This chapter describes a final sequence of 3 activities that unfolded over Lessons 13 through 18 (see Figure 8.1). Broadly speaking, these activities extended the explorations of Phase 3 by aiming to move students toward developing a sense of distribution that is intimately bound up with quantifying the variability among sample percents. Specifically, Activities 8 and 9 engaged students in systematically exploring relationships between sampling variability, population parameter, and sample size. Activity 10 sought to have students develop a statistical interpretation of the notion of margin of error.
Phase 4: Move to quantify variability and extend distribution
Lesson Activity (A) Duration
13 (09/03) Preliminary discussion to A8: quantifying variability 12 m.
14 (09/07) A8: Investigating effect of p on sampling variability—Discussion 1 25 m.
15 (09/09) A8 revisited—Discussion 2
A9: Investigating effect of n on sampling variability—Discussion 1
25 m.
25 m.
16 (09/10) A9 revisited—Discussion 2 50 m.
17 (09/13) A10: Margin of error—Discussion 1 16 m.
18 (09/14) A10 revisited—Discussion 2 22 m.
Figure 8.1. Chronological overview of instructional activities of Phase 4.
The chapter begins by elaborating a conception of distribution targeted in instruction, highlighting how it extends ideas of distribution that emerged in Phase 3 and providing a rationale for Activities 8 and 9. The chapter then details Activities 8 through 10, characterizing discussions that unfolded in their temporal order, and highlighting students’ thinking that emerged within them. The chapter concludes with analyses of students’ written work on post- instruction assessment tasks.
Prelude to Phase 4
The first two activities detailed in this chapter were designed for use in a previous teaching experiment (Saldanha & Thompson, 2002). They were re-employed in Phase 4 of the current experiment as an extension to the activities of Phase 3. Recall that in Phase 3, emergent ideas of
a collection’s variability, accuracy, or dispersion were largely informal and grounded in perceptual features discerned from dot plots and cluster diagrams of sample percents. The research team reasoned that these images might provide an imagistic underpinning for a more elaborate sense of distribution—an operational conception of distribution.
By “operational conception of distribution” I mean a structuring of a collection of randomly generated data values that emerges from partitioning the collection’s range into classes, each of which is determined by the proportion of the values that are contained within some sub-range of it or within some vicinity of the sampled population parameter. Another way to think of such a structuring is as the quantification of the variability among the collection’s values. By
determining what fraction of all values are contained within various sub-ranges of the whole, or within intervals around the parameter, one is essentially giving a measure of the collection’s dispersion relative to a reference range, or value, respectively. Thus, distribution and quantified variability can be seen as conceptual isomorphs (Thompson & Saldanha, 2003), in that each one induces a sense of the other; having one in mind essentially leads to having the other in mind as well.
Hereafter, I shall refer to this “operational conception of distribution” simply as distribution.
The activities of this phase moved to support students’ developing this sense of distribution by engaging them in quantifying the variability among sample percents. The context of these activities framed the task as one of investigating the effects of two factors on the long-run behavior of sample percents: Activity 9 investigated the effects of the sampled population percent, and Activity 10 investigated the effects of sample size.
A Preliminary Discussion
Activity 8 was preceded by a preliminary discussion—lasting approximately 12 minutes during the later part of Lesson 13—in which the instructor introduced the idea of quantifying the variability among a collection of sample percents. The aim of this discussion was to orient students to the possibility of moving beyond perceptually-based judgments of a collection’s variability and toward measurement-based judgments.
The preliminary discussion focused on two dot plots (Figure 8.2), each showing the
dispersion of a collection of sample percents. The instructor had spontaneously sketched these on
the blackboard and deliberately constructed each to have different numbers of elements. This was intended to problematize comparing their variability on the basis of visual judgments.
Figure 8.2. The distribution of each of two collections of sample percents.
The instructor then seeded the discussion by asking the class to judge, on the basis of a rough visual estimate, which collection is more variable. Several key ideas emerged among students in the course of the ensuing discussion. They are summarized below:
• The two collections have different numbers of sample percents
• The two collections have the same range, so if we go strictly by range they appear to be equally variable
• Collection A looks more spread out. If we go by how far apart the individual sample percents are, then Collection A is more variable
• Collection B contains more sample percents, and more of them are farther away from the center. So if we go by how many percents are far away from the center, Collection B is more variable
• Nicole observed that each collection is “evenly dispersed throughout the range”, leading her to believe that the two collections were equally variable
The instructor took Nicole’s observation (in the last bullet) as occasion to introduce a
conventional way of quantifying variability, that is to look at what fraction of an entire collection of sample percents are contained within various ranges. He drew a common interval around each distribution’s center (Figure 8.3) and asked students to determine what fraction of each
collection’s percents were contained within that interval.
Figure 8.3. The distribution of two collections of sample percents, each having some fraction of it contained within a common interval around the center.
On the basis of a quick calculation, the class determined that in both collections a little less than half of the percents were contained within the common interval. The instructor then stressed that by this conventional measure, the two collections were about equally variable.
The discussion concluded with the instructor highlighting that the proportions students had just calculated and compared were measures of the variability of each collection. While students did not object to the idea that a collection’s density within regions around a location provided a measure of its variability, there was also no direct evidence that they generally appropriated this idea as a way to measure variability. The concluding segment of the discussion does, however, suggest that two particular students were broaching this idea. For instance, Peter was mindful of the attribute being measured as akin to the tightness of a collection’s clustering relative to its midpoint. In addition, both Peter and Nicole believed that the tighter the clustering then the smaller its measure would be, thereby expressing an intuitive sense that variability and its measure are inversely related.
Activity 8: Investigating Effects of the Population Parameter on Sampling Variability Activity 8 was the research team’s attempt to engage students in systematically quantifying the dispersion of sample percents relative to the population percent. It comprised the table shown in the left panel of Figure 8.4 and four displays like the one shown in the right panel. The
histograms and data lists corresponded to collections of samples drawn from four populations (having, respectively, 57%, 60%, 65%, and 32% “yes” on an issue, e.g. “Do you believe you can be President?”) from which 2000, 2000, 2000, and 3000 samples were drawn, respectively. All samples contained 500 individuals. The histograms and lists showed how “percent of a sample who responded yes” were distributed.
Figure 8.4. One of four distributions of sample percents (right panel), drawn from a simulated population, for which students investigated and quantified the variability (one row of table in left panel).
Students were to fill in the rows of the table shown in the left panel of Figure 8.4 with information drawn from the histograms and data lists like that in the right panel.
Each row of the table in the left panel corresponded to one of the populations. For each row students were to specify the population percentage and number of samples selected. They were to then fill in a row’s remaining 4 cells with the percent of sample percents, for samples drawn from that population, which were within one, two, three, and four percentages points of the population percentage.
The aim of this set-up was to structure students’ investigation of the sampling data so as to orient them to patterns in the data’s dispersion around the population parameter. These patterns might then facilitate quantifying the density of the dispersion in the conventional way proposed in the preliminary discussion—that is, by determining what fraction of sample percent’s values are contained within small intervals around the population percent’s value. Moreover, by examining these patterns of dispersion across various populations, students might discern a relationship between sampling variability and the population parameter. Indeed, an instructional endpoint of the task was to have students realize that for a given sample size, the sampling variability—that is, the patterns of dispersion that emerge—across the four populations is relatively constant, thus suggesting a general relationship: for samples of a given size, sampling variability is relatively unaffected by underlying population percentage.1 Evidence for this generalization can be seen in Figure 8.5, which shows the completed table.
1This last point was not formally asked of students in the written activity guide. Rather, it was a culminating part of the activity that the instructor raised during classroom discussions. Also, while this generalization is not technically true, the research team took it as a suitable line of reasoning for pedagogical purposes. In fact, the dispersion will