mathematics and the way it was To conclude this article, recommendations are
taught in his mathematics classes. Students who are not offered to teachers for developing and supporting
the quickest to get the correct answers may learn, albeit students’ positive identities as mathematics learners—
erroneously, that they are not capable of learning members of a community that develops the practices of
→
Figure 1. The four faces of identity
Rick Anderson 11
6.
mathematics learning. The four faces of identity can also be organized to encourage discussion, sharing,
described here are used to understand how students see and collaboration (Boaler & Greeno, 2000). In this
themselves as mathematics learners in relation to their type of classroom setting, teachers “pull knowledge
experiences in the mathematics classroom and through out” (Ladson-Billings, 1995, p. 479) of students and
the ways these experiences fit into broader life make the construction of knowledge part of the
experience. Students’ experiences will not necessarily learning experience.
reflect just one of the four faces described (Gee, 2001). With respect to imagination, the development of
In fact, some experiences may be stretched over two or students’ identities as mathematics learners requires
more faces. For example, learning advanced long-term effort on the part of teachers across
mathematics in high school can contribute to a disciplines. The various images students have of
students’ identity in two ways: (a) through imagination themselves and of mathematics extending outside the
with the image of math as an important subject for classroom—in the past, present, or future—may be
entrance to higher education and (b) through alignment contradictory and change over time. Teachers and
since advanced mathematics is required to attend some others in schools can consistently reinforce that
colleges. Taken together, however, we can see that a mathematics is an interesting body of knowledge worth
focus on a particular face of identity suggests particular studying, an intellectual tool for other disciplines, and
experiences that can help to develop strong positive an admission ticket for colleges and careers.
identities as a mathematics learner in all students. The Since students’ identity development through
engagement face of identity is developed through imagination extends beyond the classroom, teachers
students’ experiences with mathematics and, for most can provide students with opportunities to see
high school students, their mathematics experiences themselves as mathematics learners away from the
occur in the mathematics classroom. Therefore, the classroom. For example, working professionals from
most significant potential to influence students’ outside the school can be invited to discuss ways they
identities exists in the mathematics classroom. To use mathematics in their professional lives; many
develop students’ identities as mathematics learners students may not be aware of the work of engineers,
through engagement, teachers should consider actuaries, or statisticians. Another suggestion is to
mathematical tasks and classroom structures where require students to keep a log and record the ways in
students are actively involved in the creation of which they use mathematics in their daily lives in order
mathematics while learning to be “people who study in to become aware of the usefulness of mathematics
school” (Lampert, 2001). That is, students must feel (Masingila, 2002). This activity could provide an
the mathematics classroom is their scholarly home and opportunity for assessing students’ views of
that the ideas they contribute are valued by the class mathematics and discussing the connections between
(Wenger, 1998). As indicated earlier, teacher-led the mathematics taught in school and that used outside
classrooms with students working independently on the classroom.
single-answer exercises can cause students to learn that Although many of students’ mathematical
mathematics is not a vibrant and useful subject to requirements are beyond the control of teachers and
study. Boaler (2000), for example, identified students, teachers can foster the alignment face of
monotony, lack of meaning, and isolation as themes identity. Teachers can hold their students to high
that emerged from a study of students and their expectations so that these expectations become as
mathematics experiences. As a result many of these strong as requirements. Also, knowledge of
students were alienated from mathematics and learned mathematics requirements for post-secondary
that they are not valuable members of the mathematics education and careers can help students decide to
community. enroll in other mathematics courses. Because students
Hence, mathematical tasks that engage students in are known to cite post-secondary education and careers
doing mathematics, making meaning, and generating as reasons for studying mathematics (Anderson, 2006;
their own solutions to complex mathematical problems Martin, 2000; Sfard & Prusak, 2005), teachers can
can be beneficial in engaging students and supporting facilitate this alignment face by keeping students
their identity as a mathematics learner (NCTM, 2000). abreast of the mathematics requirements for entrance to
A good starting point is open-ended mathematical college and careers.
tasks, questions or projects that have multiple Students may commonly reference the nature face
responses or one response with multiple solution paths of identity, but this face is the least useful—and
(Kabiri & Smith, 2003). The mathematics classroom potentially the most detrimental—for supporting
12 Four Faces of Identity
7.
students as they become mathematics learners. As
mentioned earlier, the ability to learn mathematics is Teachers need to be aware of the four faces of
not determined by genetics or biology (Lakoff & identity of mathematics learners and of how their
Núñez, 2000). All students can become mathematics students see themselves as mathematics learners and
learners, identifying themselves and being recognized doers. Detailed recommendations for developing
by others as capable of doing mathematics. Thinking students’ identities as mathematics learners are
about the tetrahedron model of identity, if the other provided in Figure 2.
faces are strong and at the fore, the nature face can be The four faces of identity discussed in this article
turned to the back As suggested above, the other three contribute to our understanding of how students come
faces of identity can sustain mathematics learners’ to be mathematics learners. Through consistent and
identities—through engaging students with sustained efforts by mathematics teachers to develop
mathematics in the classroom, developing positive positive identities in their students, more students can
images of students and mathematics, and establishing come to study advanced mathematics and improve
high expectations and requirements—regardless of their identities as mathematics learners. As I have
students’ beliefs in an innate mathematical ability. Gee pointed out throughout this article, identities are
(2001) points out that the nature face of identity will developed in relationships with others, including their
always collapse into other sorts of identities. … teachers, parents, and peers. We cannot assume that all
When people (and institutions) focus on them as students will develop positive identities if they have
“natural” or “biological,” they often do this as a experiences that run to the contrary. We must take
way to “forget” or “hide” (often for ideological action so each face of identity mutually supports the
reasons) the institutional, social-interactional, or others in developing all students’ identities as
group work that is required to create and sustain mathematics learners.
them. (p. 102)
Developing and Supporting Students’ Identities as Mathematics Learners
Engagement
• Use mathematical tasks that allow students to develop strategies for solving problems and meanings for
mathematical tools.
• Organize mathematics classrooms that allow students to express themselves creatively and communicate their
meanings of mathematical concepts to their peers and teacher.
• Focus on the process and explanations of problem solving rather than emphasize quick responses to single-answer
exercises.
Imagination
• Make explicit the ways mathematics is part of students’ daily lives. That is, help students identify ways they create
and use mathematics in their work and play.
• Have working professionals discuss with high school students ways in which they use mathematics in their
professional lives, emphasizing topics beyond arithmetic.
• Include mathematics topics in classes that relate to occupations, for example, geometric concepts that are part of
factory work or carpentry (e.g., see Smith, 1999; Masingila, 1994).
Alignment
• Maintain expectations that all students will enroll in mathematics courses every year of high school.
• Take an active role in keeping students informed of mathematics requirements for careers and college and university
entrance.
Figure 2. Recommendations
Rick Anderson 13
8.
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Others have used the idea of “faces” to convey the many
Lampert, M. (2001). Teaching problems and the problems of interrelated aspects of a topic. For example, Devlin (2000a)
teaching. New Haven, CT: Yale University Press. describes “The Four Faces of Mathematics.”
14 Four Faces of Identity
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