This results in a characterization of the educational experiment as a sequence of interconnected instructional activities that unfold in synergy with the emergence of students' ideas. Analyzes of the meanings and interpretations of probability have been frequently repeated in the psychological, historical, and educational research literature. The authors reported their interpretations of students' understanding of survey purposes, sampling bias, randomization, and stratification.
Analyzes of these data by Schwartz et al. ibid.) to find that the student's understanding of the part-whole relationship between the sample and the population improved after the instruction.6 The authors hypothesized that the real world. The interviewer drew students' attention to the difference between this distribution and the population distribution (also shown on the screen), emphasizing the difference in their variances. In the first session of the study, the students completed the pretest, received training, and completed the first posttest.
A preliminary report on the teaching experiment (Saldanha & Thompson, 2002) elaborated a conception of sample and sampling that emerged from analyzes of student data. A few students drifted in and out of the course during the transition period, in the first two weeks. The ideas of didactic objects and didactic models (Thompson, 2002) are reflected in my descriptions and analyzes of the teaching activities with which the students engaged in the teaching experiment.
Rather, they were conducted in tandem whenever such data were obtained in the chronological unfolding of the lessons.
ORIENTATION TO STATISTICAL PREDICTION AND DISTRIBUTIONAL REASONINGDISTRIBUTIONAL REASONING
The instructor used this event as a natural opportunity to ask a question about the probability of the sampling outcome (line 13). Responses like Peter's above were not uncommon among students in the early stages of the experiment. The transition from the first to the second part of the activity caused a shift in class discourse and attention from individual sample results to collections of results.
In this segment, the instructor drew on the students' experiences in the second part of the course. Immediately following the discussion in the third part of Episode 3 of Lesson 3, the instructor had the students repeat the second part of the sampling activity. In the first outcome, 5 of the 8 samples containing a maximum of 2 red candies contained only 1 red candy.
In the second outcome, 8 samples also contained at most 2 red candies, but only 1 of those 8 contained 1 red candy. W” in the Element Marks field and for 216 in each of the corresponding How Many slots. A sequence (from left to right) of the approximate results of simulated candy sampling experiments.
The table showing result 2 (see Figure 5.5) showed that 8 of the 10 samples contained 3 or more white candies. The next and final discussion segment in Activity 2 illustrates how Nicole interprets the information in the Analysis window. The result of the fifth iteration of the simulated candy sampling experiment (left) and a close approximation of it (right).
I conclude these comments on the discussions of Activity 2 by reminding the reader of the agenda underlying the activity in the first place. The question was not addressed in the class discussion in this phase of the experiment, nor will I address it here. This leads me to believe that the majority of the bag is white and not red.
Students moved toward structuring a collection of sample outcomes in terms of the relative number of samples that contained a majority of one outcome (eg, white candy). In the case of the candy sampling experiments, this structuring seems to have formed the basis of students' . inferences to the sample population.
MOVE TO CONCEPTUALIZE PROBABILISTIC SITUATIONS AND STATISTICAL UNUSUALNESSSTATISTICAL UNUSUALNESS
Based on the results of their investigation, draw conclusions about the question that arises in the situation. Indeed, this purpose was part of the underlying learning rationale for the activity. Thus, samples of size n = 250 taken from a population of 30,000 will tend to be representative of the sampled population.
2% of all pairs of skates distributed to US retailers, i.e. 2% of the sampled population. This shows the difficulty in placing and coordinating all the information given in the script. Students could not easily operationalize expectations in terms of relative frequency—the fraction of time that an event of interest may occur.
During their comparison, the students noticed that in most samples most of the sample percentages were close to the population percentages. After examining the percentages of the first sample, students immediately noticed how much farther they were from the population percentages than those in samples of size 10,000. It was intended to orient students to consider accuracy from a different perspective in the rest of the activity.
Nicole thus felt that the scatter or clustering of sample percentages indicated the accuracy of the sample. In the later part of the discussion, Sarah read aloud, almost verbatim, her written response to the activity task. I compared the results of different sample sizes and their accuracy with the original population experiment.
However, towards the end of these discussions it emerged that many students had problematic interpretations of the values in the 'real'. The move to involve students in the next activity of the series came from the. The corresponding values in the right column were from the outcome of each sample (i.e. the number of “Yes” responses in that sample of 500 people who asked the same question).
Chelsea (reading her description): A histogram showing the number of teens who answered “yes” to the question “do you believe you can become president?” out of 2000 samples with a size of 500. Discussions around Part 3 of Activity 6 revealed that most students had tremendous difficulty understanding the activity.
MOVE TO QUANTIFY VARIABILITY AND EXTEND DISTRIBUTION
By "operational conception of distribution" I mean a structuring of a collection of randomly generated data values that results from dividing the collection's range into classes, each of which is determined by the proportion of values that fall within some subrange of it is contained. or within some range of the sampled population parameter. Another way to think of such a structuring is to quantify the variability between the collection's values. Dispersion and quantified variability can therefore be seen as conceptual isomorphs (Thompson & Saldanha, 2003), in that each induces a sense of the other; having one in mind essentially leads to having the other in mind as well.
The context of these activities framed the task as one of investigating the effects of two factors on the long-term behavior of sample percentages: Activity 9 investigated the effects of the population percentage selected, and Activity 10 investigated the effects of sample size. Based on a quick calculation, the class determined that in both collections slightly less than half of the percentages were contained within the common range. The discussion ended with the instructor emphasizing that the proportions the students had just calculated and compared were measures of the variability of each collection.
The concluding segment of the discussion, however, suggests that two particular students were spreading the idea. For example, Peter was aware of the attribute that measured the tightness of the grouping of a collection with respect to its midpoint. Activity 8: Investigating the Effects of the Population Parameter on Sampling Variability Activity 8 was the research team's attempt to engage students in systematically determining the distribution of sample percentages relative to the population percentage.
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 the one in the right panel. Each row in the table in the left panel corresponded to one of the populations. They then had to fill in a row's remaining 4 cells with the percentage of sample percentages for samples drawn from that population that were within one, two, three, and four percentage points of the population percentage.
The purpose of this set-up was to structure the students' examination of the sample data in order to orient them to patterns in the data's spread around the population parameter. These patterns can then facilitate the quantification of the density of the spread in the conventional way suggested in the initial discussion—that is, by determining what fraction of the sample percentage values are contained in small intervals around the population percentage value. Rather, it was a culminating part of the activity raised by the instructor during classroom discussions.
