Much attention is currently being given to the intended curriculum for school mathematics along with the great emphasis placed on student learning as measured on achievement tests. The mathematics education research community, however, has many unanswered questions regarding the enacted curriculum, the complex site in which the intended curriculum and students come into contact, thus influencing what students learn (Stein, Remillard, & Smith, 2007). One aspect of the enacted curriculum that has been studied is the cognitive demand of mathematical tasks. Stein and her colleagues (Stein, Grover, & Henningsen, 1996; Stein & Lane, 1996) found that implementing tasks with high levels of cognitive demand, and maintaining those levels throughout enactment, related positively to student learning. According to sociocultural perspectives on learning (e.g., Lave & Wenger, 1991), it is also important to consider the ways in which students participate in the mathematics classroom community. Indeed, the enacted curriculum involves individual thought processes occurring simultaneously and in dialogue with collective processes. This suggests the complementary nature of a framework of cognitive demand and a framework of student participation when examining the enacted curriculum. The present study of middle school algebra is a correlational analysis between the enacted curriculum and student learning. The study builds on existing research in three ways. First, cognitive demand is included as an aspect of the enacted curriculum to replicate past work (i.e., Stein & Lane, 1996). Second, types of student participation are included as a complementary dimension to cognitive demand. Third, whereas past research has often focused on tasks as written or set-up by teachers, this study gives particular attention to the summary phase of mathematical task enactments because this phase typically involves whole-class participation structures. The goal is to gain insight into the nature of cognitive demand and student participation throughout the phases of mathematical task enactments and to investigate how this relates to student learning.

METHOD Data come from eight middle school algebra classrooms; four located in an East coast state and four in a Southern state. In each classroom, all lessons involving the introduction of variables were videotaped and transcribed. This particular content area was chosen because of the foundational role of variable in future mathematical domains. The mathematical content was, therefore, constant across the classrooms while other characteristics such as teacher experience, textbook, and school setting varied. The classroom observations, together with artifacts such as textbook sections and handouts, constitute the enacted curriculum data. All mathematical tasks enacted within these lessons were identified and divided into phases—the task as written, as set-up by the teacher, as implemented by the students, and as summarized by the teacher and students together. Following Stein, Grover, and Henningsen (1996), each phase of each enacted task was then coded for level of cognitive demand—doing mathematics (high), procedures with connections to meaning (high), procedures without connections to meaning (low), or memorization (low). Trajectories of cognitive demand (e.g., low to high, high maintenance) were then identified for each task enactment. Additionally, each phase of each enacted task was coded for the type of student participation (see Figure 1). Although the participation types are not ordinal, trajectories were still identified (e.g., low non-mathematical to high semi-mathematical).

Figure 1. Analytic framework for participation within task enactments.

Using trajectories of cognitive demand and student participation as predictor variables, together with other control variables (e.g., textbook series), regression analysis (Agresti & Finlay, 1997) will be performed with aggregated student gain scores on a pre/post-test as the outcome variable. This assessment was developed by the American Association for the Advancement of Science for the specific purpose of measuring student learning of the introduction to variables. Furthermore, a separate analysis of the cognitive demand and participation of the task summaries in particular will be conducted.

RESULTS Analysis is ongoing, so the full results of the regression analysis can not yet be reported. From the data that has been coded thus far, however, there is potential confirmation of Stein and Lane (1996). In particular, there is support that cognitive demand, when traced through mathematical task enactments, is an important component of the enacted curriculum with respect to student learning as evidenced on a pre/post-test. Preliminary findings also suggest that the participation dimension is distinct from cognitive demand. For example, instances were found of the same level of cognitive demand having distinct forms of participation. Conversely, instances were found of the same participation type but distinct levels of cognitive demand. Such variation is promising in terms of parsing the two dimensions in regression. Moreover, a wide range of practices with the task summary phase of the enacted curriculum were observed. This will be used to explore whether and how the whole-class discussion at the conclusion of a mathematical task relate to gain scores. Overall, this study will contribute to our understanding of the relationship between the enacted curriculum and student learning in the area of variable and will build on past work around cognitive demand by complementing it with a sociocultural perspective, considering both the thought processes that students are engaged in together with their forms of participation.

REFERENCES

Agresti, A., & Finlay, B. (1997). Statistical methods for the social sciences (3rd ed.). Upper Saddle River, NJ: Prentice Hall.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455–488.

Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2, 50–80.

Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte, NC: Information Age.