Statistics in most countries in the world, including South Africa, is not a separate subject in school curricula, but is included as a content area in the mathematics curriculum. Statistical thinking however differs from mathematical thinking. Cobb and Moore (1997) explain that this difference results from the focus in statistics on variability and the all-important role of context:

Statistics requires a different kind of thinking, because data are not just numbers, they are numbers with a context. In mathematics, context obscures structure. In data analysis, context provides meaning. (p. 801)

Variability can be defined as the quality of an entity to vary, including under uncertainty. Bakker (2004a) elaborates on the relationship between variability and uncertainty: ‘Uncertainty and variability are closely related: because there is variability, we live in uncertainty, and because not everything is determined or certain, there is variability’ (p. 14). The interchangeable use of the terms variation and variability often causes confusion. Reading and Shaughnessy (2004) elucidate the difference between the two terms by defining variability as ‘the [varying] characteristic of the entity’ and variation as ‘the describing or measuring of that characteristic’ (p. 202). For instance, when learners write a test one can expect their marks to vary without being certain about how the marks will vary (the degree of dispersion of the marks). In order to describe the degree of dispersion, one needs some measure of how widely the marks vary in relation to, for example, the mean of the marks.

Variability is omnipresent in the world around us and affects all facets of our lives: ‘It is variation that makes the results of actions unpredictable, that makes questions of cause and effect difficult to resolve, that makes it hard to uncover mechanisms’ (Wild & Pfannkuch, 1999, p. 235). Variability is fundamental to statistical thinking and reasoning and the presence of variability in the world necessitated the development of statistical methods to make sense of data (Cobb & Moore, 1997; Franklin et al., 2005; Moore, 1990; Shaughnessy, 2007; Watson, 2006; Wild & Pfannkuch, 1999). It is self-evident that the development of statistical thinking and reasoning should be one of the major goals in statistics education and should take into account the omnipresence of variability in data (Franklin et al., 2005).

The South African school mathematics curriculum requires learners to be able to ‘predict with reasons the relative frequency of the possible outcomes for a series of trials based on probability’ (Department of Basic Education, 2011c, p. 36). Wessels and Nieuwoudt (2011) point out that the wording of the learning outcome and assessment standard quoted does not explicitly state the kind of skills needed for the statistical thinking and reasoning necessary in the statistical problem solving process. The Curriculum and Assessment Policy Statement (CAPS) documents for mathematics in the Senior Phase and Further Education and Training Phase do not use the words variation or variability in the context of data handling and probability (Department of Basic Education, 2011a, 2011b, 2011c), nor do they refer to the essence thereof in statistics. This is a serious shortcoming as variability is fundamental to and hence plays an all-important role in the statistical process (Franklin et al., 2005). Watson, Kelly, Callingham and Shaughnessy (2003, p. 1) state that ‘statistics requires variation for its existence’. Data handling instruction without the inclusion of activities to develop learners’ understanding of the role of variability is therefore questionable, and without doubt undesirable. A strong case could therefore be made for the pertinent inclusion of the fundamental statistical idea of variability, together with certain ideas of describing and measuring variation, in the South African mathematics curriculum.

Learners specifically need to develop the ability to reason about variability in samples and acquire a sense for expected variability to be able to predict results: ‘When random selection is used, differences between samples will be due to chance. Understanding this chance variation is what leads to the predictability of results’ (Franklin et al., 2005, p. 21). Exposure to appropriate activities to develop this understanding should therefore be frequent and well planned:

[T]o improve students’ feel for the expected variability in a sampling situation, students need considerable hands-on experience in first predicting the results of samples, and then drawing actual samples, graphing the results, comparing their predictions to the actual data, and discussing observed variability n the distribution. (Shaughnessy, Ciancetta & Canada, 2004, p. 184)

Research on teachers’ statistical reasoning has up to now received little attention in South Africa (Wessels & Nieuwoudt, 2011). Many teachers lack a sound background in and proper understanding of statistics in general, and the ideas of variability and variance in particular; even if they did take statistics courses as part of their teacher education, such courses traditionally tend to be procedurally rather than conceptually inclined (Nieuwoudt & Golightly, 2006, p. 109), leaving many teachers lacking the critical proficiency to apply their statistical knowledge in practical settings (Wessels & Nieuwoudt, 2011). As a result of such ‘incomplete’ statistical learning experience, teachers do not feel confident about their own statistical reasoning (Shaughnessy, 2007) and lack pedagogical statistical knowledge. We concur that teachers have to understand how different fundamental statistical concepts are scaffolded in a child’s mind to be able to prepare statistically literate learners who will be able to be critical consumers of the data they are deluged with every day (Wessels, 2009, p. 4). Opportunities should therefore purposely be created for teachers to develop their own reasoning about variability, and specifically about variability in sampling situations, to equip them better for the development of learners’ awareness of variability present in samples or data in general.

FIGURE 1: The Lollie Task.

Different versions of the task had been administered to thousands of learners from Grade 3–12, mainly in the United States and Australia (Kelly & Watson, 2002; Reading & Shaughnessy, 2000; Shaughnessy et al., 2004; Shaughnessy, 2007; Torok & Watson, 2000).

Reading and Shaughnessy (2000, 2004) categorised learner responses in interview tasks according to measures of centre as high or low, and measures of spread as wide, narrow and reasonable. Some learners predicted, for the expected numbers of reds pulled from the mix of 20 yellow, 30 blue and 50 red candies in the bowl, all low numbers (all below 5) or all high numbers (all above 5), whereas others predicted a wide (range ≥ 8) or a narrow (≤ 1) range of numbers. Learners tend to give more reasonable predictions if they get the chance to draw their own samples from the candy mix (Shaughnessy, 2007).

A ‘demonstrated questionnaire’ of the Candy Task was developed for use in South Africa with Grade 6, 8 and 10 learners (Reading, Wessels & Wessels, 2005).

Based on the System of Observed Learning Outcomes (SOLO) taxonomy (Biggs & Collis, 1982), a conceptual model for the categorisation of reasoning in repeated sampling tasks emerged in the research on variability in sampling situations (Canada, 2004; Kelly & Watson, 2002; Shaughnessy, 2007; Shaughnessy et al., 2004). The SOLO taxonomy is a neo-Piagetian framework for the analysis of the level of sophistication or complexity of a response on a specific task (Biggs & Collis, 1982, 1991). According to this model, learner responses in the Candy Bowl Task display four distinctive patterns of reasoning, following a progression from iconic, to additive, to proportional and finally to distributional reasoning (Shaughnessy, 2007). Iconic reasoning, such as relating personal stories and using physical circumstances, is usually evident in younger learners’ responses (Kelly & Watson, 2002). Examples of iconic reasoning do not refer to the actual contents of the candy bowl or the proportions of the candy mix in it. Such responses might refer to luck: ‘Maybe they are lucky and will get all the reds’ or the physical act of pulling out the candies: ‘They might get more reds because their hand could find them’ (Shaughnessy, 2007). Additive responses are characterised by reasoning where no acknowledgement is given to the role of proportions in the mixture; reasoning is just about absolute numbers or frequencies of reds in the candy mix. Implicit proportional reasoning focuses on ratio, percentage or probability of reds whilst referring back to the original composition of the mixture. Explicit proportional reasoning involves reasoning about sample proportions, population proportions, probabilities or percentages. Finally, distributional responses give evidence of reasoning about centres as well as the variation around the centres. Shaughnessy et al. (2004) categorise responses of secondary school learners in repeated sampling tasks in a chance setting into only three broad groups: additive reasoning, explicit and implicit proportional reasoning, and distributional reasoning. These authors regard proportional reasoning as the cornerstone of statistical inference and call for more opportunities for learners to improve their proportional reasoning skills. They furthermore emphasise that

the power of proportional reasoning in statistical situations needs to be identified much more explicitly in order for our students to evoke the connections of proportional thinking to statistical settings. (p. 4)

Grade 1–9 teachers and were also coupled with research projects on the statistical thinking and reasoning of mathematics teachers (Burrill, Franklin, Godbold & Young, 2003; Canada, 2004; Friel & Bright, 1998; Makar & Confrey, 2002; Shaughnessy, Barrett, Billstein, Kranendonk & Peck, 2004; Shaughnessy & Chance, 2005; Watson, 2006).

In South Africa however, professional development in statistics was mostly aimed at Grade 10–12 teachers (North & Scheiber, 2008; Zewotir & North, 2011; Wessels & Nieuwoudt, 2011). Grade 1–9 teachers however also need training and support as they have to prepare learners for Grade 10–12 mathematics and statistics. Little evidence can be found in the literature of professional development initiatives to promote understanding of fundamental ideas in statistics education for Grade 1–9 teachers or of research on their statistical thinking and reasoning about these ideas (Wessels, 2009; Wessels & Nieuwoudt, 2011). Embedded in the professional development initiatives in statistics should be opportunities for teachers to conduct statistical investigations themselves to develop their own statistical literacy as well as appropriate competent statistical thinking and reasoning skills (Shaughnessy, 2007). To facilitate the transfer of the knowledge acquired through their own engagement in such statistical investigations, teachers should be supported in the classroom by researchers, advisors and colleagues after such professional development.

A problem-centred approach was used as point of departure for the series of workshops, focusing on statistical knowledge for teaching, which consists of content knowledge and pedagogical content knowledge in statistics. This model of statistical knowledge for teaching is based on the construct of mathematics knowledge for teaching developed by Ball, Thames and Phelps (2008), and includes:

• knowledge of statistics content and of relationships between statistical ideas (specialised content knowledge)
• knowledge of how students understand statistics concepts and develop statistical thinking and reasoning (knowledge of content and students)
• knowledge of how the statistics content should be facilitated and taught (knowledge of content and teaching)
• knowledge and interpretation of the curriculum
• knowledge of the use of technology to develop statistical thinking and reasoning (technological pedagogical content knowledge)
• the development of competent statistical thinking and reasoning skills of teachers.

Seven of the workshop sessions lasted two hours each; a four-hour session on the use of the computer data exploration software Tinkerplots® (Konold & Miller, 2005) in a computer laboratory was also included. The first six workshops were presented twice a week, one on a weekday with a repeat on Saturdays for teachers who could not attend during the week because of full schedules. During the workshops teachers’ statistical knowledge as well as statistical thinking and reasoning skills were developed through rich learning experiences that included all components of the statistical investigation process: posing problems, collecting data, analysing them, drawing conclusions and making predictions. The following topics were addressed in the workshops (Wessels, 2009):

• Data generation, representation and analysis of single categorical data sets to develop the language, argumentation skills and mindset for exploratory data analysis as preparation for the comparison of multiple data sets later in the programme.

• The influence of variability in measurement of a single person on interpretation of findings for the whole class.

• Identifying trends in data amidst variability.

• Chance as a precursor to probability.

• Repeated sampling from a known and then an unknown population.

• Probability.

• Using the Internet and statistics education software in the development of statistical thinking and reasoning.

• Informal inferential reasoning – comparing data sets of the same and different sizes.

Most of the previous research on variability in a repeated sampling context dealt with the statistical reasoning of students and pre-service teachers in such situations (Canada, 2004; Reading et al., 2005; Shaughnessy et al., 2004; Torok & Watson, 2000; Kelly & Watson, 2002). The study described in this article focused on variability reasoning in a repeated sampling context of a different population, namely of in-service teachers.

FIGURE 2: The Candy Bowl Task.

This question aimed to determine whether teachers expected variation when taking a sample. In this task two teachers expected something other than six candies in the handful of candies drawn from the bowl, with responses ‘any number, more red than yellow’ and ’± more than half more than yellow’. Three teachers acknowledged variability and indicated a range rather than one number of candies with responses such as ‘4–6’ and ‘0 ≤ red ≤ 10’. More than half of the teachers (57%) however answered that they expected six red candies to be drawn (Table 1). This result concurs with other research findings on learner and pre-service teachers’ reasoning about variability in the Candy Bowl Task (Shaughnessy et al., 2004; Watson & Shaughnessy, 2004). A probable reason for this focus on centre and not possible variability could be that the teachers’ experiences with and understanding of theoretical probability in their mathematics and statistics education or in professional development fostered limiting constructions and hindered their understanding of variability in a sampling context (Shaughnessy et al., 2004). Their understanding of what mathematics is could also have prompted a single value point answer, not considering possible variability present in repeated samples. They might even have considered possible variability but felt that they had to give a single point answer.

In this second part reasons for expectations were teased out. The responses were categorised with codes and code descriptors distinguishing between iconic, additive, proportional and distributional reasoning (Shaughnessy et al., 2004). Results are summarised in Table 2 and discussions are elucidated with examples of participants’ reasoning.

Additive reasoning focuses on frequencies and not relative frequencies. No acknowledgement is given to the role of proportions in the mixture; reasoning is just about absolute numbers or frequencies of reds in the candy mix. Three teachers responded with additive reasoning, for example ‘20 more red candies than yellows’ and ‘possibility of getting more reds than yellows’.

Implicit proportional reasoning attends to ratio, percentage or probability of reds whilst referring back to the original composition of the mixture. One teacher’s response showed implicit proportional reasoning, referring to the composition of the mixture with a fraction: ’Will be red, that is the grouping in the container’. It is incorrect to say that are red, but the proportion of yellow to red candies is 40:60, which relates to 2:3 and might have prompted this assumption. This comment could however be indicative of centre as is more than half.

Explicit proportional reasoning involves explicit reasoning about sample proportions, population proportions, probabilities or percentages. Six teachers used explicit proportional reasoning for this question: ‘60% is red candies and 40% is yellow’ and ‘Ratio for red and yellow is 6:4’.

Distributional reasoning involves reasoning about centres as well as the variation around the centres. None of the teachers used distributional reasoning. An example of distributional reasoning is: ‘The number of red candies will be 6, but also spread out around 6.’

In summary: six out of the 14 participants responded to Question 1 with iconic reasoning or additive reasoning. Seven teachers employed implicit proportional reasoning or explicit proportional reasoning. Only three considered a range of possible outcomes for a handful of ten candies drawn from the bowl.

Of the latter three teachers, one backed up her expected number of reds (4–6) with iconic reasoning (‘The candies are all mixed, you put your hand in and take any 10 candies. You cannot feel whether they are red or yellow’). The second teacher’s expectation for the number of red pulled from the bowl was ‘10 ≤ red ≥ 10’; she supported her choice with iconic reasoning: ‘Can’t be more than 10 or less than 10’. It can be argued that this teacher was referring to a range, but because of teachers’ busy schedules they were not available for individual interviews and therefore responses had to be analysed as presented. The third teacher also expected a range, answering ‘±6’, but used explicit proportional reasoning: ‘Ratio 100:10 and red, yellow’.

None of the teachers used distributional reasoning, showing that their ability to reason about variability in a repeated sampling context may not be well developed. One of the reasons for this situation may be a lack of exposure to variability in repeated sampling situations before the intervention alluded to in the group interview. Too much emphasis on a procedural rather than a conceptual approach when engaging with the ideas in previous training could also be at the root of the observed phenomenon.

Question 2 aimed to determine whether teachers expected variability in samples. Responses to Question 2 were analysed according to the codes and descriptors developed by Shaughnessy et al. (2004) and are summarised in Table 3.

Ten of the 13 teachers indicated that the number of reds pulled out will not be the same every time. These responses however were on different levels because of the reasons given. Of these ’no’ answers, eight showed level 1 reasoning: ‘Just a chance’, and ‘We have red and yellow candies – chance that there is many yellow or red’. One teacher’s answer gave implicit indication of variation with no further specific mention of the distribution and was categorised as a level 2 response: ‘Not a chance that you will draw the same number of reds every time’.

One teacher’s response was considered level 3. She answered that the number of reds pulled would be ’more or less’ the same each time and motivated her answer with ‘ratio 6:4’. One of the teachers responded that the number of reds would not be same every time, her reason being: ‘Random, can be red or yellow. Average colours will indeed be more red than yellow’. The reasoning does not explicitly use proportions but may be indicative of a consideration of variability and uncertainty and might be considered an intermediary stage to proportional thinking.

This question aimed to elicit teachers’ predictions for the results of repeated samples and responses were analysed using a four-point rubric as shown in Table 4 (Shaughnessy et al., 2004).

The results in Table 5 show that half of the teachers giving an appropriate range used additive reasoning whilst the other half displayed proportional and distributional reasoning. Conversely, all teachers using proportional and distributional reasoning suggested a reasonable range for the number of reds pulled from the mix. This result concurs with the research of Watson and Shaughnessy (2004) who point out that in their research participants using explicit proportional reasoning ‘were more likely to suggest a reasonable amount of variation around the expected mean of the samples’ (p. 108).

To assess if teachers had a better understanding of variability after the intervention and were able to apply this understanding to related tasks, several tasks were given to them two weeks after the two sessions on variability. These tasks (called post intervention tasks) included questions concerning a 50:50 (black:white) spinner, the tossing of a die, the wait times at two movie theatres and the comparison spelling test scores of two classes. Results of only the first three questions about the spinner will be presented here because the questions concerning the spinner were similar to those in the Candy Bowl Task, except that they focused on probability. Instead of sampling from a mix of candies, teachers this time had to predict the number of times a 50:50 (black:white) spinner would land on black in 50 spins and six sets of 50 spins. The task, used by Canada (2004) in his research with pre-service teachers as participants, consisted of three questions (Figure 3).

FIGURE 3: The Spinner Task.

The teacher who misunderstood the question gave an incomplete reason for her expectations. Of the other seven, one teacher displayed explicit proportional reasoning, bordering on distributional reasoning: ‘Big variation where it will land. Therefore I have made my choice between 40–60% of the total’. The other six teachers all showed distributional reasoning: ‘It is more or less half of 50’ and ‘It gives me the 50% chance, but with some variation’.

These results of the post intervention questions clearly show an increased awareness of variation in multiple trials and a shift from iconic reasoning to proportional and distributional reasoning after the intervention on variability.

The results of this study point to teachers’ lack of familiarity with and understanding of variability in a repeated sampling context before the intervention. Analysis of the post intervention questions shows that teachers’ understanding of the concept of variability grew during the intervention and that 75% of the eight teachers were able to transfer this increased understanding to related tasks.

Reasoning about centre and spread in a data set needs to be scaffolded carefully through well-chosen tasks and rich discussions, supported by a classroom culture conducive to independent thinking and shared understanding. Although reasoning about repeated trials in the face of probability is mentioned in the curriculum (Department of Basic Education, 2011c, p. 36), no mention is made of the crucial underlying concept of variability in repeated sampling. Another crucial aspect, proportional reasoning, is regarded as the ‘cornerstone of statistical inference’ (Shaughnessy et al., 2004, p. 184) and connects statistical reasoning in data and probability (Watson & Shaughnessy, 2004). Proportional reasoning is likewise not mentioned in connection with statistics in the curriculum.

The development of learners’ notions of these important concepts is dependent on teachers’ in-depth and in-breadth knowledge and well-developed thinking and reasoning skills about these concepts. It is therefore crucial that teachers’ awareness and understanding of variability and uncertainty in specific situations, such as repeated sampling and probability situations, is developed through well-planned activities and discussion opportunities that can be provided during professional development experiences. Teachers find the development of reasoning about variability in samples and distributions challenging (Bakker, 2004b) and might avoid teaching it.

Most teachers who participated in the study had experienced a traditional education in statistics, emphasising procedural rather than conceptual competence, and had typically not been exposed to activities that could build their understanding of and reasoning about repeated sampling (Wessels & Nieuwoudt, 2011). Research about variability in a repeated sampling environment that included an intervention (Canada, 2006) has shown an improvement of participants’ descriptions of what they expected (description) as well as of reasons for their expectations (causality). After this intervention, participants increasingly appreciated how variation occurred in multiple trials whilst reasons for their expectations improved, and progressively emphasised proportional reasoning coupled with a realisation of what is likely in the presence of variation. The results of our study concur with the results reported by Canada. Canada emphasises that to be effective, teacher education programs need to include an environment where teachers ‘can learn in a similar way that they themselves will aim to teach’ (p. 61). Teachers must get the opportunity to draw real samples and discuss differences and similarities in the distributions to develop their own skills about this topic so as to enable them to facilitate the development of proportional reasoning skills and a sense for expected variability in learners. The value of proportional thinking in repeated sampling situations also needs to be made explicit (Shaughnessy et al., 2004; Watson & Shaughnessy, 2004).

The fact that teachers’ time to participate was limited made individual interviews impossible – a fact that curtailed conclusions from their responses. Reasons for their responses could not be probed in depth; for example, in Question 1 of the Candy Bowl Task, one teacher gave the range ‘1, 2, 4, 5, 6, 7’, leaving out 3 as a possible number of reds to be pulled from the candy bowl. Did she intentionally break the sequence to impart that chances are that some numbers will not be drawn? She responded with a range (4–6) to Question 3, backed up by iconic reasoning, responded with explicit proportional reasoning on Question 2 and then said she guessed the six possible numbers for reds in successive trials. It is only possible to determine the realistic level of her reasoning through an in-depth interview. The question can also be asked about what notions of proportionality teachers held at the beginning of this research project.

The importance of variability as a fundamental concept in statistics necessitates more research about teachers’ and learners’ reasoning about variability, and especially their reasoning about variability in a repeated sampling context. Interview tasks to probe participants’ thinking more thoroughly should be an integral part of such a study.

Beliefs and conceptions about data and chance are very difficult to change, and research has suggested that empirical experiments and simulations must be systematically built into instruction over a longer period of time in order to change the patterns of students’ intuitive conceptions. (p. 976)

A greater emphasis on variability and repeated sampling is necessary in statistics education in South African schools to develop learners’ statistical reasoning skills and prepare them adequately for citizenship in our knowledge-driven society. Pre-service teacher education programs as well as professional development experiences in statistics of in-service teachers need to include ample opportunities for developing competence with regard to proportional thinking and an appreciation of the role of variability and uncertainty in statistics in order to equip them for this task. To this end, purposeful and effective opportunities should be created for teachers to engage collaboratively with activities and relevant materials to gain such competence and statistics teaching proficiency.

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