30 HOW CONFIDENT ARE YOU?

The term ‘uncertainty’ is unquestionably fraught with misinterpretation—especially by non-scientists. I’d prefer the phrase ‘how confident am I?’, the reciprocal of un- certainty. (Gibbs et al., 2013, p. 6).

2.1 Overview

The concept of inference in statistics refers to “drawing conclusions about popu- lations or processes based on sample data” (Zieffler, Garfield, delMas, & Reading, 2008, p. 40). Formal statistical inference that students encounter in advanced levels includes certain techniques to draw conclusions from data, such as estimation and hypothesis testing. Given the problematic nature of understanding these formal ideas in the context of making inferences by older students (Zieffler et al., 2008), there has been an attempt to begin to develop the foundation for these ideas early on, working with young students to use statistical inference informally (Makar & Rubin, 2009).

Informal statistical inference can be viewed as a way of making informal conclu- sions (using statistical and probabilistic knowledge) about a population or process from which the data come. The notion of uncertainty plays an important role in mak- ing such judgments using data. According to the framework described by Makar and Rubin (2009) informal statistical inference includes making a generalization be- yond data, using data as evidence, and using probabilistic language in describing the generalization. So, at the heart of statistical inference is “the process of mak- ing probabilistic generalizations from (evidenced with) data that extend beyond the data collected” (Makar & Rubin, 2009, p. 83), which inherently involves features of uncertainty. Thus, developing the language and understanding of probability in the sense of “statistical tendency, and/or level of confidence or uncertainty in a predic- tion” (Makar & Rubin, 2009, p. 87) is crucial in reasoning and making decisions based on uncertain data.

This chapter presents how young students articulate uncertainty in the context of fairness in games of chance as they test their hypotheses and update their level of confidence on the basis of the data collected both through physical experiments and computer simulations in TinkerPlots™ (Konold & Miller, 2011). Within this context, the main goal of this chapter is to examine in what ways the combination of using TinkerPlots™ and peer-to-peer dialogic interactions supports students’ reason- ing about uncertainty in making informal inferences about chance situations through a Bayesian-like approach.

The present study is part of a larger design-based research project which has the overall aim of investigating how to develop young students’ conceptual understand- ing of key ideas in statistics and probability in the context of informal statistical inference through the mediating roles of technological tools and students’ talk. The study was conducted during a 9-hour mathematics enrichment program with six pri- mary school students, ages 10–11, in Exeter, UK. The teaching experiment discussed in this paper is one of the initial iterations of the design-based study.

PROBLEM 31 2.2 Problem

In recognition of the role that working with data and making judgments under uncer- tainty play in daily life and in various fields, statistics and probability have become part of the mainstream school mathematics curricula in primary grades for more than two decades (e.g., Department for Education and Employment, 1999; National Coun- cil of Teachers of Mathematics, 2000). The emergence of teaching statistics within school mathematics has led Exploratory Data Analysis (EDA; Tukey, 1977), which involves real data analysis through looking for and describing patterns or trends in data, to become the basis for the data-handling strand in the pre-tertiary mathematics curricula (Biehler, 1986; Shaughnessy, Garfield, & Greer, 1996). In the meantime, as Ainley and Pratt (2001) point out, EDA provided an opportunity for open-ended data exploration by students, using basic concepts of descriptive statistics, while “fore- grounding data and making the mathematical model, probability, subsidiary.” (p. 7). This has created an artificial separation between data and chance topics, both relevant to uncertainty, in both research and instruction when in fact they are closely related (see Konold & Kazak, 2008). So, one way to build a strong connection between data and topics related to chance is to encourage students to make informal conclusions based on data (Moore, 1990). As it can also be seen in the development of themes of the Statistical Reasoning, Thinking, and Literacy (SRTL) forums in the past decade, the focus has recently shifted towards informal statistical inference, which is an im- portant component of statistical thinking, at all grade levels (K–College). A special issue of the journal Mathematical Thinking and Learning, for instance, was devoted to the papers presented at the SRTL-6 forum with a focus on the role of context in developing students’ reasoning in informal statistical inference (Makar & Ben-Zvi, 2011).

Since informal statistical inference is considered an end product, the underlying reasoning process leading to that is called informal inferential reasoning (Makar, Bakker, & Ben-Zvi, 2011). According to Fisherian inference, this reasoning process is based on the concept of likelihood which entails: (1) formulating a hypothesis (i.e., null-hypothesis or model), (2) making a judgment that if the hypothesis or model were true, the observed data would have been very unlikely (i.e., intuitively comput- ing a p-value), and (3) rejecting the initial hypothesis or model based on the con- ditional probability that the observed data would occur by chance (Rossman, 2008). Rossman argues that students do not seem to spontaneously apply this common form of reasoning when making statistical inferences. An alternative approach to statisti- cal inference, which is distinct from that of Fisher, is based on a Bayesian frame- work. Bayesian inference uses a subjectivist interpretation of probability (Rossman, 2008). In this approach, one would start with a priori probabilities associated with a hypothesis or model based on a belief or previous data, then update these probabil- ities as new information or data is obtained. It is argued that this form of deductive reasoning seems to be more intuitive than that of Fisher (Albert, 2002; Rossman, 2008). Because of this, there has been a tendency to shift the focus of inference in undergraduate-level statistics courses to a Bayesian framework (Albert, 2002; D´ıaz, 2010).