Contributed by:

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

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

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

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

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

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

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

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.
References Lave, J., & Wenger, E. (1991). Situated learning: Legitimate

peripheral participation. Cambridge, UK: Cambridge

Anderson, R. (2006). Mathematics, meaning, and identity: A study University Press.

of the practice of mathematics education in a rural high

Martin, D. B. (2000). Mathematics success and failure among

school. Unpublished doctoral dissertation, Portland State

African-American youth: The roles of sociohistorical context,

University, Oregon.

community forces, school influences, and individual agency.

Boaler, J. (2000). Mathematics from another world: Traditional Mahwah, NJ: Lawrence Erlbaum.

communities and the alienation of learners. Journal of

Masingila, J. O. (1994). Mathematics practice in carpet laying.

Mathematical Behavior, 18, 379–397.

Anthropology & Education Quarterly, 25, 430–462.

Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing

Masingila, J. O. (2002). Examining students’ perceptions of their

in mathematics worlds. In J. Boaler (Ed.), Multiple

everyday mathematics practice. In M. E. Brenner & J. N.

perspectives on mathematics teaching and learning (pp. 171–

Moschkovich (Eds.), Everyday and academic mathematics in

200). Westport, CT: Ablex.

the classroom (Journal for Research in Mathematics Education

Boaler, J., & Humphreys, C. (2005). Connecting mathematical Monograph No. 11, pp. 30–39). Reston, VA: National Council

ideas: Middle school video cases to support teaching and of Teachers of Mathematics.

learning. Portsmouth, NH: Heinemann.

Mendick, H. (2003). Choosing maths/doing gender: A look at why

D’Ambrosio, U. (1990). The role of mathematics education in there are more boys than girls in advanced mathematics

building a democratic and just society. For the Learning of classes in England. In L. Burton (Ed.), Which way social

Mathematics, 10, 20–23. justice in mathematics education? (pp. 169–187). Westport,

Devlin, K. (2000a). The four faces of mathematics. In M. J. Burke CT: Praeger.

& F. R. Curcio (Eds.), Learning mathematics for a new Moses, R. P., & Cobb, C. E. (2001). Radical equations: Civil rights

century (pp. 16–27). Reston, VA: NCTM. from Mississippi to the Algebra Project. Boston, MA: Beacon

Devlin, K. (2000b). The math gene: How mathematical thinking Press.

evolved and why numbers are like gossip. New York: Basic Nasir, N. S. (2002). Identity, goals, and learning: Mathematics in

Books. cultural practice. Mathematical Thinking & Learning, 4, 213–

Gee, J. P. (2001). Identity as an analytic lens for research in 247.

education. Review of Research in Education, 25, 99–125. National Council of Teachers of Mathematics. (2000). Principles

Kabiri, M. S., & Smith, N. L. (2003). Turning traditional textbook and standards for school mathematics. Reston, VA: Author.

problems into open-ended problems. Mathematics Teaching in Sfard, A., & Prusak, A. (2005). Telling identities: In search of an

the Middle School, 9, 186–192. analytic tool for investigating learning as a culturally shaped

Kirshner, D. (2002). Untangling teachers’ diverse aspirations for activity. Educational Researcher, 34(4), 14–22.

student learning: A crossdisciplinary strategy for relating Smith, J. P. (1999). Tracking the mathematics of automobile

psychological theory to pedagogical practice. Journal for production: Are schools failing to prepare students for work?

Research in Mathematics Education, 33, 46–58. American Educational Research Journal, 36, 835–878.

Ladson-Billings, G. (1995). Toward a theory of culturally relevant Wenger, E. (1998). Communities of practice: Learning, meaning,

pedagogy. American Educational Research Journal, 32, 465– and identity. Cambridge, UK: Cambridge University Press.

491.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes

from: How the embodied mind brings mathematics into being.

New York: Basic Books. 1

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

peripheral participation. Cambridge, UK: Cambridge

Anderson, R. (2006). Mathematics, meaning, and identity: A study University Press.

of the practice of mathematics education in a rural high

Martin, D. B. (2000). Mathematics success and failure among

school. Unpublished doctoral dissertation, Portland State

African-American youth: The roles of sociohistorical context,

University, Oregon.

community forces, school influences, and individual agency.

Boaler, J. (2000). Mathematics from another world: Traditional Mahwah, NJ: Lawrence Erlbaum.

communities and the alienation of learners. Journal of

Masingila, J. O. (1994). Mathematics practice in carpet laying.

Mathematical Behavior, 18, 379–397.

Anthropology & Education Quarterly, 25, 430–462.

Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing

Masingila, J. O. (2002). Examining students’ perceptions of their

in mathematics worlds. In J. Boaler (Ed.), Multiple

everyday mathematics practice. In M. E. Brenner & J. N.

perspectives on mathematics teaching and learning (pp. 171–

Moschkovich (Eds.), Everyday and academic mathematics in

200). Westport, CT: Ablex.

the classroom (Journal for Research in Mathematics Education

Boaler, J., & Humphreys, C. (2005). Connecting mathematical Monograph No. 11, pp. 30–39). Reston, VA: National Council

ideas: Middle school video cases to support teaching and of Teachers of Mathematics.

learning. Portsmouth, NH: Heinemann.

Mendick, H. (2003). Choosing maths/doing gender: A look at why

D’Ambrosio, U. (1990). The role of mathematics education in there are more boys than girls in advanced mathematics

building a democratic and just society. For the Learning of classes in England. In L. Burton (Ed.), Which way social

Mathematics, 10, 20–23. justice in mathematics education? (pp. 169–187). Westport,

Devlin, K. (2000a). The four faces of mathematics. In M. J. Burke CT: Praeger.

& F. R. Curcio (Eds.), Learning mathematics for a new Moses, R. P., & Cobb, C. E. (2001). Radical equations: Civil rights

century (pp. 16–27). Reston, VA: NCTM. from Mississippi to the Algebra Project. Boston, MA: Beacon

Devlin, K. (2000b). The math gene: How mathematical thinking Press.

evolved and why numbers are like gossip. New York: Basic Nasir, N. S. (2002). Identity, goals, and learning: Mathematics in

Books. cultural practice. Mathematical Thinking & Learning, 4, 213–

Gee, J. P. (2001). Identity as an analytic lens for research in 247.

education. Review of Research in Education, 25, 99–125. National Council of Teachers of Mathematics. (2000). Principles

Kabiri, M. S., & Smith, N. L. (2003). Turning traditional textbook and standards for school mathematics. Reston, VA: Author.

problems into open-ended problems. Mathematics Teaching in Sfard, A., & Prusak, A. (2005). Telling identities: In search of an

the Middle School, 9, 186–192. analytic tool for investigating learning as a culturally shaped

Kirshner, D. (2002). Untangling teachers’ diverse aspirations for activity. Educational Researcher, 34(4), 14–22.

student learning: A crossdisciplinary strategy for relating Smith, J. P. (1999). Tracking the mathematics of automobile

psychological theory to pedagogical practice. Journal for production: Are schools failing to prepare students for work?

Research in Mathematics Education, 33, 46–58. American Educational Research Journal, 36, 835–878.

Ladson-Billings, G. (1995). Toward a theory of culturally relevant Wenger, E. (1998). Communities of practice: Learning, meaning,

pedagogy. American Educational Research Journal, 32, 465– and identity. Cambridge, UK: Cambridge University Press.

491.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes

from: How the embodied mind brings mathematics into being.

New York: Basic Books. 1

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