**The Study**

Cobb, Paul, Gresalfi, Melissa, and Liao Hodge, Lynn. “An Interpretive Scheme for Analyzing the Identities That Students Develop in Mathematics Classrooms.” *Journal for Research in Mathematics Education *40 (January 2009): 40–68.

**Question**

How is authority distributed and to whom are students accountable in a mathematics classroom? What prerogatives do students legitimately exercise in particular classrooms? How do students’ identities change when the same students interact with each other and the teacher in contrasting classrooms where teachers use different styles of instruction? In this article, the authors propose an interpretive scheme for analyzing the identities that students develop in mathematics classrooms that can inform instructional design and teaching.

**Context**

In recent years references to the notion of identity have become increasingly common in mathematics education research literature. Advocates of this construct contend that it enables researchers to broaden the scope of their analyses beyond an exclusive focus on the nature of students’ mathematical reasoning. The issues addressed in investigations of identity include the ways that students think about themselves in relation to mathematics and the extent to which they have developed a commitment to, and have come to see value in, mathematics as it is presented in the classroom.

For proponents, the notion of identity therefore encompasses a range of issues, including students’ persistence and interest in mathematics and their motivation to learn mathematics.

Despite these arguments, identity has not become a central focus of research in mathematics education research. Detractors can argue that the construct is vague and ill-defined. In addition, its relevance to mathematics educators’ traditional focus on improving the learning and teaching of central mathematical ideas has not always been apparent.

The authors address the apparent vagueness of this notion by proposing a concrete approach for analyzing the identities that students are developing in particular classrooms. They present a specific case that illustrates the relevance of focusing on the identities that students are developing as they engage in classroom activities.

**Who Was Studied?**

To illustrate the analytic approach, the authors focused on 11 eighth-grade students, each of who was a member of two contrasting mathematics classrooms, in which what it meant to know and do mathematics differed significantly. The students attended an urban middle school whose population was approximately 40% African American and 60% Caucasian. Seven of the 11 students were African American, 3 were Caucasian, and 1 was Asian American. The two participating teachers taught different subjects—a regular algebra class and a design experiment that focused on statistics. The algebra teacher had 25 years of classroom experience and was considered to be a successful teacher by both administrators and her peers. The design experiment teacher had 15 years of classroom experience and was a co-researcher on the research project.

**Details**

Paul Cobb, Melissa Gresalfi, and Lynn Liao Hodge conducted the study and report their findings in an article in the January 2009 issue of the *Journal for Research in Mathematics Education* (see below for full citation).

The authors determined their findings by studying the nature of mathematical activity in two middle school mathematics classrooms. They examined how students came to understand what it means to know and do mathematics in the two classrooms, and whether and to what extent they came to value the two contrasting forms of mathematical activity.

The design experiment class focused on statistical data analysis and met during the last period of the school day. A single member of the research team served as the teacher in all 41 classroom sessions during the 14-week experiment. The algebra class was conducted by the students’ regular mathematics teacher earlier in the school day. The study included classroom observations by the research team.

Prior investigations have documented that the extent to which students identify, merely cooperate, or resist in mathematics classrooms can differ significantly from one classroom to another. The previous findings indicated the need for an interpretive scheme that focuses directly on the relationship between the cultures established in particular classrooms and the identities that students are developing in those classrooms.

The two central constructs of the interpretive scheme were the normative identity for being an effective mathematics student in a particular classroom, and the personal identity that a student is actually developing through participation in classroom activities. As the authors define it, normative identity comprises both the general and specifically mathematical obligations that delineate the role of an effective student in a particular classroom. In contrast, personal identity concerns the extent to which individual students identify with, merely comply with, or resist their classroom obligations, and thus what it means to know and do mathematics in their classroom. For this article, the authors enlarged the meaning of *identify* to include both how the students understand what it means to do mathematics in their classroom and whether and to what extent they associate or affiliate themselves with that activity.

The goal of the study was to document the normative identities as doers of mathematics established in the two contrasting classrooms and examine, by empirical analysis, the personal identities the students were developing in the two classrooms.

The study looked at the students’ normative identities in each classroom in relation to the following:

- Distribution of authority and students’ prerogatives to act and think for themselves
- Power differentials between teacher and students; effects of the teacher’s censure or approval
- Power differentials among students; the weight of the approval and censure of some students relative to those of others

The authors examined the students’ personal identities in relation to the following:

- Degree to which students valued their classroom obligations and turned them into obligations to oneself
- Students’ perceptions of their own and their peers’ relative capabilities
- Issues of status and power in the classroom

**Findings **

The teacher of the algebra class typically divided each classroom session into three parts. First, the teacher and the students reviewed homework problems from the previous class. The teacher assisted students who found problems difficult by calling on volunteers to present the solutions, and then answered students’ questions.

In the second part, the teacher introduced the problems that would be assigned as homework later in the current session. The teacher demonstrated a solution method, asked students to solve a second equation, and then called on individual students to provide particular steps of the solution.

During the third phase, the students worked on problems the teacher assigned for homework, individually or in groups, for the remainder of the session.

In this class, authority resided primarily with the teacher, who was responsible for determining both the methods that students should use to solve particular tasks and the legitimacy of their responses. Students’ opportunities to initiate contributions were generally restricted to asking clarifying questions about demonstrated solution methods. Mathematical activity was limited to producing answers that the teacher determined were legitimate by enacting prescribed calculational steps written in algebraic notations.

In contrast, the teacher of the design experiment class provided instructional activities that call on the students to analyze data sets rather than to collect data.

In the first phase of an instructional activity, the teacher and students talked through how they could generate data that would enable them to address a particular problem or issue. Against this background, the teacher introduced the data the students were to analyze.

In the second phase, the students analyzed the data individually or in small groups. In the final phase, the whole class discussed the students’ analyses in which a computer projection system was frequently used to support the students’ explanations.

In the design experiment class, authority was distributed more widely than in the algebra class, because the teacher and students jointly determined the legitimacy of contributions. Further, the students frequently exercised conceptual prerogatives as they developed and explained their analyses and as they challenged others’ analyses and asked clarifying questions.

The analysis of the students’ general and specifically mathematical obligations in the algebra and design experiment classrooms illustrates a concrete approach for documenting the normative identity as a doer of mathematics established in particular classrooms. The sample analyses revealed that the normative identities established in the two classes differed significantly.

In the algebra class, the students needed to identify with a form of mathematical activity in which their only prerogative was to exercise self-discipline as they enacted prescribed calculational steps on written algebraic notations.

In contrast, the standards for mathematical argument in the design experiment class obliged the students to justify why a solution gave insight into the question at hand, allowing students the opportunity to exercise a conceptual prerogative.

The study also found significant differences in the personal identities the students were developing in the two classrooms. Interview questions posed to the students focused on their interpretations of classroom activities, with a particular emphasis on their understandings and valuations of their general and specifically mathematical obligations, and their assessments of their own and others’ mathematical competence.

Students in the algebra class saw their role in relation to the general and mathematical obligations as passive and believed that authority rested solely with the teacher. They were required to produce correct answers by following a specific sequence of steps. All 11 students merely cooperated with the algebra teacher and none identified with mathematical activity in this classroom. In evaluating their own and other students’ competence in the algebra class, all 11 students identified specific students as successful, although only 4 described themselves that way.

In contrast, the students in the design experiment classroom saw their general and specifically mathematical obligations as shared with the teacher, in that they contributed to decisions about the reasonableness and legitimacy of solutions. All 11 students identified with mathematical activity in this classroom and all 11 viewed themselves and all the other students as successful.

In conclusion, the authors find that discussions in which the teacher judiciously supports students’ attempts to articulate their task interpretations can be highly supportive settings for mathematical learning.

**Caveat**

The authors acknowledge that their purpose in documenting the normative and personal identities of the students in the two classrooms is illustrative rather than evaluative, since the two teachers were performing under different constraints and standards of accountability in the school.

**Other Resources**

Boaler, J., and J. G. Greeno. “Identity, Agency, and Knowing in Mathematical Worlds.” In *Multiple Perspectives on Mathematics Teaching and Learning*, edited by J.Boaler, pp.45–82. Stamford, Conn.: Ablex, 2000.

Cobb, P., and L. L. Hodge. “A Relational Perspective on Issues of Cultural Diversity and Equity as They Play Out in the Mathematics Classroom.” *Mathematical Thinking and Learning *4 (July 2002): 249–284.

Sfard, A., and A. Prusak. “Telling Identities: In a Search of an Analytic Tool for Investigating Learning as a Culturally Shaped Activity.” *Educational Researcher *34 (May 2005): 14–22.