Relationships between Student Participation, Task Summaries, and Algebra
Learning
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.