34 HOW CONFIDENT ARE YOU?

Designing Relevant Tasks. The research indicates that older students have diffi- culties in understanding the concepts and reasoning related to the common formal statistical inference methods (Zieffler et al., 2008) and this formal inferential reason- ing process does not come naturally to students (Rossman, 2008). It has also been argued that a Bayesian approach to making a statistical inference is more intuitive and better reflects the commonsense thinking about uncertainty in daily life (Albert, 2002). Even young students use probabilistic language (e.g., more likely, possible, impossible, always, and rare) to express different levels of uncertainty. Subjective probability, to which Bayesian inference is closely related, is a way of assigning quantities between 0 and 1 to these different levels of uncertainty with beliefs chang- ing based on new evidence (Albert, 2002). The findings of Huber and Huber (1987) suggest that even young children can use personal knowledge or beliefs in the tasks involving ordinal comparison of subjective probabilities about given events in the contexts of sports and gambling. In addition, it is pointed out that children’s subjec- tive probability evaluations of events tend to show more stability in the gambling task because of the availability of the objective probabilities, (i.e., the areas in the spinner device used in the task; Huber & Huber, 1987). This suggests that young students’ use of subjective probability in making informal inferences may be supported by enabling them to estimate the likelihood of events from other sources as well (e.g., the symmetry in the mechanism of chance setup and relative frequencies). To do so, we need a task which allows students to use subjective, empirical, and theoretical estimates of the probability.

The Bayesian viewpoint seems to be often consistent with people’s way of de- veloping intuitions based on learning from their experiences and revising their be- liefs as new information is obtained (Falk & Konold, 1992). Furthermore, Hawkins and Kapadia (1984) suggest that subjective probability is utilized to complement the traditional classical and frequentist approaches in teaching probability. Therefore, to support students’ informal inferential reasoning, a task was designed using Bayesian- like thinking to develop informal inference where students were expected to state their initial hypothesis (prediction) about the fairness of chance games, provide an explanation, and rate their level of confidence in their hypothesis on a 0%–100% scale. After generating their hypotheses, students were asked to both physically play the game and simulate results from the game using TinkerPlots™ to gather data to support or revise their initial hypothesis and to update their level of confidence. Af- terwards, students used the possible outcomes for the combined events to provide a theoretical model for data. They were then expected to be able to explain using this theoretical model how their previously collected empirical results could be used to support their final hypothesis. Through this process, the aim was to highlight the inherent relationship between probability and informal statistical inference in the context of chance games.

Using Appropriate Computer Tools. Several studies have investigated young stu- dents’ reasoning processes relevant to inference through technology-enhanced tasks. Pratt, Johnston-Wilder, Ainley, and Mason (2008) found that when guessing the hid- den numbers in the sides of a die on a computer simulation tool, called Inference

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Maker, 10–11-year old students tended not to focus on emergent, aggregate charac- teristics of data. Accordingly, students failed to see the relevance of larger samples in making inferences with a greater confidence, which then would lead to the idea of the Law of Large Numbers. Others reported on effective use of technological tools, in particular TinkerPlots™, to support these understandings with middle school stu- dents (Ben-Zvi, 2006; Fitzallen & Watson, 2010). For instance, Fitzallen and Watson (2010) reported that when using TinkerPlots™, students (ages 10–12) generated dif- ferent kinds of plots that appeared meaningful to them and used these effectively in making their conclusions from data. In their study, the software also facilitated students’ thinking processes that involved moving back and forth between making hypotheses and constructing plots in making sense of the data. Additionally, Ben-Zvi (2006) indicated that students used TinkerPlots™ not only as a representation tool, but also as an argumentation tool in expressing their ideas to others.

Other studies (Konold, Harradine, & Kazak, 2007; Konold & Kazak, 2008) con- ducted using the development version of TinkerPlots™ also revealed how the new probability simulation feature could support middle school students’ development of an integrated set of statistical and probabilistic ideas. Findings from Konold and Kazak (2008) suggested that the TinkerPlots™ environment facilitated students’ vi- sual reasoning via dynamic graphs where the results accumulated as they were gen- erated by the Sampler (a tool within TinkerPlots™ to model probabilistic processes). The combination of observing the simulation data from multiple trials and sketching only the overall shape and relative heights of stacks seen in the plot enabled students to explore the fit between the expected distribution based on the sample space and the empirical data. These observations led students to perceive ‘data as signal and noise’, which is a key idea in dealing with situations involving uncertainty (Konold & Pollatsek, 2002). More recent studies presented in the SRTL-8 forum (Ainley, Ari- dor, Ben-Zvi, Braham, & Pratt, 2013; Braham, Ben-Zvi, & Aridor, 2013; Harradine & Konold, 2013) further documented the benefits of using TinkerPlots™’s simula- tion and modeling features in promoting young students’ statistical understanding and reasoning about uncertainty in the context of informal inference.

Using Talk as Scaffolding in Peer Group Interaction. It is suggested that en- couraging talk in a mathematics classroom helps students reflect on their thinking, explore, and form new understandings (Wickham, 2008). Mercer (1996) identified three different types of talk when students engaged in small group work in class- rooms: (1) disputational talk in which a lot of disagreement between children, indi- vidualized decision making and a competitive, rather than cooperative, relationship can be seen; (2) cumulative talk in which children tend to simply build on what the other has said in a shared, supportive but an uncritical way; and (3) exploratory talk in which children listen to each other actively, ask questions, challenge ideas in a critical but constructive way, and give explicit reasons for challenges. Several stud- ies using the Thinking Together approach, a dialogue-based pedagogy to develop children’s collective thinking and learning (Dawes, Mercer, & Wegerif, 2000), found exploratory talk effective in promoting young students’ mathematical reasoning and