Sostenibilidad de las zonas de pesca

the expectation that different samples taken from a population vary from each other and do not match the population (Rubin, Bruce, & Tenney, 1991). Students that hold the first idea have almost an absolute certainty in relation to the sample representa- tiveness of the population. Students that hold the second idea have a big uncertainty in relation to the sample representativeness of the population. Rubin et al. (1991) showed that senior high school students did not integrate these ideas in their rea- soning about distributions of sample outcomes, but held instead one idea at a time depending on the given task.

To integrate these contradicting ideas, students need to envision a process of re- peated sampling (Shaughnessy, 2007) with the result of a sampling distribution—the idea “that the values of a statistic are distributed somehow with a range of possi- bilities” (Thompson, Liu, & Saldanha, 2007, p. 209). Engaging students with sam- pling distributions might support an emergence of a probabilistic view—the ability to make probability statements about sample statistics in order to control or quantify uncertainty.

However, sampling distribution is one of the most difficult concepts in learning statistics (Saldanha & Thompson, 2002). As a result of failing to develop a deep un- derstanding of sampling distribution, students often develop a procedural knowledge of statistical inference. Garfield, delMas, and Chance (2005) listed what students should understand about sampling distributions including, for example, as sample size (n) gets larger, variability of the sample means gets smaller, and students should be able to interpret or apply areas under curve as probability statements about sample statistics. They also listed what students should be able to do with this knowledge about sampling distributions, such as describe the size of the standard error of the mean and the likelihood of different values of the sample mean.

We suggest that in order to develop deep understanding of informal statistical inference, students should be exposed informally first to ideas of sampling variability and sampling distribution over several years starting at early age. Learning these complex ideas in early years is enabled nowadays with technological advancements. Next, we situate the rationale of the learning environment of this study by reviewing a technological tool that guided the design of students’ activities, and two types of settings used in previous studies to develop and study students’ IIR.

3.3.4 Learning in a Technology-Enhanced Environment

Technological advancements have led to numerous changes in statistical instruction, including new school curricula that introduce advanced statistical concepts as early as the elementary level (Franklin & Garfield, 2006). Technology enables students to organize and represent data dynamically with less emphasis on calculations. Thus, class discussions or activities may focus on “what if” questions by manipulating graphs and instantly seeing the results (Chance, Ben-Zvi, Garfield, & Medina, 2007). Using technology also enables students to experience and participate in the statistical processes in tangible and dynamic ways, which are not available without technology (Biehler, Ben-Zvi, Bakker, & Maker, 2013). For example, simulations can offer ways to understand ideas of long-run patterns and random processes (Garfield, Chance, &

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Snell, 2000). Several studies have demonstrated the advantage of dynamic and in- novative technological tools, such as Fathom®_{and TinkerPlots™ (Konold & Miller,} 2011) in developing students’ statistical reasoning and in supporting their compe- tence in making general arguments via data-based evidence (e.g., Ben-Zvi, 2000; Paparistodemou & Meletiou-Mavrotheris, 2008).

3.3.5 Research on Students’ Informal Inferential Reasoning

Two main types of settings have been used in the research literature to examine young students’ informal inferential reasoning. The first, Exploratory Data Analysis (EDA) is a learning environment in which students are engaged in real world data investi- gations where they create surveys to study some question of interest (e.g., Ben-Zvi, 2006; Pfannkuch, 2006; Makar et al., 2011; Makar & Rubin, 2009). In a study by Ben-Zvi (2006), fifth grade students collected and investigated real data about them- selves using TinkerPlots™. Following the growing samples instructional heuristic (Bakker, 2004; Ben-Zvi et al., 2012; Konold & Pollatsek, 2002), the students were gradually introduced to increasing sample sizes in order to support their reasoning about informal inference and sampling. The growing samples task design supported students’ informal inferential and sampling reasoning by observing aggregate fea- tures of distributions, identifying signals out of noise, accounting for the constraints of their inferences, and providing persuasive data-based arguments.

The second setting is probability-based learning environments (e.g., Konold et al., 2011; Pratt, 2000; Pratt, Johnston-Wilder, Ainley, & Mason, 2008), in which students are engaged in manipulating chance devices, such as coins, spinners and dice. Such settings emphasize how probability is used by statisticians in problem solving. For example, 10-year old students who worked with the Chance-Maker microworld were able to understand how empirical probability, theoretical probability and sample size are related to drawing valid inferences (Pratt, 2000). In a study by Konold, Harradine, and Kazak (2007) students built models using computer-based simulations, in order to create reasonable approximations of phenomena, ones that take into account signal and noise.

The first setting has a big potential to improve students’ use of data as evidence to draw conclusions: When students work on topics close to their world, which makes the task authentic and relevant, they can gain important insights into how statisti- cal tools can be used to argue, investigate, and communicate foundational statis- tical ideas. However, those settings might lack probabilistic considerations, which are important for understanding the relationship between samples and populations. The second setting might encourage and develop students’ probabilistic reasoning: When students manipulate chance devices, they can easily build probability mod- els of the expected distribution and observe simulation data of the model. Then, they can compare simulation data and empirical data to draw conclusions. This strategy of comparing simulated and empirical data introduces students to the logic of statistical inference and emphasizes the key role played by chance variation in statistical infer- ence. Probability settings, however, might lack aspects of authentic data exploration and might exclude the relevance of the situation.