Sostenibilidad de las zonas de pesca

The subjective viewpoint of probability is closely connected to Bayesian infer- ence, and is often equated to the degree of belief. Despite its common use in every- day reasoning, subjective estimates of probability have been neglected in the school mathematics curricula (Jones, Langrall, & Mooney, 2007). Hence, there has been a lack of interest in researching this aspect of probability. Given this gap in the lit- erature, and that Bayesian thinking and reasoning tend to be more intuitive than the frequentist perspective in statistical inference, a Bayesian-like approach was adopted to study 10–11 year old students’ reasoning about uncertainty in making informal in- ference in the context of random chance events. More specifically, the Chips Game Task described in this chapter is designed to understand students’ articulation of un- certainty as they evaluate the fairness of chance games by making an initial hypoth- esis and expressing their confidence in the likelihood of a particular game actually being fair (or not). Students then revise both their initial hypothesis and their level of confidence as new information is obtained from the data collected through physical experiments and computer simulations. The notion of uncertainty in this task relates to both students’ probability assessments and their personal degree of confidence in judging the fairness of the games. Moreover, a link to making informal inference is established by focusing on the probability estimates through experimental data in the task (Konold et al., 2011). Within this context, the following research questions are investigated: (1) How does the combination of using TinkerPlots™ and dialogic in- teractions in small groups promote students’ reasoning about uncertainty in making informal inferences about random events? (2) What are the dialogic mechanisms that help support students’ reasoning in the joint activities?

2.3 Literature and Background

Probability is the science of quantification of uncertainty in random processes. The approach used in this study to examine reasoning about uncertainty has roots in the historical development of probability. Hacking (1975) noted that the concept of prob- ability has historically had a dual characteristic: On the one hand is an epistemic notion of probability understood as degree of support by evidence, and on the other hand is a statistical notion of probability concerned with stable frequencies of occur- rences of certain outcomes during statistical processes like tossing a coin repeatedly many times. Similarly, Hald (2003) distinguished the two kinds of probability as: subjective probability “used for measuring the degree of belief in a proposition war- ranted by evidence” (p. 28) and derived from our imperfect knowledge, and objective probability “used for describing properties of random mechanisms or experiments, such as games of chance, and for describing chance events in populations, such as the chance of a male birth” (p. 28).

An implication of this dual nature of probability mentioned in Hacking (1975) and Hald (2003) is twofold. First, the epistemic and subjective notions of probability em- phasize personal probabilities relative to our background knowledge and beliefs, and thus enable us to represent learning from experience. Second, the statistical and ob- jective notions of probability are based on the symmetry in the mechanisms of chance

LITERATURE AND BACKGROUND 33

setups, such as equally likely outcomes or the stability of relative frequencies from experiments in the long run. Furthermore, Bernoulli distinguished between theoreti- cal “probabilities which can be calculated a priori (deductively, from considerations of symmetry) and [empirical probabilities] which can be calculated only a posteriori (inductively, from relative frequencies)” (Hald, 2003, p. 247). He then proved the first limit theorem of probability (“Bernoulli Theorem”) stating that the probability of a large difference between the empirical probability and the theoretical probabil- ity tends to zero as the number of trials increases (Stohl, 2005). The idea that the long-run relative frequency of an event should be very close to the probability of that event is an important corollary of this theorem.

For educational purposes, these different views of probability concepts suggest that when we deal with uncertainty in chance events, we draw upon a variety of evi- dence, such as personal knowledge or belief, empirical results, and theoretical knowl- edge. It is also implied that as one learns to appeal to evidence, symmetry of chance setups, and running simulations, one begins to link subjective, empirical, and theo- retical estimates of the probability. In particular, young students’ personal and expe- riential knowledge about the world plays an important role in their understanding of probability. Therefore, in this study the students started with formulating a hypoth- esis about the fairness of a chance game based on their personal knowledge/belief and updated it with the new data from a Bayesian viewpoint (where certainty level is changeable). The assumption was that the simulation of chance experiments would help students interpret probability of events as the relative frequency of outcomes in the long run (where certainty level increases as the number of trials get larger). Then students were expected to provide evidence for the observed results through theoret- ical analysis of chance events based on the sample space (where certainty level about their hypothesis is the highest).

2.3.1 Theoretical Background

Based on the idea that inference is an end product of inferential reasoning, Makar et al. (2011) recognized the need for understanding and supporting the informal in- ferential reasoning process that leads to informal statistical inference. Drawing upon their review of relevant literature and analysis of three sixth graders’ informal in- ferential reasoning, Makar et al. claim that informal statistical inference needs to be embedded in informal inferential reasoning, “nurtured by statistical knowledge, knowledge about the problem context, useful norms and habits developed over time, and supported by an inquiry-based environment (tasks, tools, scaffolds)” (p. 171). Within the aim of this study, the design of the learning environment suggested by Makar et al. is seen as particularly relevant to support young students’ reasoning about uncertainty. In this research, the design element involves relevant tasks, appro- priate computer tools, and talk as scaffolding in peer group interaction. Then these are used to understand how to promote students’ emerging reasoning in making in- formal inferences in the context of chance events in the current research data.