Appropriating Mathematical Practices: a Case Study of Learning to Use and Explore Functions through Interaction with a Tutor

This case study uses a sociocultural perspective and the concept of appropriation (Newman, Griffin and Cole, 1989; Rogoff, 1990) to describe how a student learned to work with linear functions. The analysis describes in detail the impact that interaction with a tutor had on a learner, how the learner appropriated goals, actions, perspectives, and meanings that are part of mathematical practices, and how the learner was active in transforming several of the goals that she appropriated. The paper describes how a learner appropriated two aspects of mathematical practices that are crucial for working with functions (Breidenbach, Dubinsky, Nichols and Hawks, 1992; Even, 1990; Moschkovich, Schoenfeld and Arcavi, 1993; Schwarz and Yerushalmy, 1992; Sfard, 1992): a perspective treating lines as objects and the action of connecting a line to its corresponding equation in the form y = mx + b. I use examples from the analysis of two tutoring sessions to illustrate how the tutor introduced three tasks (estimating y-intercepts, evaluating slopes, and exploring parameters) that reflect these two aspects of mathematical practices in this domain and describe how the student appropriated goals, actions, meanings, and perspectives for carrying out these tasks. I describe how appropriation functioned in terms of the focus of attention, the meaning for utterances, and the goals for these three tasks. I also examine how the learner did not merely repeat the goals the tutor introduced but actively transformed some of these goals. How does a newcomer to a mathematical domain learn from a tutor? One response to this question is that the tutors explicitly tell newcomers information and show newcomers their expert skills, and that newcomers acquire new skills through imitation. This study presents an alternative to this scenario using a sociocultural view of how a student learned from a tutor by participating in joint problem solving and appropriating aspects of mathematical practices. The case study shows how a student learned from a tutor not by the explicit teaching, telling, or showing of skills and information but through participation in joint problem solving. I also describe how the student learned important mathematical ideas not by imitation but by appropriating (and thus transforming) some of the tutor’s ways of seeing, talking, and acting. This study describes how appropriation functions in detail and examines how a student appropriated aspects of the mathematical practices crucial for working with functions. The study focuses on two aspects of appropriation: what it is that learners appropriate and how learners actively Educational Studies in Mathematics 55: 49–80, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 50 JUDIT N. MOSCHKOVICH transform what they appropriate. I describe in detail how one student appropriated ways of seeing, talking, and acting (or perspectives, meanings, actions, and goals) that reflect expertise in working with functions: a perspective treating lines as objects and the action of connecting a line to its corresponding equation (in this case in the form y = mx + b). This paper builds on previous work on appropriation and addresses questions specific to the appropriation of mathematical practices: What particular aspects of mathematical practices does a learner appropriate as they gain expertise in this domain? How does a learner appropriate actions, goals, perspectives, and meanings? What are the central features of the appropriation process? How does a learner actively transform what he or she appropriates? This case study describes how this learner, by solving problems jointly with a tutor, appropriated the focus of attention for tasks, meanings for utterances, and the actions and goals for carrying out new tasks. The focus of attention, meanings, and goals were not evident in the interactions as isolated pieces of tutor knowledge that were stated explicitly by the tutor. Rather, ways of seeing, talking, and acting were implicitly embedded in mathematical activity as the focus of attention, the meanings of utterances, the perspectives, and the actions and goals used during joint activity. I describe two aspects of mathematical practices, a perspective treating lines as objects and the action of connecting a line to its corresponding equation. These two aspects of expert mathematical practices for working with functions were evident in where the tutor and student focused their attention, in how they interpreted the meanings of utterances (especially questions by the tutor), and in the goals and actions the tutor and student set and carried out to accomplish tasks. This analysis builds on previous work addressing mathematical practices (Cobb, Stephan, McClain and Gravemeijer, 2001) and describing central perspectives in this domain (Moschkovich et al. 1993). While work using the concept of appropriation (Newman et al., 1989; Rogoff, 1990; Radford, 2001; Rosebery, Warren, and Conant, 1992) provides a foundation, it also leaves room for elaborating on the notion of appropriation by providing more detailed analyses of how learners appropriate particular mathematical practices in different domains. This study elaborates on the notion of appropriation by describing in detail what it was that a learner appropriated as she learned to work with linear functions (where to focus, what utterances mean, how to use new goals, and how to carry out new actions) and how the learner actively transformed several of the goals she appropriated. The sociocultural perspective used here is a Neo-Vygotskian one that assumes that learning is mediated by social interaction. Appropriation (RoAPPROPRIATING MATHEMATICAL PRACTICES 51 goff, 1990; Wells, 1999) is a central Neo-Vygotskian concept that has been used to describe how learning is mediated by interaction with others and how children learn when adults guide or teach them (Newman et al., 1989; Rogoff, 1990; Radford, 2001; Rosebery et al., 1992; Wells, 1999). Although literature in mathematics education sometimes refers to the appropriation of cultural or mathematical practices, it is not always clear what we mean by ‘appropriation’ and only a few studies describe how appropriation functions in detail for advanced mathematical topics (for one example, see Radford, 2001). If we use the notion of ‘appropriation’ within a sociocultural theoretical framework, then the term has a deeper meaning than the dictionary’s definition, ‘using something for one’s own purposes’ and the concept has central characteristics that distinguish it from other constructs for describing learning. Appropriation involves joint productive activity, a shared focus of attention, and shared meanings (Rogoff, 1990). Appropriation also involves taking what someone else produces during joint activity for one’s own use in subsequent productive activity while using new meanings for words, new perspectives, and new goals and actions. The analysis provided here uses this meaning of ‘appropriation’ to describe in detail how a student and a tutor developed a common focus of attention, shared meanings for utterances, and new goals and actions while engaged in joint productive activity working with lines and equations. I use an interpretation of the concept of appropriation that is compatible with a view of learners as actively constructing knowledge. Appropriation “should not be viewed as limited to the process by which the child (novice) learns from the adult (expert) via a static process of imitation, internalizing observed behaviors in an untransformed manner” (Brown et al., 1993, p. 193). Appropriation does not imply that the learner merely repeats or imitates what she appropriates. Rather, learners use appropriated meanings, actions, or goals for their own purposes and are actively involved in appropriation by transforming what they appropriate. The examples of appropriation presented here show that this student did not merely imitate the goals set through interaction with the tutor; instead, she transformed these goals. The case study examines how a student appropriated two aspects of mathematical practices through interaction with a tutor. The tutor and student worked together as the student explored the algebraic and graphical representations of functions using a computer.1 The analysis of two videotaped sessions traced how goals were originally set through interaction with the tutor and how the student came to independently initiate, set, and use these goals. This paper focuses on describing the appropriation 52 JUDIT N. MOSCHKOVICH process and examining how the student was active in transforming goals. This study complements the cognitive analysis presented in Schoenfeld, Arcavi, and Smith (1993) describing learning in this domain. This student learned many things during these tutoring sessions: she corrected previous knowledge, added new pieces of knowledge, and made new connections between pieces of knowledge. However, the focus of this study is not what she learned but how she learned through interactions with the tutor (for a detailed analysis of what this student learned, see Schoenfeld, et al., 1993). The analysis presented here thus extends this work by describing in detail how this student’s learning was mediated by interactions with the tutor. This study also builds on previous work describing two perspectives of linear functions, treating functions as objects and processes (Moschkovich et al., 1993). One difference from this previous work is that these perspectives are not seen as purely cognitive or individual. Instead, they are described as aspects of mathematical practices that a learner appropriates through interaction with a tutor.