## 1. Introduction

This paper concerns the staged development of the big ideas of statistical inference over a period of years. It was prompted by the authors’ need to make inferential ideas accessible to New Zealand school students aged approximately 14–17 years but much of its discussion is also relevant to adult education and introductory statistics courses at colleges and universities. The audiences that we most wish to engage include academic and professional statisticians. The other desired audiences for this paper are researchers in statistics education and teachers. Our biggest difficulty in trying to engage academic and professional statisticians on topics such as this is a common attitude that says, ‘We don't care about school stuff’. We shall confront this before proceeding further.

We have often heard academic statisticians complain that statistics at school level is taught poorly, that most mathematics teachers have little or no training in statistics and that the statistics that is taught at school turns students off. ‘What's the point? It all has to be redone from scratch at university anyway. Schools should just concentrate on laying solid mathematical foundations that we can build on at university.’ There are good reasons why academic and professional statisticians should care deeply about building more and better statistics into school curricula. Change is afoot and major curriculum developments are under way in many countries. We are not talking about continuing ‘business as usual’. What statisticians should be heavily involved in is reconceiving what a more exciting and valuable school level statistics course could look like.

Why should statisticians care? First, there is the inherent but underexploited value of our product. There is a treasury of life skills lessons within statistics of value in the future lives of students regardless of what they end up doing. Second, there are dangers for society in statisticians failing to engage. Can any statistician really believe that it is desirable that society and its decision makers be made up of people whose minds have been conditioned by years of relentless determinism and who have no facility in stochastic thinking and no appreciation of its benefits? Third, there is the future of the discipline. Graduate programmes in statistics have traditionally relied on the conversion of people who started in mathematics undergraduate programmes. The declining numbers entering mathematics programmes in many countries mean that this strategy can no longer be relied on. We must build interest in statistics before students decide what to major in at university. In most jurisdictions that means that we must build interest in statistics while students are still at school. If we do not do this, then we must somehow grab the attention of someone who has no awareness of what statistics has to offer, and who is probably already planning to go into another area, and then reverse that decision in our favour. We have not shown any particular talent for such conversions in the past, so why would we bet the future of our discipline on such an implausible long shot?

Technology provides exciting possibilities for changing the landscape of statistics education in schools in ways that could make it unrecognizable. The inspirational ‘Technology, entertainment, design’ lectures of Hans Rosling (which are available from http://www.ted.com/), in which complex stories involving multi-dimensional data were made accessible to a general audience by using clever graphics, made this abundantly clear. In the same vein, work done in the SMART Centre of Durham University (Ridgway *et al.*, 2007a,b) shows that, with suitable visualization tools, ordinary teenagers can uncover and understand patterns involving interactions in four- or five-dimensional data. For statistics education, technology is the ultimate game changer. Its biggest pedagogical implications come from the fact that it allows us to conceptualize, in ways that were previously unavailable, potentially providing access to concepts at much earlier stages of development. With creative approaches, school level statistics can become much more ambitious, exciting and useful. Determining what a changed landscape could look like will, however, require the creative engagement of both academia and the profession. Although there are kernels of truth behind the objections some academic statisticians raise about school level statistics, these truths are simply evidence that there are difficulties which will require a large amount of creativity to overcome, that we need more and deeper engagement from more and better thinkers. The difficulties should not be taken as a justification for abandoning the battle field.

In a paper read before the Royal Statistical Society, Holmes (2003) brilliantly chronicled the history of statistics teaching in English schools and extracted important lessons to be learned from it. Holmes's main interest was in the journey towards statistics as a *practical subject* taught in *practical contexts* for *practical use*, to use some of his recurring phrases. This is statistics taught to enable students to understand better the real world that they live in, and sooner rather than later, in contrast with a rarefied enterprise that simply lays mathematical building blocks for future use. A prescient report of a Royal Statistical Society committee chaired by E. S. Pearson (Royal Statistical Society, 1952) was a major early milestone on this journey. Unfortunately, it had to wait nearly 30 years for any serious implementation via the Schools Council Project on Statistical Education (1975–1980), which was led by Peter Holmes himself. The Schools Council Project in turn helped to inform the American Statistical Association's influential Quantitative Literacy Project of the 1980s, which was a watershed for parallel developments in the USA (R. L. Scheaffer, personal communication; see also Scheaffer (1990)). The report of the Statistics Focus Group sponsored by the Mathematical Association of America's Curriculum Action Project in 1991 was similarly a watershed for related developments for introductory courses at universities. The major thrust of the Focus Group's recommendations have survived to form the basis of the six recommendations that were fleshed out in the American Statistical Association's 2005 ‘Guidelines for assessment and instruction in statistics education’ (GAISE) college report, namely: emphasize statistical literacy and develop statistical thinking; use real data; stress conceptual understanding rather than mere knowledge of procedures; foster active learning in the classroom; use technology for developing conceptual understanding and analysing data; and use assessments to improve and evaluate student learning. These sentiments also pervade the GAISE pre-K-12 report (Franklin *et al.*, 2007). Both reports are available from http://www.amstat.org/education/gaise/. We see our developments as next steps on this same journey.

Current realities fall far short of the worthy goals that were developed in the Royal Statistical Society and American Statistical Association. School level statistics for the bulk of students has suffered and stagnated for many years under a computational mentality pejoratively termed ‘meanmedianmode’ and the ‘construct a graph’ syndrome (Friel *et al.*, 2006). This has further been compounded by ‘univariatitis’ (Shaughnessy, 1997; Wild, 2006) and a focus on the construction of the tools of statistics rather than statistical reasoning processes resulting in a discipline that is perceived by many students and teachers as boring with little intellectual substance (Ridgway *et al.*, 2007a). Often descriptive statistics has been the only diet for students up to the penultimate year of high school, to be followed by an attempt to force-feed statistical inference, with its mathematical underpinnings, concepts and reasoning in the final year. It has not been entirely this way at all times in all places, but this has been the general tendency apart from ‘an occasional creative oasis in a largely empty desert’ (adapting Scheaffer (2002)).

The increased use of real data addressing interesting problems and multivariate data sets that permit students themselves to come up with interesting differences and other relationships to investigate are important new trends. The fostering of student engagement in data exploration as a ‘data detective’ (exploratory data analysis) is a hugely positive development in statistics education that partially obviates the problems above. Within it, however, lie the seeds of a new problem. When investigating interesting questions, relationships seen in data lead naturally to wanting to draw conclusions that apply to a universe beyond the data. Put more concisely, data addressing *motivationally compelling* questions beg inferences. Preventing inferential extrapolation makes the whole statistical exercise seem pointless. But, although good data and good questions make students want to make inferential claims, they currently have no rational bases on which to do so until they finally encounter formal inference. Moreover, the research on ‘informal inference’ that was reviewed in Wild *et al.* (2010) shows that students tend to grasp at it in incoherent ways. When the students do make claims they, and often their teachers, have no clear idea about whether they concern the data or a parent population. Additionally, research strongly suggests (e.g. Chance *et al.* (2004)) that large numbers of students fail to comprehend formal statistical inference when they do meet it at either school or introductory university level, and that they will continue to do so unless a much better job is done of laying essential conceptual foundations over a period of years before any attempt to teach formal inference is made. Otherwise there are simply too many ideas to be comprehended and interlinked all at once.

Work on informal inference has been going on in the statistics education research community, as a result of the statistical reasoning, thinking and literacy series of biennial international research forums that were initiated by Joan Garfield and Dani Ben-Zvi in 1999. Initially the forums addressed different types of statistical reasoning but the researchers at the 2005 forum came to a consensus that students should be learning to make inferences, initially informally. Consequently, the fifth forum in 2007 was focused on informal statistical inference (see, for example, the articles by Pratt and Ainley, Rossman, Pratt *et al.*, Zieffler *et al.*, Watson, Paparistodemou and Meletiou-Mavrotheris, Beyth-Marom *et al.* and Bakker *et al.* in volume 7, number 2, 2008, of the *Statistics Education Research Journal* (http://www.stat.auckland.ac.nz/serj) and by Makar and Rubin in volume 8, number 1, 2009). Konold and Kazak (2008) commented that the recognition that students need deeper understandings of inference is a move towards an acceptance that chance or sampling behaviour must be addressed. It is also noteworthy in view of the developments of this paper that the 2011 forum is to focus on new approaches to developing reasoning about samples and sampling in the context of informal statistical inference. For a review of this literature and some of its antecedents, see Wild *et al.* (2010).

Our own challenge, which has been made urgent by the demands of the imminent roll-out of the new statistics curriculum in New Zealand, has been to devise simpler versions of statistical inference in a way that lays solid conceptual foundations on which to build more formal inference in the longer term while giving students simple inferential tools, with reasonable operating properties, that they can use immediately. Moreover, we wanted the concepts to be built in a staged manner over several years so that there is time for them to be revisited several times to begin to bed in properly.

The remainder of this paper does not attempt to address broad aspects of statistics, of what should be taught and when, nor to illustrate exploratory data analysis with real data. It concerns educational experiences that *specifically target* statistical inference and, within that context, the development of integrated conceptual schema and tools that will assist students in making inferences when they are exploring real and interesting data.