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The term "quantitative literacy" is used in this article to reinforce its importance in catering for reform-based curricula in Australia. The medium chosen for the problems discussed in this paper is the newspaper. The purpose of this article is to tie together traditional views of the importance of problem solving with the current aims of numeracy and quantitative literacy. In doing this, the author considers another angle on the development of quantitative literacy. Watson (1997) suggested a hierarchy for the related field of statistical literacy, which is easily adapted in the wider milieu. The hierarchy suggests three tiers or steps in the development of the goal of quantitative literacy: (1) The terminology needs to be understood; (2) This terminology then needs to be understood within the social (or scientific) context in which it is used; and (3) There needs to be a critical awareness that prompts questioning of claims in context that are made without proper justification. (Contains 5 figures.)

This paper focuses on the curriculum links between statistics and science that teachers need to understand and apply in order to be effective teachers of the two fields of study. Meaningful statistics does not exist without context and science is the context for this paper. Although curriculum documents differ from country to country, this paper uses extracts from three countries: Australia, New Zealand and the USA. The statistical ideas from the "Australian Curriculum: Mathematics," the "New Zealand Mathematics and Statistics" Curriculum and the US "Common Core State Standards for Mathematics" are linked to the relevant parts of the "Australian Curriculum: Science," the New Zealand "Science" Curriculum and the US "Next Generation Science Standards for States, by States." Teachers of mathematics need to be aware of the potential of science to provide meaningful contexts within which to set statistical investigations. Similarly, teachers of science, who are developing methods for implementing investigations and experiments in their classrooms, need to be aware of the close ties to statistical tools for decision-making.

Statistical literacy increasingly is considered an important outcome of schooling. There is little information, however, about appropriate expectations of students at different stages of schooling. Some progress towards this goal was made by Watson and Callingham (2005), who identified an empirical 6-level hierarchy of statistical literacy and the distribution of middle school students across the levels, using archived data from 1993-2000. There is interest in reconsidering these outcomes a decade later, during which statistics and probability has become a recognised strand of the Australian mathematics curriculum. Using a new data-set of over 7000 student responses from middle-years students in different parts of Australia during the period 2007-2009, the nature of the hierarchy was confirmed. Longitudinal analysis identified how students performed across time against the hierarchy. Suggestions are made for systems and teachers about realistic expectations for middle-years students, and possible curriculum challenges.

This report is based on an activity completed by 89 Grade 6 students carrying out a statistical investigation with a focus on posing an initial question given a context and then refining it for a chosen data set. Having reached a conclusion based on evidence from their data analysis, students were given additional information on the context and asked to reconsider their conclusions and degree of certainty associated with them. Data from student workbooks were used to assess students' capacity to engage in the extensions of a statistical investigation not usually encountered at the school level. Results showed approximately 70% of students were able to carry out the complete investigation and justify a meaningful decision.

Statistics is an increasingly important component of the mathematics curriculum. "StatSmart" was a project intended to influence middle-years students' learning outcomes in statistics through the provision of appropriate professional learning opportunities and technology to teachers. Participating students in grade 5/6 to grade 9 undertook three tests, a pre-test, a post-test and a longitudinal retention test over a period of 2 years. Their teachers completed a survey that included items measuring pedagogical content knowledge (PCK) for teaching statistics. Despite the development of valid instruments to measure both student and teacher content knowledge and teachers' PCK, linking teachers' knowledge directly to students' learning outcomes has proved elusive. Multilevel modelling of results from 789 students for whom there were 3 completed tests and measures from their teachers indicated that students' outcomes were influenced positively by their initial teacher's PCK. Extended participation of teachers in the project also appeared to reduce negative effects of changing teachers.

As an extension to an activity introducing Year 5 students to the practice of statistics, the software "TinkerPlots" made it possible to collect repeated random samples from a finite population to informally explore students' capacity to begin reasoning with a distribution of sample statistics. This article provides background for the sampling process and reports on the success of students in making predictions for the population from the collection of simulated samples and in explaining their strategies. The activity provided an application of the numeracy skill of using percentages, the numerical summary of the data, rather than graphing data in the analysis of samples to make decisions on a statistical question. About 70% of students made what were considered at least moderately good predictions of the population percentages for five yes-no questions, and the correlation between predictions and explanations was 0.78.

The authors analyzed the development of 4th-grade students' understanding of the transition from experimental relative frequencies of outcomes to theoretical probabilities with a focus on the foundational statistical concepts of variation and expectation. After observing the decreasing variation from the theoretical probability as the sample size increased, students developed a deeper understanding of the relationship between relative frequency of outcomes and theoretical probability as well as their respective associations with variation and expectation.

The practice of statistics is the focus of the world in which professional statisticians live. To understand meaningfully what this practice is about, students need to engage in it themselves. Acknowledging the limitations of a genuine classroom setting, this study attempted to expose four classes of year 5 students (n?=?91) to an authentic experience of the practice of statistics. Setting an overall context of people's habits that are considered environmentally friendly, the students sampled their class and set criteria for being environmentally friendly based on questions from the Australian Bureau of Statistics "CensusAtSchool" site. They then analysed the data and made decisions, acknowledging their degree of certainty, about three populations based on their criteria: their class, year 5 students in their school and year 5 students in Australia. The next step was to collect a random sample the size of their class from an Australian Bureau of Statistics "population", analyse it and again make a decision about Australian year 5 students. At the end, they suggested what further research they might do. The analysis of students' responses gives insight into primary students' capacity to appreciate and understand decision-making, and to participate in the practice of statistics, a topic that has received very little attention in the literature. Based on the total possible score of 23 from student workbook entries, 80% of students achieved at least a score of 11.

Background: This study was based on the premise that variation is the foundation of statistics and statistical investigations. The study followed the development of fourth-grade students' understanding of variation through participation in a sequence of two lessons based on measurement. In the first lesson all students measured the arm span of one student, revealing pathways students follow in developing understanding of variation and linear measurement (related to research question 1). In the second lesson each student's arm span was measured once, introducing a different aspect of variation for students to observe and contrast. From this second lesson, students' development of the ability to compare their representations for the two scenarios and explain differences in terms of variation was explored (research question 2). Students' documentation, in both workbook and software formats, enabled us to monitor their engagement and identify their increasing appreciation of the need to observe, represent, and contrast the variation in the data. Following the lessons, a written student assessment was used for judging retention of understanding of variation developed through the lessons and the degree of transfer of understanding to a different scenario (research question 3). Results: The results were based either on the application of the hierarchical SOLO model or on non-hierarchical clustering of responses to individual questions in the student workbooks. Students' progress throughout the lessons displayed a wide range of explanations for the estimate of a single student's arm span, general surprise at the variation in measurements, and a large variety of hand-drawn representations based on the values or frequencies of measurements. Many different representations were also created in the software for the single student measurements and for the comparison of measurements for the two scenarios. Although the students' interpretations of their plots were generally more basic than sophisticated, the results of the assessment indicated that many students had developed the ability to transfer their appreciation of variation to another context and could clearly explain the meaning of variation. Conclusions: The findings highlight the importance of an early focus on variation and distribution, with meaningful activities that motivate students to conduct and observe measurements, together with creating both hand-drawn and software representations to relate their experiences.

This article compares the definition of "box plot" as used in the "Australian Curriculum: Mathematics" with other definitions used in the education community; describes the difficulties students experience when dealing with box plots; and discusses the elaboration that is necessary to enable teachers to develop the knowledge necessary to use them effectively. The box plot is 40 years old. Like many 40-year-olds the box plot has changed its shape and varied its personal descriptors over the years. Its power to inform exploratory data analysis has gained it a place in statistics curricula at various levels in many countries. The "Australian Curriculum: Mathematics" has placed it at Year 10 in the Statistics and Probability strand of the learning area. As the curriculum document is sparse in its description and elaboration of the term, three issues relating to the needs of the teachers required to implement the curriculum are discussed in this article: (1) there is not one universally accepted definition of a box plot and teachers need to be aware that they and their students may encounter contradictions in different contexts where different conventions apply; (2) teachers need to be aware of the difficulties their students encounter in understanding box plots; and, (3) there is minimal advice in the curriculum elaborations about how the box plot can be used effectively in a statistical investigation. (Contains 12 figures, 1 table and 1 footnote.)

Developing statistical literacy for older school students includes recognizing and interpreting conditional statements in the media. Data on couples' success in predicting whether their partners have cheated provides a motivating context for teachers to lead their students through this process.

To capture aspects of pedagogical content knowledge (PCK) not illuminated in an earlier written survey, an interview protocol was used with 40 middle school teachers. The scenarios were intended to elicit teachers' understanding of the big ideas, ability to anticipate students' answers, and intervention strategies for the classroom. This was expected to be a straight-forward journey based on teachers' responses to three context-based scenarios regarding students' answers to questions. Instead we were surprised by teachers' responses that revealed their perceptions that their experiences teaching mathematics and teaching statistics are very different. This led to further analysis of the PCK tasks and a suggestion that the mathematics embedded in the tasks was sometimes an impediment for the teachers, especially in relation to intervention strategies in the classroom. (Contains 2 figures.)

This study considers the evolving influence of variation and expectation on the development of school students' appreciation of distribution as displayed in their construction of graphical representations of data sets. Three interview protocols are employed, presenting different contexts within which 109 students, ranging in age from 6 to 15 years, could display and interpret their understanding. Responses are analyzed within a hierarchical cognitive framework. It is hypothesized from the analysis that, contrary to the order in which expectation and variation are introduced in the school curriculum, the natural tendency for students is to acknowledge variation first and then expectation. (Contains 8 tables and 21 figures.)

This paper considers the development of school students' ability to define three terms that are fundamental to statistical literacy: sample, random, and variation. A total of 738 students in grades 3, 5, 7, and 9 were asked in a survey to define and give an example for the word "sample." Of these, 379 students in grades 7 and 9 were also asked about the words "random" and "variation." Responses were used to describe developmental levels overall and to document differences across grades on the understanding of these terms. Changes in performance were also monitored after lessons on chance and data, emphasizing variation for 335 students. After 2 years, 132 of these students and a further 209 students who were surveyed originally but did not take part in specialized lessons, were surveyed again. The difference after 2 years between the performance of students who experienced the specialized lessons and those who did not was considered, revealing no differences in performance longitudinally. For students in grades 7 and 9, the association of performance on the three terms was explored. Implications for mathematics and literacy educators are discussed.

This study documented efforts to facilitate ideas of beginning inference in novice grade 7 students. A design experiment allowed modified teaching opportunities in light of observation of components of a framework adapted from that developed by Pfannkuch for teaching informal inference with box plots. Box plots were replaced by hat plots, a feature available with the software TinkerPlots[TM]. Data in TinkerPlots files were analyzed on four occasions and observed responses to tasks were categorized using a hierarchical model. The observed outcomes provided evidence of change in students' appreciation of beginning inference over the four sessions. Suggestions for change are made for the use of the framework in association with the intervention and the software to enhance understanding of beginning inference. (Contains 9 figures and 4 tables.)

The literature that is available on the topic of representations in mathematics is vast. One commonly discussed item is graphical representations. From the history of mathematics to modern uses of technology, a variety of graphical forms are available for middle school students to use to represent mathematical ideas. The ideas range from algebraic relationships to summaries of data sets. Traditionally, textbooks delineate the rules to be followed in creating conventional graphical forms, and software offers alternatives for attractive presentations. This article presents a new data representation tool called the hat plot, which is a featured tool of the data analysis software TinkerPlots Dynamic Data Exploration (Konold and Miller 2005). TinkerPlots software gives students considerable freedom to create graphs to tell the stories of their data sets. Here, the authors demonstrate the value of the hat plot in representing and interpreting data sets. (Contains 11 figures.)

Several tasks used in research studies are presented with assessment rubrics and examples of the development of student understanding. The tasks focus on students' appreciation of variation in several contexts and illustrate the need to discuss variation in the classroom and to ask students specifically about it during assessment.

Two strands of research motivate this study. One is the interest in school students' development of understanding of the concept of average, historically part of the mathematics curriculum and prominent in the statistics curriculum introduced in the early 1990s. The other is the belief of some educators that students learn meaningfully when experiencing cognitive conflict that challenges incorrect or incomplete understandings. This study presented 58 students in Grades 3, 6, and 9 with a series of questions about the concept of average. After initial levels of response were observed, students were presented with alternative responses on video from other school students and asked to choose which best resolved the task at hand. Initial responses confirmed the levels of understanding in an earlier study based on the same questions. Responses after the experience of cognitive conflict were either at the same level as before or higher, with no student finally agreeing with a lower level response. The results are compared with longitudinal change in relation to average and change resulting from cognitive conflict in relation to other areas of chance and data. Implications of the research are considered.

This study presents the results of a partial credit Rasch analysis of in-depth interview data exploring statistical understanding of 73 school students in 6 contextual settings. The use of Rasch analysis allowed the exploration of a single underlying variable across contexts, which included probability sampling, representation of temperature change, beginning inference, independent events, the relationship of sample and population, and description of variation. Interpretation of the demands of increasing code levels for the resulting variable revealed an increasing appreciation of and interaction between the ideas of variation and expectation. Student progression in understanding is illustrated with kidmaps, and educational implications are considered.

This study uses the context of the weather to explore the development of students' intuitive ideas of variation from pre-Grade 1 to Grade 9. Three aspects of understanding these intuitions associated with variation are explored in individual videotaped interviews with 73 students: explanations, suggestions of data, and graphing. The development of these three aspects across grades is explored, as well as the associations among them. Fifty-eight of the students also answered a general question on the definitions of "variation" and "variable, " and these responses are discussed and compared with responses to the weather task. The interview protocol may prove useful for teachers, particularly with younger children, to appreciate students' developing understanding of variation and provide starting points for classroom work of a more specific nature, either with respect to weather or other contextual topics.

This article presents two statistical tasks, one on comparing data sets, another on repeated sampling, both of which have powerful connections to proportional reasoning. Students' work and reasoning on these tasks are included, as well as suggestions for ways to use these tasks with middle school students to connect statistical thinking to proportional reasoning. (Contains 2 figures.)

Comprehension of chance language, such as is found in newspapers, is a fundamental aspect of statistical literacy. In this study, students' understandings of chance language were explored through responses to two items in surveys administered to 2,726 students from grades 5 to 11. One item involved evaluating the chance expressed in phrases from newspaper headlines using a number line, and responses were described in four levels of chance language evaluation. The other item involved interpreting, in context, an expression of percent chance, and responses were described in four levels of chance language interpretation. Students in higher grades were more likely to demonstrate higher levels of both evaluation and interpretation. The association between levels of evaluation and interpretation was further explored generally and in relation to one of the headlines involving percent. Implications for mathematics educators in relation to chance language in the curriculum across the years of schooling are discussed.

Presents a questionnaire devised to assess students' understanding of statistical variation. Carries out three stages of the analysis of outcomes: (1) a hierarchical coding scheme; (2) a Rasch analysis; and (3) a holistic model of development. Illustrates the range of student responses and possible rubrics for teacher use. (Author/KHR)

Reports on a study designed to provide baseline data in the area of conjunction and conditional events. Cross-sectional and longitudinal analyses revealed improvement with grade in expressing probability numerically and distinguishing conditional events, but no change in incidence of conjunction errors. (Author/MM)

Investigates the factors that influence the effectiveness of collaboration on open-ended mathematical tasks. Identifies various phenomena that influence cognitive "lifting", "hovering", and "falling"; that is, improvement, no change, and reduction in levels of functioning, respectively. (Author/MM)

Presents an instrument designed as a profile of teacher achievement and teacher needs with regard to the probability and statistics strands in the mathematics curriculum. Discusses uses for the instrument and further development possibilities. (MM)

Considers the circumstances surrounding instances where help is sought and/or provided in a collaborative problem-solving situation. Analyzed videotaped observations of nine groups of three grade 5-6 students working over three 45-minute periods on an open-ended task. Defines hierarchical levels for questions, answers, and outcomes. Indicates some differences associated with gender. (Contains 36 references.) (Author/ASK)

Uses a particularly contentious newspaper report that makes a cause-and-effect claim as the basis for discussing the important aspects of statistical understanding. Suggests a hierarchy to help teachers plan for and assess student learning in this area, and closes with a plea for teachers to cooperate across subjects in order to achieve results. (Contains 12 references.) (ASK)

Explores the development of understanding of the concept of average with students from grades 3 to 9 through interviews. Observed six levels of response based on an hierarchical model of cognitive functioning. Documents usage of ideas associated with the three standard measures of central tendency and representation as strategies for problem solving. (Contains 36 references.) (Author/ASK)

Focuses on students' ideas of average in differing contexts from two perspectives, one of which reflects the structural complexity of concepts as they are expressed and the other which documents the prevalence of ideas associated with three common mathematical measures of average--mean, median, and mode. (Contains 35 references.) (ASK)

Explores the development of elementary school students' (n=88) understanding of comparing two datasets through the responses of students in individual interview settings. Strategies observed within the developmental cycles were visual, numerical, or a combination of the two. Contains 26 references. (Author/ASK)