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).

Data, Analysis, Setting, and Classroom Organization

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.

The design experiment was conducted over 14 weeks in the fall of 1998 in an urban middle school. Cobb et al. conducted a similar design experiment the previous year with 29 7th graders (McGatha, Cobb, & McClain, 1999), but due to conflicts only 11 continued in the 8th grade experiment. Kay McClain served as the primary teacher in the experiment, with help from Paul Cobb. Lessons, lasting one or more days, typically began as a whole group to discuss the origins of the data being analyzed before breaking into small group work at computers and finally finishing with whole-group discussion. Work in the prior year showed the importance of students understanding the context of their datasets and how the data were created (Latour, 1987; Lehrer & Romberg, 1996; Roth, 1997; Tzou, 2000). Students were tasked with analyzing the data using specialized computer minitools and using their results to produce policy recommendations. Whole-class discussions focused not just on sharing recommendations and student reasoning, but to questions about reasoning that might inform the learning trajectory.

The Hypothetical Learning Trajectory

The primary potential endpoint envisioned by Cobb et al. was for students to see bivariate data as something distributed across a two-dimensional space of values, as in a scatterplot (Wilensky, 1997). A conversation with Clifford Konold suggested to the researchers that students frequently read scatterplots diagonally rather than vertically, leading them to focus on the distance between points and a possible line of best fit rather than deviations in the y-direction. Cobb et al. conjectured that students should be able to mentally organize the graph into vertical slices that represents univariate distributions of the y variable for a particular x variable (see also Noss, Pozzi, & Hoyles, 1999).

For starting points Cobb et al. looked at their previous design experiment (Cobb, 1999; McClain & Cobb (2001b); McClain, Cobb, & Gravemeijer, 2000) conducted when the students were working with univariate data as 7th graders. One of the minitools students had worked with allowed students to view and partition two univariate distributions, one above the other. As 7th graders, it became normative for students to qualitatively describe proportions of the data with terms like the majority and most and speaking of the hill in the distribution. Nineteen of the 29 students could make data-based arguments when the points in the distribution were hidden, leaving only partition markers (such as quartile boundaries), and 8 of those 19 were involved in the 8th grade design experiment.

The conjectured learning route began with activities that gave students choice in how they chose to represent bivariate data. While Cobb et al. expected some students would construct two-dimensional graphs similar to scatterplots, they also expected students to make double-bar graphs or other representations. Researchers looked for the degree to which students' incriptions of the data helped explain how one quantity varied as the other increased. Cobb et al. also looked to see how well students understood that each data point was associated with two measures. From this initial phase of the experiment, researchers introduced a new minitool to the students that inscribed the bivariate data as a scatterplot. The new tool showed perpendicular lines through any clicked point, to emphasize the point's position relative to both measures, and also provided partitioning guides in the forms of a moveable cross, a grid, a two equal groups partition that created vertical slices of the data in which each slice was divided by the slice's median value, and a four equal groups partition that divided vertical slices into quartiles. With these representations, Cobb et al. saw a challenge for the teacher to have students see distributions of y-values for points constrained in an interval of x. Fitting lines of best fit were left for later in the design experiment, after the reading of scatter plots as bivariate distributions became normative. Across the hypothetical learning trajectory, the sophistication of student reasoning was conjectured to increase as students worked with increasingly sophisticated ways of inscribing data (Biehler, 1993; de Lange, van Reeuwijk, Burrill, & Romberg, 1993; Lehrer & Romberg, 1996; Roth & McGinn, 1998), a view that saw tools as reorganizers of activity rather than amplifiers of activity (Dörfler, 1993; Kaput, 1994; Meira, 1998; Pea, 1993).

The Actual Learning Trajectory

Comparing Univariate Datasets

The first phase of the actual learning trajectory occurred in the first 8 of the 41 sessions and focused on a hypothetical scenario in which two ambulance companies were competition for a contract to serve the school. The initial focus was on the generation of the data and potentially important factors in the school's decision to choose a company, which led to the key variable of response time. The researchers found as before (Tzou, 2000) that students would not engage in the analysis without first making sense of the phenomenon they were asked to study, which was seen as the violation of a key norm. Students then worked in pairs using a minitool that displays two univariate distributions to make dot plots of each ambulance company's data. Some students used a "two equal groups" option in the minitool to explain how 50 percent of one company's response times were faster than the other, an argument which was accepted as legitimate by the class. Another pair of students used the "four equal groups" option to make a similar argument, comparing the similar times for 75 percent of one company's times versus 50 percent of the other company's times, which was also seen as legitimate by the class. Together, researchers felt these results supported their conjectures about the starting point of the trajectory.

Inscribing Bivariate Data

The second phase of the actual learning trajectory, lasting 6 sessions, examined the relationship between the speed a car is driven and the amount of carbon dioxide that is emitted. Again, students engaged in questions about the data and how it is generated before proceeding with the analysis. Students were asked to draw a diagram or a graph that they could use to recommend a speed limit for an interstate highway. As anticipated, about half the students constructed double bar graphs, with some treating speed as nominal/categorical rather than a continuous quantity. Two pairs of students constructed orthogonal axes and plotted points, but differed in their choice of axes for speed and carbon dioxide. Those students were able to explain that a point represented both a measurement of speed and carbon dioxide. Another pair also constructed orthogonal axes, but used them to construct a bar graph where speed was represented as categories on the horizontal and the height of the bar represented carbon dioxide emissions. The whole class discussion focused on the scatter plots from the two pairs of students and the meaning of the points, and the representation of a point as both measures was seen as normative. One of the students who constructed the bar graph explained how their graph was the "same thing" and the class accepted his explanation as legitimate. Researchers believed that their data supported their conjecture about interpreting a dot in a scatter plot as two measures.

Cite

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}