The Australian Curriculum: Mathematics v. 9.0 (ACARA 2022) is structured around six content strands: Number, Algebra, Measurement, Space, Statistics and Probability. An expectation of mathematical proficiency has been embedded into curriculum content across all strands. It is expected that students will develop and apply mathematical understanding, fluency, reasoning and problem-solving as they learn mathematical content. It is these areas that typically receive the most attention in mathematics classrooms, particularly as there are requirements to assess and report on students’ progress in these strands. The Australian Curriculum v. 9.0 also identifies seven general capabilities, which encompass knowledge, skills, behaviours, and dispositions, and three cross-curriculum priorities: Aboriginal and Torres Strait Islander histories and cultures, Asia and Australia’s engagement with Asia, and Sustainability. However, Atweh, Miller and Thornton (2012) contend that these areas receive minimal reference in the content descriptions and elaborations, leading to the impression that they are only given lip service. This chapter will provide an overview of the general capabilities and cross-curriculum priorities, and will also identify ways in which these aspects of the curriculum can be enacted into authentic mathematical experiences for students.

This chapter discusses international humanitarian law (IHL) and international human rights law (IHRL) which apply concurrently in armed conflict, but differ in scope and purpose. Their relationship is reviewed by looking at the right to life, the right to a fair trial, arbitrary detention and military responses to terrorism. The chapter then considers derogations from IHRL and the issue of the IHRL obligations of armed groups, particularly in NIACs. Similar issues are discussed regarding peacekeeping operations.

Critical sampling issues were discussed, including the heavy reliance on a narrow range and type of subjects. In general, selecting and including a diverse sample ought to be the default position. In clinical research of course, the sample often is determined by patient characteristics of interest. Several different designs were discussed. Clinical trials and their variations used to evaluate interventions (e.g., in medicine, psychology, education, and other areas with applied foci) were also presented. Quasi-experimental (nonequivalent control group) designs were highlighted. In these designs, subjects cannot be assigned randomly to conditions. Multiple-treatment designs were also discussed. In these designs, each subject receives all the conditions (e.g., more than one treatment or treatment and control conditions). Counterbalancing is designed to ensure that the effects of the treatments can be separated from the order in which they appear.

By this last chapter, you should have a clear biological and statistical model, well-designed data collection, and careful interpretation of models fitted to your data. The challenge now is to communicate what may be some complex analyses to a range of audiences. This aspect of data analysis has been relatively neglected, but we now have audiences less tolerant of unclear or unengaging communication, coupled with the challenge of describing complex analyses. We advocate bringing a storytelling approach to presenting results and being very clear how the data support our larger biological story. We also introduce basic ideas about how to report complex analyses and offer suggestions for improving the clarity of supporting graphics. Two important recommendations are that biologists learn more about how to communicate quantitative information and alter our data communication dramatically to match the mode of delivery and the target audience.

Construct validity pertains to interpreting the basis for the causal relation between the independent variable and the dependent variable. The investigator may conclude that the experimental manipulation was responsible for group differences (internal validity was well handled), but the study may not permit a conclusion about why the effect occurred. Other factors embedded in the manipulation alone or in combination with the manipulation might account for the findings (construct validity is in question). Data-evaluation validity refers to those aspects of the study that affect the quantitative evaluation and can lead to misleading or false conclusions. Several concepts basic to statistical evaluation were mentioned and include the probability of accepting and rejecting the null hypothesis, the probability of making such decisions when they are false, and effect size. Major factors that commonly serve as threats to data-evaluation validity include low statistical power, subject heterogeneity, variability in the procedures of an investigation, unreliability of the measures, restricted range of the measure, and multiple statistical comparisons and their error rates. All four types of validity including internal, external, construct, and data-evaluation validity need to be considered at the design stage of an investigation.

We use proportional reasoning every day, often without being aware that we are reasoning in terms of two quantities that vary in relation to each other – that is, as one quantity increases or decreases, so does the other. I may decide to buy two tins of tomatoes. The price of each tin is the same, so if I purchase double the number of tins, the amount I pay also doubles. Despite using this thinking informally quite regularly, it is surprising how many people have trouble with this concept. Doubling or trebling a quantity is one thing, but what about wanting one-and-a-half times, or only needing one-fifth of something? These calculations can become very tricky. Often we make some kind of estimate and end up with either too little or too much of something.

Proportional reasoning is used widely to solve a range of everyday problems from ‘best buys’ to understanding data presented in tables. It underpins scaling problems such as scale drawings of house plans and currency conversions, and appears in many other situations, including the Australian electoral system.

This book is grounded in empirically evidenced developmental models and linked closely to practical classroom practice. While many classrooms have been resourced with equipment such as base-10 materials, counters, shape kits, mobile devices, dice kits, drawing tools and interactive whiteboard (IWB) technology, and even a laptop trolley in some cases, extensive professional development is required to enable the range of classroom resources to be transformed into teaching tools. The difficulty faced by the teaching profession is in integrating a wide range of teaching approaches and resources to weave a pedagogically sound learning sequence. This book provides mathematics teachers and pre-service teachers with detailed teaching activities that are designed and informed by research-based practices. The aim is to provide you with a sensible and achievable integration of available educational tools, with research-based approaches to mathematical development that provide for the mathematical needs of all learners. It is intended for primary pre-service teachers, and teachers looking for ways to enhance their teaching of primary mathematics, to assist them to design student tasks that are meaningful and to use educationally sound ways to improve their mathematics teaching.

This chapter looks at appropriate early childhood pedagogy, particularly as it applies to the early learning of mathematics. The importance of play and recognition of children’s prior learning is emphasised throughout. Although many of the experiences and learning documented focus on number, we acknowledge that children’s early mathematical learning extends beyond number into areas such as geometry, measurement and spatial awareness.

This chapter presented information to supplement statistical significance testing. Among this is the inclusion of a measure of the strength or magnitude of the relation whenever significant tests are presented. Completer analyses, intent-to-treat, and models of imputation were discussed as ways of handling missing data in research where there are repeated measures and participants drop out before all measures are completed. Outliers in the data and deleting data were also discussed. Multiple comparison tests and controlling for error rates were presented as well. Many decision points in the data presentation and analyses can influence the findings. I discussed robustness of findings and encouraged evaluating the results in different ways to see if critical decisions influenced the results and conclusions. Also, the use exploratory data analyses was encouraged to examine a variety of relations and special findings that might generate hypotheses for future research. Meta-analysis was also discussed. This now is very commonly used as a way of combining studies to summarize a literature and to ask novel questions that usually go beyond what any single study has examined. Finally, secondary data analyses were discussed. This is conducting studies based on data that other people have collected. There are rich databases that provide special opportunities for further evaluation beyond the original goals of the study.

This chapter discusses command responsibility as an IHL doctrine. It commences by discussing the concept of responsible command before addressing the scope of application of the doctrine of command responsibility and the elements of the doctrine of command responsibility. The underlying offence is then discussed. The criteria for the existence of a superior-subordinate relationship, the requirement for effective control, the criteria for the mens rea requirement and the criteria with regard to failure to adopt necessary and ressonable measures are considered. The chapter then discusses successor command responsibility and the issue of causation before concluding with analysis of the nature of command responsibility

Null hypothesis significance testing is the dominant method to analyze the results of research. Statistical tests use probability levels to make the decision to accept or reject the null hypothesis (there is no difference statistically). Issues critical to statistical evaluation were discussed, including significance levels (alpha), power, sample size, and significance and magnitude of effects. Statistical power has received extensive discussion in research in part because repeated evaluations have shown that the majority of studies are designed in such a way as to have weak power. Multiple ways of increasing power were presented. There have been many sources of dissatisfaction with statistical significance testing including the fact that null hypothesis and statistical significance testing give us arbitrary cutoff points to make binary decisions (accept or reject the null hypothesis), and most importantly do not provide the critical information we would like (e.g., direct tests of our hypotheses and information about the strengths of our interventions). Null hypothesis statistical testing is not the only way of approaching the data and data analyses. Bayesian data analyses were highlighted as an alternative to null hypothesis statistical tests.

Ethical issues refer to the investigator’s moral, professional, and legal responsibilities in relation to the care of research participants. Many issues can arise such as deception, invasion of privacy, and informed consent. Each issue has its guidelines and requirements. For example, informed consent is a central issue that encompasses many ethical concerns and means of protecting participants in experimentation. Informed consent requires that the participant be capable of providing consent (competence), is aware of the procedures, risks, and benefits (knowledge), and willingly agrees to participate (volition). More generally, ethical issues are so critical that they usually need to be suitably addressed in a proposal or plan for research prior to running any subject. The many ethical issues raised in research have prompted guidelines and regulations designed to protect the rights of individual subjects. The guidelines apprise investigators of their obligations and the priority of ensuring protection of the subject at all times. Regulations are more demanding and put into policy, law, and federally mandated requirements to more strongly ensure that rights are protected. The concern over protection of participants is an international focus and international organizations as well individual countries have guidelines. Also, many guidelines span multiple countries (e.g., Declaration of Helsinki).

Australian classrooms are becoming increasingly diverse. In addition to a wide range of mathematical abilities, primary teachers of mathematics must meet the needs of children from varied cultural backgrounds, many of whom have had different mathematical experiences. Children who have physical, intellectual, social or emotional difficulties may be included in the mainstream classroom. There may also be children who are classified as gifted and talented in one or more domains. All have the capacity to learn mathematics, and the right to experience mathematics suitable to their learning needs. Although this chapter addresses issues relating to inclusion in primary mathematics classrooms, it does not pretend to provide a special education focus. The needs of children with specific disabilities can be highly technical, and it is well beyond the scope of this chapter to try to deal with all the detailed requirements and concerns that may be encountered. Rather, this chapter aims to help the primary teacher deal with the reality of mathematics teaching in modern classrooms, where there may be children with very diverse learning and mathematical needs. Many students are not categorised as having a disability but have information processing delays. These students will benefit from the same approaches as those with recognised disabilities.

There is a daunting array of statistical “methods” out there – regression, ANOVA, loglinear models, GLMMS, ANCOVA, etc. They often are treated as different data analysis approaches. We take a more holistic view. Most methods biologists use are variations on a central theme of generalized linear models – relating a biological response to a linear combination of predictor variables. We show how several common “named” methods are related, based on classifying biological response and predictor variables as continuous or categorical. We use simple regression, single-factor ANOVA, logistic regression, and two-dimensional contingency tables to show how these methods all represent generalized linear models with a single predictor. We describe how we fit these models and outline their assumptions.

The chapter discusses the application of the principle of distinction to combatants and civilians, the consequences of that distinction and the issue of those civilians who directly participate in hostilities both in IACs and NIACs. It then discusses the application of the concept in cyber warfare. The chapter then considers the positions of members of private military security companies, unlawful combatants and UN peacekeepers.

Communication of results of research is a critical step in science and entails all the other topics we have covered in relation to methodology. This communication usually is a written report (e.g., for a thesis or dissertation, granting agency, or journal article) or for presentation (e.g., poster session, conference presentation). In these different formats, there are common goals and requirements. In each format, the researcher’s challenge is to convey why the question that guides research is important and the way in which it has been addressed in the study is suitable. Methodology plays major roles throughout the processes of planning, conducting, interpreting, and communicating research results. Three interrelated tasks are involved in preparing a manuscript whether for a thesis, dissertation, presentation or journal article. These were described as description, explanation, and contextualization of the study. The writing we are routinely taught in science focuses on description, but the other portions are central as well and determine whether a study not only appears to be important but also in fact actually is. Recommendations were made regarding what to address and how to incorporate description, explanation, and contextualization within the different sections of a manuscript (e.g., Introduction, Method). In addition, questions were provided to direct the researcher to the types of issues reviewers are likely to ask about a manuscript.