Mejora de la competitividad en el sector pesquero

The activities of this study were designed according to the “Integrated Modeling Approach” (described below) to help students in understanding the relationship be- tween sample and population. This approach intends to integrate these two types of settings. In other words, IMA aims to support students’ IIR on authentic data while taking into account probabilistic considerations.

The topic of informal inferential reasoning is not yet sufficiently examined in the literature, and specifically lacks studies on the combination of probabilistic reasoning and making ISIs in authentic contexts. This is the focus of the current study.

3.4 Method

This case study focuses on the question: How can students’ articulations of uncer- tainty emerge while informally exploring sampling distributions in the integrated modeling approach? In order to address this question, we closely followed the ar- ticulations of a pair of seventh grade students (age 13) as they examined a sampling distribution using the sampler in TinkerPlots™. This study is part of the longitu- dinal design and research Connections Project (2005–2015; Gil & Ben-Zvi, 2011) aiming to develop and study children’s statistical reasoning in an inquiry-based and technology-enhanced environment for learning statistics in grades 4–9.

3.4.1 Participants

This study involved a pair of students (grade 7, aged 13), Shay and Liron, in a private school in northern Israel. We selected them since they had high communication and thinking skills which can provide a window to their statistical reasoning. They had already participated in two Connections Project experiments. In fifth grade (age 11, 2010), they collected and investigated data about their peers using the first version of TinkerPlots™. Following the growing samples heuristic (Ben-Zvi et al., 2012), the students were introduced gradually to samples of increasing sizes, in order to support their reasoning about ISI and sampling. In sixth grade (age 12, 2011), they engaged in both real world data investigations and model-based investigations using Tinker- Plots™ chance devices in order to support their reasoning about ISI and sampling. The first co-author observed and guided the students during eight sessions (about 80 minutes each) over a four-week period.

3.4.2 The “Integrated Modeling Approach” (IMA)

The IMA was developed to guide the design and analysis of experimental tasks (as part of Manor’s Ph.D. study) to help students learn about the relationship between sample and population. It is comprised of data and model worlds. In the data world, students collect a real sample, frequently through a random sampling process, in or- der to study a particular phenomenon in the population. Students choose a research theme, pose questions, select attributes, collect and analyze data, make informal in- ferences about a population and express their level of confidence in the data. Students

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also begin to model real world phenomena using statistics by moving from real world questions to statistical ones. However, they do not necessarily account for probabilis- tic considerations (e.g., the chance variability that stems from the random sampling process). In the model world, students build a model (a probability distribution) of an explored (hypothetical) population and produce data of random samples from this model. Hence, they pay attention to a model and to the random process that produces the outcomes of samples from this model. Due to randomness, the details vary from sample to sample, but the variability is controlled. That is, given a certain distribution of the population, the likelihood of certain results can be estimated.

In the IMA learning trajectory, students iteratively create connections between the two worlds by working on the same problem context in both worlds. They begin their exploration in the data world (the first dotted trajectory in Figure3.1) by choosing a meaningful research theme, formulating a question, making an initial conjecture based on their contextual knowledge, building a questionnaire, planning how to draw a sample and collecting real small sample data (represented by a small dotted circle in Figure3.1). While exploring the sample data they start making sense of it and search for typical characteristics or trends in the data to make ISIs. In the end of this part, they make a second version of their conjecture (the second triangle in Figure

3.1) about the population based both on their contextual knowledge and the sample data results.

As a motivation to move to the model world, the students are asked to express their level of confidence in the sample data that they had collected in relation to the second version of their conjecture and to consider what is the minimal sample size needed to draw reliable inferences about the population with a reasonable confidence level. At this point, the students are first introduced to the model world. They are told: “Imagine you were almighty and could know what characterizes the population. What do you think a random sample from this population would look like? Could you find in this imaginary world the minimal sample size that could represent the population well?”

To do that, the students begin their exploration in the model world (the first lined trajectory in Figure3.1). They build in TinkerPlots™ a model of the hypothetical population according to their second version of their conjecture and then they sim- ulate sample data from this model. They explore the variability between simulated samples, compare them to the model and gradually enlarge sample size to reduce the variability between samples. Agreeing on the minimal sample size by which they can draw conclusions with confidence, they move again to the data world (the sec- ond dotted trajectory in Figure3.1) to collect real sample data of that size. In the data world, they collect more sample data and formulate the third version of their conjecture (the third triangle in Figure3.1) about the population based on both their contextual knowledge and the second sample data. They also explain their level of confidence in the data based on what they have learned in the model world.

The continuous trajectory in Figure3.1describes integrative transitions between the worlds which might occur to improve the model in relation to different issues, like the dependency between attributes in the model, the shape of distributions of attributes in the model. Our hypothesis is that the IMA can support students’ de-