Cobb, McClain, & Gravemeijer (2003)

Learning About Statistical Covariation

• Author: Paul Cobb, Kay McClain, and Koeno Gravemeijer
• Journal: Cognition and Instruction
• Year: 2003
• Source: http://www.tandfonline.com/doi/abs/10.1207/S1532690XCI2101\_1

Abstract

In this article, we report on a design experiment conducted in an 8th grade classroom that focused on students' analysis of bivariate data. Our immediate goal is to document both the actual learning trajectory of the classroom community and the diversity in the students' reasoning as they participated in the classroom mathematical practices that constituted this trajectory. On a broader level, we also focus on the learning of the research team by documenting the conjectures about the students' statistical learning and the means of supporting it that the research team generated, tested, and revised on-line while the experiment was in progress. In the final part of the article, we synthesize the results of this learning by proposing a revised learning trajectory that can inform design and instruction in other classrooms. In doing so, we make a contribution to the cumulative development of a domain-specific instructional theory for statistical data analysis.

Outline of Headings

• The Design Experiment Methodology
• Data Sources and Method of Analysis
• The Setting of the Design Experiment
• Classroom Organization
• The Hypothetical Learning Trajectory
  • Potential Endpoints
  • Starting Points
  • The Conjectured Learning Route and Means of Support
    • Cross
    • Grids
    • Two Equal Groups
    • Four Equal Groups
• The Actual Learning Trajectory
  • Social and Sociomathematical Norms
  • Comparing Univariate Datasets
  • Inscribing Bivariate Data
  • Reducing Scatter Plots to Lines
  • Negotiating the Median
  • Reading Stacks and Slices as Distributions
• What We Learned from the Design Experiments
  • Structuring and Organizing Bivariate Data
    • Starting points: univariate datasets as distributions
    • Developing ways of inscribing bivariate data
    • Stacked data as bivariate distributions
    • Scatter plots as bivariate distributions
  • Initial Steps Toward Statistical Inference
• Conclusions

Summary

Cobb, McClain, and Gravemeijer conducted a 14-week design experiment with a group of 8th grade students to yield a learning trajectory for statistical covariation. The researchers looked both at the learning of the classroom community as well as individual students, and considered their own learning as integral in the experiment. Cobb et al. conducted the design experiment in three parts: planning for the experiment, experimenting in the classroom, and conducting a retrospective analysis (Cobb, 2000; Confrey & Lachance, 2000; Simon, 2000). Gravemeijer (1994) provides the basis for the preparation stage, with thought experiments about mathematical activity and discourse that might take place under various instructional designs. These conjectures about the trajectories for student learning and the means to support it compose what Simon (1995) called a hypothetical learning trajectory. The goal of the experiment is not to see if the proposed instruction is effective, but to iteratively test and modify the conjectures as the experiment progresses (Brown, 1992; Cobb, 2001; Collins, 1999; Suter & Frechtling, 2000). The researchers did this in what Gravemeijer (1994) calls minicycles of design occur almost daily. The retrospective analysis of the planned versus actual learning trajectory helps inform a domain-specific instructional theory that can then be used to plan instruction in other classrooms. These local instructional theories describe a demonstrated plan for learning significant mathematical ideas and the means by which learning is supported and organized. Although the design experiment is conducted in the unique constraints of a single classroom, the local instructional theory is what can make the research generalizable (Steffe & Thompson, 2000) and the basis for further theory refinement (Stigler & Hiebert, 1999).

Cobb et al. collected video data from 41 classes, student work, two sets of field notes, and audio recordings of the research meetings. The unit of analysis was the microculture of the classroom community rather than the thinking of individual students, which varies too greatly. Analysis tended to three types of classroom norms, simplified from Cobb, Stephan, McClain, & Gravemeijer (2001): classroom social norms, sociomathematical norms, and normative mathematical meanings. Social norms (Erickson, 1986; Lampert, 1990) included participation structure, such as norms for justifying reasoning or indicating misunderstanding. Sociomathematical norms "focus on regularities in classroom actions and interactions that are specific to mathematics" (p. 5; Hershkowitz & Schwartz, 1999; McClain & Cobb, 2001a; Sfard, 2001a; Simon & Blume, 1996; Voigt, 1995; Yackel & Cobb, 1996), such as criteria for what counts as a mathematical solution or mathematical efficiency. Normative mathematical meanings are more specific to mathematical ideas and subject matter knowledge; in this study, this refers to normative ways of talking and reasoning about bivariate data. Due to the goals of the research and the volume of data, Cobb et al. focused less on short moments of student or classroom reasoning and more on significant mathematical ideas that developed over weeks. Inferences about each of these norms are taken as disprovable conjectures using methods similar to Glaser and Strauss's (1967) constant comparison method that are adapted for design research (Cobb & Whitenack, 1996). Two passes were made over the data: the first chronologically to conjecture about normative activity at a particular time, and the second to test each conjecture from the first pass using the entire data set.

References

• Cobb, McClain, & Gravemeijer (2003) 75.jpg

  Atkinson-Collins

• Cobb, McClain, & Gravemeijer (2003) 76.jpg

  Confrey-McGatha

• Cobb, McClain, & Gravemeijer (2003) 77.jpg

  McGatha-Taylor

• Cobb, McClain, & Gravemeijer (2003) 78.jpg

  Thompson-Yackel

Corrolary

APA

Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1–78. doi:10.1207/S1532690XCI2101_1

BibTeX

@article{Cobb2003,
abstract = {In this article, we report on a design experiment conducted in an 8th grade classroom that focused on students' analysis of bivariate data. Our immediate goal is to document both the actual learning trajectory of the classroom community and the diversity in the students' reasoning as they participated in the classroom mathematical practices that constituted this trajectory. On a broader level, we also focus on the learning of the research team by documenting the conjectures about the students' statistical learning and the means of supporting it that the research team generated, tested, and revised on-line while the experiment was in progress. In the final part of the article, we synthesize the results of this learning by proposing a revised learning trajectory that can inform design and instruction in other classrooms. In doing so, we make a contribution to the cumulative development of a domain-specific instructional theory for statistical data analysis.},
author = {Cobb, Paul and McClain, Kay and Gravemeijer, Koeno},
doi = {10.1207/S1532690XCI2101\_1},
journal = {Cognition and Instruction},
number = {1},
pages = {1--78},
title = {{Learning about statistical covariation}},
url = {http://www.tandfonline.com/doi/abs/10.1207/S1532690XCI2101\_1},
volume = {21},
year = {2003}