All statistical models have assumptions, and violation of these assumptions can affect the reliability of any conclusions we draw. Before we fit any statistical model, we need to explore the data to be sure we fit a valid model. Are relationships assumed to be a straight line really linear? Does the response variable follow the assumed distribution? Are variances consistent? We outline several graphical techniques for exploring data and introduce the analysis of model residuals as a powerful tool. If assumptions are violated, we consider two solutions, transforming variables to satisfy assumptions and using models that assume different distributions more consistent with the raw data and residuals. The exploratory stage can be extensive, but it is essential. At this pre-analysis stage, we also consider what to do about missing observations.

This chapter delves into the challenges and rewards of working in remote areas of countries such as Australia and small Pacific nations. Teaching strategies are presented to assist in maintaining a positive learning environment in remote and small Pacific-nation classrooms. The importance of the relationships among and between parents, students, teachers and other community members is explored, along with practical suggestions for making the most of the available resources. This chapter explores strategies for making the most of available resources and the invaluable professional experience of working in these areas.

Until recently, algebra was regarded as the domain of the secondary school years in most countries. In addition, it was often regarded in quite narrow ways by non-mathematics teachers, parents and students as being concerned with the manipulation of symbols according to tightly prescribed rules. Recent attention to algebra in the primary school has not regarded it as appropriate that such a narrow view of algebra be taken, leading to the use of terms such as ‘pre-algebra’ or ‘early algebra’ to describe the mathematics involved.

In this chapter, it is recognised that students’ understanding of algebra in the secondary school rests on foundations that are laid in the primary school, as reflected in the Australian Curriculum: Mathematics v. 9.0. These foundations are concerned with key algebraic ideas about patterns and generalisations, rather than with symbolic representations of these, such as x and y. This chapter explores developmental models associated with patterns and algebraic concepts, with a focus on developing algebraic thinking.

We don’t always have a single response variable, and disciplines like community ecology or the new “omics” bring rich datasets. Chapters 14–16 introduce the treatment of these multivariate data, with multiple variables recorded for each unit or “object.” We start with how we measure association between variables and use eigenanalysis to reduce the original variables to a smaller number of summary components or functions while retaining most of the variation. Then we look at the broad range of measures of dissimilarity or distance between objects based on the variables. Both approaches allow examination of relationships among objects and can be used in linear modeling when response and predictor variables are identified. We also highlight the important role of transformations and standardizations when interpreting multivariate analyses.

Single-case experimental designs refer to arrangements that allow experimentation with the individual subject as well as with groups. The methodology is different from the usual group research and relies on ongoing assessment over time, assessment of baseline (pre-intervention functioning), and the use of multiple phases in which performance is evaluated and altered. Three major design strategies, ABAB, multiple-baseline, and changing-criterion designs, are highlighted. The designs vary in the way in which intervention effects are demonstrated and the requirements for experimental evaluation. However, the logic of the designs in demonstrating causal relations is the same in which ongoing assessment across different phases is used to describe, predict, and test predictions as changes are made in the conditions to which participants are exposed. Data evaluation of the results of single-case designs usually relies on nonstatistical methods referred to as visual inspection. Multiple criteria to invoke this method are detailed. Single-case designs have special strengths. These include the ability to: evaluate interventions in everyday settings without the need of control groups or random assignment, provide feedback on the effectiveness of an intervention while that intervention is underway, permit beginning the intervention on a small scale before any larger scale extension, permit evaluation of whether an intervention genuinely is effective with a given individual, and study rare conditions for which group studies are not feasible.

In observational studies, the investigator evaluates the variables of interest by selecting groups rather than experimentally manipulating the variable of interest. Case-control studies were identified and include those investigations in which groups that vary in the outcome or characteristic of interest are delineated. Cohort studies are quite useful in delineating the timeline, that is, that some conditions are antecedent to and in fact predict occurrence of the later outcome. Birth-cohort studies have been a special case that has generated fascinating results related to physical and mental health, educational outcomes, and criminal and social behavior. The cohort usually is followed for decades and that allows investigators to evaluate outcomes at different developmental periods. Data from cohort studies often are used to classify, select, and predict an outcome. Sensitivity and specificity were discussed as key concepts related to the accurate identification of individuals who will show an outcome (sensitivity or true positives) as well as the accurate identification of individuals who will not show an outcome (specificity or true negatives). Critical issues in designing and interpreting observational studies were discussed including the importance of specifying the construct that will guide the study, selecting case and control groups, addressing possible confounds in the design and data analyses, and drawing causal inferences.

The extent to which an experiment rules out as explanations those factors that otherwise might account for the results is referred to as internal validity. Aside from evaluating the internal validity of an experiment, it is important to understand the extent to which the findings can be generalized to populations, settings, measures, experimenters, and other circumstances than those used in the original investigation. The generality of findings is referred to as external validity. Internal and external validity address central aspects of the logic of experimentation and scientific research more generally. The purpose of research is to structure the situation in such a way that inferences can be drawn about the effects of the variable of interest (internal validity) and to establish relations that extend beyond the highly specific circumstances in which the variable was examined (external validity). Internal and external validity are concepts to include in a methodological thinking tool kit. These are central to the evaluation of any study in all areas of scientific research.

Schools are technology rich. Teachers routinely now use digital tools for reporting, communications within school and with parents, for maintaining class records, for preparing materials and so on. Some schools use online teaching programs or electronic textbooks. With NAPLAN moving to become fully online (see Chapter 19) there is a need for both teachers and students across the primary years to be confident and creative users of digital technology. Each chapter in this book has included examples and strategies for integrating digital tools into the teaching of mathematics across a range of mathematical content areas.

This chapter considers the range of possible education support roles in the mathematics classroom that a teacher may work with. This chapter presents effective ways of working collaboratively with education support workers and explores positive planning of learning experiences which considers the affordances of various education support workers’ roles.

This chapter discusses the protection of civilians against direct attack and the effects of military operations, and the protections should they fall into the hands of the enemy in IACs and NIACs under IHL and IHRL . The scope of the protection afforded is reviewed. The chapter goes on to consider its application in specific cases; the prohibition of arbitrary punishment, detention, internal displacement and deportation, sexual violence in armed conflict and SEA on the part of UN peacekeepers. The protection afforded to specific groups to wit children, journalists, UN peacekeepers and PMSC personnel is then examined.

Biological data commonly involve multiple predictors. This chapter starts expanding our models to include multiple categorical predictors (factors) when they are in factorial designs. These designs allow us to introduce synergistic effects – interactions. Two- and three-factor designs are used to illustrate the estimation and interpretation of interactions. Our approach is first to consider the most complex interactions and use them to decide whether it is helpful to continue examining simple interactions. Main effects – single predictors acting independently of each other – are the last to be considered. We also deal with problems caused by missing observations (unbalanced designs) and missing cells (fractional and incomplete factorials) and discuss how to estimate and interpret them.

This chapter examines suitable statistics questions for investigation by children of different ages, using a cycle of problem, plan, data, analysis and conclusion (PPDAC) (Wild & Pfannkuch 1999). The importance of variation in data and different types of variables and the difference between a population and a sample are investigated. Readers will explore different ways of displaying data to ‘tell a story’. The importance of drawing inferences from data and the uncertainty associated with these inferences are discussed. Readers will engage in activities that use technology to support the development of statistical understanding.

Several types of measures were covered that are used in clinical psychological research. These included inventories, questionnaires, scales, global ratings, interviews, projective measures, direct observations of behavior, psychobiological measures, computerized, technology-based and web-based assessment, and unobtrusive measures. Reactivity of assessment and the use of unobtrusive measures and their advantages were also discussed. In general, it is useful to rely upon multiple measures rather than a single measure. It is useful to demonstrate that changes in the construct of interest (e.g., anxiety) are not restricted to only one method of assessment.

It’s surprisingly common for biologists to combine crossed and nested factors. These designs are partly nested or split-plot designs. They are nearly always mixed models, usually a random nested effect and at least two fixed effects. We describe the analysis of these designs, starting with a simple three-factor design with a single between-plot and a single within-plot effect, extending this analysis to include multiple effects, including interactions at this level, and adding continuous predictors (covariates).