6.3 Literature and Background
Research on informal statistical reasoning has been carried out with students from different levels of the educational system (Arnold et al., 2013; Bakker, Ben-Zvi, Makar, & Kurvers, 2013; Ben-Zvi et al., 2012; Garfield & Ben-Zvi, 2008; Manor, Ben-Zvi, & Aridor, 2013; Metz, 1998; Pfannkuch, 2011). However, informal sta- tistical reasoning seems to be a topic of little interest within teachers’ professional development programs. This lack of interest could be grounded in the belief that teachers are already able to guide experiences on informal statistical inference in their teaching.
In the field of cognitive development, the work of Piaget and Inhelder (1975) has strongly influenced the research on uncertainty. They presented children from 5 to 14 years old with a sequence of different physical tasks. The results of their extended study showed that ideas about uncertainty begin to appear in children around seven years old, who prior to this age assume a deterministic causality. In contrast, other researchers (Fay & Klahr, 1996; Kuzmak & Gelman, 1986) found that preschool age children have an understanding about uncertainty.
The literature on decision-making has also contributed to the understanding of people’s reasoning about uncertainty. This field of research has documented the dif- ficulties adults face making probabilistic judgments. Usually, a typical adult fails to make any probabilistic distinction between determinacy and indeterminacy (Konold, 1991) and most of the time assigns deterministic behaviors to phenomena that are regulated by chance (Kahneman & Tversky, 1982). Consequently, the adults fail to recognize the extent to which chance contributes to what they experience about the world.
Educational studies have provided an interesting view on students’ reasoning. A particular study focused on the development of students’ expressions of uncer- tainty in reasoning from samples (Ben-Zvi et al., 2012). The researchers were able to show the evolution of fifth graders’ probabilistic language. Another influential study (Makar & Rubin, 2009) developed a theoretical framework for how people reason about informal statistical inference. In the proposed framework, the authors highlight three principles that appear to be essential in informal statistical inference: Generalization, use of data as evidence and use of probabilistic language. The study highlighted that the correct use of probabilistic language is important to avoid deter- ministic claims. However, in the classes the authors studied, little attention was paid to probabilistic language.
Many research studies have documented the difficulties encountered when learn- ing about statistical inference. This literature contains studies that disclose different errors and misuses in reasoning about inference (e.g., Watson, 2002). It also focuses on the exploration of how to develop students’ reasoning about statistical inference (Ben-Zvi et al., 2012; Franklin et al., 2007; Garfield & Ben-Zvi, 2008; Pfannkuch, 2005; Pfannkuch & Wild, 2000; Wild & Pfannkuch, 1999). One method of develop- ing students’ reasoning about uncertainty that has shown promise in this literature is the focus on informal statistical inference.
LITERATURE AND BACKGROUND 167 6.3.1 Informal Statistical Inference
Informal statistical inference is described in the literature as the process of making probabilistic generalizations from a sample to a population without running a formal statistical test. It is the act of looking beyond the data to cases outside of the sample at hand (Makar & Rubin, 2009) and the cognitive activity involved in drawing con- clusions or making predictions about “some wider universe” (Garfield & Ben-Zvi, 2008). Informal statistical inference takes into account multiple dimensions such as data, distributions, measurements, representations, and statistical models.
In making informal statistical inferences, language is essential since it articulates the level of confidence or uncertainty in the prediction (Ben-Zvi et al., 2012; Makar & Rubin, 2009). One of the tools in statistical inference is the information gathered from samples. However, the results from a sample might lead the learner to think that the sample reflects the behavior of the population (Rubin, Bruce, & Tenney, 1991). This is known as over-reliance on sample representativeness.
In contrast, the learner might doubt the information given from the sample be- cause of the variability intrinsic in every sample. The learner might conclude that the sample does not give relevant information about the population and attribute the results exclusively to chance. This is known as over-reliance on sample variability. These two methods of judging results from a sample reflect either a deterministic or a relativistic view of the learner that influences the language used in the informal inference. According to Rubin and colleagues (1991), the information from a sample should not tell everything or nothing, but something about the subjacent population. A possible educational approach that might help to support the development of in- formal statistical inference is the statistical investigations inspired in the “investiga- tive cycle” (Wild & Pfannkuch, 1999). In this approach, the participants are involved in the solution of a problem that takes them through the stages of the investigation process (questioning, planning, gathering data, analysis, and interpretation). The in- vestigative cycle is also highlighted in the Guidelines for Assessment and Instruc- tion in Statistics Education (Franklin et al., 2007). This document, endorsed by the American Statistical Association, emphasizes that the statistical question is a very important beginning of the investigation.
At the Eighth International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL-8), the results from several different research studies related to people’s reasoning about uncertainty were presented. For example, there were stud- ies looking at new ideas of uncertainty emerging in students during an introductory statistics course focused on bootstrapping and randomization methods (Arnold et al., 2013). Others were interested in students’ web of actions and reasons involved in reducing uncertainty in solving a real problem (Bakker et al., 2013). While oth- ers were interested in confronting how confident students were with their inferences after working with sampling distributions (Manor et al., 2013). Different ways to approach the study of uncertainty lead to different ideas about uncertainty. Whereas the research conducted by Arnold et al. (2013) and (Bakker et al., 2013) was focused on aspects of uncertainty related to study design and hypothesis testing, the research