Constructing competence: an analysis of student participation in the activity systems of mathematics classrooms Melissa Gresalfi & Taylor Martin & Victoria Hand & James Greeno Published online: 3 September 2008 # Springer Science + Business Media B.V. 2008 Abstract This paper investigates the construction of systems of competence in two middle school mathematics classrooms. Drawing on analyses of discourse from videotaped classroom sessions, this paper documents the ways that agency and accountability were distributed in the classrooms through interactions between the teachers and students as they worked on mathematical content. In doing so, we problematize the assumption that competencies are simply attributes of individuals that can be externally defined. Instead, we propose a concept of individual competence as an attribute of a person's participation in an activity system such as a classroom. In this perspective, what counts as ���competent��� gets constructed in particular classrooms, and can therefore look very different from setting to setting. The implications of the ways that competence can be defined are discussed in terms of future research and equitable learning outcomes. Keywords Classroom culture . Discourse analysis . Competence . Middle school . Urban schools Educ Stud Math (2009) 70:49���70 DOI 10.1007/s10649-008-9141-5 This research was supported by a grant from the U.S. National Science Foundation. We are grateful that this project afforded collaboration with Frank Davis, Mary Maxwell West, and Robert Moses. M. Gresalfi (*) Indiana University, Bloomington, IN, USA e-mail: mgresalf@indiana.edu T. Martin University of Texas, Austin, TX, USA V. Hand University of Colorado, Boulder, CO, USA J. Greeno University of Pittsburgh, Pittsburgh, PA, USA

1 Introduction The purpose of this paper is to offer a framework for considering the process of competence construction in mathematics classrooms. In doing so, we seek to highlight the constructed nature of competence, in order to problematize the customary usage of the term in discussions of learning and teaching, and in particular in American state and national standards documents. In these discussions, competence refers to a collection of skills or abilities that are attributed to individuals apart from the specific contexts in which they participate. For example, we talk about the mathematical competencies that we want all students to acquire, or the level or proficiency that we hope students will attain (Kilpatrick, Swafford and Findell 2001 National Council of Teachers of Mathematics 2000 National Mathematics Advisory Council 2008 Schoenfeld 1999). There is reason to worry about the content that students engage in their classrooms, and we do not want to brush those concerns aside. The aim of this paper, however, is to problematize the assumption that competencies are simply attributes of individuals. Instead, we propose a concept of competence as an attribute of participation in an activity system (Engestr��m 1993 Greeno 2006 Lemke 1990) such as a classroom. In this perspective, what counts as ���competent��� gets constructed in particular classrooms, and can therefore look very different from setting to setting. Viewing competence as constructed has important implications for both theory and practice, as it requires shifting the focus of analysis from the individual to the classroom in which the individual is a participant. As a consequence, whether or not a student is deemed to be ���competent��� is no longer seen as a trait of that student, but rather an interaction between the opportunities that a student has to participate competently and the ways that individual takes up those opportunities. Studying the classroom as an ���activity system,��� which we define as a social organization that may contain learners, teachers, curriculum materials, software tools, and the physical environment, is appropriate for understanding how all these aspects interact in the construction of competence. This work contributes to a body of literature that has sought to establish that classrooms are social systems that are organized through regularities of shared practice (Bauersfeld 1992 Brown 1994 Cobb 1999, 2000 Yackel and Cobb 1996). In particular, these systems of meaning shape the ways that individuals are expected, entitled, and obligated to participate, but also the meanings that members make of particular acts of participation (Gee 1999 Holland, Skinner, Lachicotte and Cain 1998 Jungwirth 1991). In this paper, we address a specific aspect of the classroom system���the ways that competence is constructed amongst participants in particular classroom contexts. This focus is important because it enables an illustration of the ways that students' participation���and in particular, attributions of participation���truly are a function of the relationship between what participants do given what they have opportunities to do (Gresalfi 2004). More specifically, we focus on two aspects of classrooms that support the ways that systems of competence are constructed���the ways that agency and accountability are distributed in the classroom. By highlighting these two aspects of classroom participation structures in particular, we seek to provide both an analytic lens for future research, and a tool for practitioners as a means of better understanding the complexities of classroom systems. 1.1 Competence, redefined Competence, as we use the term, refers generally to what students need to know or do in order to be considered successful by the teacher and other students in the classroom. This is related to the notion of normative identity, as discussed by Cobb, Gresalfi and Hodge 50 M. Gresalfi et al.

(2008). Both terms refer to the system of expectations that is established around accepted mathematical practice. In this paper we focus on competence in an effort to highlight not just the kinds of practices that are considered normative (accepted aspects of collective activity), but those activities that are specifically associated with designations of success. In examining classroom systems at this level, we seek to demonstrate a method for better understanding the mechanisms underlying the ways that normative constructions of competence, or normative identities more broadly, become established. One version of competence involves what students need to know or do in order to be ���correct,��� but there can be other versions. For example, sharing mistakes or misunder- standings can be a useful learning activity, and the person who has shared a mistaken idea can be considered to be competent (Hiebert, Carpenter, Fennema, Fuson, Wearne and Murray 1997 Lampert 2001). This distinction highlights the constructed nature of competence as it is locally and individually defined through participation between the teacher and students. As a consequence, the competencies that are constructed in a classroom impact both what counts as mathematics, and who gets to say (Amit and Fried 2005 Cobb, Wood and Yackel 1993 Cobb, Wood, Yackel and McNeal 1992 Yackel and Cobb 1996). In turn, their negotiation can impact and change which behaviors are sanctioned as competent. The process of negotiating what it means to ���do mathematics,��� and which elements of ���doing mathematics��� are reinforced and highlighted does not happen in one moment, or in just one way. A behavior, way of thinking, or contribution gets positioned as competent most often by the teacher, who generally holds power in classrooms to determine what is both correct and acceptable. For example, a teacher's response to a student's contribution, such as ���nice job!��� quickly and clearly positions that contribution as competent1. It is not only correct responses that can be counted as competent, however. At times a teacher may intentionally reinforce a particular way of doing mathematics that may not be comfortable for students, such as by explicitly positioning a student's mistaken solution as an important contribution for the learning of the class (c.f. Lampert 1990, 2001). In that context, the act of sharing any solution is positioned as competent, as it serves to push the thinking of the class. Activity systems such as classrooms are not defined by one member, no matter how powerful that member may be. Rather, they are ���dynamic, open, semiotic system(s) of meaningful actions and meaning-making processes��� (Lemke 1990: p. 191). Thus, the teacher is not the only participant who is able to shape the construction of competence in a classroom students also play a role in this negotiation. In the same example above, students might thwart a teacher's attempt to support the idea that making mistakes is valuable if they mock or belittle a student who has made a mistake. Students may also play a more positive role in constructing competence, from taking up opportunities to participate in ways advocated for by the teacher, to consistently asking particular students for help or checking with particular students to make sure that their own answer is correct. Both kinds of acts��� from identifying to openly resisting���contribute to the ongoing negotiation of the classroom culture (Cobb, Gresalfi and Hodge 2008). 1 This move simultaneously serves to reinforce the relative power difference between teacher and students by indexing the teacher's position relative to the student as the person who is responsible for determining the flow of activity and the direction of the discussion (Lemke 1990). Although students can also utter the same words, the meaning of such an utterance is slightly different as it does not (usually) promise the same kinds of potential outcomes (namely, good grades). In some rare classrooms this structure can be disrupted, but it is difficult to imagine a complete reversal of these positions. Constructing competence 51